sbml_import.py

"""
SBML Import
-----------
This module provides all necessary functionality to import a model specified
in the `Systems Biology Markup Language (SBML) <http://sbml.org/Main_Page>`_.
"""
import copy
import itertools as itt
import logging
import math
import os
import re
import warnings
from pathlib import Path
from typing import (Any, Callable, Dict, Iterable, List, Optional, Tuple,
                    Union)

import libsbml as sbml
import sympy as sp

from . import has_clibs
from .constants import SymbolId
from .import_utils import (RESERVED_SYMBOLS,
                           _check_unsupported_functions,
                           _get_str_symbol_identifiers,
                           _parse_special_functions,
                           generate_measurement_symbol,
                           generate_regularization_symbol,
                           noise_distribution_to_cost_function,
                           noise_distribution_to_observable_transformation,
                           smart_subs, smart_subs_dict, toposort_symbols)
from .logging import get_logger, log_execution_time, set_log_level
from .ode_export import (
    ODEExporter, ODEModel, symbol_with_assumptions, _default_simplify
)


class SBMLException(Exception):
    pass


SymbolicFormula = Dict[sp.Symbol, sp.Expr]


default_symbols = {
    symbol: {} for symbol in SymbolId
}

ConservationLaw = Dict[str, Union[str, sp.Expr]]
NoiseDistDict = Dict[str, Union[str, list]]

logger = get_logger(__name__, logging.ERROR)


class SbmlImporter:
    """
    Class to generate AMICI C++ files for a model provided in the Systems
    Biology Markup Language (SBML).

    :ivar show_sbml_warnings:
        indicates whether libSBML warnings should be
        displayed

    :ivar symbols:
        dict carrying symbolic definitions

    :ivar sbml_reader:

        The libSBML sbml reader

        .. warning::
           Not storing this may result in a segfault.

    :ivar sbml_doc:
        document carrying the sbml definition

        .. warning::
           Not storing this may result in a segfault.

    :ivar sbml:
        SBML model to import

    :ivar compartments:
        dict of compartment ids and compartment volumes

    :ivar stoichiometric_matrix:
        stoichiometric matrix of the model

    :ivar flux_vector:
        reaction kinetic laws

    :ivar flux_ids:
        identifiers for elements of flux_vector

    :ivar _local_symbols:
        model symbols for sympy to consider during sympification
        see `locals`argument in `sympy.sympify`

    :ivar species_assignment_rules:
        Assignment rules for species.
        Key is symbolic identifier and value is assignment value

    :ivar compartment_assignment_rules:
        Assignment rules for compartments.
        Key is symbolic identifier and value is assignment value

    :ivar parameter_assignment_rules:
        assignment rules for parameters, these parameters are not permissible
        for sensitivity analysis

    :ivar initial_assignments:
        initial assignments for parameters, these parameters are not
        permissible for sensitivity analysis

    :ivar sbml_parser_settings:
        sets behaviour of SBML Formula parsing

    """

    def __init__(self,
                 sbml_source: Union[str, Path, sbml.Model],
                 show_sbml_warnings: bool = False,
                 from_file: bool = True) -> None:
        """
        Create a new Model instance.

        :param sbml_source:
            Either a path to SBML file where the model is specified,
            or a model string as created by sbml.sbmlWriter(
            ).writeSBMLToString() or an instance of `libsbml.Model`.

        :param show_sbml_warnings:
            Indicates whether libSBML warnings should be displayed.

        :param from_file:
            Whether `sbml_source` is a file name (True, default), or an SBML
            string
        """
        if isinstance(sbml_source, sbml.Model):
            self.sbml_doc: sbml.Document = sbml_source.getSBMLDocument()
        else:
            self.sbml_reader: sbml.SBMLReader = sbml.SBMLReader()
            if from_file:
                sbml_doc = self.sbml_reader.readSBMLFromFile(str(sbml_source))
            else:
                sbml_doc = self.sbml_reader.readSBMLFromString(sbml_source)
            self.sbml_doc = sbml_doc

        self.show_sbml_warnings: bool = show_sbml_warnings

        # process document
        self._process_document()

        self.sbml: sbml.Model = self.sbml_doc.getModel()

        # Long and short names for model components
        self.symbols: Dict[SymbolId, Dict[sp.Symbol, Dict[str, Any]]] = {}

        self._local_symbols: Dict[str, Union[sp.Expr, sp.Function]] = {}
        self.compartments: SymbolicFormula = {}
        self.compartment_assignment_rules: SymbolicFormula = {}
        self.species_assignment_rules: SymbolicFormula = {}
        self.parameter_assignment_rules: SymbolicFormula = {}
        self.initial_assignments: SymbolicFormula = {}

        self._reset_symbols()

        # http://sbml.org/Software/libSBML/5.18.0/docs/python-api/classlibsbml_1_1_l3_parser_settings.html#abcfedd34efd3cae2081ba8f42ea43f52
        # all defaults except disable unit parsing
        self.sbml_parser_settings = sbml.L3ParserSettings(
            self.sbml, sbml.L3P_PARSE_LOG_AS_LOG10,
            sbml.L3P_EXPAND_UNARY_MINUS, sbml.L3P_NO_UNITS,
            sbml.L3P_AVOGADRO_IS_CSYMBOL,
            sbml.L3P_COMPARE_BUILTINS_CASE_INSENSITIVE, None,
            sbml.L3P_MODULO_IS_PIECEWISE
        )

    def _process_document(self) -> None:
        """
        Validate and simplify document.
        """
        # Ensure we got a valid SBML model, otherwise further processing
        # might lead to undefined results
        self.sbml_doc.validateSBML()
        _check_lib_sbml_errors(self.sbml_doc, self.show_sbml_warnings)

        # apply several model simplifications that make our life substantially
        # easier
        if self.sbml_doc.getModel().getNumFunctionDefinitions():
            convert_config = sbml.SBMLFunctionDefinitionConverter()\
                .getDefaultProperties()
            self.sbml_doc.convert(convert_config)

        convert_config = sbml.SBMLLocalParameterConverter().\
            getDefaultProperties()
        self.sbml_doc.convert(convert_config)

        # If any of the above calls produces an error, this will be added to
        # the SBMLError log in the sbml document. Thus, it is sufficient to
        # check the error log just once after all conversion/validation calls.
        _check_lib_sbml_errors(self.sbml_doc, self.show_sbml_warnings)

    def _reset_symbols(self) -> None:
        """
        Reset the symbols attribute to default values
        """
        self.symbols = copy.deepcopy(default_symbols)
        self._local_symbols = {}

    def sbml2amici(
            self,
            model_name: str,
            output_dir: Union[str, Path] = None,
            observables: Dict[str, Dict[str, str]] = None,
            event_observables: Dict[str, Dict[str, str]] = None,
            constant_parameters: Iterable[str] = None,
            sigmas: Dict[str, Union[str, float]] = None,
            event_sigmas: Dict[str, Union[str, float]] = None,
            noise_distributions: Optional[
                Dict[str, Union[NoiseDistDict, Callable]]] = None,
            event_noise_distributions: Dict[str, Union[str, Callable]] = None,
            verbose: Union[int, bool] = logging.ERROR,
            assume_pow_positivity: bool = False,
            compiler: str = None,
            allow_reinit_fixpar_initcond: bool = True,
            compile: bool = True,
            compute_conservation_laws: bool = True,
            simplify: Optional[Callable] = _default_simplify,
            cache_simplify: bool = False,
            log_as_log10: bool = True,
            generate_sensitivity_code: bool = True,
    ) -> None:
        """
        Generate and compile AMICI C++ files for the model provided to the
        constructor.

        The resulting model can be imported as a regular Python module (if
        `compile=True`), or used from Matlab or C++ as described in the
        documentation of the respective AMICI interface.

        Note that this generates model ODEs for changes in concentrations, not
        amounts unless the `hasOnlySubstanceUnits` attribute has been
        defined for a particular species.

        Sensitivity analysis for local parameters is enabled by creating
        global parameters _{reactionId}_{localParameterName}.

        :param model_name:
            name of the model/model directory

        :param output_dir:
            see :meth:`amici.ode_export.ODEExporter.set_paths`

        :param observables:
            dictionary( observableId:{'name':observableName
            (optional), 'formula':formulaString)}) to be added to the model

        :param event_observables:
            dictionary( eventObservableId:{'name':eventObservableName
            (optional), 'event':eventId, 'formula':formulaString)}) to be
            added to the model

        :param constant_parameters:
            list of SBML Ids identifying constant parameters

        :param sigmas:
            dictionary(observableId: sigma value or (existing) parameter name)

        :param event_sigmas:
            dictionary(eventObservableId: sigma value or (existing) parameter
            name)

        :param noise_distributions:
            dictionary(observableId: noise type).
            If nothing is passed for some observable id, a normal model is
            assumed as default. Either pass a noise type identifier, or a
            callable generating a custom noise string.

        :param event_noise_distributions:
            dictionary(eventObservableId: noise type).
            If nothing is passed for some observable id, a normal model is
            assumed as default. Either pass a noise type identifier, or a
            callable generating a custom noise string.

        :param verbose:
            verbosity level for logging, ``True``/``False`` default to
            ``logging.Error``/``logging.DEBUG``

        :param assume_pow_positivity:
            if set to ``True``, a special pow function is
            used to avoid problems with state variables that may become
            negative due to numerical errors

        :param compiler:
            distutils/setuptools compiler selection to build the
            python extension

        :param allow_reinit_fixpar_initcond:
            see :class:`amici.ode_export.ODEExporter`

        :param compile:
            If ``True``, compile the generated Python package,
            if ``False``, just generate code.

        :param compute_conservation_laws:
            if set to ``True``, conservation laws are automatically computed
            and applied such that the state-jacobian of the ODE
            right-hand-side has full rank. This option should be set to
            ``True`` when using the Newton algorithm to compute steadystate
            sensitivities.
            Conservation laws for constant species are enabled by default.
            Support for conservation laws for non-constant species is
            experimental and may be enabled by setting an environment variable
            ``AMICI_EXPERIMENTAL_SBML_NONCONST_CLS`` to either ``demartino``
            to use the algorithm proposed by De Martino et al. (2014)
            https://doi.org/10.1371/journal.pone.0100750, or to any other value
            to use the deterministic algorithm implemented in
            ``conserved_moieties2.py``. In some cases, the ``demartino`` may
            run for a very long time. This has been observed for example in the
            case of stoichiometric coefficients with many significant digits.

        :param simplify:
            see :attr:`amici.ODEModel._simplify`

        :param cache_simplify:
                see :meth:`amici.ODEModel.__init__`

        :param log_as_log10:
            If ``True``, log in the SBML model will be parsed as ``log10``
            (default), if ``False``, log will be parsed as natural logarithm
            ``ln``

        :param generate_sensitivity_code:
            If ``False``, the code required for sensitivity computation will
            not be generated
        """
        set_log_level(logger, verbose)

        ode_model = self._build_ode_model(
            observables=observables,
            event_observables=event_observables,
            constant_parameters=constant_parameters,
            sigmas=sigmas,
            event_sigmas=event_sigmas,
            noise_distributions=noise_distributions,
            event_noise_distributions=event_noise_distributions,
            verbose=verbose,
            compute_conservation_laws=compute_conservation_laws,
            simplify=simplify,
            cache_simplify=cache_simplify,
            log_as_log10=log_as_log10,
        )

        exporter = ODEExporter(
            ode_model,
            model_name=model_name,
            outdir=output_dir,
            verbose=verbose,
            assume_pow_positivity=assume_pow_positivity,
            compiler=compiler,
            allow_reinit_fixpar_initcond=allow_reinit_fixpar_initcond,
            generate_sensitivity_code=generate_sensitivity_code
        )
        exporter.generate_model_code()

        if compile:
            if not has_clibs:
                warnings.warn('AMICI C++ extensions have not been built. '
                              'Generated model code, but unable to compile.')
            exporter.compile_model()

    def _build_ode_model(
            self,
            observables: Dict[str, Dict[str, str]] = None,
            event_observables: Dict[str, Dict[str, str]] = None,
            constant_parameters: Iterable[str] = None,
            sigmas: Dict[str, Union[str, float]] = None,
            event_sigmas: Dict[str, Union[str, float]] = None,
            noise_distributions: Optional[
                Dict[str, Union[NoiseDistDict, Callable]]] = None,
            event_noise_distributions: Dict[str, Union[str, Callable]] = None,
            verbose: Union[int, bool] = logging.ERROR,
            compute_conservation_laws: bool = True,
            simplify: Optional[Callable] = _default_simplify,
            cache_simplify: bool = False,
            log_as_log10: bool = True,
    ) -> ODEModel:
        """Generate an ODEModel from this SBML model.

        See :py:func:`sbml2amici` for parameters.
        """
        constant_parameters = list(constant_parameters) \
            if constant_parameters else []

        if sigmas is None:
            sigmas = {}

        if event_sigmas is None:
            event_sigmas = {}

        if noise_distributions is None:
            noise_distributions = {}

        if event_noise_distributions is None:
            event_noise_distributions = {}

        self._reset_symbols()
        self.sbml_parser_settings.setParseLog(
            sbml.L3P_PARSE_LOG_AS_LOG10 if log_as_log10 else
            sbml.L3P_PARSE_LOG_AS_LN
        )
        self._process_sbml(constant_parameters)
        if self.symbols.get(SymbolId.EVENT, False):
            if compute_conservation_laws:
                logger.warning(
                    'Conservation laws are currently not supported for models '
                    'with events, and will be turned off.'
                )
            compute_conservation_laws = False

        self._process_observables(
            observables,
            sigmas,
            noise_distributions
        )
        self._process_event_observables(
            event_observables,
            event_sigmas,
            event_noise_distributions
        )
        self._replace_compartments_with_volumes()

        self._clean_reserved_symbols()
        self._process_time()

        ode_model = ODEModel(
            verbose=verbose,
            simplify=simplify,
            cache_simplify=cache_simplify,
        )
        ode_model.import_from_sbml_importer(
            self, compute_cls=compute_conservation_laws)
        return ode_model

    @log_execution_time('importing SBML', logger)
    def _process_sbml(self, constant_parameters: List[str] = None) -> None:
        """
        Read parameters, species, reactions, and so on from SBML model

        :param constant_parameters:
            SBML Ids identifying constant parameters
        """
        self.check_support()
        self._gather_locals()
        self._process_parameters(constant_parameters)
        self._process_compartments()
        self._process_species()
        self._process_reactions()
        self._process_rules()
        self._process_initial_assignments()
        self._process_species_references()
        self._process_events()

    def check_support(self) -> None:
        """
        Check whether all required SBML features are supported.
        Also ensures that the SBML contains at least one reaction, or rate
        rule, or assignment rule, to produce change in the system over time.
        """

        # Check for required but unsupported SBML extensions
        if self.sbml_doc.getLevel() != 3 \
                and hasattr(self.sbml, 'all_elements_from_plugins') \
                and self.sbml.all_elements_from_plugins.getSize():
            raise SBMLException('SBML extensions are currently not supported!')

        if self.sbml_doc.getLevel() == 3:
            # the "required" attribute is only available in SBML Level 3
            for i_plugin in range(self.sbml.getNumPlugins()):
                plugin = self.sbml.getPlugin(i_plugin)
                if plugin.getPackageName() in ('layout',):
                    # 'layout' plugin does not have the 'required' attribute
                    continue
                if hasattr(plugin, 'getRequired') and not plugin.getRequired():
                    # if not "required", this has no impact on model
                    #  simulation, and we can safely ignore it
                    continue
                # Check if there are extension elements. If not, we can safely
                #  ignore the enabled package
                if plugin.getListOfAllElements():
                    raise SBMLException(
                        f'Required SBML extension {plugin.getPackageName()} '
                        f'is currently not supported!')

        if any(not rule.isAssignment() and not isinstance(
                    self.sbml.getElementBySId(rule.getVariable()),
                    (sbml.Compartment, sbml.Species, sbml.Parameter)
                ) for rule in self.sbml.getListOfRules()):
            raise SBMLException('Algebraic rules are currently not supported, '
                                'and rate rules are only supported for '
                                'species, compartments, and parameters.')

        if any(not (rule.isAssignment() or rule.isRate())
                and isinstance(
                    self.sbml.getElementBySId(rule.getVariable()),
                    (sbml.Compartment, sbml.Species, sbml.Parameter)
                ) for rule in self.sbml.getListOfRules()):
            raise SBMLException('Only assignment and rate rules are '
                                'currently supported for compartments, '
                                'species, and parameters!')

        if any(r.getFast() for r in self.sbml.getListOfReactions()):
            raise SBMLException('Fast reactions are currently not supported!')

        # Check events for unsupported functionality
        self.check_event_support()

    def check_event_support(self) -> None:
        """
        Check possible events in the model, as AMICI does currently not support

        * delays in events
        * priorities of events
        * events fired at initial time

        Furthermore, event triggers are optional (e.g., if an event is fired at
        initial time, no trigger function is necessary).
        In this case, warn that this event will have no effect.
        """
        for event in self.sbml.getListOfEvents():
            event_id = event.getId()
            # Check for delays in events
            delay = event.getDelay()
            if delay is not None:
                try:
                    delay_time = float(self._sympy_from_sbml_math(delay))
                    if delay_time != 0:
                        raise ValueError
                # `TypeError` would be raised in the above `float(...)`
                # if the delay is not a fixed time
                except (TypeError, ValueError):
                    raise SBMLException('Events with execution delays are '
                                        'currently not supported in AMICI.')
            # Check for priorities
            if event.getPriority() is not None:
                raise SBMLException(f'Event {event_id} has a priority '
                                    'specified. This is currently not '
                                    'supported in AMICI.')

            # check trigger
            trigger_sbml = event.getTrigger()
            if trigger_sbml is None:
                logger.warning(f'Event {event_id} trigger has no trigger, '
                               'so will be skipped.')
                continue
            if trigger_sbml.getMath() is None:
                logger.warning(f'Event {event_id} trigger has no trigger '
                               'expression, so a dummy trigger will be set.')

            if not trigger_sbml.getPersistent():
                raise SBMLException(
                    f'Event {event_id} has a non-persistent trigger.'
                    'This is currently not supported in AMICI.'
                )

    @log_execution_time('gathering local SBML symbols', logger)
    def _gather_locals(self) -> None:
        """
        Populate self.local_symbols with all model entities.

        This is later used during sympifications to avoid sympy builtins
        shadowing model entities as well as to avoid possibly costly
        symbolic substitutions
        """
        self._gather_base_locals()
        self._gather_dependent_locals()

    def _gather_base_locals(self):
        """
        Populate self.local_symbols with pure symbol definitions that do not
        depend on any other symbol.
        """

        special_symbols_and_funs = {
            # oo is sympy infinity
            'INF': sp.oo,
            'NaN': sp.nan,
            'rem': sp.Mod,
            'time': symbol_with_assumptions('time'),
            # SBML L3 explicitly defines this value, which is not equal
            # to the most recent SI definition.
            'avogadro': sp.Float(6.02214179e23),
            'exponentiale': sp.E,
        }
        for s, v in special_symbols_and_funs.items():
            self.add_local_symbol(s, v)

        for c in itt.chain(self.sbml.getListOfSpecies(),
                           self.sbml.getListOfParameters(),
                           self.sbml.getListOfCompartments()):
            if not c.isSetId():
                continue

            self.add_local_symbol(c.getId(), _get_identifier_symbol(c))

        for x_ref in _get_list_of_species_references(self.sbml):
            if not x_ref.isSetId():
                continue
            if x_ref.isSetStoichiometry() and not \
                    self.is_assignment_rule_target(x_ref):
                value = sp.Float(x_ref.getStoichiometry())
            else:
                value = _get_identifier_symbol(x_ref)

            ia_sym = self._get_element_initial_assignment(x_ref.getId())
            if ia_sym is not None:
                value = ia_sym

            self.add_local_symbol(x_ref.getId(), value)

        for r in self.sbml.getListOfReactions():
            for e in itt.chain(r.getListOfReactants(), r.getListOfProducts()):
                if isinstance(e, sbml.SpeciesReference):
                    continue

                if not (e.isSetId() and e.isSetStoichiometry()) or \
                        self.is_assignment_rule_target(e):
                    continue

                self.add_local_symbol(e.getId(),
                                      sp.Float(e.getStoichiometry()))

    def _gather_dependent_locals(self):
        """
        Populate self.local_symbols with symbol definitions that may depend on
        other symbol definitions.
        """
        for r in self.sbml.getListOfReactions():
            if not r.isSetId():
                continue
            self.add_local_symbol(
                r.getId(),
                self._sympy_from_sbml_math(r.getKineticLaw())
            )

    def add_local_symbol(self, key: str, value: sp.Expr):
        """
        Add local symbols with some sanity checking for duplication which
        would indicate redefinition of internals, which SBML permits,
        but we don't.

        :param key:
            local symbol key

        :param value:
            local symbol value
        """
        if key in self._local_symbols.keys():
            raise SBMLException(
                f'AMICI tried to add a local symbol {key} with value {value}, '
                f'but {key} was already instantiated with '
                f'{self._local_symbols[key]}. This means that there '
                f'are multiple SBML elements with SId {key}, which is '
                f'invalid SBML. This can be fixed by renaming '
                f'the elements with SId {key}.'
            )
        if key in {'True', 'False', 'true', 'false', 'pi'}:
            raise SBMLException(
                f'AMICI tried to add a local symbol {key} with value {value}, '
                f'but {key} is a reserved symbol in AMICI. This can be fixed '
                f'by renaming the element with SId {key}.'
            )
        self._local_symbols[key] = value

    @log_execution_time('processing SBML compartments', logger)
    def _process_compartments(self) -> None:
        """
        Get compartment information, stoichiometric matrix and fluxes from
        SBML model.
        """
        compartments = self.sbml.getListOfCompartments()
        self.compartments = {}
        for comp in compartments:
            init = sp.Float(1.0)

            if comp.isSetVolume():
                init = self._sympy_from_sbml_math(comp.getVolume())

            ia_sym = self._get_element_initial_assignment(comp.getId())
            if ia_sym is not None:
                init = ia_sym

            self.compartments[_get_identifier_symbol(comp)] = init

    @log_execution_time('processing SBML species', logger)
    def _process_species(self) -> None:
        """
        Get species information from SBML model.
        """
        if self.sbml.isSetConversionFactor():
            conversion_factor = symbol_with_assumptions(
                self.sbml.getConversionFactor()
            )
        else:
            conversion_factor = 1

        for s in self.sbml.getListOfSpecies():
            if self.is_assignment_rule_target(s):
                continue
            self.symbols[SymbolId.SPECIES][_get_identifier_symbol(s)] = {
                'name': s.getName() if s.isSetName() else s.getId(),
                'compartment': _get_species_compartment_symbol(s),
                'constant': s.getConstant() or s.getBoundaryCondition(),
                'amount': s.getHasOnlySubstanceUnits(),
                'conversion_factor': symbol_with_assumptions(
                    s.getConversionFactor()
                )
                if s.isSetConversionFactor()
                else conversion_factor,
                'index': len(self.symbols[SymbolId.SPECIES]),
            }

        self._convert_event_assignment_parameter_targets_to_species()
        self._process_species_initial()
        self._process_rate_rules()

    @log_execution_time('processing SBML species initials', logger)
    def _process_species_initial(self):
        """
        Extract initial values and initial assignments from species
        """
        for species_variable in self.sbml.getListOfSpecies():
            initial = get_species_initial(species_variable)

            species_id = _get_identifier_symbol(species_variable)
            # If species_id is a target of an AssignmentRule, species will be
            # None, but we don't have to account for the initial definition
            # of the species itself and SBML doesn't permit AssignmentRule
            # targets to have InitialAssignments.
            species = self.symbols[SymbolId.SPECIES].get(species_id, None)

            ia_initial = self._get_element_initial_assignment(
                species_variable.getId()
            )
            if ia_initial is not None:
                if species and species['amount'] \
                        and 'compartment' in species:
                    ia_initial *= self.compartments.get(
                        species['compartment'], species['compartment']
                    )
                initial = ia_initial
            if species:
                species['init'] = initial

        # don't assign this since they need to stay in order
        sorted_species = toposort_symbols(self.symbols[SymbolId.SPECIES],
                                          'init')
        for species in self.symbols[SymbolId.SPECIES].values():
            species['init'] = smart_subs_dict(species['init'],
                                              sorted_species,
                                              'init')

    @log_execution_time('processing SBML rate rules', logger)
    def _process_rate_rules(self):
        """
        Process rate rules for species, compartments and parameters.
        Compartments and parameters with rate rules are implemented as species.
        Note that, in the case of species, rate rules may describe the change
        in amount, not concentration, of a species.
        """
        rules = self.sbml.getListOfRules()
        # compartments with rules are replaced with constants in the relevant
        # equations during the _replace_in_all_expressions call inside
        # _process_rules
        for rule in rules:
            if rule.getTypeCode() != sbml.SBML_RATE_RULE:
                continue

            variable = symbol_with_assumptions(rule.getVariable())
            formula = self._sympy_from_sbml_math(rule)
            if formula is None:
                continue

            # Species rules are processed first, to avoid processing
            # compartments twice (as compartments with rate rules are
            # implemented as species).
            ia_init = self._get_element_initial_assignment(rule.getVariable())
            if variable in self.symbols[SymbolId.SPECIES]:
                init = self.symbols[SymbolId.SPECIES][variable]['init']
                name = None

            if variable in self.compartments:
                init = self.compartments[variable]
                name = str(variable)
                del self.compartments[variable]

            elif variable in self.symbols[SymbolId.PARAMETER]:
                init = self._sympy_from_sbml_math(
                    self.symbols[SymbolId.PARAMETER][variable]['value'],
                )
                name = self.symbols[SymbolId.PARAMETER][variable]['name']
                del self.symbols[SymbolId.PARAMETER][variable]

            # parameter with initial assignment, cannot use
            # self.initial_assignments as it is not filled at this
            # point
            elif ia_init is not None:
                init = ia_init
                par = self.sbml.getElementBySId(rule.getVariable())
                name = par.getName() if par.isSetName() else par.getId()

            self.add_d_dt(formula, variable, init, name)

    def add_d_dt(
            self,
            d_dt: sp.Expr,
            variable: sp.Symbol,
            variable0: Union[float, sp.Expr],
            name: str,
    ) -> None:
        """
        Creates or modifies species, to implement rate rules for
        compartments and species, respectively.

        :param d_dt:
            The rate rule (or, right-hand side of an ODE).

        :param variable:
            The subject of the rate rule.

        :param variable0:
            The initial value of the variable.

        :param name:
            Species name, only applicable if this function generates a new
            species
        """
        if variable in self.symbols[SymbolId.SPECIES]:
            # only update dt if species was already generated
            self.symbols[SymbolId.SPECIES][variable]['dt'] = d_dt
        else:
            # update initial values
            for species_id, species in self.symbols[SymbolId.SPECIES].items():
                variable0 = smart_subs(variable0, species_id, species['init'])

            for species in self.symbols[SymbolId.SPECIES].values():
                species['init'] = smart_subs(species['init'],
                                             variable, variable0)

            # add compartment/parameter species
            self.symbols[SymbolId.SPECIES][variable] = {
                'name': name,
                'init': variable0,
                'amount': False,
                'conversion_factor': 1.0,
                'constant': False,
                'index': len(self.symbols[SymbolId.SPECIES]),
                'dt': d_dt,
            }

    @log_execution_time('processing SBML parameters', logger)
    def _process_parameters(self,
                            constant_parameters: List[str] = None) -> None:
        """
        Get parameter information from SBML model.

        :param constant_parameters:
            SBML Ids identifying constant parameters
        """

        if constant_parameters is None:
            constant_parameters = []

        # Ensure specified constant parameters exist in the model
        for parameter in constant_parameters:
            if not self.sbml.getParameter(parameter):
                raise KeyError('Cannot make %s a constant parameter: '
                               'Parameter does not exist.' % parameter)

        fixed_parameters = [
            parameter
            for parameter in self.sbml.getListOfParameters()
            if parameter.getId() in constant_parameters
        ]
        for parameter in fixed_parameters:
            if self._get_element_initial_assignment(parameter.getId()) is not \
                    None or self.is_assignment_rule_target(parameter) or \
                    self.is_rate_rule_target(parameter):
                raise SBMLException(
                    f'Cannot turn parameter {parameter.getId()} into a '
                    'constant/fixed parameter since it either has an '
                    'initial assignment or is the target of an assignment or '
                    'rate rule.'
                )

        parameters = [
            parameter for parameter
            in self.sbml.getListOfParameters()
            if parameter.getId() not in constant_parameters
            and self._get_element_initial_assignment(parameter.getId()) is None
            and not self.is_assignment_rule_target(parameter)
        ]

        loop_settings = {
            SymbolId.PARAMETER: {'var': parameters, 'name': 'parameter'},
            SymbolId.FIXED_PARAMETER: {'var': fixed_parameters,
                                       'name': 'fixed_parameter'}
        }

        for partype, settings in loop_settings.items():
            for par in settings['var']:
                self.symbols[partype][_get_identifier_symbol(par)] = {
                    'name': par.getName() if par.isSetName() else par.getId(),
                    'value': par.getValue()
                }

    @log_execution_time('processing SBML reactions', logger)
    def _process_reactions(self):
        """
        Get reactions from SBML model.
        """
        reactions = self.sbml.getListOfReactions()
        # nr (number of reactions) should have a minimum length of 1. This is
        # to ensure that, if there are no reactions, the stoichiometric matrix
        # and flux vector multiply to a zero vector with dimensions (nx, 1).
        nr = max(1, len(reactions))
        nx = len(self.symbols[SymbolId.SPECIES])
        # stoichiometric matrix
        self.stoichiometric_matrix = sp.SparseMatrix(sp.zeros(nx, nr))
        self.flux_vector = sp.zeros(nr, 1)
        # Use reaction IDs as IDs for flux expressions (note that prior to SBML
        #  level 3 version 2 the ID attribute was not mandatory and may be
        #  unset)
        self.flux_ids = [
            f"flux_{reaction.getId()}" if reaction.isSetId()
            else f"flux_r{reaction_idx}"
            for reaction_idx, reaction in enumerate(reactions)
        ] or ['flux_r0']

        reaction_ids = [
            reaction.getId() for reaction in reactions
            if reaction.isSetId()
        ]

        for reaction_index, reaction in enumerate(reactions):
            for element_list, sign in [(reaction.getListOfReactants(), -1),
                                       (reaction.getListOfProducts(), 1)]:
                for element in element_list:
                    stoichiometry = self._get_element_stoichiometry(
                        element
                    )
                    sbml_species = self.sbml.getSpecies(element.getSpecies())
                    if self.is_assignment_rule_target(sbml_species):
                        continue
                    species_id = _get_identifier_symbol(sbml_species)
                    species = self.symbols[SymbolId.SPECIES][species_id]

                    if species['constant']:
                        continue

                    # Division by species compartment size (to find the
                    # rate of change in species concentration) now occurs
                    # in the `dx_dt` method in "ode_export.py", which also
                    # accounts for possibly variable compartments.
                    self.stoichiometric_matrix[species['index'],
                                               reaction_index] += \
                        sign * stoichiometry * species['conversion_factor']
            if reaction.isSetId():
                sym_math = self._local_symbols[reaction.getId()]
            else:
                sym_math = self._sympy_from_sbml_math(reaction.getKineticLaw())

            self.flux_vector[reaction_index] = sym_math
            if any(
                str(symbol) in reaction_ids
                for symbol in self.flux_vector[reaction_index].free_symbols
            ):
                raise SBMLException(
                    'Kinetic laws involving reaction ids are currently'
                    ' not supported!'
                )

    @log_execution_time('processing SBML rules', logger)
    def _process_rules(self) -> None:
        """
        Process Rules defined in the SBML model.
        """
        for rule in self.sbml.getListOfRules():
            # rate rules are processed in _process_species
            if rule.getTypeCode() == sbml.SBML_RATE_RULE:
                continue

            sbml_var = self.sbml.getElementBySId(rule.getVariable())
            sym_id = symbol_with_assumptions(rule.getVariable())
            formula = self._sympy_from_sbml_math(rule)
            if formula is None:
                continue

            if isinstance(sbml_var, sbml.Species):
                self.species_assignment_rules[sym_id] = formula

            elif isinstance(sbml_var, sbml.Compartment):
                self.compartment_assignment_rules[sym_id] = formula
                self.compartments[sym_id] = formula

            elif isinstance(sbml_var, sbml.Parameter):
                self.parameter_assignment_rules[sym_id] = formula

            self.symbols[SymbolId.EXPRESSION][sym_id] = {
                'name': str(sym_id),
                'value': formula
            }

        self.symbols[SymbolId.EXPRESSION] = toposort_symbols(
            self.symbols[SymbolId.EXPRESSION], 'value'
        )

        # expressions must not occur in definition of x0
        for species in self.symbols[SymbolId.SPECIES].values():
            species['init'] = self._make_initial(
                smart_subs_dict(species['init'],
                                self.symbols[SymbolId.EXPRESSION],
                                'value')
            )

    def _process_time(self) -> None:
        """
        Convert time_symbol into cpp variable.
        """
        sbml_time_symbol = symbol_with_assumptions('time')
        amici_time_symbol = symbol_with_assumptions('t')
        self.amici_time_symbol = amici_time_symbol

        self._replace_in_all_expressions(sbml_time_symbol, amici_time_symbol)

    def _convert_event_assignment_parameter_targets_to_species(self):
        """
        Convert parameters that are targets of event assignments to species.

        This is for the convenience of only implementing event assignments for
        "species".
        """
        parameter_targets = \
            _collect_event_assignment_parameter_targets(self.sbml)
        for parameter_target in parameter_targets:
            # Parameter rate rules already exist as species.
            if parameter_target in self.symbols[SymbolId.SPECIES]:
                continue
            if parameter_target in self.parameter_assignment_rules:
                raise SBMLException(
                    'AMICI does not currently support models with SBML events '
                    'that affect parameters that are also the target of '
                    'assignment rules.'
                )
            parameter_def = None
            for symbol_id in {SymbolId.PARAMETER, SymbolId.FIXED_PARAMETER}:
                if parameter_target in self.symbols[symbol_id]:
                    # `parameter_target` should only exist in one of the
                    # `symbol_id` dictionaries.
                    if parameter_def is not None:
                        raise AssertionError(
                            'Unexpected error. The parameter target of an '
                            'event assignment was processed twice.'
                        )
                    parameter_def = \
                        self.symbols[symbol_id].pop(parameter_target)
            if parameter_def is None:
                # this happens for parameters that have initial assignments
                # or are assignment rule targets
                par = self.sbml.getElementBySId(str(parameter_target))
                ia_init = self._get_element_initial_assignment(
                    par.getId()
                )
                parameter_def = {
                    'name': par.getName() if par.isSetName() else par.getId(),
                    'value': par.getValue() if ia_init is None else ia_init
                }
            # Fixed parameters are added as species such that they can be
            # targets of events.
            self.symbols[SymbolId.SPECIES][parameter_target] = {
                'name': parameter_def['name'],
                'init': sp.Float(parameter_def['value']),
                # 'compartment': None,  # can ignore for amounts
                'constant': False,
                'amount': True,
                # 'conversion_factor': 1.0,  # can be ignored
                'index': len(self.symbols[SymbolId.SPECIES]),
                'dt': sp.Float(0),
            }

    @log_execution_time('processing SBML events', logger)
    def _process_events(self) -> None:
        """Process SBML events."""
        events = self.sbml.getListOfEvents()

        def get_empty_bolus_value() -> sp.Float:
            """
            Used in the event update vector for species that are not affected
            by the event.
            """
            return sp.Symbol('AMICI_EMTPY_BOLUS')

        # Used to update species concentrations when an event affects a
        # compartment.
        concentration_species_by_compartment = {
            symbol_with_assumptions(c.getId()): []
            for c in self.sbml.getListOfCompartments()
        }
        for species, species_def in self.symbols[SymbolId.SPECIES].items():
            if (
                    # Species is a concentration
                    not species_def.get('amount', True) and
                    # Species has a compartment
                    'compartment' in species_def
            ):
                concentration_species_by_compartment[
                    species_def['compartment']
                ].append(species)

        for ievent, event in enumerate(events):
            # get the event id (which is optional unfortunately)
            event_id = event.getId()
            if event_id is None or event_id == '':
                event_id = f'event_{ievent}'
            event_sym = sp.Symbol(event_id)

            # get and parse the trigger function
            trigger_sbml = event.getTrigger()
            trigger_sym = self._sympy_from_sbml_math(trigger_sbml)
            trigger = _parse_event_trigger(trigger_sym)

            # Currently, all event assignment targets must exist in
            # self.symbols[SymbolId.SPECIES]
            state_vector = list(self.symbols[SymbolId.SPECIES].keys())

            # parse the boluses / event assignments
            bolus = [get_empty_bolus_value() for _ in state_vector]
            event_assignments = event.getListOfEventAssignments()
            compartment_event_assignments = set()
            for event_assignment in event_assignments:
                variable_sym = \
                    symbol_with_assumptions(event_assignment.getVariable())
                if event_assignment.getMath() is None:
                    # Ignore event assignments with no change in value.
                    continue
                formula = self._sympy_from_sbml_math(event_assignment)
                try:
                    # Try to find the species in the state vector.
                    index = state_vector.index(variable_sym)
                    bolus[index] = formula
                except ValueError:
                    raise SBMLException(
                        'Could not process event assignment for '
                        f'{str(variable_sym)}. AMICI currently only allows '
                        'event assignments to species; parameters; or, '
                        'compartments with rate rules, at the moment.'
                    )
                try:
                    # Try working with the formula now to detect errors
                    # here instead of at multiple points downstream.
                    _ = formula - variable_sym
                except TypeError:
                    raise SBMLException(
                        'Could not process event assignment for '
                        f'{str(variable_sym)}. AMICI only allows symbolic '
                        'expressions as event assignments.'
                    )
                if variable_sym in concentration_species_by_compartment:
                    compartment_event_assignments.add(variable_sym)

                for comp, assignment in \
                        self.compartment_assignment_rules.items():
                    if variable_sym not in assignment.free_symbols:
                        continue
                    compartment_event_assignments.add(comp)

            # Update the concentration of species with concentration units
            # in compartments that were affected by the event assignments.
            for compartment_sym in compartment_event_assignments:
                for species_sym in concentration_species_by_compartment[
                        compartment_sym
                ]:
                    # If the species was not affected by an event assignment
                    # then the old value should be updated.
                    if (
                            bolus[state_vector.index(species_sym)]
                            == get_empty_bolus_value()
                    ):
                        species_value = species_sym
                    # else the species was affected by an event assignment,
                    # hence the updated value should be updated further.
                    else:
                        species_value = bolus[state_vector.index(species_sym)]
                    # New species value is old amount / new volume.
                    bolus[state_vector.index(species_sym)] = (
                        species_value * compartment_sym / formula
                    )

            # Subtract the current species value from each species with an
            # update, as the bolus will be added on to the current species
            # value during simulation.
            for index in range(len(bolus)):
                if bolus[index] != get_empty_bolus_value():
                    bolus[index] -= state_vector[index]
                bolus[index] = bolus[index].subs(get_empty_bolus_value(),
                                                 sp.Float(0.0))

            self.symbols[SymbolId.EVENT][event_sym] = {
                'name': event_id,
                'value': trigger,
                'state_update': sp.MutableDenseMatrix(bolus),
                'initial_value':
                    trigger_sbml.getInitialValue() if trigger_sbml is not None
                    else True,
            }

    @log_execution_time('processing SBML observables', logger)
    def _process_observables(
        self,
        observables: Union[Dict[str, Dict[str, str]], None],
        sigmas: Dict[str, Union[str, float]],
        noise_distributions: Dict[str, NoiseDistDict]
    ) -> None:
        """
        Perform symbolic computations required for observable and objective
        function evaluation.

        :param observables:
            dictionary(observableId: {'name':observableName
            (optional), 'formula':formulaString})
            to be added to the model

        :param sigmas:
            dictionary(observableId: sigma value or (existing)
            parameter name)

        :param noise_distributions:
            dictionary(observableId: {'type': noise type, 'parameters':
            list with parameter values})
            See :py:func:`sbml2amici`.
        """

        _validate_observables(observables, sigmas, noise_distributions,
                              events=False)

        # add user-provided observables or make all species, and compartments
        # with assignment rules, observable
        if observables:
            # gather local symbols before parsing observable and sigma formulas
            for obs in observables.keys():
                self.add_local_symbol(obs, symbol_with_assumptions(obs))

            self.symbols[SymbolId.OBSERVABLE] = {
                symbol_with_assumptions(obs): {
                    'name': definition.get('name', f'y{iobs}'),
                    'value': self._sympy_from_sbml_math(
                        definition['formula']
                    ),
                    'transformation':
                        noise_distribution_to_observable_transformation(
                            noise_distributions.get(obs, {'type': 'normal',
                                                          'parameters': []})
                        )
                }
                for iobs, (obs, definition) in enumerate(observables.items())
            }
            # check for nesting of observables (unsupported)
            observable_syms = set(self.symbols[SymbolId.OBSERVABLE].keys())
            for obs in self.symbols[SymbolId.OBSERVABLE].values():
                if any(sym in observable_syms
                       for sym in obs['value'].free_symbols):
                    raise ValueError(
                        "Nested observables are not supported, "
                        f"but observable `{obs['name']} = {obs['value']}` "
                        "references another observable."
                    )
        elif observables is None:
            self._generate_default_observables()

        _check_symbol_nesting(self.symbols[SymbolId.OBSERVABLE],
                              'eventObservable')

        self._process_log_likelihood(sigmas, noise_distributions)

    @log_execution_time('processing SBML event observables', logger)
    def _process_event_observables(
            self,
            event_observables: Dict[str, Dict[str, str]],
            event_sigmas: Dict[str, Union[str, float]],
            event_noise_distributions: Dict[str, str]
    ) -> None:
        """
        Perform symbolic computations required for observable and objective
        function evaluation.

        :param event_observables:
            See :py:func:`sbml2amici`.

        :param event_sigmas:
            See :py:func:`sbml2amici`.

        :param event_noise_distributions:
            See :py:func:`sbml2amici`.
        """
        if event_observables is None:
            return

        _validate_observables(event_observables, event_sigmas,
                              event_noise_distributions,
                              events=True)

        # gather local symbols before parsing observable and sigma formulas
        for obs, definition in event_observables.items():
            self.add_local_symbol(obs, symbol_with_assumptions(obs))
            # check corresponding event exists
            if sp.Symbol(definition['event']) not in \
                    self.symbols[SymbolId.EVENT]:
                raise ValueError(
                    'Could not find an event with the event identifier '
                    f'{definition["event"]} for the event observable with name'
                    f'{definition["name"]}.'
                )

        self.symbols[SymbolId.EVENT_OBSERVABLE] = {
            symbol_with_assumptions(obs): {
                'name': definition.get('name', f'z{iobs}'),
                'value': self._sympy_from_sbml_math(
                    definition['formula']
                ),
                'event': sp.Symbol(definition.get('event')),
                'transformation':
                    noise_distribution_to_observable_transformation(
                        event_noise_distributions.get(obs, 'normal')  # TODO
                    )
            }
            for iobs, (obs, definition) in
            enumerate(event_observables.items())
        }

        wrong_t = sp.Symbol('t')
        for eo in self.symbols[SymbolId.EVENT_OBSERVABLE].values():
            if eo['value'].has(wrong_t):
                warnings.warn(f'Event observable {eo["name"]} uses `t` in '
                              'it\'s formula which is not the time variable. '
                              'For the time variable, please use `time` '
                              'instead!')

        # check for nesting of observables (unsupported)
        _check_symbol_nesting(self.symbols[SymbolId.EVENT_OBSERVABLE],
                              'eventObservable')

        self._process_log_likelihood(event_sigmas, event_noise_distributions,
                                     events=True)
        self._process_log_likelihood(event_sigmas, event_noise_distributions,
                                     events=True, event_reg=True)

    def _generate_default_observables(self):
        """
        Generate default observables from species, compartments and
        (initial) assignment rules.
        """
        self.symbols[SymbolId.OBSERVABLE] = {
            symbol_with_assumptions(f'y{species_id}'): {
                'name': specie['name'],
                'value': species_id
            }
            for species_id, specie
            in self.symbols[SymbolId.SPECIES].items()
        }

        for variable, formula in itt.chain(
                self.parameter_assignment_rules.items(),
                self.initial_assignments.items(),
                self.compartment_assignment_rules.items(),
                self.species_assignment_rules.items(),
                self.compartments.items()
        ):
            symbol = symbol_with_assumptions(f'y{variable}')
            # Assignment rules take precedence over compartment volume
            # definitions, so they need to be evaluated first.
            # Species assignment rules always overwrite.
            if symbol in self.symbols[SymbolId.OBSERVABLE] \
                    and variable not in self.species_assignment_rules:
                continue
            self.symbols[SymbolId.OBSERVABLE][symbol] = {
                'name': str(variable), 'value': formula
            }

    def _process_log_likelihood(self,
                                sigmas: Dict[str, Union[str, float]],
                                noise_distributions: Dict[str, NoiseDistDict],
                                events: bool = False,
                                event_reg: bool = False):
        """
        Perform symbolic computations required for objective function
        evaluation.

        :param sigmas:
            See :py:func:`SBMLImporter._process_observables`

        :param noise_distributions:
            See :py:func:`SBMLImporter._process_observables`

        :param events:
            indicates whether the passed definitions are for observables
            (False) or for event observables (True).

        :param event_reg:
            indicates whether log-likelihoods definitons should be processed
            for event observable regularization (Jrz). If this is activated,
            measurements are substituted by 0 and the observable by the
            respective regularization symbol.
        """

        if events:
            if event_reg:
                obs_symbol = SymbolId.EVENT_OBSERVABLE
                sigma_symbol = SymbolId.SIGMAZ
                llh_symbol = SymbolId.LLHRZ
            else:
                obs_symbol = SymbolId.EVENT_OBSERVABLE
                sigma_symbol = SymbolId.SIGMAZ
                llh_symbol = SymbolId.LLHZ
        else:
            assert not event_reg
            obs_symbol = SymbolId.OBSERVABLE
            sigma_symbol = SymbolId.SIGMAY
            llh_symbol = SymbolId.LLHY

        for obs_id, obs in self.symbols[obs_symbol].items():
            obs['measurement_symbol'] = generate_measurement_symbol(obs_id)
            if event_reg:
                obs['reg_symbol'] = generate_regularization_symbol(obs_id)

        if not event_reg:
            self.symbols[sigma_symbol] = {
                symbol_with_assumptions(f'sigma_{obs_id}'): {
                    'name': f'sigma_{obs["name"]}',
                    'value': self._sympy_from_sbml_math(
                        sigmas.get(str(obs_id), '1.0')
                    )
                }
                for obs_id, obs in self.symbols[obs_symbol].items()
            }

        self.symbols[llh_symbol] = {}
        for (obs_id, obs), (sigma_id, sigma) in zip(
                self.symbols[obs_symbol].items(),
                self.symbols[sigma_symbol].items()
        ):
            symbol = symbol_with_assumptions(f'J{obs_id}')
            dist = noise_distributions.get(
                str(obs_id), {'type': 'normal', 'parameters': []})
            cost_fun = noise_distribution_to_cost_function(dist)(obs_id)
            value = sp.sympify(cost_fun, locals=dict(zip(
                _get_str_symbol_identifiers(obs_id),
                (obs_id, obs['measurement_symbol'], sigma_id)
            )))
            if event_reg:
                value = value.subs(obs['measurement_symbol'], 0.0)
                value = value.subs(obs_id, obs['reg_symbol'])
            self.symbols[llh_symbol][symbol] = {
                    'name': f'J{obs["name"]}',
                    'value': value,
                    'dist': dist['type'],
                }

    @log_execution_time('processing SBML initial assignments', logger)
    def _process_initial_assignments(self):
        """
        Accounts for initial assignments of parameters and species
        references. Initial assignments for species and compartments are
        processed in :py:func:`amici.SBMLImporter._process_initial_species` and
        :py:func:`amici.SBMLImporter._process_compartments` respectively.
        """
        for ia in self.sbml.getListOfInitialAssignments():
            identifier = _get_identifier_symbol(ia)
            if identifier in itt.chain(self.symbols[SymbolId.SPECIES],
                                       self.compartments):
                continue

            sym_math = self._get_element_initial_assignment(ia.getId())
            if sym_math is None:
                continue

            sym_math = self._make_initial(smart_subs_dict(
                sym_math, self.symbols[SymbolId.EXPRESSION], 'value'
            ))
            self.initial_assignments[_get_identifier_symbol(ia)] = sym_math

        # sort and flatten
        self.initial_assignments = toposort_symbols(self.initial_assignments)
        for ia_id, ia in self.initial_assignments.items():
            self.initial_assignments[ia_id] = smart_subs_dict(
                ia, self.initial_assignments
            )

        for identifier, sym_math in list(self.initial_assignments.items()):
            self._replace_in_all_expressions(identifier, sym_math)

    @log_execution_time('processing SBML species references', logger)
    def _process_species_references(self):
        """
        Replaces species references that define anything but stoichiometries.

        Species references for stoichiometries are processed in
        :py:func:`amici.SBMLImporter._process_reactions`.
        """
        # doesnt look like there is a better way to get hold of those lists:
        species_references = _get_list_of_species_references(self.sbml)
        for species_reference in species_references:
            if hasattr(species_reference, 'getStoichiometryMath') and \
                    species_reference.getStoichiometryMath() is not None:
                raise SBMLException('StoichiometryMath is currently not '
                                    'supported for species references.')
            if species_reference.getId() == '':
                continue

            stoich = self._get_element_stoichiometry(species_reference)
            self._replace_in_all_expressions(
                _get_identifier_symbol(species_reference),
                self._sympy_from_sbml_math(stoich)
            )

    def _make_initial(self, sym_math: Union[sp.Expr, None, float]
                      ) -> Union[sp.Expr, None, float]:
        """
        Transforms an expression to its value at the initial time point by
        replacing species by their initial values.

        :param sym_math:
            symbolic expression
        :return:
            transformed expression
        """

        if not isinstance(sym_math, sp.Expr):
            return sym_math

        for species_id, species in self.symbols[SymbolId.SPECIES].items():
            if 'init' in species:
                sym_math = smart_subs(sym_math, species_id, species['init'])

        sym_math = smart_subs(sym_math, self._local_symbols['time'],
                              sp.Float(0))

        return sym_math

    def process_conservation_laws(self, ode_model) -> None:
        """
        Find conservation laws in reactions and species.

        :param ode_model:
            ODEModel object with basic definitions
        """
        conservation_laws = []

        # Create conservation laws for constant species
        species_solver = _add_conservation_for_constant_species(
            ode_model, conservation_laws
        )
        # Non-constant species processed here
        if "AMICI_EXPERIMENTAL_SBML_NONCONST_CLS" in os.environ \
                or "GITHUB_ACTIONS" in os.environ:
            species_solver = list(set(
                self._add_conservation_for_non_constant_species(
                    ode_model, conservation_laws)) & set(species_solver))

        # Check, whether species_solver is empty now. As currently, AMICI
        # cannot handle ODEs without species, CLs must be switched off in this
        # case
        if not len(species_solver):
            conservation_laws = []
            species_solver = list(range(ode_model.num_states_rdata()))

        # prune out species from stoichiometry and
        self.stoichiometric_matrix = \
            self.stoichiometric_matrix[species_solver, :]

        # add the found CLs to the ode_model
        for cl in conservation_laws:
            ode_model.add_conservation_law(**cl)

    def _get_conservation_laws_demartino(
            self,
            ode_model: ODEModel,
    ) -> List[Tuple[int, List[int], List[float]]]:
        """Identify conservation laws based on algorithm by DeMartino et al.
        (see conserved_moieties.py).

        :param ode_model: Model for which to compute conserved quantities
        :returns: List of one tuple per conservation law, each containing:
            (0) the index of the (solver-)species to eliminate,
            (1) (solver-)indices of all species engaged in the conserved
            quantity (including the eliminated one)
            (2) coefficients for the species in (1)
        """
        from .conserved_quantities_demartino \
            import compute_moiety_conservation_laws

        try:
            stoichiometric_list = [
                float(entry) for entry in self.stoichiometric_matrix.T.flat()
            ]
        except TypeError:
            # Due to the numerical algorithm currently used to identify
            #  conserved quantities, we can't have symbols in the
            #  stoichiometric matrix
            warnings.warn("Conservation laws for non-constant species in "
                          "combination with parameterized stoichiometric "
                          "coefficients are not currently supported "
                          "and will be turned off.")
            return []

        if any(rule.getTypeCode() == sbml.SBML_RATE_RULE
               for rule in self.sbml.getListOfRules()):
            # see SBML semantic test suite, case 33 for an example
            warnings.warn("Conservation laws for non-constant species in "
                          "models with RateRules are not currently supported "
                          "and will be turned off.")
            return []

        cls_state_idxs, cls_coefficients = compute_moiety_conservation_laws(
            stoichiometric_list, *self.stoichiometric_matrix.shape,
            rng_seed=32,
            species_names=[str(x.get_id()) for x in ode_model._states]
        )

        # Sparsify conserved quantities
        #  ``compute_moiety_conservation_laws`` identifies conserved quantities
        #  with positive coefficients. The resulting system is, therefore,
        #  often non-sparse. This leads to circular dependencies in the
        #  state expressions of eliminated states. The identified conserved
        #  quantities are linearly independent. We can construct `A` as in
        #  `A * x0 = total_cl` and bring it to reduced row echelon form. The
        #  pivot species are the ones to be eliminated. The resulting state
        #  expressions are sparse and void of any circular dependencies.
        A = sp.zeros(len(cls_coefficients), len(ode_model._states))
        for i_cl, (cl, coefficients) in enumerate(zip(cls_state_idxs,
                                                      cls_coefficients)):
            for i, c in zip(cl, coefficients):
                A[i_cl, i] = sp.Rational(c)
        rref, pivots = A.rref()

        raw_cls = []
        for i_cl, target_state_model_idx in enumerate(pivots):
            # collect values for species engaged in the current CL
            state_idxs = [i for i, coeff in enumerate(rref[i_cl, :])
                          if coeff]
            coefficients = [coeff for coeff in rref[i_cl, :] if coeff]
            raw_cls.append((target_state_model_idx, state_idxs,
                            coefficients),)
        return raw_cls

    def _get_conservation_laws_rref(
            self
    ) -> List[Tuple[int, List[int], List[float]]]:
        """Identify conservation laws based on left nullspace of the
        stoichiometric matrix, computed through (numeric) Gaussian elimination

        :returns: List of one tuple per conservation law, each containing:
            (0) the index of the (solver-)species to eliminate,
            (1) (solver-)indices of all species engaged in the conserved
            quantity (including the eliminated one)
            (2) coefficients for the species in (1)
        """
        import numpy as np
        from numpy.linalg import matrix_rank
        from .conserved_quantities_rref import nullspace_by_rref, rref

        try:
            S = np.asarray(self.stoichiometric_matrix, dtype=float)
        except TypeError:
            # Due to the numerical algorithm currently used to identify
            #  conserved quantities, we can't have symbols in the
            #  stoichiometric matrix
            warnings.warn("Conservation laws for non-constant species in "
                          "combination with parameterized stoichiometric "
                          "coefficients are not currently supported "
                          "and will be turned off.")
            return []

        if any(rule.getTypeCode() == sbml.SBML_RATE_RULE
               for rule in self.sbml.getListOfRules()):
            # see SBML semantic test suite, case 33 for an example
            warnings.warn("Conservation laws for non-constant species in "
                          "models with RateRules are not currently supported "
                          "and will be turned off.")
            return []

        # Determine rank via SVD
        rank = matrix_rank(S) if S.shape[0] else 0
        if rank == S.shape[0]:
            return []
        kernel = nullspace_by_rref(S.T)
        # Check dimensions - due to numerical errors, nullspace_by_rref may
        #  fail in certain situations
        if kernel.shape[0] != S.shape[0] - rank:
            raise AssertionError(
                "Failed to determine all conserved quantities "
                f"(found {kernel.shape[0]}, expected {S.shape[0] - rank}). "
                "Try another algorithm, disable detection of conservation "
                "laws, or submit a bug report along with the model."
            )
        kernel = rref(kernel)
        raw_cls = []
        for row in kernel:
            state_idxs = [i for i, coeff in enumerate(row) if coeff]
            coefficients = [coeff for coeff in row if coeff]
            raw_cls.append((state_idxs[0], state_idxs, coefficients),)

        return raw_cls

    def _add_conservation_for_non_constant_species(
        self,
        ode_model: ODEModel,
        conservation_laws: List[ConservationLaw]
    ) -> List[int]:
        """Add non-constant species to conservation laws

        :param ode_model:
            ODEModel object with basic definitions
        :param conservation_laws:
            List of already known conservation laws
        :returns:
            List of species indices which later remain in the ODE solver
        """
        # indices of retained species
        species_solver = list(range(ode_model.num_states_rdata()))

        algorithm = os.environ.get("AMICI_EXPERIMENTAL_SBML_NONCONST_CLS", "")
        if algorithm.lower() == "demartino":
            raw_cls = self._get_conservation_laws_demartino(ode_model)
        else:
            raw_cls = self._get_conservation_laws_rref()

        if not raw_cls:
            # no conservation laws identified
            return species_solver

        species_to_be_removed = {x[0] for x in raw_cls}

        # keep new conservations laws separate until we know everything worked
        new_conservation_laws = []
        # previously removed constant species
        eliminated_state_ids = {cl['state'] for cl in conservation_laws}

        all_state_ids = [x.get_id() for x in ode_model._states]
        all_compartment_sizes = [
            sp.Integer(1)
            if self.symbols[SymbolId.SPECIES][state_id]['amount']
            else self.compartments[
                self.symbols[SymbolId.SPECIES][state_id]['compartment']
            ]
            for state_id in all_state_ids
        ]

        # iterate over list of conservation laws, create symbolic expressions,
        for target_state_model_idx, state_idxs, coefficients in raw_cls:
            if all_state_ids[target_state_model_idx] in eliminated_state_ids:
                # constants state, already eliminated
                continue
            # collect values for species engaged in the current CL
            state_ids = [all_state_ids[i_state] for i_state in state_idxs]
            compartment_sizes = [all_compartment_sizes[i] for i in state_idxs]

            target_state_id = all_state_ids[target_state_model_idx]
            total_abundance = symbol_with_assumptions(f'tcl_{target_state_id}')

            new_conservation_laws.append({
                'state': target_state_id,
                'total_abundance': total_abundance,
                'coefficients': {
                     state_id: coeff * compartment
                     for state_id, coeff, compartment
                     in zip(state_ids, coefficients, compartment_sizes)
                },
            })
            species_to_be_removed.add(target_state_model_idx)

        conservation_laws.extend(new_conservation_laws)

        # list of species that are not determined by conservation laws
        return [ix for ix in species_solver if ix not in species_to_be_removed]

    def _replace_compartments_with_volumes(self):
        """
        Replaces compartment symbols in expressions with their respective
        (possibly variable) volumes.
        """
        for comp, vol in self.compartments.items():
            if comp in self.symbols[SymbolId.SPECIES]:
                # for comps with rate rules volume is only initial
                for species in self.symbols[SymbolId.SPECIES].values():
                    if isinstance(species['init'], sp.Expr):
                        species['init'] = smart_subs(species['init'],
                                                     comp, vol)
                continue
            self._replace_in_all_expressions(comp, vol)

    def _replace_in_all_expressions(self,
                                    old: sp.Symbol,
                                    new: sp.Expr,
                                    replace_identifiers=False) -> None:
        """
        Replace 'old' by 'new' in all symbolic expressions.

        :param old:
            symbolic variables to be replaced

        :param new:
            replacement symbolic variables
        """
        fields = [
            'stoichiometric_matrix', 'flux_vector',
        ]
        for field in fields:
            if field in dir(self):
                self.__setattr__(field, smart_subs(
                    self.__getattribute__(field), old, new
                ))

        dictfields = [
            'compartment_assignment_rules', 'parameter_assignment_rules',
            'initial_assignments'
        ]
        for dictfield in dictfields:
            d = getattr(self, dictfield)

            # replace identifiers
            if old in d and replace_identifiers:
                d[new] = d[old]
                del d[old]

            if dictfield == 'initial_assignments':
                tmp_new = self._make_initial(new)
            else:
                tmp_new = new

            # replace values
            for k in d:
                d[k] = smart_subs(d[k], old, tmp_new)

        # replace in identifiers
        if replace_identifiers:
            for symbol in [SymbolId.EXPRESSION, SymbolId.SPECIES]:
                # completely recreate the dict to keep ordering consistent
                if old not in self.symbols[symbol]:
                    continue
                self.symbols[symbol] = {
                    smart_subs(k, old, new): v
                    for k, v in self.symbols[symbol].items()
                }

            for symbol in [SymbolId.OBSERVABLE, SymbolId.LLHY,
                           SymbolId.SIGMAY]:
                if old not in self.symbols[symbol]:
                    continue
                self.symbols[symbol][new] = self.symbols[symbol][old]
                del self.symbols[symbol][old]

        # replace in values
        for symbol in [SymbolId.OBSERVABLE, SymbolId.LLHY, SymbolId.LLHZ,
                       SymbolId.SIGMAY, SymbolId.SIGMAZ, SymbolId.EXPRESSION,
                       SymbolId.EVENT, SymbolId.EVENT_OBSERVABLE]:
            if not self.symbols.get(symbol, None):
                continue
            for element in self.symbols[symbol].values():
                element['value'] = smart_subs(element['value'], old, new)

        # replace in event state updates (boluses)
        if self.symbols.get(SymbolId.EVENT, False):
            for event in self.symbols[SymbolId.EVENT].values():
                for index in range(len(event['state_update'])):
                    event['state_update'][index] = \
                        smart_subs(event['state_update'][index], old, new)

        if SymbolId.SPECIES in self.symbols:
            for species in self.symbols[SymbolId.SPECIES].values():
                species['init'] = smart_subs(species['init'],
                                             old, self._make_initial(new))

                fields = ['dt']
                if replace_identifiers:
                    fields.append('compartment')

                for field in ['dt']:
                    if field in species:
                        species[field] = smart_subs(species[field], old, new)

        # Initial compartment volume may also be specified with an assignment
        # rule (at the end of the _process_species method), hence needs to be
        # processed here too.
        self.compartments = {smart_subs(c, old, new) if replace_identifiers
                             else c:
                             smart_subs(v, old, self._make_initial(new))
                             for c, v in self.compartments.items()}

    def _clean_reserved_symbols(self) -> None:
        """
        Remove all reserved symbols from self.symbols
        """
        for sym in RESERVED_SYMBOLS:
            old_symbol = symbol_with_assumptions(sym)
            new_symbol = symbol_with_assumptions(f'amici_{sym}')
            self._replace_in_all_expressions(old_symbol, new_symbol,
                                             replace_identifiers=True)
            for symbols_ids, symbols in self.symbols.items():
                if old_symbol in symbols:
                    # reconstitute the whole dict in order to keep the ordering
                    self.symbols[symbols_ids] = {
                        new_symbol if k is old_symbol else k: v
                        for k, v in symbols.items()
                    }

    def _sympy_from_sbml_math(self, var_or_math: [sbml.SBase, str]
                              ) -> Union[sp.Expr, float, None]:
        """
        Sympify Math of SBML variables with all sanity checks and
        transformations

        :param var_or_math:
            SBML variable that has a getMath() function or math string
        :return:
            sympfified symbolic expression
        """
        if isinstance(var_or_math, sbml.SBase):
            math_string = sbml.formulaToL3StringWithSettings(
                var_or_math.getMath(),
                self.sbml_parser_settings
            )
            ele_name = var_or_math.element_name
        else:
            math_string = var_or_math
            ele_name = 'string'
        math_string = replace_logx(math_string)
        try:
            try:
                formula = sp.sympify(_parse_logical_operators(
                    math_string
                ), locals=self._local_symbols)
            except TypeError as err:
                if str(err) == 'BooleanAtom not allowed in this context.':
                    formula = sp.sympify(_parse_logical_operators(
                        math_string
                    ), locals={'true': sp.Float(1.0), 'false': sp.Float(0.0),
                               **self._local_symbols})
                else:
                    raise
        except (sp.SympifyError, TypeError, ZeroDivisionError) as err:
            raise SBMLException(f'{ele_name} "{math_string}" '
                                'contains an unsupported expression: '
                                f'{err}.')

        if isinstance(formula, sp.Expr):
            formula = _parse_special_functions_sbml(formula)
            _check_unsupported_functions_sbml(formula,
                                              expression_type=ele_name)
        return formula

    def _get_element_initial_assignment(self,
                                        element_id: str) -> Union[sp.Expr,
                                                                  None]:
        """
        Extract value of sbml variable according to its initial assignment

        :param element_id:
            sbml variable name
        :return:

        """
        assignment = self.sbml.getInitialAssignment(
            element_id
        )
        if assignment is None:
            return None
        sym = self._sympy_from_sbml_math(assignment)
        # this is an initial assignment so we need to use
        # initial conditions
        sym = self._make_initial(sym)
        return sym

    def _get_element_stoichiometry(self,
                                   ele: sbml.SBase) -> sp.Expr:
        """
        Computes the stoichiometry of a reactant or product of a reaction

        :param ele:
            reactant or product
        :return:
            symbolic variable that defines stoichiometry
        """
        if ele.isSetId():
            sym = self._get_element_initial_assignment(ele.getId())
            if sym is not None:
                return sym

            if self.is_assignment_rule_target(ele):
                return _get_identifier_symbol(ele)

        if ele.isSetStoichiometry():
            stoichiometry: float = ele.getStoichiometry()
            return sp.Integer(stoichiometry) if stoichiometry.is_integer() \
                else sp.Float(stoichiometry)

        return sp.Integer(1)

    def is_assignment_rule_target(self, element: sbml.SBase) -> bool:
        """
        Checks if an element has a valid assignment rule in the specified
        model.

        :param element:
            SBML variable

        :return:
            boolean indicating truth of function name
        """
        a = self.sbml.getAssignmentRuleByVariable(element.getId())
        return a is not None and self._sympy_from_sbml_math(a) is not None

    def is_rate_rule_target(self, element: sbml.SBase) -> bool:
        """
        Checks if an element has a valid assignment rule in the specified
        model.

        :param element:
            SBML variable

        :return:
            boolean indicating truth of function name
        """
        a = self.sbml.getRateRuleByVariable(element.getId())
        return a is not None and self._sympy_from_sbml_math(a) is not None


def _check_lib_sbml_errors(sbml_doc: sbml.SBMLDocument,
                           show_warnings: bool = False) -> None:
    """
    Checks the error log in the current self.sbml_doc.

    :param sbml_doc:
        SBML document

    :param show_warnings:
        display SBML warnings
    """
    num_warning = sbml_doc.getNumErrors(sbml.LIBSBML_SEV_WARNING)
    num_error = sbml_doc.getNumErrors(sbml.LIBSBML_SEV_ERROR)
    num_fatal = sbml_doc.getNumErrors(sbml.LIBSBML_SEV_FATAL)

    if num_warning + num_error + num_fatal:
        for i_error in range(sbml_doc.getNumErrors()):
            error = sbml_doc.getError(i_error)
            # we ignore any info messages for now
            if error.getSeverity() >= sbml.LIBSBML_SEV_ERROR \
                    or (show_warnings and
                        error.getSeverity() >= sbml.LIBSBML_SEV_WARNING):
                logger.error(f'libSBML {error.getCategoryAsString()} '
                             f'({error.getSeverityAsString()}):'
                             f' {error.getMessage()}')

    if num_error + num_fatal:
        raise SBMLException(
            'SBML Document failed to load (see error messages above)'
        )


def _parse_event_trigger(trigger: sp.Expr) -> sp.Expr:
    """
    Recursively translates a boolean trigger function into a real valued
    root function

    :param trigger:
    :return: real valued root function expression
    """
    # Events can be defined without trigger, i.e., the event will never fire.
    # In this case, set a dummy trigger:
    if trigger is None:
        return sp.Float(1.0)
    if trigger.is_Relational:
        root = trigger.args[0] - trigger.args[1]
        _check_unsupported_functions_sbml(root, 'sympy.Expression')

        # convert relational expressions into trigger functions
        if isinstance(trigger, (sp.core.relational.LessThan,
                                sp.core.relational.StrictLessThan)):
            # y < x or y <= x
            return -root
        if isinstance(trigger, (sp.core.relational.GreaterThan,
                                sp.core.relational.StrictGreaterThan)):
            # y >= x or y > x
            return root

    # or(x,y): any of {x,y} is > 0: sp.Max(x, y)
    if isinstance(trigger, sp.Or):
        return sp.Max(*[_parse_event_trigger(arg) for arg in trigger.args])
    # and(x,y): all out of {x,y} are > 0: sp.Min(x, y)
    if isinstance(trigger, sp.And):
        return sp.Min(*[_parse_event_trigger(arg) for arg in trigger.args])

    raise SBMLException(
        'AMICI can not parse piecewise/event trigger functions with argument '
        f'{trigger}.'
    )


def _parse_logical_operators(math_str: Union[str, float, None]
                             ) -> Union[str, float, None]:
    """
    Parses a math string in order to replace logical operators by a form
    parsable for sympy

    :param math_str:
        str with mathematical expression
    :param math_str:
        parsed math_str
    """
    if not isinstance(math_str, str):
        return math_str

    if ' xor(' in math_str or ' Xor(' in math_str:
        raise SBMLException('Xor is currently not supported as logical '
                            'operation.')

    return (math_str.replace('&&', '&')).replace('||', '|')


def assignmentRules2observables(sbml_model: sbml.Model,
                                filter_function: Callable = lambda *_: True):
    """
    Turn assignment rules into observables.

    :param sbml_model:
        Model to operate on

    :param filter_function:
        Callback function taking assignment variable as input and returning
        ``True``/``False`` to indicate if the respective rule should be
        turned into an observable.

    :return:
        A dictionary(observableId:{
        'name': observableName,
        'formula': formulaString
        })
    """
    observables = {}
    for rule in sbml_model.getListOfRules():
        if rule.getTypeCode() != sbml.SBML_ASSIGNMENT_RULE:
            continue
        parameter_id = rule.getVariable()
        if (p := sbml_model.getParameter(parameter_id)) \
                and filter_function(p):
            observables[parameter_id] = {
                'name': p.getName() if p.isSetName() else parameter_id,
                'formula': sbml_model.getAssignmentRuleByVariable(
                    parameter_id
                ).getFormula()
            }

    for parameter_id in observables:
        sbml_model.removeRuleByVariable(parameter_id)
        sbml_model.removeParameter(parameter_id)

    return observables


def _add_conservation_for_constant_species(
        ode_model: ODEModel,
        conservation_laws: List[ConservationLaw]
) -> List[int]:
    """
    Adds constant species to conservations laws

    :param ode_model:
        ODEModel object with basic definitions

    :param conservation_laws:
        List of already known conservation laws

    :returns species_solver:
        List of species indices which remain later in the ODE solver
    """

    # decide which species to keep in stoichiometry
    species_solver = list(range(ode_model.num_states_rdata()))

    # iterate over species, find constant ones
    for ix in reversed(range(ode_model.num_states_rdata())):
        if ode_model.state_is_constant(ix):
            # dont use sym('x') here since conservation laws need to be
            # added before symbols are generated
            target_state = ode_model._states[ix].get_id()
            total_abundance = symbol_with_assumptions(f'tcl_{target_state}')
            conservation_laws.append({
                'state': target_state,
                'total_abundance': total_abundance,
                'coefficients': {target_state: 1.0},
            })
            # mark species to delete from stoichiometric matrix
            species_solver.pop(ix)

    return species_solver


def _get_species_compartment_symbol(species: sbml.Species) -> sp.Symbol:
    """
    Generate compartment symbol for the compartment of a specific species.
    This function will always return the same unique python object for a
    given species name.

    :param species:
        sbml species
    :return:
        compartment symbol
    """
    return symbol_with_assumptions(species.getCompartment())


def _get_identifier_symbol(var: sbml.SBase) -> sp.Symbol:
    """
    Generate identifier symbol for a sbml variable.
    This function will always return the same unique python object for a
    given entity.

    :param var:
        sbml variable
    :return:
        identifier symbol
    """
    return symbol_with_assumptions(var.getId())


def get_species_initial(species: sbml.Species) -> sp.Expr:
    """
    Extract the initial concentration from a given species

    :param species:
        species index

    :return:
        initial species concentration
    """
    if species.isSetInitialConcentration():
        conc = species.getInitialConcentration()
        if species.getHasOnlySubstanceUnits():
            return sp.Float(conc) * _get_species_compartment_symbol(species)
        else:
            return sp.Float(conc)

    if species.isSetInitialAmount():
        amt = species.getInitialAmount()
        if math.isnan(amt):
            return sp.Float(0.0)

        if species.getHasOnlySubstanceUnits():
            return sp.Float(amt)
        else:
            return sp.Float(amt) / _get_species_compartment_symbol(species)

    return sp.Float(0.0)


def _get_list_of_species_references(sbml_model: sbml.Model) \
        -> List[sbml.SpeciesReference]:
    """
    Extracts list of species references as SBML doesn't provide a native
    function for this.

    :param sbml_model:
        SBML model instance

    :return:
        ListOfSpeciesReferences
    """
    return [
        reference
        for element in sbml_model.all_elements
        if isinstance(element, sbml.ListOfSpeciesReferences)
        for reference in element
    ]


def replace_logx(math_str: Union[str, float, None]) -> Union[str, float, None]:
    """
    Replace logX(.) by log(., X) since sympy cannot parse the former

    :param math_str:
        string for sympification

    :return:
        sympifiable string
    """
    if not isinstance(math_str, str):
        return math_str

    return re.sub(
        r'(^|\W)log(\d+)\(', r'\g<1>1/ln(\2)*ln(', math_str
    )


def _collect_event_assignment_parameter_targets(sbml_model: sbml.Model):
    targets = set()
    sbml_parameters = sbml_model.getListOfParameters()
    sbml_parameter_ids = [p.getId() for p in sbml_parameters]
    for event in sbml_model.getListOfEvents():
        for event_assignment in event.getListOfEventAssignments():
            target_id = event_assignment.getVariable()
            if target_id in sbml_parameter_ids:
                targets.add(_get_identifier_symbol(
                    sbml_parameters[sbml_parameter_ids.index(target_id)]
                ))
    return targets


def _check_unsupported_functions_sbml(sym: sp.Expr,
                                      expression_type: str,
                                      full_sym: Optional[sp.Expr] = None):
    try:
        _check_unsupported_functions(sym, expression_type, full_sym)
    except RuntimeError as err:
        raise SBMLException(str(err))


def _parse_special_functions_sbml(sym: sp.Expr,
                                  toplevel: bool = True) -> sp.Expr:
    try:
        return _parse_special_functions(sym, toplevel)
    except RuntimeError as err:
        raise SBMLException(str(err))


def _validate_observables(
    observables: Union[Dict[str, Dict[str, str]], None],
    sigmas: Dict[str, Union[str, float]],
    noise_distributions: Dict[str, NoiseDistDict],
    events: bool = False
) -> None:

    if observables is None or not observables:
        return

    # Ensure no non-existing observableIds have been specified
    # (no problem here, but usually an upstream bug)
    unknown_ids = set(sigmas.keys()) - set(observables.keys())
    if unknown_ids:
        raise ValueError(
            f"Sigma provided for unknown "
            f"{'eventO' if events else 'o'}bservableIds: "
            f"{unknown_ids}.")

    # Ensure no non-existing observableIds have been specified
    # (no problem here, but usually an upstream bug)
    unknown_ids = set(noise_distributions.keys()) - \
        set(observables.keys())
    if unknown_ids:
        raise ValueError(
            f"Noise distribution provided for unknown "
            f"{'eventO' if events else 'o'}bservableIds: "
            f"{unknown_ids}.")


def _check_symbol_nesting(symbols: Dict[sp.Symbol, Dict[str, sp.Expr]],
                          symbol_type: str):
    observable_syms = set(symbols.keys())
    for obs in symbols.values():
        if any(sym in observable_syms
               for sym in obs['value'].free_symbols):
            raise ValueError(
                "Nested observables are not supported, "
                f"but {symbol_type} `{obs['name']} = {obs['value']}` "
                "references another observable."
            )