Web Algebra Formal Semantics

Abstract Type System

Primitive Types

URI        = Abstract URI reference
Literal    = Abstract literal value with optional datatype and language
BNode      = Abstract blank node identifier
Graph      = Abstract RDF graph
Term       = URI + Literal + BNode

Collection Types

Sequence α     = [α]                    -- Ordered list of elements
Result         = SPARQL SELECT result with variables and bindings  
ResultRow      = Single binding row from Result
VariableStack  = [Dict[String, Any]]    -- Stack of variable scopes
Context        = Any                    -- Current execution context (varies by operation)

Operation Types

Operation      = Abstract operation that can be executed
Expression     = Operation + Literal + Integer  -- Expressions for filtering

Concrete Python Type System

RDFLib Types

URI            = rdflib.URIRef
Literal        = rdflib.Literal  
BNode          = rdflib.BNode
Graph          = rdflib.Graph
Term           = Union[rdflib.URIRef, rdflib.Literal, rdflib.BNode]

Collection Types

Sequence       = List[Any]
Result         = rdflib.query.Result
ResultRow      = rdflib.query.ResultRow
VariableStack  = List[Dict[str, Any]]
Context        = Any

Execution Architecture

# Triple execution pattern - all operations implement:
def execute(*args: RDFLib_types) -> RDFLib_type           # Pure function with RDFLib terms
def execute_json(arguments: dict, variable_stack: list) -> Any  # JSON processing 
def mcp_run(arguments: dict, context: Any = None) -> Any   # MCP interface

Operation Catalog

Core System Operations

Value - Access variables and context values

Abstract: String × Context × VariableStack → Any
Python:   def execute(self, name: str, context: Any, variable_stack: List[Dict[str, Any]]) -> Any

Variable - Set variables in current scope (XSLT-style)

Abstract: String × Any × VariableStack → ⊥
Python:   def execute(self, name: str, value: Any, variable_stack: List[Dict[str, Any]]) -> None

Current - Return current context item

Abstract: Any → Any  
Python:   def execute(self, current_item: Any) -> Any

Execute - Execute nested operation

Abstract: Operation → Any
Python:   def execute(self, operation: Any) -> Any

URI - Convert term to URI reference

Abstract: Term → URI
Python:   def execute(self, term: rdflib.term.Node) -> rdflib.URIRef

ForEach - Map operation over sequence (sequence → sequence semantics)

Abstract: Sequence α × Operation → Sequence β
Python:   def execute(self, select_data: Union[List[Any], rdflib.query.Result], operation: Any) -> List[Any]

Filter - Filter sequences or select from results

Abstract: (Sequence α × Expression → α) + (Result × Expression → Result)
Python:   def execute(self, input_data: Any, expression: Any) -> Union[list, Any]

Bindings - Extract binding sequence from SPARQL results

Abstract: Result → Sequence ResultRow
Python:   def execute(self, table: rdflib.query.Result) -> List[Dict[str, Any]]

String Operations

Str - Convert any term to string literal

Abstract: Term → Literal
Python:   def execute(self, term: rdflib.term.Node) -> rdflib.Literal

Replace - Replace patterns in strings using regex

Abstract: Literal × Literal × Literal → Literal  
Python:   def execute(self, input_str: rdflib.Literal, pattern: rdflib.Literal, replacement: rdflib.Literal) -> rdflib.Literal

EncodeForURI - URL-encode strings for URI usage

Abstract: Literal → Literal
Python:   def execute(self, input_str: Literal) -> Literal

STRUUID - Generate random UUID string

Abstract: () → Literal
Python:   def execute(self) -> Literal

SPARQL Operations

SELECT - Execute SPARQL SELECT query

Abstract: URI × Literal → Result
Python:   def execute(self, endpoint: rdflib.URIRef, query: rdflib.Literal) -> rdflib.query.Result

CONSTRUCT - Execute SPARQL CONSTRUCT query

Abstract: URI × Literal → Graph
Python:   def execute(self, endpoint: rdflib.URIRef, query: rdflib.Literal) -> rdflib.Graph

DESCRIBE - Execute SPARQL DESCRIBE query

Abstract: URI × Literal → Graph  
Python:   def execute(self, endpoint: rdflib.URIRef, query: rdflib.Literal) -> rdflib.Graph

Substitute - Replace variables in SPARQL queries

Abstract: Literal × Literal × Term → Literal
Python:   def execute(self, query: Literal, var: Literal, binding_value: Any) -> Literal

SPARQLString - Generate SPARQL queries from natural language

Abstract: Literal → Literal
Python:   def execute(self, question: Literal) -> Literal

HTTP Operations

GET - Retrieve RDF data via HTTP GET

Abstract: URI → Graph
Python:   def execute(self, url: rdflib.URIRef) -> Graph

POST - Submit RDF data via HTTP POST

Abstract: URI × Graph → Result
Python:   def execute(self, url: rdflib.URIRef, data: rdflib.Graph) -> Result

PUT - Replace RDF data via HTTP PUT

Abstract: URI × Graph → Result
Python:   def execute(self, url: rdflib.URIRef, data: rdflib.Graph) -> Result

PATCH - Update RDF data via HTTP PATCH with SPARQL Update

Abstract: URI × Literal → Result
Python:   def execute(self, url: URIRef, update: Literal) -> Result

LinkedDataHub Operations

ldh-CreateContainer - Create LinkedDataHub container document

Abstract: URI × Literal × Maybe Literal × Maybe Literal → Result
Python:   def execute(self, parent_uri: rdflib.URIRef, title: rdflib.Literal, slug: rdflib.Literal = None, description: rdflib.Literal = None) -> Result

ldh-CreateItem - Create LinkedDataHub item document

Abstract: URI × Literal × Maybe Literal → Result
Python:   def execute(self, container_uri: rdflib.URIRef, title: rdflib.Literal, slug: Optional[rdflib.Literal] = None) -> Result

ldh-List - List LinkedDataHub resources

Abstract: URI × URI → List[Dict]
Python:   def execute(self, url: URIRef, endpoint: URIRef) -> list[dict]

ldh-AddView - Add view to LinkedDataHub document

Abstract: URI × URI × Literal × Maybe Literal × Maybe Literal × Maybe URI → Any  
Python:   def execute(self, url: URIRef, query: URIRef, title: Literal, description: Literal = None, fragment: Literal = None, mode: URIRef = None) -> Any

ldh-AddResultSetChart - Add result set chart to LinkedDataHub document

Abstract: URI × URI × Literal × URI × Literal × Literal × Maybe Literal × Maybe Literal → Any
Python:   def execute(self, url: URIRef, query: URIRef, title: Literal, chart_type: URIRef, category_var_name: Literal, series_var_name: Literal, description: Literal = None, fragment: Literal = None) -> Any

ldh-AddSelect - Add SPARQL SELECT service to LinkedDataHub

Abstract: URI × Literal × Literal × Maybe Literal × Maybe Literal × Maybe URI → Any
Python:   def execute(self, url: URIRef, query: Literal, title: Literal, description: Literal = None, fragment: Literal = None, service: URIRef = None) -> Any

ldh-AddGenericService - Add generic SPARQL service to LinkedDataHub

Abstract: URI × URI × Literal × Maybe Literal × Maybe Literal × Maybe URI × Maybe Literal × Maybe Literal → Any
Python:   def execute(self, url: URIRef, endpoint: URIRef, title: Literal, description: Literal = None, fragment: Literal = None, graph_store: URIRef = None, auth_user: Literal = None, auth_pwd: Literal = None) -> Any

ldh-AddObjectBlock - Add object content block to LinkedDataHub document

Abstract: URI × URI × Maybe Literal × Maybe Literal × Maybe Literal × Maybe URI → Any
Python:   def execute(self, url: URIRef, value: URIRef, title: Literal = None, description: Literal = None, fragment: Literal = None, mode: URIRef = None) -> Any

ldh-AddXHTMLBlock - Add XHTML content block to LinkedDataHub document

Abstract: URI × Literal × Maybe Literal × Maybe Literal × Maybe Literal → Any
Python:   def execute(self, url: URIRef, value: Literal, title: Literal = None, description: Literal = None, fragment: Literal = None) -> Any

ldh-RemoveBlock - Remove content block from LinkedDataHub document

Abstract: URI × Maybe URI → Any  
Python:   def execute(self, url: URIRef, block: URIRef = None) -> Any

Schema Operations

ExtractClasses - Extract RDF classes from graph

Abstract: URI → Graph
Python:   def execute(self, endpoint: URIRef) -> rdflib.Graph

ExtractDatatypeProperties - Extract datatype properties from graph

Abstract: URI → Graph  
Python:   def execute(self, endpoint: URIRef) -> rdflib.Graph

ExtractObjectProperties - Extract object properties from graph

Abstract: URI → Graph
Python:   def execute(self, endpoint: URIRef) -> rdflib.Graph

Utility Operations

Merge - Merge multiple RDF graphs into one

Abstract: Sequence Graph → Graph
Python:   def execute(self, graphs: List[rdflib.Graph]) -> rdflib.Graph

ResolveURI - Resolve relative URI against base URI

Abstract: URI × Literal → URI
Python:   def execute(self, base: URIRef, relative: Literal) -> URIRef

Type System Properties

Strict Type Checking

All operations enforce strict input type checking
TypeError raised for mismatched input types with informative messages
No automatic type casting or conversion
RDFLib types must match exactly as specified in signatures

Execution Architecture

execute(): Pure functions operating on RDFLib types only
execute_json(): JSON processing layer that calls execute() with type validation
mcp_run(): MCP interface layer that calls execute() with plain arguments

Sequence Semantics

ForEach: Maps from sequences to sequences (sequence → sequence)
Context: In ForEach, context is the current sequence item (ResultRow for JSONResult)
Filter: Can operate on both sequences and JSONResult tables
Automatic Application: Single-item operations applied element-wise to sequences in execution layer

Variable System

Lexical Scoping: Variables follow XSLT-style lexical scoping rules
Variable Stack: Maintains nested scopes for variable resolution
Syntax: $variableName for variable references, variableName for context access
Variable: Sets variables in current scope, Variable operation manages the stack

Context System

Context Type: Any - varies by operation and execution context
ForEach Context: Current sequence item (ResultRow for SPARQL results)
Current Operation: Returns the current context item unchanged
Value Operation: Accesses both context values and variables from stack

URI Resolution

JSON-LD Parsing: All operations that parse JSON-LD use base parameter to resolve relative URIs
Fragment URIs: Fragment identifiers like #service resolve against target document URI
Base URI: Set to the target document URI for correct resolution of relative references
Implementation: Uses rdflib.Graph.parse(data=json_data, format="json-ld", base=target_url)

Provide feedback

Saved searches

Use saved searches to filter your results more quickly

Web Algebra Formal Semantics

Abstract Type System

Primitive Types

Collection Types

Operation Types

Concrete Python Type System

RDFLib Types

Collection Types

Execution Architecture

Operation Catalog

Core System Operations

String Operations

SPARQL Operations

HTTP Operations

LinkedDataHub Operations

Schema Operations

Utility Operations

Type System Properties

Strict Type Checking

Execution Architecture

Sequence Semantics

Variable System

Context System

URI Resolution

FilesExpand file tree

formal-semantics.md

Latest commit

History

formal-semantics.md

File metadata and controls

Web Algebra Formal Semantics

Abstract Type System

Primitive Types

Collection Types

Operation Types

Concrete Python Type System

RDFLib Types

Collection Types

Execution Architecture

Operation Catalog

Core System Operations

String Operations

SPARQL Operations

HTTP Operations

LinkedDataHub Operations

Schema Operations

Utility Operations

Type System Properties

Strict Type Checking

Execution Architecture

Sequence Semantics

Variable System

Context System

URI Resolution