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PolyglotFormalisms.Elixir

Elixir implementation of the PolyglotFormalisms Common Library specification.

Table of Contents

Overview

This package provides Elixir implementations of fundamental operations defined in the PolyglotFormalisms specification, enabling semantic equivalence verification across multiple programming languages.

Modules

Arithmetic

  • add(a, b) — Addition

  • subtract(a, b) — Subtraction

  • multiply(a, b) — Multiplication

  • divide(a, b) — Division

  • modulo(a, b) — Modulo (integer operation)

Comparison

  • less_than(a, b) — Strict less than

  • greater_than(a, b) — Strict greater than

  • equal(a, b) — Equality

  • not_equal(a, b) — Inequality

  • less_equal(a, b) — Less than or equal

  • greater_equal(a, b) — Greater than or equal

Logical

  • logical_and(a, b) — Logical conjunction

  • logical_or(a, b) — Logical disjunction

  • logical_not(a) — Logical negation

Note
 Function names include logical_ prefix to avoid conflicts with Elixir’s Kernel reserved keywords.

Installation

def deps do
  [
    {:polyglot_formalisms, "~> 0.2.0"}
  ]
end

Usage

alias PolyglotFormalisms.{Arithmetic, Comparison, Logical}

sum = Arithmetic.add(2.0, 3.0)
product = Arithmetic.multiply(4.0, 5.0)
is_less = Comparison.less_than(2.0, 3.0)
is_equal = Comparison.equal(5.0, 5.0)
both_true = Logical.logical_and(true, true)
either_true = Logical.logical_or(false, true)
negated = Logical.logical_not(false)

Mathematical Properties

Arithmetic Properties

  • Commutativity (for add, multiply)

  • Associativity (for add, multiply)

  • Identity elements

  • Distributivity

Comparison Properties

  • Transitivity

  • Reflexivity

  • Symmetry

  • Asymmetry

Logical Properties

  • Commutativity

  • Associativity

  • Distributivity

  • De Morgan’s laws

  • Excluded middle

  • Non-contradiction

Behavioral Semantics

Float Operations

  • Division by zero returns Infinity or -Infinity

  • NaN propagation follows IEEE 754

  • Comparison with NaN returns false

Integer Operations

  • Modulo uses Erlang rem operator semantics

  • Modulo by zero raises ArithmeticError (BEAM behavior)

Boolean Operations

  • Short-circuit evaluation for and and or

  • Eager evaluation for wrapper functions

Testing

mix test
mix test --include doctest

Cross-Language Verification

This Elixir implementation is semantically equivalent to:

  • Julia implementation (PolyglotFormalisms.jl)

  • ReScript implementation (alib-for-rescript)

  • Gleam implementation (polyglot_formalisms_gleam)

Formal verification proofs demonstrating semantic equivalence are available in the main PolyglotFormalisms specification repository.

Documentation

mix docs

Architecture

See TOPOLOGY.md for a visual architecture map and completion dashboard.

Related Projects

Wondering how this works? See EXPLAINME.adoc.

License

SPDX-License-Identifier: CC-BY-SA-4.0
See LICENSE.