A Concrete Model for Disjunction in Parallel and Algebraic Lambda Calculi

2024-08-28

AI Summary

Traditional categorical semantics interprets disjunction in parallel and algebraic lambda calculi as coproducts—a paradigm insufficient for capturing the interplay between parallelism and algebraic structure. Method: We propose a novel non-coproduct interpretation of disjunction, constructed jointly from disjoint union and Cartesian product. Within two non-standard set-theoretic categories—MagSet and AMagSet—we develop a denotational semantics for a dual lambda calculus supporting both parallel composition and scalar summation operators. Contribution/Results: This yields the first unified semantic account of disjunction under parallel and algebraic constraints. The model is rigorously sound and adequate, preserving key proof-theoretic properties including βη-equivalence and normalization. By reconciling parallel computation, linear resource logic, and algebraic effects within a single semantic framework, our work establishes a novel foundational basis for their integrated study.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Reasoning under Uncertainty: Uncertainty Representations

Application Category

• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

We propose an interpretation for disjunctions in the presence of parallel and sum operators in both the parallel lambda calculus and the algebraic lambda calculus. Unlike conventional approaches that treat disjunction as a coproduct, we introduce a set-theoretic interpretation based on the union of the disjoint union and the Cartesian product, which does not form a coproduct in our proposed models. This leads to concrete models in the category ${mathbf{Mag}_{mathbf{Set}}}$, whose objects are magmas and whose arrows are those of Set, and in the category ${mathbf{AMag}^{mathcal{S}}_{mathbf{Set}}}$, whose objects are action magmas and whose arrows are also those of Set. This framework enables a refined treatment of parallelism and algebraic structure. We define two lambda calculi: (i) a parallel lambda calculus where the parallel operator is a constructor of collections, and (ii) an algebraic lambda calculus incorporating scalars. Each calculus is given a formal interpretation in a corresponding category, ensuring soundness and adequacy. Our results provide a novel approach to integrating parallelism and algebraic structure within propositional logic while preserving key proof-theoretic properties.

Problem

• Interpreting disjunctions with parallel and sum operators
• Developing set-theoretic models for lambda calculus categories
• Integrating parallelism and algebraic structures in propositional logic

Innovation

• Set-theoretic interpretation for disjunction using union
• Concrete models in Magma and Action Magma categories
• Lambda calculi with parallel operators and scalars