[Paper Review] Compositional Semantics for the Procedural Interpretation of Logic

Van Emden|arXiv (Cornell University)|Feb 28, 2006

Logic, programming, and type systems16 references5 citations

TL;DR

This paper proposes a compositional semantics for logic programs by completing the abstract syntax of the procedural interpretation of logic, enabling meaning to be built from constituent parts via mathematical operations. It establishes equivalence between this compositional semantics and the standard immediate-consequence operator, thereby providing a rigorous, syntax-driven foundation for logic programming semantics.

ABSTRACT

Semantics of logic programs has been given by proof theory, model theory and by fixpoint of the immediate-consequence operator. If clausal logic is a programming language, then it should also have a compositional semantics. Compositional semantics for programming languages follows the abstract syntax of programs, composing the meaning of a unit by a mathematical operation on the meanings of its constituent units. The procedural interpretation of logic has only yielded an incomplete abstract syntax for logic programs. We complete it and use the result as basis of a compositional semantics. We present for comparison Tarski's algebraization of first-order predicate logic, which is in substance the compositional semantics for his choice of syntax. We characterize our semantics by equivalence with the immediate-consequence operator.

Motivation & Objective

• To address the lack of a complete abstract syntax for the procedural interpretation of logic programs.
• To develop a compositional semantics that builds program meaning from constituent parts using mathematical operations.
• To establish equivalence between the proposed compositional semantics and the standard immediate-consequence operator in logic programming.
• To complete the abstract syntax framework so that it supports full compositional meaning assignment.

Proposed method

• Completing the abstract syntax of the procedural interpretation of logic to support compositional meaning construction.
• Using Tarski’s algebraization of first-order predicate logic as a benchmark for compositional semantics.
• Defining a meaning function that composes the semantics of program components via algebraic operations.
• Establishing a formal equivalence between the compositional semantics and the immediate-consequence operator.
• Applying the semantics to clausal logic as a programming language, ensuring consistency with established model-theoretic foundations.
• Using fixpoint theory to validate that the compositional semantics aligns with the standard semantics of logic programs.

Experimental results

Research questions

• RQ1 How can a compositional semantics be defined for logic programs based on the procedural interpretation of logic?
• RQ2 What abstract syntax is required to support a fully compositional semantics in logic programming?
• RQ3 How does the proposed compositional semantics relate to the standard immediate-consequence operator?
• RQ4 Can the compositional semantics be formally shown equivalent to the fixpoint semantics of logic programs?
• RQ5 What role does Tarski’s algebraization of first-order logic play in validating the proposed semantics?

Key findings

• The paper successfully completes the abstract syntax for the procedural interpretation of logic, enabling compositional meaning assignment.
• The proposed compositional semantics is formally equivalent to the immediate-consequence operator, validating its correctness.
• The semantics is grounded in algebraic operations that compose meanings of program constituents, ensuring modularity and clarity.
• Tarski’s algebraization of first-order logic serves as a foundational reference, demonstrating the feasibility of compositional semantics in logical systems.
• The approach provides a syntax-driven, mathematically rigorous semantics for logic programs, filling a gap in procedural interpretation.