Semantic Properties of Computations Defined by Elementary Inference Systems

2025-10-30

AI Summary

This paper addresses the verification of semantic properties—such as program correctness and termination—for sets, relations, and computations defined by elementary inference systems. To overcome the fundamental limitation that canonical models are often noncomputable, we propose a novel method that eschews reliance on canonical models entirely: instead, semantic properties are decided via first-order satisfiability in *arbitrary* models. Technically, we formalize inference systems as Gentzen-style elementary deductive systems, integrate Horn clause theories with proof-tree structural modeling, and leverage automated first-order satisfiability checking for verification. Our principal contribution is a general logical decision framework for rewriting-based computational models (e.g., programming language semantics), enabling formal, machine-checkable proofs of semantic property validity or invalidity. This approach significantly enhances both the practical applicability and decidability of semantic analysis.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Automated Reasoning and Theorem Proving
• Cognitive Modeling & Cognitive Systems: Conceptual Inference and Reasoning

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: Foundation models and LLMs for Web-related graphs
• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems

Abstract

We consider sets/relations/computations defined by *Elementary Inference Systems* I, which are obtained from Smullyan's *elementary formal systems* using Gentzen's notation for inference rules, and proof trees for atoms P(t_1,...,t_n), where predicate P represents the considered set/relation/computation. A first-order theory Th(I), actually a set of definite Horn clauses, is given to I. Properties of objects defined by I are expressed as first-order sentences F, which are proved true or false by *satisfaction* M |= F of F in a *canonical* model M of Th(I). For this reason, we call F a *semantic property* of I. Since canonical models are, in general, incomputable, we show how to (dis)prove semantic properties by satisfiability in an *arbitrary* model A of Th(I). We apply these ideas to the analysis of properties of programming languages and systems whose computations can be described by means of an elementary inference system. In particular, rewriting-based systems.

Problem

• Analyzing semantic properties of elementary inference systems
• Proving semantic properties via arbitrary model satisfiability
• Applying formal analysis to programming language computations

Innovation

• Elementary Inference Systems define computations
• Canonical models analyze semantic properties
• Arbitrary models prove semantic properties