Finite-Context Semantics in Finitely Supported Structures

2026-09-25

AI Summary

This study addresses the semantic incompleteness of systems with names and data, arising from the non-complete lattice structure of finitely supported predicate spaces, which impedes fixed-point existence guarantees. To overcome this, the work develops a theory of finitely supported structures over arbitrary permutation groups, introducing support-shifting bounds and uniform finiteness. By integrating group theory, invariant logic, and ultrahomogeneous atomic structures with abstract interpretation and resource rewriting techniques, it establishes a rigorous formalization framework. The authors prove that monotone transformers admit least and greatest fixed points under specific conditions, revealing a novel mechanism whereby Boolean predicates achieve finite iterative convergence even with infinitely many orbits, thereby decoupling semantic existence, convergence, and computation. These results are successfully applied to automata theory and operational semantics, validating the effectiveness of the proposed approach.

Abstract

Semantics for systems with names and data often depends on finitely many distinguished values. The theory of finitely supported structures treats this dependence through invariance under permutations fixing a finite context. We develop this approach to semantics over arbitrary permutation groups and arbitrary infinite sets of atoms. The central difficulty is that finitely supported predicate spaces need not be complete lattices. We prove that, for predicates valued in a complete lattice with trivial atom action, every monotone finitely supported transformer nevertheless has least and greatest fixed points. These lie in the complete lattice determined by the transformer's context and coincide with the fixed points of every compatible monotone ambient extension equivariant under its stabilizer. Support-transfer bounds track dependencies through semantic constructions, while uniform finiteness yields finite convergence. For Boolean predicates, $m$ context-stabilizer orbits on the carrier suffice for convergence after at most $m$ iterations, even when the supported predicate lattice is orbit-infinite. We apply these results to automata, operational and modal semantics, abstract interpretation, and resource rewriting. Ultrahomogeneous atom structures in finite relational signatures yield finite cell representations, illustrated by an authorization monitor. The resulting account separates semantic existence, finite convergence, and effective computation.

Problem

• finitely supported structures
• finite-context semantics
• complete lattices
• fixed points
• permutation groups

Innovation

• Finitely Supported Structures
• Fixed-Point Theory
• Support-Transfer Bounds
• Finite Convergence
• Ultrahomogeneous Atom Structures