A Logical 3-valued Semantics for Nondeterministic Choice

2026-07-22

AI Summary

This work addresses the formal challenges posed by errors and nondeterministic choices in reactive nondeterministic systems by introducing a novel three-valued symmetric nondeterministic disjunction operator. The proposed approach integrates Kleene’s tolerant semantics with Bochvar’s symmetric error propagation mechanism, thereby avoiding the asymmetry inherent in sequential evaluation strategies such as McCarthy logic. Logical connectives are defined within the framework of nondeterministic matrices (Nmatrices), yielding a semantics that supports genuinely symmetric nondeterministic behavior even in the presence of mixed error conditions. Crucially, the resulting logic strictly preserves commutativity and operational symmetry, enabling an accurate characterization of nondeterministic choice in concurrent computation.

Abstract

We propose a logical formalisation of computational errors in reactive, nondeterministic systems. To this aim, we introduce a new three-valued symmetric nondeterministic disjunction, designed to provide a faithful logical representation of the nondeterministic choice arising in concurrent computations. The connective is defined within the framework of nondeterministic matrices (Nmatrices) and derives from a minimal combination of Kleene's tolerant semantics and Bochvar's symmetric error persistence, thereby eliminating the residual asymmetry induced by sequential evaluation strategies such as McCarthy's logic. The resulting semantics admits genuinely nondeterministic outcomes in mixed cases involving errors, while preserving commutativity and operational symmetry.

Problem

• nondeterministic choice
• computational errors
• three-valued semantics
• reactive systems
• logical formalisation

Innovation

• three-valued semantics
• nondeterministic choice
• nondeterministic matrices
• symmetric disjunction
• computational errors