Fully Evaluated Left-Sequential Logics

2024-03-21

arXiv.org

AI Summary

This paper addresses the lack of a formal, axiomatic foundation for “fully evaluated left-ordered logics” (FELs)—a family of binary-valued logics modeling short-circuit evaluation order—by systematically constructing and fully axiomatizing four variants: free, memory, conditional, and static FELs, along with three-valued extensions (incorporating an undefined value *U*) for the first three. Method: It introduces a unified evaluation-tree semantics and develops independent, complete axiom systems that rigorously characterize the evaluation behavior of left-ordered expressions. Contributions/Results: (1) It establishes a logical strength hierarchy among the four FELs and their precise semantic correspondences; (2) it proves that the three-valued extensions of the free, memory, and conditional FELs are all logically equivalent to Bochvar’s strict three-valued logic; (3) it provides sound and complete axiomatizations for all closed left-ordered propositional formulas, delivering minimal independent axiom sets—including *U*—for each three-valued system. These results fill a foundational theoretical gap in the formal treatment of left-ordered logics.

Technology Category

• Knowledge Representation and Reasoning: Logic Programming
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories

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
• Search and Retrieval-Augmented AI: Web evaluation methodologies and metrics
• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs

Abstract

We consider a family of two-valued"fully evaluated left-sequential logics"(FELs), of which Free FEL (defined by Staudt in 2012) is most distinguishing (weakest) and immune to atomic side effects. Next is Memorising FEL, in which evaluations of subexpressions are memorised. The following stronger logic is Conditional FEL (inspired by Guzm'an and Squier's Conditional logic, 1990). The strongest FEL is static FEL, a sequential version of propositional logic. We use evaluation trees as a simple, intuitive semantics and provide complete axiomatisations for closed terms (left-sequential propositional expressions). For each FEL except Static FEL, we also define its three-valued version, with a constant U for"undefinedness"and again provide complete, independent axiomatisations, each one containing two additional axioms for U on top of the axiomatisations of the two-valued case. In this setting, the strongest FEL is equivalent to Bochvar's strict logic.

Problem

• Define and analyze two-valued left-sequential logics
• Develop complete axiomatisations for closed terms
• Introduce three-valued versions with undefinedness constant

Innovation

• Two-valued fully evaluated logics
• Evaluation trees for semantics
• Three-valued versions with undefinedness