[Paper Review] Infinitary and Cyclic Proof Systems for Transitive Closure Logic

Liron Cohen, Reuben N. S. Rowe|arXiv (Cornell University)|Jan 1, 2018

Logic, programming, and type systems19 references1 citations

TL;DR

This paper introduces an infinitary proof system for Transitive Closure Logic (TC logic) based on infinite descent, which is cut-free complete for standard semantics and subsumes explicit induction systems. It further shows that restricting to cyclic (regular) proofs yields an effective, automatable system for inductive reasoning, with a syntactic criterion ensuring completeness under Henkin semantics.

ABSTRACT

Transitive closure logic is a known extension of first-order logic obtained by introducing a transitive closure operator. While other extensions of first-order logic with inductive definitions are a priori parametrized by a set of inductive definitions, the addition of the transitive closure operator uniformly captures all finitary inductive definitions. In this paper we present an infinitary proof system for transitive closure logic which is an infinite descent-style counterpart to the existing (explicit induction) proof system for the logic. We show that, as for similar systems for first-order logic with inductive definitions, our infinitary system is complete for the standard semantics and subsumes the explicit system. Moreover, the uniformity of the transitive closure operator allows semantically meaningful complete restrictions to be defined using simple syntactic criteria. Consequently, the restriction to regular infinitary (i.e. cyclic) proofs provides the basis for an effective system for automating inductive reasoning.

Motivation & Objective

• To develop a complete, infinitary proof system for Transitive Closure Logic (TC logic) under standard semantics.
• To demonstrate that infinitary proofs based on infinite descent subsume explicit induction systems.
• To identify syntactic restrictions (regular/cyclic proofs) that yield effective, automatable systems for inductive reasoning.
• To establish completeness of cyclic proofs under Henkin semantics using simple syntactic criteria.
• To explore the relationship between implicit (infinitary) and explicit (LKID-style) induction in the context of TC logic.

Proposed method

• Proposes an infinitary proof system for TC logic using infinite-height, non-well-founded trees with infinite descent conditions.
• Employs the principle of infinite descent, where every infinite path in a proof traces terms or formulas to elements of a well-founded set.
• Introduces a uniform treatment of all inductive definitions via a single transitive closure operator, avoiding bespoke induction rules.
• Restricts infinitary proofs to cyclic (regular) proofs, which can be finitely represented as graphs, enabling automation.
• Uses syntactic criteria—specifically, cycle structure and term descent—to define a complete fragment under Henkin semantics.
• Compares the system to LKID and CLKIDω, showing TC logic subsumes LKID's inductive machinery and offers a more uniform framework.

Experimental results

Research questions

• RQ1 Can a cut-free, infinitary proof system for TC logic achieve completeness under standard semantics?
• RQ2 How does the infinitary system relate to explicit induction systems like LKID?
• RQ3 Can cyclic proofs in this system be effectively automated while preserving completeness?
• RQ4 Is there a syntactic criterion that ensures completeness of cyclic proofs under Henkin semantics?
• RQ5 What is the relative expressiveness of TC logic compared to LKID, and how do their proof systems compare?

Key findings

• The infinitary proof system is cut-free and complete for standard semantics, subsuming explicit induction systems.
• The system is sound and complete for Henkin semantics when restricted to cyclic proofs with a simple syntactic criterion.
• Cyclic proofs in the system can be finitely represented as graphs, enabling effective automation of inductive reasoning.
• The transitive closure operator uniformly captures all finitary inductive definitions, eliminating the need for bespoke induction rules.
• The system subsumes LKID-style formalizations, suggesting TC logic as a more uniform foundation for inductive reasoning.
• The relationship between RTCG and CRTCωG remains open, as does the equivalence of cyclic and non-cyclic systems in general.