[Paper Review] Labeled Natural Deduction Systems for a Family of Tense Logics

Luca Viganò, Marco Volpe|ArXiv.org|Mar 21, 2008

Advanced Algebra and Logic13 references4 citations

TL;DR

This paper presents labeled natural deduction systems for a family of tense logics extending basic linear tense logic Kl, using labeled formulas and relational rules with full first-order syntax to capture temporal properties like linearity and irreflexivity. The system is proven sound and complete, with derivations normalizing to a form that satisfies the subformula property, and is extendable to richer logics such as fragments of LTL.

ABSTRACT

We give labeled natural deduction systems for a family of tense logics extending the basic linear tense logic Kl. We prove that our systems are sound and complete with respect to the usual Kripke semantics, and that they possess a number of useful normalization properties (in particular, derivations reduce to a normal form that enjoys a subformula property). We also discuss how to extend our systems to capture richer logics like (fragments of) LTL.

Motivation & Objective

• To develop a uniform, Gentzen-style natural deduction framework for tense logics extending Kl, overcoming limitations of traditional modal labeling techniques.
• To address the need for non-Horn relational rules (e.g., disjunctions and universal quantifiers) to capture temporal properties like linearity and irreflexivity.
• To establish soundness and completeness of the labeled system with respect to Kripke semantics for tense logic.
• To prove normalization and subformula properties in derivations, ensuring proof-theoretic clarity and usability.
• To extend the system to richer temporal logics such as fragments of LTL, enabling practical reasoning about linear time properties.

Proposed method

• Labeled formulas of the form x:A are used to represent that formula A holds at world x, with relational formulas x<y expressing accessibility between time points.
• The system employs full first-order logic for relational reasoning, including disjunctions (e.g., x<y ⊔ x=y ⊔ y<x) and universal quantifiers to express linearity and irreflexivity.
• A universal falsum (⊥) is introduced to allow bidirectional propagation of contradictions between formula and relational components, breaking the strict separation seen in modal logic systems.
• Derivations are constructed using standard natural deduction rules for propositional connectives and temporal operators (G and H), with specialized rules for relational reasoning.
• Normalization is achieved via a track-based analysis of derivations, ensuring that all eliminations precede introductions and maximal formulas are eliminated.
• The system integrates relational and formula derivations by allowing lwffs to depend on rwffs and vice versa through the universal falsum, enabling completeness in richer logics.

Experimental results

Research questions

• RQ1 Can labeled natural deduction systems be designed for tense logics that require non-Horn relational rules, such as disjunctions and universal quantifiers?
• RQ2 How can the separation between formula and relational reasoning be maintained or relaxed to ensure completeness in the presence of complex temporal axioms?
• RQ3 Can normalization and the subformula property be preserved in labeled natural deduction systems for tense logic, despite the inclusion of full first-order relational reasoning?
• RQ4 What modifications are needed to extend such systems to capture richer temporal logics like fragments of LTL?
• RQ5 How does the introduction of a universal falsum enable proof-theoretic control in systems where formula and relational derivations are interdependent?

Key findings

• The labeled natural deduction system N(Kl) is proven sound and complete with respect to Kripke semantics for the basic linear tense logic Kl.
• Derivations in the system normalize to a normal form that satisfies the subformula property, ensuring that all formulas in a derivation are subformulas of the conclusion or assumptions.
• The system successfully captures linearity and irreflexivity of time points using full first-order relational axioms, such as ∀x,y. (x<y) ⊔ (x=y) ⊔ (y<x).
• The use of a universal falsum allows for bidirectional contradiction propagation between formula and relational components, enabling completeness in the presence of interdependent derivations.
• The system is extendable to richer logics, including fragments of LTL, by incorporating additional temporal operators and relational constraints.
• Normalization is achieved through a track-based analysis of derivations, where elimination rules are applied before introduction rules, and maximal formulas are eliminated, ensuring proof-theoretic coherence.