[Paper Review] A Logical Framework for Convergent Infinite Computations

Wei Li, Shilong Ma+2 more|ArXiv.org|May 10, 2001

Logic, Reasoning, and Knowledge15 references3 citations

TL;DR

This paper proposes a logical framework for convergent infinite computations by extending first-order logic using Cauchy sequences to represent limits of infinite sequences of theories and models. It introduces infinitary terms and real Herbrand models, establishes a distance-based limit definition analogous to $C\epsilon$-$N$ convergence, and proves that the limit of a convergent sequence of Horn logic programs admits a real Herbrand model, enabling sound computation of infinite limits.

ABSTRACT

Classical computations can not capture the essence of infinite computations very well. This paper will focus on a class of infinite computations called convergent infinite computations}. A logic for convergent infinite computations is proposed by extending first order theories using Cauchy sequences, which has stronger expressive power than the first order logic. A class of fixed points characterizing the logical properties of the limits can be represented by means of infinite-length terms defined by Cauchy sequences. We will show that the limit of sequence of first order theories can be defined in terms of distance, similar to the $ε-N$ style definition of limits in real analysis. On the basis of infinitary terms, a computation model for convergent infinite computations is proposed. Finally, the interpretations of logic programs are extended by introducing real Herbrand models of logic programs and a sufficient condition for computing a real Herbrand model of Horn logic programs using convergent infinite computation is given.

Motivation & Objective

• To address the limitations of classical first-order logic in capturing the semantics of infinite computations, particularly those that converge to a limit.
• To formalize the notion of convergence in infinite computations using a metric-based limit definition for sequences of first-order theories.
• To extend logic program semantics by introducing real Herbrand models that capture the limits of infinite sequences of Herbrand models.
• To establish a sufficient condition under which convergent infinite computation can correctly compute the real Herbrand model of a Horn logic program.
• To provide a computation model based on infinitary terms defined via Cauchy sequences, enabling symbolic computation over infinite structures.

Proposed method

• Extends first-order theories using Cauchy sequences of terms to represent infinite-length terms and model limits.
• Defines the limit of a sequence of first-order theories via a distance-based $C\epsilon$-$N$ style criterion, analogous to real analysis.
• Introduces infinitary terms as limits of Cauchy sequences of finite terms, enabling representation of fixed points and limit behaviors.
• Proposes a computation model based on these infinitary terms, supporting continuous symbolic computation over infinite domains.
• Introduces real Herbrand models as limits of Cauchy sequences of standard Herbrand models, extending logic program semantics.
• Uses a Lipschitz-like condition on substitutions to ensure that head terms form Cauchy sequences when body terms do, preserving convergence.

Experimental results

Research questions

• RQ1 How can the semantics of infinite computations that converge to a limit be formally captured using logical frameworks?
• RQ2 Can the limit of a sequence of first-order theories be defined using a metric-based, $C\epsilon$-$N$ style convergence analogous to real analysis?
• RQ3 How can logic programs be semantically extended to model infinite computations whose Herbrand models converge?
• RQ4 Under what conditions does the limit of a sequence of Horn logic programs admit a real Herbrand model that is the limit of their individual models?
• RQ5 What is the role of Cauchy sequences of substitutions and terms in ensuring convergence of logical computations?

Key findings

• The limit of a sequence of first-order theories can be formally defined using a distance metric in an $C\epsilon$-$N$ style, generalizing real analysis convergence to logical theories.
• A class of fixed points characterizing limit properties can be represented using infinitary terms constructed as limits of Cauchy sequences of finite terms.
• The sequence of least Herbrand models of a convergent sequence of Horn logic programs forms a Cauchy sequence, and its limit is a real Herbrand model of the limit program.
• The limit of a convergent sequence of Horn logic programs admits a real Herbrand model, and this model can be computed via convergent infinite computation.
• The convergence of body terms under a substitution implies the convergence of head terms, satisfying a Lipschitz-like condition with constant 1.
• In the example, $\lim_{k\to\infty} M_k = \{p(f^\infty(a))\}$, and this limit is a real Herbrand model of $\Gamma = \{p(f(x)) \leftarrow p(x)\}$, confirming the framework's correctness.