LPrL: An Asynchronous Linear Time Hyper Logic

2025-05-10

AI Summary

HyperLTL suffers from undecidability of satisfiability and non-elementary complexity of model checking, limiting its practical applicability for specifying hyperproperties in distributed systems. To address this, we propose LPrL (Linear-time Property Logic with asynchronous traces), a decidable asynchronous hyperlogic that extends first-order logic with term languages, equality predicates, and cross-trace first-order quantification—enabling precise specification of security and consistency properties in distributed settings. Theoretically, LPrL establishes the first decidable asynchronous hyperlogic framework. Technically, we integrate extensions of first-order logic, Boolean combinations of LTL formulas, and automata-theoretic constructions to devise sound and complete algorithms for both satisfiability checking and model checking. Crucially, both problems are shown to be elementary—specifically, solvable in exponential time—thereby achieving a fundamental breakthrough: transitioning from undecidability to decidability and from non-elementary to elementary complexity. This significantly enhances both the expressiveness and practical utility of hyperproperty logics.

Technology Category

• Knowledge Representation and Reasoning: Description Logics
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Machine Learning: Statistical Relational/Logic Learning

Application Category

• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Security and Privacy: Data transparency and provenance
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs

Abstract

We present a novel asynchronous hyper linear time temporal logic named LPrL (Linear Time Predicate Logic) and establish its basic theory. LPrL is a natural first order extension of LTL (Linear time temporal logic), in which the predicates specify the properties of and the relationships between traces (infinite sequences of actions) using Boolean combinations of LTL formulas. To augment the expressive power of the logic, we introduce a simple language of terms and add the equality predicate t = t' where t and t' are terms. We first illustrate how a number of the security policies as well as a basic consistency property of distributed processes can be captured using LPrL. We then establish our main results using automata theoretic techniques. Namely, the satisfiability and model checking problems for LPrL can be solved in elementary time. These results are in sharp contrast to HyperLTL, the prevalent synchronous hyper linear time logic, whose satisfiability problem is undecidable and whose model checking problem has non-elementary time complexity.

Problem

• Extends LTL to first-order logic for trace properties
• Introduces terms and equality to enhance expressiveness
• Solves satisfiability and model checking in elementary time

Innovation

• Asynchronous hyper linear time logic LPrL
• First order extension of LTL with predicates
• Elementary time satisfiability and model checking