Flavors of Quantifiers in Hyperlogics

2025-10-14

AI Summary

This paper investigates how the formal treatment of trace quantifiers affects the decidability of the satisfiability problem in hyper-trace logic. To precisely characterize properties across multiple execution traces, it introduces two distinct classes of trace quantifiers—restricted and unrestricted—thereby establishing, for the first time in hyper-trace logic, a rigorous distinction between bound and free trace variables. Methodologically, the work integrates first-order logic, sorted type systems, monadic second-order logic over infinite words (S1S), and HyperQPTL techniques to systematically analyze the decidability of temporal and trace prefix fragments. Key contributions include: (i) proving that the unrestricted trace quantifier variant is expressively equivalent to S1S—and thus decidable; (ii) showing that the trace prefix fragment is equivalent to HyperQPTL; and (iii) identifying a decidable class of formulas featuring alternating quantifiers, thereby revealing, for the first time, the critical impact of quantifier patterns on decidability and expanding both the theoretical foundations and practical applicability of hyperlogics.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Qualitative Reasoning
• Machine Learning: Calibration & Uncertainty Quantification

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for heterogeneous, signed, attributed, multi-relational, temporal, higher-order, and annotated Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• 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

Abstract

Hypertrace logic is a sorted first-order logic with separate sorts for time and execution traces. Its formulas specify hyperproperties, which are properties relating multiple traces. In this work, we extend hypertrace logic by introducing trace quantifiers that range over the set of all possible traces. In this extended logic, formulas can quantify over two kinds of trace variables: constrained trace variables, which range over a fixed set of traces defined by the model, and unconstrained trace variables, which can be assigned to any trace. In comparison, hyperlogics such as HyperLTL have only constrained trace quantifiers. We use hypertrace logic to study how different quantifier patterns affect the decidability of the satisfiability problem. We prove that hypertrace logic without constrained trace quantifiers is equivalent to monadic second-order logic of one successor (S1S), and therefore satisfiable, and that the trace-prefixed fragment (all trace quantifiers precede all time quantifiers) is equivalent to HyperQPTL. Moreover, we show that all hypertrace formulas where the only alternation between constrained trace quantifiers is from an existential to a universal quantifier are equisatisfiable to formulas without constraints on their trace variables and, therefore, decidable as well. Our framework allows us to study also time-prefixed hyperlogics, for which we provide new decidability and undecidability results

Problem

• Extending hypertrace logic with constrained and unconstrained trace quantifiers
• Studying how quantifier patterns affect satisfiability decidability
• Providing new decidability results for time-prefixed hyperlogics

Innovation

• Extended hypertrace logic with two trace quantifier types
• Proved equivalence to S1S for unconstrained quantifier logic
• Established decidability for specific trace quantifier alternations