249 research outputs found
LIPIcs, Volume 251, ITCS 2023, Complete Volume
LIPIcs, Volume 251, ITCS 2023, Complete Volum
An Infinite Needle in a Finite Haystack: Finding Infinite Counter-Models in Deductive Verification
First-order logic, and quantifiers in particular, are widely used in
deductive verification. Quantifiers are essential for describing systems with
unbounded domains, but prove difficult for automated solvers. Significant
effort has been dedicated to finding quantifier instantiations that establish
unsatisfiability, thus ensuring validity of a system's verification conditions.
However, in many cases the formulas are satisfiable: this is often the case in
intermediate steps of the verification process. For such cases, existing tools
are limited to finding finite models as counterexamples. Yet, some quantified
formulas are satisfiable but only have infinite models. Such infinite
counter-models are especially typical when first-order logic is used to
approximate inductive definitions such as linked lists or the natural numbers.
The inability of solvers to find infinite models makes them diverge in these
cases. In this paper, we tackle the problem of finding such infinite models.
These models allow the user to identify and fix bugs in the modeling of the
system and its properties. Our approach consists of three parts. First, we
introduce symbolic structures as a way to represent certain infinite models.
Second, we describe an effective model finding procedure that symbolically
explores a given family of symbolic structures. Finally, we identify a new
decidable fragment of first-order logic that extends and subsumes the
many-sorted variant of EPR, where satisfiable formulas always have a model
representable by a symbolic structure within a known family. We evaluate our
approach on examples from the domains of distributed consensus protocols and of
heap-manipulating programs. Our implementation quickly finds infinite
counter-models that demonstrate the source of verification failures in a simple
way, while SMT solvers and theorem provers such as Z3, cvc5, and Vampire
diverge
LIPIcs, Volume 261, ICALP 2023, Complete Volume
LIPIcs, Volume 261, ICALP 2023, Complete Volum
Linear-Time Temporal Answer Set Programming
[Abstract]: In this survey, we present an overview on (Modal) Temporal Logic Programming in view of its application to Knowledge Representation and Declarative Problem Solving. The syntax of this extension of logic programs is the result of combining usual rules with temporal modal operators, as in Linear-time Temporal Logic (LTL). In the paper, we focus on the main recent results of the non-monotonic formalism called Temporal Equilibrium Logic (TEL) that is defined for the full syntax of LTL but involves a model selection criterion based on Equilibrium Logic, a well known logical characterization of Answer Set Programming (ASP). As a result, we obtain a proper extension of the stable models semantics for the general case of temporal formulas in the syntax of LTL. We recall the basic definitions for TEL and its monotonic basis, the temporal logic of Here-and-There (THT), and study the differences between finite and infinite trace length. We also provide further useful results, such as the translation into other formalisms like Quantified Equilibrium Logic and Second-order LTL, and some techniques for computing temporal stable models based on automata constructions. In the remainder of the paper, we focus on practical aspects, defining a syntactic fragment called (modal) temporal logic programs closer to ASP, and explaining how this has been exploited in the construction of the solver telingo, a temporal extension of the well-known ASP solver clingo that uses its incremental solving capabilities.Xunta de Galicia; ED431B 2019/03We are thankful to the anonymous reviewers for their thorough work and their useful
suggestions that have helped to improve the paper. A special thanks goes to Mirosaw
Truszczy´nski for his support in improving the quality of our paper. We are especially
grateful to David Pearce, whose help and collaboration on Equilibrium Logic was the
seed for a great part of the current paper. This work was partially supported by MICINN,
Spain, grant PID2020-116201GB-I00, Xunta de Galicia, Spain (GPC ED431B 2019/03),
R´egion Pays de la Loire, France, (projects EL4HC and etoiles montantes CTASP), European
Union COST action CA-17124, and DFG grants SCHA 550/11 and 15, Germany
The Alternating-Time \mu-Calculus With Disjunctive Explicit Strategies
Alternating-time temporal logic (ATL) and its extensions, including the
alternating-time -calculus (AMC), serve the specification of the strategic
abilities of coalitions of agents in concurrent game structures. The key
ingredient of the logic are path quantifiers specifying that some coalition of
agents has a joint strategy to enforce a given goal. This basic setup has been
extended to let some of the agents (revocably) commit to using certain named
strategies, as in ATL with explicit strategies (ATLES). In the present work, we
extend ATLES with fixpoint operators and strategy disjunction, arriving at the
alternating-time -calculus with disjunctive explicit strategies (AMCDES),
which allows for a more flexible formulation of temporal properties (e.g.
fairness) and, through strategy disjunction, a form of controlled
nondeterminism in commitments. Our main result is an ExpTime upper bound for
satisfiability checking (which is thus ExpTime-complete). We also prove upper
bounds QP (quasipolynomial time) and NP coNP for model checking under
fixed interpretations of explicit strategies, and NP under open interpretation.
Our key technical tool is a treatment of the AMCDES within the generic
framework of coalgebraic logic, which in particular reduces the analysis of
most reasoning tasks to the treatment of a very simple one-step logic featuring
only propositional operators and next-step operators without nesting; we give a
new model construction principle for this one-step logic that relies on a
set-valued variant of first-order resolution.Comment: Full version with appendix as well as corrected set-valued resolution
metho
Proof-theoretic Semantics for Intuitionistic Multiplicative Linear Logic
This work is the first exploration of proof-theoretic semantics for a substructural logic. It focuses on the base-extension semantics (B-eS) for intuitionistic multiplicative linear logic (IMLL). The starting point is a review of Sandqvist’s B-eS for intuitionistic propositional logic (IPL), for which we propose an alternative treatment of conjunction that takes the form of the generalized elimination rule for the connective. The resulting semantics is shown to be sound and complete. This motivates our main contribution, a B-eS for IMLL
, in which the definitions of the logical constants all take the form of their elimination rule and for which soundness and completeness are established
A decidable temporal DL-Lite logic with undecidable first-order and datalog-rewritability of ontology-mediated atomic queries
We design a logic in the temporal DL-Lite family (with non-Horn role inclusions and restricted temporalised roles), for which answering ontology-mediated atomic queries (OMAQs) can be done in ExpSpace
and even in PSpace for ontologies without existential quantification in the rule heads but determining FO-rewritability or (linear) Datalog-rewritability of OMAQs is undecidable. On the other hand, we
show (by reduction to monadic disjunctive Datalog) that deciding FO-rewritability of OMAQs in the non-temporal fragment of our logic can be done in 3NExpTime
Towards a logical foundation of randomized computation
This dissertation investigates the relations between logic and TCS in the probabilistic setting. It is motivated by two main considerations. On the one hand, since their appearance in the 1960s-1970s, probabilistic models have become increasingly pervasive in several fast-growing areas of CS. On the other, the study and development of (deterministic) computational models has considerably benefitted from the mutual interchanges between logic and CS. Nevertheless, probabilistic computation was only marginally touched by such fruitful interactions. The goal of this thesis is precisely to (start) bring(ing) this gap, by developing logical systems corresponding to specific aspects of randomized computation and, therefore, by generalizing standard achievements to the probabilistic realm. To do so, our key ingredient is the introduction of new, measure-sensitive quantifiers associated with quantitative interpretations.
The dissertation is tripartite. In the first part, we focus on the relation between logic and counting complexity classes. We show that, due to our classical counting propositional logic, it is possible to generalize to counting classes, the standard results by Cook and Meyer and Stockmeyer linking propositional logic and the polynomial hierarchy. Indeed, we show that the validity problem for counting-quantified formulae captures the corresponding level in Wagner's hierarchy.
In the second part, we consider programming language theory. Type systems for randomized \lambda-calculi, also guaranteeing various forms of termination properties, were introduced in the last decades, but these are not "logically oriented" and no Curry-Howard correspondence is known for them. Following intuitions coming from counting logics, we define the first probabilistic version of the correspondence.
Finally, we consider the relationship between arithmetic and computation. We present a quantitative extension of the language of arithmetic able to formalize basic results from probability theory. This language is also our starting point to define randomized bounded theories and, so, to generalize canonical results by Buss
Alternating (In)Dependence-Friendly Logic
Hintikka and Sandu originally proposed Independence Friendly Logic ([Formula presented]) as a first-order logic of imperfect information to describe game-theoretic phenomena underlying the semantics of natural language. The logic allows for expressing independence constraints among quantified variables, in a similar vein to Henkin quantifiers, and has a nice game-theoretic semantics in terms of imperfect information games. However, the [Formula presented] semantics exhibits some limitations, at least from a purely logical perspective. It treats the players asymmetrically, considering only one of the two players as having imperfect information when evaluating truth, resp., falsity, of a sentence. In addition, truth and falsity of sentences coincide with the existence of a uniform winning strategy for one of the two players in the semantic imperfect information game. As a consequence, [Formula presented] does admit undetermined sentences, which are neither true nor false, thus failing the law of excluded middle. These idiosyncrasies limit its expressive power to the existential fragment of Second Order Logic ([Formula presented]). In this paper, we investigate an extension of [Formula presented], called Alternating Dependence/Independence Friendly Logic ([Formula presented]), tailored to overcome these limitations. To this end, we introduce a novel compositional semantics, generalising the one based on trumps proposed by Hodges for [Formula presented]. The new semantics (i) allows for meaningfully restricting both players at the same time, (ii) enjoys the property of game-theoretic determinacy, (iii) recovers the law of excluded middle for sentences, and (iv) grants [Formula presented] the full descriptive power of [Formula presented]. We also provide an equivalent Herbrand-Skolem semantics and a game-theoretic semantics for the prenex fragment of [Formula presented], the latter being defined in terms of a determined infinite-duration game that precisely captures the other two semantics on finite structures
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