Showing papers in "ACM Transactions on Computational Logic in 2021"

Action Logic is Undecidable

Abstract: Action logic is the algebraic logic (inequational theory) of residuated Kleene lattices. One of the operations of this logic is the Kleene star, which is axiomatized by an induction scheme. For a stronger system that uses an -rule instead (infinitary action logic), Buszkowski and Palka (2007) proved -completeness (thus, undecidability). Decidability of action logic itself was an open question, raised by Kozen in 1994. In this article, we show that it is undecidable, more precisely, -complete. We also prove the same undecidability results for all recursively enumerable logics between action logic and infinitary action logic, for fragments of these logics with only one of the two lattice (additive) connectives, and for action logic extended with the law of distributivity.

Slanted Canonicity of Analytic Inductive Inequalities

Abstract: We prove an algebraic canonicity theorem for normal LE-logics of arbitrary signature, in a generalized setting in which the non-lattice connectives are interpreted as operations mapping tuples of elements of the given lattice to closed or open elements of its canonical extension. Interestingly, the syntactic shape of LE-inequalities which guarantees their canonicity in this generalized setting turns out to coincide with the syntactic shape of analytic inductive inequalities, which guarantees LE-inequalities to be equivalently captured by analytic structural rules of a proper display calculus. We show that this canonicity result connects and strengthens a number of recent canonicity results in two different areas: subordination algebras, and transfer results via Godel-McKinsey-Tarski translations.

Display to Labeled Proofs and Back Again for Tense Logics

Abstract: We introduce translations between display calculus proofs and labeled calculus proofs in the context of tense logics. First, we show that every derivation in the display calculus for the minimal tense logic Kt extended with general path axioms can be effectively transformed into a derivation in the corresponding labeled calculus. Concerning the converse translation, we show that for Kt extended with path axioms, every derivation in the corresponding labeled calculus can be put into a special form that is translatable to a derivation in the associated display calculus. A key insight in this converse translation is a canonical representation of display sequents as labeled polytrees. Labeled polytrees, which represent equivalence classes of display sequents modulo display postulates, also shed light on related correspondence results for tense logics.

Strategy Logic with Imperfect Information

Abstract: We introduce an extension of Strategy Logic for the imperfect-information setting, called SLii and study its model-checking problem. As this logic naturally captures multi-player games with imperfect information, this problem is undecidable; but we introduce a syntactical class of “hierarchical instances” for which, intuitively, as one goes down the syntactic tree of the formula, strategy quantifications are concerned with finer observations of the model, and we prove that model-checking SLii restricted to hierarchical instances is decidable. This result, because it allows for complex patterns of existential and universal quantification on strategies, greatly generalises the decidability of distributed synthesis for systems with hierarchical information. It allows us to easily derive new decidability results concerning strategic problems under imperfect information such as the existence of Nash equilibria or rational synthesis.To establish this result, we go through an intermediary, “low-level” logic much more adapted to automata techniques. QCTLa is an extension of CTLa with second-order quantification over atomic propositions that has been used to study strategic logics with perfect information. We extend it to the imperfect information setting by parameterising second-order quantifiers with observations. The simple syntax of the resulting logic, QCTLaii, allows us to provide a conceptually neat reduction of SLii to QCTLaii that separates concerns, allowing one to forget about strategies and players and focus solely on second-order quantification. While the model-checking problem of QCTLaii is, in general, undecidable, we identify a syntactic fragment of hierarchical formulas and prove, using an automata-theoretic approach, that it is decidable.

Stratification in Approximation Fixpoint Theory and Its Application to Active Integrity Constraints

Abstract: Approximation fixpoint theory (AFT) is an algebraic study of fixpoints of lattice operators that unifies various knowledge representation formalisms. In AFT, stratification of operators has been studied, essentially resulting in a theory that specifies when certain types of fixpoints can be computed stratum per stratum. Recently, novel types of fixpoints related to groundedness have been introduced in AFT. In this article, we study how those fixpoints behave under stratified operators.One recent application domain of AFT is the field of active integrity constraints (AICs). We apply our extended stratification theory to AICs and find that existing notions of stratification in AICs are covered by this general algebraic definition of stratification. As a result, we obtain stratification results for a large variety of semantics for AICs.

The Effects of Adding Reachability Predicates in Quantifier-Free Separation Logic

Abstract: The list segment predicate ls used in separation logic for verifying programs with pointers is well suited to express properties on singly-linked lists. We study the effects of adding ls to the full quantifier-free separation logic with the separating conjunction and implication, which is motivated by the recent design of new fragments in which all these ingredients are used indifferently and verification tools start to handle the magic wand connective. This is a very natural extension that has not been studied so far. We show that the restriction without the separating implication can be solved in polynomial space by using an appropriate abstraction for memory states, whereas the full extension is shown undecidable by reduction from first-order separation logic. Many variants of the logic and fragments are also investigated from the computational point of view when ls is added, providing numerous results about adding reachability predicates to quantifier-free separation logic.

Higher-order Recursion Schemes and Collapsible Pushdown Automata: Logical Properties

Abstract: This article studies the logical properties of a very general class of infinite ranked trees, namely, those generated by higher-order recursion schemes. We consider, for both monadic second-order logic and modal -calculus, three main problems: model-checking, logical reflection (a.k.a. global model-checking, that asks for a finite description of the set of elements for which a formula holds), and selection (that asks, if exists, for some finite description of a set of elements for which an MSO formula with a second-order free variable holds). For each of these problems, we provide an effective solution. This is obtained, thanks to a known connection between higher-order recursion schemes and collapsible pushdown automata and on previous work regarding parity games played on transition graphs of collapsible pushdown automata.

Small Circuits and Dual Weak PHP in the Universal Theory of p-time Algorithms

Abstract: We prove, under a computational complexity hypothesis, that it is consistent with the true universal theory of p-time algorithms that a specific p-time function extending bits to bits violates the dual weak pigeonhole principle: Every string equals the value of the function for some . The function is the truth-table function assigning to a circuit the table of the function it computes and the hypothesis is that every language in P has circuits of a fixed polynomial size .

Reiterman’s Theorem on Finite Algebras for a Monad

Abstract: Profinite equations are an indispensable tool for the algebraic classification of formal languages. Reiterman’s theorem states that they precisely specify pseudovarieties, i.e., classes of finite a...

Inference from Visible Information and Background Knowledge

Abstract: We provide a wide-ranging study of the scenario where a subset of the relations in a relational vocabulary is visible to a user—that is, their complete contents are known—while the remaining relations are invisible. We also have a background theory—invariants given by logical sentences—that may relate the visible relations to invisible ones, and also may constrain both the visible and invisible relations in isolation. We want to determine whether some other information, given as a positive existential formula, can be inferred using only the visible information and the background theory. This formula whose inference we are concerned with is denoted as the query. We consider whether positive information about the query can be inferred, and also whether negative information—the sentence does not hold—can be inferred. We further consider both the instance-level version of the problem, where both the query and the visible instance are given, and the schema-level version, where we want to know whether truth or falsity of the query can be inferred in some instance of the schema.

Generalized Realizability and Basic Logic

Abstract: Let V be a set of number-theoretical functions. We define a notion of absolute V-realizability for predicate formulas and sequents in such a way that the indices of functions in V are used for inte...

Strategic Knowledge Acquisition

Abstract: The article proposes a trimodal logical system that can express the strategic ability of coalitions to learn from their experience. The main technical result is the completeness of the proposed system.

Complete Abstractions for Checking Language Inclusion

Abstract: We study the language inclusion problem $L_1 \subseteq L_2$ where $L_1$ is regular or context-free. Our approach relies on abstract interpretation and checks whether an overapproximating abstraction of $L_1$, obtained by overapproximating the Kleene iterates of its least fixpoint characterization, is included in $L_2$. We show that a language inclusion problem is decidable whenever this overapproximating abstraction satisfies a completeness condition (i.e., its loss of precision causes no false alarm) and prevents infinite ascending chains (i.e., it guarantees termination of least fixpoint computations). This overapproximating abstraction of languages can be defined using quasiorder relations on words, where the abstraction gives the language of all the words "greater than or equal to" a given input word for that quasiorder. We put forward a range of such quasiorders that allow us to systematically design decision procedures for different language inclusion problems such as regular languages into regular languages or into trace sets of one-counter nets, and context-free languages into regular languages. In the case of inclusion between regular languages, some of the induced inclusion checking procedures correspond to well-known state-of-the-art algorithms like the so-called antichain algorithms. Finally, we provide an equivalent language inclusion checking algorithm based on a greatest fixpoint computation that relies on quotients of languages and, to the best of our knowledge, was not previously known.

Kripke Semantics for Intersection Formulas

Abstract: We propose a notion of the Kripke-style model for intersection logic. Using a game interpretation, we prove soundness and completeness of the proposed semantics. In other words, a formula is provable (a type is inhabited) if and only if it is forced in every model. As a by-product, we obtain another proof of normalization for the Barendregt–Coppo–Dezani intersection type assignment system.

Expressiveness and Nash Equilibrium in Iterated Boolean Games

Abstract: We define and investigate a novel notion of expressiveness for temporal logics that is based on game theoretic equilibria of multi-agent systems. We use iterated Boolean games as our abstract model of multi-agent systems [Gutierrez et al. 2013, 2015a]. In such a game, each agent has a goal , represented using (a fragment of) Linear Temporal Logic (). The goal captures agent ’s preferences, in the sense that the models of represent system behaviours that would satisfy . Each player controls a subset of Boolean variables , and at each round in the game, player is at liberty to choose values for variables in any way that she sees fit. Play continues for an infinite sequence of rounds, and so as players act they collectively trace out a model for , which for every player will either satisfy or fail to satisfy their goal. Players are assumed to act strategically, taking into account the goals of other players, in an attempt to bring about computations satisfying their goal. In this setting, we apply the standard game-theoretic concept of (pure) Nash equilibria. The (possibly empty) set of Nash equilibria of an iterated Boolean game can be understood as inducing a set of computations, each computation representing one way the system could evolve if players chose strategies that together constitute a Nash equilibrium. Such a set of equilibrium computations expresses a temporal property—which may or may not be expressible within a particular fragment. The new notion of expressiveness that we formally define and investigate is then as follows: What temporal properties are characterised by the Nash equilibria of games in which agent goals are expressed in specific fragments of ? We formally define and investigate this notion of expressiveness for a range of fragments. For example, a very natural question is the following: Suppose we have an iterated Boolean game in which every goal is represented using a particular fragment of : is it then always the case that the equilibria of the game can be characterised within ? We show that this is not true in general.

Reasoning about Petri Nets: A Calculus Based on Resolution and Dynamic Logic

Abstract: Petri Nets are a widely used formalism to deal with concurrent systems. Dynamic Logics (DLs) are a family of modal logics where each modality corresponds to a program. Petri-PDL is a logical language that combines these two approaches: it is a dynamic logic where programs are replaced by Petri Nets. In this work we present a clausal resolution-based calculus for Petri-PDL. Given a Petri-PDL formula, we show how to obtain its translation into a normal form to which a set of resolution-based inference rules are applied. We show that the resulting calculus is sound, complete, and terminating. Some examples of the application of the method are also given.

Decidability of a Sound Set of Inference Rules for Computational Indistinguishability

Abstract: Computational indistinguishability is a key property in cryptography and verification of security protocols. Current tools for proving it rely on cryptographic game transformations.We follow Bana and Comon’s approach [7, 8], axiomatizing what an adversary cannot distinguish. We prove the decidability of a set of first-order axioms that are computationally sound, though incomplete, for protocols with a bounded number of sessions whose security is based on an IND-CCA2 encryption scheme. Alternatively, our result can be viewed as the decidability of a family of cryptographic game transformations. Our proof relies on term rewriting and automated deduction techniques.

Collapsible Pushdown Parity Games

Abstract: This article studies a large class of two-player perfect-information turn-based parity games on infinite graphs, namely, those generated by collapsible pushdown automata. The main motivation for studying these games comes from the connections from collapsible pushdown automata and higher-order recursion schemes, both models being equi-expressive for generating infinite trees. Our main result is to establish the decidability of such games and to provide an effective representation of the winning region as well as of a winning strategy. Thus, the results obtained here provide all necessary tools for an in-depth study of logical properties of trees generated by collapsible pushdown automata/recursion schemes.

The Complexity of Counting Problems Over Incomplete Databases

Abstract: We study the complexity of various fundamental counting problems that arise in the context of incomplete databases, i.e., relational databases that can contain unknown values in the form of labeled nulls. Specifically, we assume that the domains of these unknown values are finite and, for a Boolean query q , we consider the following two problems: Given as input an incomplete database D , (a) return the number of completions of D that satisfy q ; or (b) return the number of valuations of the nulls of D yielding a completion that satisfies q . We obtain dichotomies between #P-hardness and polynomial-time computability for these problems when q is a self-join–free conjunctive query and study the impact on the complexity of the following two restrictions: (1) every null occurs at most once in D (what is called Codd tables ); and (2) the domain of each null is the same. Roughly speaking, we show that counting completions is much harder than counting valuations: For instance, while the latter is always in #P, we prove that the former is not in #P under some widely believed theoretical complexity assumption. Moreover, we find that both (1) and (2) can reduce the complexity of our problems. We also study the approximability of these problems and show that, while counting valuations always has a fully polynomial-time randomized approximation scheme (FPRAS), in most cases counting completions does not. Finally, we consider more expressive query languages and situate our problems with respect to known complexity classes.

EXPSPACE-Completeness of the Logics K4 × S5 and S4 × S5 and the Logic of Subset Spaces

Abstract: It is known that the satisfiability problems of the product logics K4 × S5 and S4 × S5 are NEXPTIME-hard and that the satisfiability problem of the logic SSL of subset spaces is PSPACE-hard. Furthe...

From 2-Sequents and Linear Nested Sequents to Natural Deduction for Normal Modal Logics

Abstract: We extend to natural deduction the approach of Linear Nested Sequents and of 2-Sequents. Formulas are decorated with a spatial coordinate, which allows a formulation of formal systems in the original spirit of natural deduction: only one introduction and one elimination rule per connective, no additional (structural) rule, no explicit reference to the accessibility relation of the intended Kripke models. We give systems for the normal modal logics from K to S4. For the intuitionistic versions of the systems, we define proof reduction, and prove proof normalization, thus obtaining a syntactical proof of consistency. For logics K and K4 we use existence predicates (a la Scott) for formulating sound deduction rules.

α β-Relations and the Actual Meaning of α-Renaming

Abstract: In this work we provide an alternative, and equivalent, formulation of the concept of λ-theory without introducing the notion of substitution and the sets of all, free and bound variables occurring in a term. We call α β-relations our alternative versions of λ-theories. We also clarify the actual role of α-renaming in the lambda calculus: it expresses a property of extensionality for a certain class of terms. To motivate the necessity of α-renaming, we construct an unusual denotational model of the lambda calculus that validates all structural and beta conditions but not α-renaming. The article also has a survey character.