ACM Transactions on

Computational Logic (TOCL)

Latest Articles

Central Limit Model Checking

We consider probabilistic model checking for continuous-time Markov chains (CTMCs) induced from Stochastic Reaction Networks against a fragment of Continuous Stochastic Logic (CSL) extended with reward operators. Classical numerical algorithms for CSL model checking based on uniformisation are limited to finite CTMCs and suffer from exponential... (more)

A Representation Theorem for Change through Composition of Activities

The expanding use of information systems in industrial and commercial settings has increased the need for interoperation between software systems. In... (more)

Reasoning about Cognitive Trust in Stochastic Multiagent Systems

We consider the setting of stochastic multiagent systems modelled as stochastic multiplayer games and formulate an automated verification framework... (more)

De Morgan Dual Nominal Quantifiers Modelling Private Names in Non-Commutative Logic

This article explores the proof theory necessary for recommending an expressive but decidable first-order system, named MAV1, featuring a De Morgan... (more)

Modal Resolution: Proofs, Layers, and Refinements

Resolution-based provers for multimodal normal logics require pruning of the search space for a proof to ameliorate the inherent intractability of the satisfiability problem for such logics. We present a clausal modal-layered hyper-resolution calculus for the basic multimodal logic, which divides the clause set according to the modal level at which... (more)

Probabilistic Epistemic Updates on Algebras

The present article contributes to the development of the mathematical theory of epistemic updates using the tools of duality theory. Here, we focus on Probabilistic Dynamic Epistemic Logic (PDEL). We dually characterize the product update construction of PDEL-models as a certain construction transforming the complex algebras associated with the... (more)

Incomplete SMT Techniques for Solving Non-Linear Formulas over the Integers

We present new methods for solving the Satisfiability Modulo Theories problem over the theory of Quantifier-Free Non-linear Integer Arithmetic,... (more)


TOCL welcomes submissions related to all aspects of logic as it pertains to topics in computer science. The journal is published quarterly. The first issue appeared in July 2000, and the journal is indexed by ISI beginning with the 2006 volume. About

Forthcoming Articles

Monadic Datalog, Tree Validity, and Limited Access Containment

We reconsider the problem of containment of monadic datalog (MDL) queries in unions of conjunctive queries (UCQs). Prior work has dealt with special cases of the problem, but has left the precise complexity characterization open. In addition, the complexity of one important special case, that of containment under access patterns, was not known before. We start by revisiting the connection between MDL/UCQ containment and containment problems involving regular tree languages. We then present a general approach for getting tighter bounds on the complexity of query containment, based on analysis of the number of mappings of queries into tree-like instances. We give two applications of the machinery. We first give an important special case of the MDL/UCQ containment problem that is in EXPTIME, and use this bound to show an $\exptime$ bound on containment under access patterns. Secondly we show that the same technique can be used to get a new tight upper bound for containment of tree automata in UCQs. We finally show that the new MDL/UCQ upper bounds are tight. We establish a 2EXPTIME lower bound on the MDL/UCQ containment problem, resolving an open problem from the early 1990s. We also show that changes to the conditions given in our special cases can not be eliminated, and that in particular slight variations of the problem of containment under access patterns become TWOEXPTIME-complete.

Verification Methods for the Computationally Complete Symbolic Attacker Based on Indistinguishability

In recent years, a new approach has been developed for verifying security protocols with the aim of combining the benefits of symbolic attackers and the benefits of unconditional soundness: the technique of the computationally complete symbolic attacker of Bana and Comon (BC). In this paper we argue that the real breakthrough of this technique is the recent introduction of its version for indistinguishability because, with the extensions we introduce here, for the first time, there is a computationally sound symbolic technique that is syntactically strikingly simple, to which translating standard computational security notions is a straightforward matter, and that can be effectively used for verification of not only equivalence properties, but trace properties of protocols as well. We first fully develop the core elements of this newer version by introducing several new axioms. We illustrate the power and the diverse use of the introduced axioms on simple examples first. We introduce an axiom expressing the Decisional Diffie-Hellman property. We analyze the Diffie-Hellman key exchange, both in its simplest form and an authenticated version as well. We provide computationally sound verification of real-or-random secrecy of the Diffie-Hellman key exchange protocol for multiple sessions, without any restrictions on the computational implementation other than the DDH assumption. We also show authentication for a simplified version of the station-to-station protocol using UF-CMA assumption for digital signatures. Finally, we axiomatize IND-CPA, IND-CCA1 and IND-CCA2 security properties and illustrate their usage. We have formalized the axiomatic system in an interactive theorem prover, Coq, and have machine- checked the proofs of various auxiliary theorems, and security properties of Diffie-Hellman and station-to-station protocol.

Adding successor: A transfer theorem for separation and covering

Given a class C of word languages, the C-separation problem asks for an algorithm that, given as input two regular languages, decides whether there exists a third language in C containing the first language, while being disjoint from the second. Separation is usually investigated as a means to obtain a deep understanding of the class C. In the paper, we are mainly interested in classes defined by logical formalisms. Such classes are often built on top of each other: given some logic, one builds a stronger one by adding new predicates to its signature. A natural construction is to enrich a logic with the successor relation. In this paper, we present a transfer result applying to this construction: we show that for suitable logically defined classes, separation for the logic enriched with the successor relation reduces to separation for the original logic. Our theorem also applies to a problem that is stronger than separation: covering. Moreover, we actually present two reductions: one for languages of finite words and the other for languages of infinite words.

A First-Order Logic for Reasoning about Knowledge and Probability

We present a first-order probabilistic epistemic logic, which allows combining operators of knowledge and probability within a group of possibly infinitely many agents. The proposed framework is the first order extension of the logic of Fagin and Halpern from (J.ACM 41:340-367,1994). We define its syntax and semantics, and prove the strong completeness property of the corresponding axiomatic system.

Dynamic QBF Dependencies in Reduction and Expansion

We provide the first proof complexity results for QBF dependency calculi. By showing that the reflexive resolution path dependency scheme admits exponentially shorter Q-resolution proofs on a known family of instances, we answer a question first posed by Slivovsky and Szeider in 2014. Further, we conceive a method of QBF solving in which dependency recomputation is utilised as a form of inprocessing. Formalising this notion, we introduce a new version of Q-resolution in which a dependency scheme is applied dynamically. We demonstrate the further potential of this approach beyond that of the existing static system with an exponential separation. Lastly, we show that the same picture emerges in an analogous approach to the universal expansion paradigm.

Model-Checking on Ordered Structures

We study the model-checking problem for first- and monadic second-order logic on finite relational structures. The problem of verifying whether a formula of these logics is true on a given structure is considered intractable in general, but it does become tractable on interesting classes of structures, such as on classes whose Gaifman graphs have bounded treewidth. In this paper we continue this line of research and study model-checking for first- and monadic second-order logic in the presence of an ordering on the input structure. We do so in two settings: the general ordered case, where the input structures are equipped with a fixed order or successor relation, and the order invariant case, where the formulas may resort to an ordering, but their truth must be independent of the particular choice of order. In the first setting we show very strong intractability results for most interesting classes of structures. In contrast, in the order invariant case we obtain tractability results for order-invariant monadic second-order formulas on the same classes of graphs as in the unordered case. For first-order logic, we obtain tractability of successor-invariant formulas on classes whose Gaifman graphs have bounded expansion. Furthermore, we show that model-checking for order-invariant first-order formulas is tractable on coloured posets of bounded width.

Extending Liquid Types to Arrays

A liquid type is an ordinary Hindley-Milner type annotated with a logical predicate that states the properties satisfied by the elements of that type. Liquid types are a powerful tool for program verification, since programmers can use them to specify pre- and postconditions of their programs, while the predicates of intermediate variables and auxiliary functions are inferred automatically. Type inference is feasible in this context, since the logical predicates within liquid types are constrained to a quantifier-free logic in order to maintain decidability. In this paper we extend liquid types by allowing them to contain quantified properties on arrays, so that they can be used to infer invariants on array-related programs (for example, implementations of sorting algorithms). Although quantified logic is, in general, undecidable, we restrict properties on arrays to a decidable subset introduced by Bradley et al. We describe in detail the extended type system, the verification condition generator, and the iterative weakening algorithm for inferring invariants. After proving the correctness and completeness of these two algorithms, we apply them to find invariants on a set of algorithms involving array manipulations.

Fixed-point elimination in the Intuitionistic Propositional Calculus

It is a consequence of existing literature that least and greatest xed-points of monotone polynomials on Heyting algebrasthat is, the algebraic models of the Intuitionistic Propositional Calculusalways exist, even when these algebras are not complete as lattices. The reason is that these extremal xed-points are de nable by formulas of the IPC. Consequently, the ¼-calculus based on intuitionistic logic is trivial, every ¼-formula being equivalent to a xed-point free formula. In the rst part of this paper we give an axiomatization of least and greatest xed-points of formulas, and an algorithm to compute a xed-point free formula equivalent to a given ¼-formula. The axiomatization of the greatest xed-point is simple. The axiomatization of the least xed-point is more complex, in particular every monotone formula converges to its least xed-point by Kleenes iteration in a nite number of steps, but there is no uniform upper bound on the number of iterations. The axiomatization yields a decision procedure for the ¼-calculus based on propositional intuitionistic logic. The second part of the paper deals with closure ordinals of monotone polynomials on Heyting algebras and of intuitionistic monotone formulas; these that are the least numbers of iterations needed for a polynomial/formula to converge to its least xed-point. Mirroring the elimination procedure, we show how to compute upper bounds for closure ordinals of arbitrary intuitionistic formulas. For some classes of formulas we provide tighter upper bounds that, in some cases, we prove exact.

Runtime Verification over Out-of-order Streams

We present an approach for verifying systems at runtime. Our approach targets distributed systems whose components communicate with monitors over unreliable channels, where messages can be delayed, reordered, or even lost. Furthermore, our approach handles an expressive specifications language that extends the real-time logic MTL with freeze quantifiers for reasoning about data values. The logic's main novelty is a new three-valued semantics that is well suited for runtime verification as it accounts for partial knowledge about system behavior. Based on this semantics, we present online algorithms that reason soundly and completely about streams where events can occur out of order. We also evaluate our algorithms experimentally. Depending on the specification, our prototype implementation scales to out-of-order streams with hundreds to thousands of events per second.

Idempotent Anti-Unification

In this paper we address two problems related to idempotent anti-unification. First, we show that there exists an anti-unification problem with a single idempotent symbol which has an infinite minimal complete set of generalizations. It means that anti-unification with a single idempotent symbol has infinitary or nullary generalization type, similar to anti-unification with two idem- potent symbols, shown earlier by Loı?c Pottier. Next, we develop an algorithm, which takes an arbitrary idempotent anti-unification problem and computes a representation of its solution set in the form of a regular tree grammar. The algorithm does not depend on the number of idempotent function symbols in the input terms. The language generated by the grammar is the minimal complete set of generalizations of the given anti-unification problem, which implies that idem- potent anti-unification is infinitary.

Reason-maintenance Belief Logic with Uncertain Information

In this paper, we propose a logic for reasoning about belief based on fusion of uncertain information. The resultant reason-maintenance possibilistic belief logic can represent both implicit and explicit uncertain beliefs of an agent. While implicit beliefs stipulate what are believable, explicit beliefs can trace the process of belief formation by information fusion. To set up the basic framework, we start with developing a reason-maintenance belief logic without uncertainty, present its syntax and semantics, and investigate its axiomatization and properties. We also demonstrate its applicability by using a realistic example in system accountability. Then, we extend the basic logic to accommodate the possibilistic uncertainty of information and belief, provide a complete axiomatization of the extended logic, and show that it can address the reason-maintenance issue of partially inconsistent belief.

Satisfiability of Modal Inclusion Logic: Lax and Strict Semantics

We investigate the computational complexity of the satisfiability problem of modal inclusion logic. We distinguish two variants of the problem: one for the strict and another one for the lax semantics. Both problems turn out to be EXPTIME-complete on general structures. Finally, we show how for a specific class of structures NEXPTIME-completeness for these problems under strict semantics can be achieved.

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