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## Idempotent Anti-unification

In this article, we address two problems related to idempotent anti-unification. First, we show that there exists an anti-unification problem with a single idempotent symbol that has an infinite minimal complete set of generalizations. It means that anti-unification with a single idempotent symbol has infinitary or nullary generalization type,... (more)

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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

Finite Open-World Query Answering With Number Restrictions

Open-world query answering is the problem of deciding, given a set of facts, conjunction of constraints, and query, whether the facts and constraints imply the query. This amounts to reasoning over all instances that include the facts and satisfy the constraints. We study \emph{finite open-world query answering} (FQA), which assumes that the underlying world is finite and thus only considers the finite completions of the instance. The major known decidable cases of FQA derive from the following: the guarded fragment of first-order logic, which can express referential constraints (data in one place points to data in another) but cannot express number restrictions such as functional dependencies; and the guarded fragment with number restrictions but on a signature of arity only two. In this paper, we give the first decidability results for FQA that combine both referential constraints and number restrictions for arbitrary signatures: we show that, for unary inclusion dependencies and functional dependencies, the finiteness assumption of FQA can be lifted up to taking the finite implication closure of the dependencies. Our result relies on new techniques to construct finite universal models of such constraints, for any bound on the maximal query size.

Duality between unprovability and provability in forward refutation-search for Intuitionistic Propositional Logic

The inverse method is a saturation based theorem proving technique; it relies on a forward proof-search strategy and can be applied to cut-free calculi enjoying the subformula property. Here we apply this method to derive the unprovability of a goal formula G in Intuitionistic Propositional Logic. To this aim we design a forward calculus FRJ(G) for Intuitionistic unprovability which is prone to constructively ascertain the unprovability of a formula G by providing a concise countermodel for it; in particular we prove that the generated countermodels have minimal height. Moreover, we clarify the role of the saturated database obtained as result of a failed proof-search in FRJ(G) by showing how to extract from such a database a derivation witnessing the Intuitionistic validity of the goal.

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.

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.

The Power of the Weak

A landmark result in the study of logics for formal verification is Janin & Walukiewicz's theorem, stating that the modal ?-calculus (?ML) is equivalent modulo bisimilarity to standard monadic second-order logic (here abbreviated as smso), over the class of labelled transition systems (LTSs for short). Our work proves two results of the same kind, one for the alternation-free fragment of ?ML (?DML) and one for weak mso (wmso). Whereas it was known that ?DML and wmso are equivalent modulo bisimilarity on binary trees, our analysis shows that the picture radically changes once we reason over arbitrary LTSs. The first theorem that we prove is that, over LTSs, ?DML is equivalent modulo bisimilarity to noetherian mso (nmso), a newly introduced variant of smso where second-order quantification ranges over "well-founded" subsets only. Our second theorem starts from wmso, and proves it equivalent modulo bisimilarity to a fragment of ?DML defined by a notion of continuity. Analogously to Janin & Walukiewicz's result, our proofs are automata-theoretic in nature: as another contribution, we introduce classes of parity automata characterising the expressiveness of wmso and nmso (on tree models) and of ?CML and ?DML (for all transition systems).

Intuitionistic Linear Temporal Logics

We consider intuitionistic variants of linear temporal logic with next', until' and release' based on {\em expanding posets:} partial orders equipped with an order-preserving transition function. This class of structures gives rise to a logic which we denote $ITL^e$, and by imposing additional constraints we obtain the logics $ITL^p$ of {\em persistent posets} and $ITL^{ht}$ of {\em here-and-there temporal logic,} both of which have been considered in the literature. We prove that $ITL^e$ has the effective finite model property and hence is decidable, while $ITL^p$ does not have the finite model property. We also introduce notions of bounded bisimulations for these logics and use them to show that the until' and `release' operators are not definable in terms of each other, even over the class of persistent posets.

Why liveness for timed automata is hard, and what we can do about it

The reachability problem for timed automata asks if a given automaton has a run leading to an accepting state, and the liveness problem asks if the automaton has an infinite run which visits accepting states infinitely often. Both these problems are known to be PSPACE-complete. We show that if P ? PSPACE, the liveness problem is more difficult than the reachability problem; in other words we exhibit a family of automata for which solving the reachability problem with the standard algorithm is in P but solving the liveness problem is PSPACE-hard. This leads us to revisit the algorithmics for the liveness problem. We propose a notion of a witness for the fact that a timed automaton violates a liveness property. We give an algorithm for computing such a witness and compare it with existing solutions.

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