Modal logics have traditionally played a central role in Computer Science, appearing, e.g., in the guise of temporal logics, program logics such as PDL, epistemic logics, and later as description logics. The development of modal logics has seen extensions along (at least) two axes: the enhancement of the expressive power of basic (relational) modal logic on the one hand, and the continual extension, beyond the purely relational realm, of the class of structures described using modal logics on the other hand. Hybrid logic falls into the first category, extending modal logic with the ability to reason about individual states in models. This feature, originally suggested by Prior and first studied in the context of tense logics and PDL (see [5] for references), is of particular relevance in knowledge representation languages and as such has found its way into modern description logics, where it is denoted by the letter O in the standard naming scheme [2]. Extensions along the second axis - semantics beyond Kripke structures and neighbourhood models - include various probabilistic modal logics, interpreted over probabilistic transition systems, graded modal logic over multigraphs [8], conditional logics over selection function frames [6], and coalition logic [17], interpreted over so-called game frames. As a unifying semantic bracket covering all these logics and many further ones, coalgebraic modal logic has emerged ([7] gives a survey). The scope of coalgebraic modal logic has recently been expanded to encompass nominals; we refer to the arising class of logics as coalgebraic hybrid logics.

Many problems can be specified by patterns of propositional formulae depending on a parameter, e.g. the specification of a circuit usually depends on the number of bits of its input. We define a logic whose formulae, called "iterated schemata", allow to express such patterns. Schemata extend propositional logic with indexed propositions, e.g. P_i, P_i+1, P_1, and with generalized connectives, e.g. /\i=1..n or i=1..n (called "iterations") where n is an (unbound) integer variable called a "parameter". The expressive power of iterated schemata is strictly greater than propositional logic: it is even out of the scope of first-order logic. We define a proof procedure, called DPLL*, that can prove that a schema is satisfiable for at least one value of its parameter, in the spirit of the DPLL procedure. However the converse problem, i.e. proving that a schema is unsatisfiable for every value of the parameter, is undecidable so DPLL* does not terminate in general. Still, we prove that it terminates for schemata of a syntactic subclass called "regularly nested". This is the first non trivial class for which DPLL* is proved to terminate. Furthermore the class of regularly nested schemata is the first decidable class to allow nesting of iterations, i.e. to allow schemata of the form /\i=1..n (/\j=1..n ...).

We apply proof-theoretic techniques in answer Set Programming. The main results include: 1. A characterization of continuity properties of Gelfond-Lifschitz operator for logic program. 2. A propositional characterization of stable models of logic programs (without referring to loop formulas.

Nieuwenhuis, Oliveras, and Tinelli (2006) showed how to describe enhancements of the Davis-Putnam-Logemann-Loveland algorithm using transition systems, instead of pseudocode. We design a similar framework for several algorithms that generate answer sets for logic programs: Smodels, Smodels-cc, Asp-Sat with Learning (Cmodels), and a newly designed and implemented algorithm Sup. This approach to describing answer set solvers makes it easier to prove their correctness, to compare them, and to design new systems.

Many researchers have suggested that the psychological complexity of a concept is related to the length of its representation in a language of thought. As yet, however, there are few concrete proposals about the nature of this language. This paper makes one such proposal: the language of thought allows first order quantification (quantificationover objects) more readily than second-order quantification (quantification over features). To support this proposal we present behavioral results froma concept learning study inspired by the work of Shepard, Hovland and Jenkins. Humans can learn and think about many kinds of concepts, including natural kinds such as elephant and water and nominal kinds such as grandmother and prime number.

We address the problem of belief change in (nonmonotonic) logic programming under answer set semantics. Unlike previous approaches to belief change in logic programming, our formal techniques are analogous to those of distance-based belief revision in propositional logic. In developing our results, we build upon the model theory of logic programs furnished by SE models. Since SE models provide a formal, monotonic characterisation of logic programs, we can adapt techniques from the area of belief revision to belief change in logic programs. We introduce methods for revising and merging logic programs, respectively. For the former, we study both subset-based revision as well as cardinality-based revision, and we show that they satisfy the majority of the AGM postulates for revision. For merging, we consider operators following arbitration merging and IC merging, respectively. We also present encodings for computing the revision as well as the merging of logic programs within the same logic programming framework, giving rise to a direct implementation of our approach in terms of off-the-shelf answer set solvers. These encodings reflect in turn the fact that our change operators do not increase the complexity of the base formalism.

We propose a database model that allows users to annotate data with belief statements. Our motivation comes from scientific database applications where a community of users is working together to assemble, revise, and curate a shared data repository. As the community accumulates knowledge and the database content evolves over time, it may contain conflicting information and members can disagree on the information it should store. For example, Alice may believe that a tuple should be in the database, whereas Bob disagrees. He may also insert the reason why he thinks Alice believes the tuple should be in the database, and explain what he thinks the correct tuple should be instead. We propose a formal model for Belief Databases that interprets users' annotations as belief statements. These annotations can refer both to the base data and to other annotations. We give a formal semantics based on a fragment of multi-agent epistemic logic and define a query language over belief databases. We then prove a key technical result, stating that every belief database can be encoded as a canonical Kripke structure. We use this structure to describe a relational representation of belief databases, and give an algorithm for translating queries over the belief database into standard relational queries. Finally, we report early experimental results with our prototype implementation on synthetic data.

Boolean Satisfiability (SAT) solvers are now routinely used in the verification of large industrial problems. However, their application in safety-critical domains such as the railways, avionics, and automotive industries requires some form of assurance for the results, as the solvers can (and sometimes do) have bugs. Unfortunately, the complexity of modern, highly optimized SAT solvers renders impractical the development of direct formal proofs of their correctness. This paper presents an alternative approach where an untrusted, industrial-strength, SAT solver is plugged into a trusted, formally certified, SAT proof checker to provide industrial-strength certified SAT solving. The key novelties and characteristics of our approach are (i) that the checker is automatically extracted from the formal development, (ii), that the combined system can be used as a standalone executable program independent of any supporting theorem prover, and (iii) that the checker certifies any SAT solver respecting the agreed format for satisfiability and unsatisfiability claims. The core of the system is a certified checker for unsatisfiability claims that is formally designed and verified in Coq. We present its formal design and outline the correctness proofs. The actual standalone checker is automatically extracted from the the Coq development. An evaluation of the certified checker on a representative set of industrial benchmarks from the SAT Race Competition shows that, albeit it is slower than uncertified SAT checkers, it is significantly faster than certified checkers implemented on top of an interactive theorem prover.

Action description languages, such as A and B (Gelfond and Lifschitz 1998), are expressive instruments introduced for formalizing planning domains and planning problem instances. The paper starts by proposing a methodology to encode an action language (with conditional effects and static causal laws), a slight variation of B, using Constraint Logic Programming over Finite Domains. The approach is then generalized to raise the use of constraints to the level of the action language itself. A prototype implementation has been developed, and the preliminary results are presented and discussed. To appear in Theory and Practice of Logic Programming (TPLP).

In the recent years rule-based programming in terms of decla rative logic programming has formed the basis for many Artificial In telligence (AI) applications and is well integrated in the mainstream infor mation technology capturing higher-level decision logics. Typically, the st andard rule systems and rule-based logic programming languages such as Prolog deri vatives are based on the untyped theory of predicate calculus with untyped logic al objects (untyped terms), i.e. the logical reasoning algorithms apply pure sy ntactical reasoning. From a rule engineering perspective this is a serious restri ction which lacks major Software Engineering principles such as data abstracti on or modularization, which become more and more important when rule applications grow larger and more complex. To support such principles in logic programmi ng and capture the rule engineer's intended meaning of a logic program, types a nd typed objects play an important role. Moreover, from a computational poin t of view, the use of types drastically reduces the search space, i.e. proofs c an be kept at a more abstract level and it offers the option to restrict the applic ation of rules and to control the level of generality in queries.