Logic Programming as Constructivism

The features of logic programming that seem unconventional from the viewpoint of classical logic can be explained in terms of constructivistic logic. We motivate and propose a constructivistic proof theory of non-Horn logic programming. Then, we apply this formalization for establishing results of practical interest. First, we show that 'stratification can be motivated in a simple and intuitive way. Relying on similar motivations, we introduce the larger classes of 'loosely stratified' and 'constructively consistent' programs. Second, we give a formal basis for introducing quantifiers into queries and logic programs by defining 'constructively domain independent* formulas. Third, we extend the Generalized Magic Sets procedure to loosely stratified and constructively consistent programs, by relying on a 'conditional fixpoini procedure

The prospects for mathematical logic in the twenty-first century

The four authors present their speculations about the future developments of mathematical logic in the twenty-first century. The areas of recursion theory, proof theory and logic for computer science, model theory, and set theory are discussed independently.Comment: Association for Symbolic Logi

Structure and Power: an emerging landscape

In this paper, we give an overview of some recent work on applying tools from category theory in finite model theory, descriptive complexity, constraint satisfaction, and combinatorics. The motivations for this work come from Computer Science, but there may also be something of interest for model theorists and other logicians. The basic setting involves studying the category of relational structures via a resource-indexed family of adjunctions with some process category - which unfolds relational structures into treelike forms, allowing natural resource parameters to be assigned to these unfoldings. One basic instance of this scheme allows us to recover, in a purely structural, syntax-free way: the Ehrenfeucht-Fraisse~game; the quantifier rank fragments of first-order logic; the equivalences on structures induced by (i) the quantifier rank fragments, (ii) the restriction of this fragment to the existential positive part, and (iii) the extension with counting quantifiers; and the combinatorial parameter of tree-depth (Nesetril and Ossona de Mendez). Another instance recovers the k-pebble game, the finite-variable fragments, the corresponding equivalences, and the combinatorial parameter of treewidth. Other instances cover modal, guarded and hybrid fragments, generalized quantifiers, and a wide range of combinatorial parameters. This whole scheme has been axiomatized in a very general setting, of arboreal categories and arboreal covers. Beyond this basic level, a landscape is beginning to emerge, in which structural features of the resource categories, adjunctions and comonads are reflected in degrees of logical and computational tractability of the corresponding languages. Examples include semantic characterisation and preservation theorems, and Lovasz-type results on counting homomorphisms.Comment: To appear in special issue for Trakhtenbrot centenary of Fundamenta Informaticae vol. 186 no 1-

Crossing the Undecidability Border with Extensions of Propositional Neighborhood Logic over Natural Numbers

Propositional Neighborhood Logic (PNL) is an interval temporal logic featuring two modalities corresponding to the relations of right and left neighborhood between two intervals on a linear order (in terms of Allen's relations, meets and met by). Recently, it has been shown that PNL interpreted over several classes of linear orders, including natural numbers, is decidable (NEXPTIME-complete) and that some of its natural extensions preserve decidability. Most notably, this is the case with PNL over natural numbers extended with a limited form of metric constraints and with the future fragment of PNL extended with modal operators corresponding to Allen's relations begins, begun by, and before. This paper aims at demonstrating that PNL and its metric version MPNL, interpreted over natural numbers, are indeed very close to the border with undecidability, and even relatively weak extensions of them become undecidable. In particular, we show that (i) the addition of binders on integer variables ranging over interval lengths makes the resulting hybrid extension of MPNL undecidable, and (ii) a very weak first-order extension of the future fragment of PNL, obtained by replacing proposition letters by a restricted subclass of first-order formulae where only one variable is allowed, is undecidable (in contrast with the decidability of similar first-order extensions of point-based temporal logics)

Dual Logic Concepts based on Mathematical Morphology in Stratified Institutions: Applications to Spatial Reasoning

Several logical operators are defined as dual pairs, in different types of logics. Such dual pairs of operators also occur in other algebraic theories, such as mathematical morphology. Based on this observation, this paper proposes to define, at the abstract level of institutions, a pair of abstract dual and logical operators as morphological erosion and dilation. Standard quantifiers and modalities are then derived from these two abstract logical operators. These operators are studied both on sets of states and sets of models. To cope with the lack of explicit set of states in institutions, the proposed abstract logical dual operators are defined in an extension of institutions, the stratified institutions, which take into account the notion of open sentences, the satisfaction of which is parametrized by sets of states. A hint on the potential interest of the proposed framework for spatial reasoning is also provided.Comment: 36 page

Horn fragments of the Halpern-Shoham Interval Temporal Logic

We investigate the satisfiability problem for Horn fragments of the Halpern-Shoham interval temporal logic depending on the type (box or diamond) of the interval modal operators, the type of the underlying linear order (discrete or dense), and the type of semantics for the interval relations (reflexive or irreflexive). For example, we show that satisfiability of Horn formulas with diamonds is undecidable for any type of linear orders and semantics. On the contrary, satisfiability of Horn formulas with boxes is tractable over both discrete and dense orders under the reflexive semantics and over dense orders under the irreflexive semantics but becomes undecidable over discrete orders under the irreflexive semantics. Satisfiability of binary Horn formulas with both boxes and diamonds is always undecidable under the irreflexive semantics