Representing First-Order Causal Theories by Logic Programs

Nonmonotonic causal logic, introduced by Norman McCain and Hudson Turner, became a basis for the semantics of several expressive action languages. McCain's embedding of definite propositional causal theories into logic programming paved the way to the use of answer set solvers for answering queries about actions described in such languages. In this paper we extend this embedding to nondefinite theories and to first-order causal logic.Comment: 29 pages. To appear in Theory and Practice of Logic Programming (TPLP); Theory and Practice of Logic Programming, May, 201

Computer Science and Metaphysics: A Cross-Fertilization

Computational philosophy is the use of mechanized computational techniques to unearth philosophical insights that are either difficult or impossible to find using traditional philosophical methods. Computational metaphysics is computational philosophy with a focus on metaphysics. In this paper, we (a) develop results in modal metaphysics whose discovery was computer assisted, and (b) conclude that these results work not only to the obvious benefit of philosophy but also, less obviously, to the benefit of computer science, since the new computational techniques that led to these results may be more broadly applicable within computer science. The paper includes a description of our background methodology and how it evolved, and a discussion of our new results.Comment: 39 pages, 3 figure

A Galois connection between classical and intuitionistic logics. I: Syntax

In a 1985 commentary to his collected works, Kolmogorov remarked that his 1932 paper "was written in hope that with time, the logic of solution of problems [i.e., intuitionistic logic] will become a permanent part of a [standard] course of logic. A unified logical apparatus was intended to be created, which would deal with objects of two types - propositions and problems." We construct such a formal system QHC, which is a conservative extension of both the intuitionistic predicate calculus QH and the classical predicate calculus QC. The only new connectives ? and ! of QHC induce a Galois connection (i.e., a pair of adjoint functors) between the Lindenbaum posets (i.e. the underlying posets of the Lindenbaum algebras) of QH and QC. Kolmogorov's double negation translation of propositions into problems extends to a retraction of QHC onto QH; whereas Goedel's provability translation of problems into modal propositions extends to a retraction of QHC onto its QC+(?!) fragment, identified with the modal logic QS4. The QH+(!?) fragment is an intuitionistic modal logic, whose modality !? is a strict lax modality in the sense of Aczel - and thus resembles the squash/bracket operation in intuitionistic type theories. The axioms of QHC attempt to give a fuller formalization (with respect to the axioms of intuitionistic logic) to the two best known contentual interpretations of intiuitionistic logic: Kolmogorov's problem interpretation (incorporating standard refinements by Heyting and Kreisel) and the proof interpretation by Orlov and Heyting (as clarified by G\"odel). While these two interpretations are often conflated, from the viewpoint of the axioms of QHC neither of them reduces to the other one, although they do overlap.Comment: 47 pages. The paper is rewritten in terms of a formal meta-logic (a simplified version of Isabelle's meta-logic

Complexity of Nested Circumscription and Nested Abnormality Theories

The need for a circumscriptive formalism that allows for simple yet elegant modular problem representation has led Lifschitz (AIJ, 1995) to introduce nested abnormality theories (NATs) as a tool for modular knowledge representation, tailored for applying circumscription to minimize exceptional circumstances. Abstracting from this particular objective, we propose L_{CIRC}, which is an extension of generic propositional circumscription by allowing propositional combinations and nesting of circumscriptive theories. As shown, NATs are naturally embedded into this language, and are in fact of equal expressive capability. We then analyze the complexity of L_{CIRC} and NATs, and in particular the effect of nesting. The latter is found to be a source of complexity, which climbs the Polynomial Hierarchy as the nesting depth increases and reaches PSPACE-completeness in the general case. We also identify meaningful syntactic fragments of NATs which have lower complexity. In particular, we show that the generalization of Horn circumscription in the NAT framework remains CONP-complete, and that Horn NATs without fixed letters can be efficiently transformed into an equivalent Horn CNF, which implies polynomial solvability of principal reasoning tasks. Finally, we also study extensions of NATs and briefly address the complexity in the first-order case. Our results give insight into the ``cost'' of using L_{CIRC} (resp. NATs) as a host language for expressing other formalisms such as action theories, narratives, or spatial theories.Comment: A preliminary abstract of this paper appeared in Proc. Seventeenth International Joint Conference on Artificial Intelligence (IJCAI-01), pages 169--174. Morgan Kaufmann, 200

On the Concept of a Notational Variant

In the study of modal and nonclassical logics, translations have frequently been employed as a way of measuring the inferential capabilities of a logic. It is sometimes claimed that two logics are “notational variants” if they are translationally equivalent. However, we will show that this cannot be quite right, since first-order logic and propositional logic are translationally equivalent. Others have claimed that for two logics to be notational variants, they must at least be compositionally intertranslatable. The definition of compositionality these accounts use, however, is too strong, as the standard translation from modal logic to first-order logic is not compositional in this sense. In light of this, we will explore a weaker version of this notion that we will call schematicity and show that there is no schematic translation either from first-order logic to propositional logic or from intuitionistic logic to classical logic

Complete Additivity and Modal Incompleteness

In this paper, we tell a story about incompleteness in modal logic. The story weaves together a paper of van Benthem, `Syntactic aspects of modal incompleteness theorems,' and a longstanding open question: whether every normal modal logic can be characterized by a class of completely additive modal algebras, or as we call them, V-BAOs. Using a first-order reformulation of the property of complete additivity, we prove that the modal logic that starred in van Benthem's paper resolves the open question in the negative. In addition, for the case of bimodal logic, we show that there is a naturally occurring logic that is incomplete with respect to V-BAOs, namely the provability logic GLB. We also show that even logics that are unsound with respect to such algebras do not have to be more complex than the classical propositional calculus. On the other hand, we observe that it is undecidable whether a syntactically defined logic is V-complete. After these results, we generalize the Blok Dichotomy to degrees of V-incompleteness. In the end, we return to van Benthem's theme of syntactic aspects of modal incompleteness

First-Order Logical Duality

From a logical point of view, Stone duality for Boolean algebras relates theories in classical propositional logic and their collections of models. The theories can be seen as presentations of Boolean algebras, and the collections of models can be topologized in such a way that the theory can be recovered from its space of models. The situation can be cast as a formal duality relating two categories of syntax and semantics, mediated by homming into a common dualizing object, in this case 2. In the present work, we generalize the entire arrangement from propositional to first-order logic. Boolean algebras are replaced by Boolean categories presented by theories in first-order logic, and spaces of models are replaced by topological groupoids of models and their isomorphisms. A duality between the resulting categories of syntax and semantics, expressed first in the form of a contravariant adjunction, is established by homming into a common dualizing object, now \Sets, regarded once as a boolean category, and once as a groupoid equipped with an intrinsic topology. The overall framework of our investigation is provided by topos theory. Direct proofs of the main results are given, but the specialist will recognize toposophical ideas in the background. Indeed, the duality between syntax and semantics is really a manifestation of that between algebra and geometry in the two directions of the geometric morphisms that lurk behind our formal theory. Along the way, we construct the classifying topos of a decidable coherent theory out of its groupoid of models via a simplified covering theorem for coherent toposes.Comment: Final pre-print version. 62 page

Formal Semantics and the Grammar of Predication

Linguistic