Solving Practical Reasoning Poblems with Extended Disjunctive Logic Programming

We present a definition of stable generated models for extended generalized logic programs (EGLP) which a) subsumes the definition of the answer set semantics for extended normal logic programs [GL91]; and b) does not refer to negation-as-failure by allowing for arbitrary quantifier free formulas in the body and in the head of as rule (i.e. does not depend on the presence of any specific connective, nor any specific syntax of rules). We show how to solve classical ATP problems in the framework of extended disjunctive logic programming (EDLP) where neither Contraposition nor the Law of the Excluded Middle are admitted principles of inference. Besides being able to solve classical ATP problems in a monotonic reasoning mode, EDLP can as well treat commonsense reasoning problems by employing its intrinsic nonmonotonic inference capabilities based on stable generated models. EDLP thus proves itself as a general-purpose AI reasoning system

Linear logic for constructive mathematics

We show that numerous distinctive concepts of constructive mathematics arise automatically from an interpretation of "linear higher-order logic" into intuitionistic higher-order logic via a Chu construction. This includes apartness relations, complemented subsets, anti-subgroups and anti-ideals, strict and non-strict order pairs, cut-valued metrics, and apartness spaces. We also explain the constructive bifurcation of classical concepts using the choice between multiplicative and additive linear connectives. Linear logic thus systematically "constructivizes" classical definitions and deals automatically with the resulting bookkeeping, and could potentially be used directly as a basis for constructive mathematics in place of intuitionistic logic.Comment: 39 page

Some model theory for the modal μ\mu-calculus: syntactic characterisations of semantic properties

This paper contributes to the theory of the modal $\mu$-calculus by proving some model-theoretic results. More in particular, we discuss a number of semantic properties pertaining to formulas of the modal $\mu$-calculus. For each of these properties we provide a corresponding syntactic fragment, in the sense that a $\mu$-formula $\xi$ has the given property iff it is equivalent to a formula $\xi'$ in the corresponding fragment. Since this formula $\xi'$ will always be effectively obtainable from $\xi$, as a corollary, for each of the properties under discussion, we prove that it is decidable in elementary time whether a given $\mu$-calculus formula has the property or not. The properties that we study all concern the way in which the meaning of a formula $\xi$ in a model depends on the meaning of a single, fixed proposition letter $p$. For example, consider a formula $\xi$ which is monotone in $p$; such a formula a formula $\xi$ is called continuous (respectively, fully additive), if in addition it satisfies the property that, if $\xi$ is true at a state $s$ then there is a finite set (respectively, a singleton set) $U$ such that $\xi$ remains true at $s$ if we restrict the interpretation of $p$ to the set $U$. Each of the properties that we consider is, in a similar way, associated with one of the following special kinds of subset of a tree model: singletons, finite sets, finitely branching subtrees, noetherian subtrees (i.e., without infinite paths), and branches. Our proofs for these characterization results will be automata-theoretic in nature; we will see that the effectively defined maps on formulas are in fact induced by rather simple transformations on modal automata. Thus our results can also be seen as a contribution to the model theory of modal automata

Model checking: Algorithmic verification and debugging

Turing Lecture from the winners of the 2007 ACM A.M. Turing Award.In 1981, Edmund M. Clarke and E. Allen Emerson, working in the USA, and Joseph Sifakis working independently in France, authored seminal papers that founded what has become the highly successful field of model checking. This verification technology provides an algorithmic means of determining whether an abstract model-representing, for example, a hardware or software design-satisfies a formal specification expressed as a temporal logic (TL) formula. Moreover, if the property does not hold, the method identifies a counterexample execution that shows the source of the problem.The progression of model checking to the point where it can be successfully used for complex systems has required the development of sophisticated means of coping with what is known as the state explosion problem. Great strides have been made on this problem over the past 28 years by what is now a very large international research community. As a result many major hardware and software companies are beginning to use model checking in practice. Examples of its use include the verification of VLSI circuits, communication protocols, software device drivers, real-time embedded systems, and security algorithms.The work of Clarke, Emerson, and Sifakis continues to be central to the success of this research area. Their work over the years has led to the creation of new logics for specification, new verification algorithms, and surprising theoretical results. Model checking tools, created by both academic and industrial teams, have resulted in an entirely novel approach to verification and test case generation. This approach, for example, often enables engineers in the electronics industry to design complex systems with considerable assurance regarding the correctness of their initial designs. Model checking promises to have an even greater impact on the hardware and software industries in the future.-Moshe Y. Vardi, Editor-in-Chief

Neuere Entwicklungen der deklarativen KI-Programmierung : proceedings

The field of declarative AI programming is briefly characterized. Its recent developments in Germany are reflected by a workshop as part of the scientific congress KI-93 at the Berlin Humboldt University. Three tutorials introduce to the state of the art in deductive databases, the programming language Gödel, and the evolution of knowledge bases. Eleven contributed papers treat knowledge revision/program transformation, types, constraints, and type-constraint combinations

Constructive Geometry and the Parallel Postulate

Euclidean geometry consists of straightedge-and-compass constructions and reasoning about the results of those constructions. We show that Euclidean geometry can be developed using only intuitionistic logic. We consider three versions of Euclid's parallel postulate: Euclid's own formulation in his Postulate 5; Playfair's 1795 version, and a new version we call the strong parallel postulate. These differ in that Euclid's version and the new version both assert the existence of a point where two lines meet, while Playfair's version makes no existence assertion. Classically, the models of Euclidean (straightedge-and-compass) geometry are planes over Euclidean fields. We prove a similar theorem for constructive Euclidean geometry, by showing how to define addition and multiplication without a case distinction about the sign of the arguments. With intuitionistic logic, there are two possible definitions of Euclidean fields, which turn out to correspond to the different versions of the parallel axiom. In this paper, we completely settle the questions about implications between the three versions of the parallel postulate: the strong parallel postulate easily implies Euclid 5, and in fact Euclid 5 also implies the strong parallel postulate, although the proof is lengthy, depending on the verification that Euclid 5 suffices to define multiplication geometrically. We show that Playfair does not imply Euclid 5, and we also give some other independence results. Our independence proofs are given without discussing the exact choice of the other axioms of geometry; all we need is that one can interpret the geometric axioms in Euclidean field theory. The proofs use Kripke models of Euclidean field theories based on carefully constructed rings of real-valued functions.Comment: 114 pages, 39 figure

Kripke Semantics for Dependent Type Theory and Realizability Interpretations

Abstract Constructive reasoning has played an increasingly important role in the development of provably correct software. Both typed and type-free frameworks stemming from ideas of Heyting, Kleene, and Curry have been developed for extracting computations from constructive specifications. These include Realizability, and Theories based on the Curry-Howard isomorphism. Realizability -in its various typed and type-free formulations -brings out the algorithmic content of theories and proofs and supplies models of the "recursive universe". Formal systems based on the propositions-as-types paradigm, such as Martin-Löf's dependent type theories, incorporate term extraction into the logic itself. Another, major tradition in constructive semantics originated in the model theory developed by Gödel, Herbrand and Tarski, resulting in the interpretations developed by Kripke and Beth, and in subsequent categorical generalizations. They provide a complete semantics for constructive logic. These models are a powerful tool for building counterexamples and establishing independence and conservativity results, but they are often less constructive and less computationally oriented. It is highly desirable to combine the power of these approaches to constructive semantics, and to elucidate some connections between them. We define modified Kripke and Beth models for syntactic Realizability and Dependent Type theory, in particular for the one-universe Intensional Martin-Löf Theory ML i 0 . These models provide a new framework for reasoning about computational evidence and the process of term-extraction. They are defined over a constructive type-free metatheory based on the Feferman-Beeson theories of abstract applicative structure. Our models have a feature which is shared by all published constructive completeness theorems for intuitionistic logic, known in the literature as "fallibility": there may be worlds in which some sentences are both false and true, a phenomenon which corresponds to the presence of empty types in various type disciplines. We also identify a natural lattice of truth values associated with type theory and realizability: the degrees of inhabitation