Panos Rondogiannis

Minimum model semantics for logic programs with negation-as-failure

We give a purely model-theoretic characterization of the semantics of logic programs with negation-as-failure allowed in clause bodies. In our semantics, the meaning of a program is, as in the classical case, the unique <i>minimum</i> model in a program-independent ordering. We use an expanded truth domain that has an uncountable linearly ordered set of truth values between <i>False</i> (the minimum element) and <i>True</i> (the maximum), with a <i>Zero</i> element in the middle. The truth values below <i>Zero</i> are ordered like the countable ordinals. The values above <i>Zero</i> have exactly the reverse order. Negation is interpreted as reflection about <i>Zero</i> followed by a step towards <i>Zero</i>; the only truth value that remains unaffected by negation is <i>Zero</i>. We show that every program has a unique minimum model <i>M</i><inf>P</inf>, and that this model can be constructed with a <i>T</i><inf>P</inf> iteration which proceeds through the countable ordinals. Furthermore, we demonstrate that <i>M</i><inf>P</inf> can alternatively be obtained through a construction that generalizes the well-known model intersection theorem for classical logic programming. Finally, we show that by collapsing the true and false values of the infinite-valued model <i>M</i><inf>P</inf> to (the classical) <i>True</i> and <i>False</i>, we obtain a three-valued model identical to the well-founded one.

Higher-order functional languages and intensional logic

In this paper we demonstrate that a broad class of higher-order functional programs can be transformed into semantically equivalent multidimensional intensional programs that contain only nullary variable definitions. The proposed algorithm systematically eliminates user-defined functions from the source program, by appropriately introducing context manipulation (i.e. intensional) operators. The transformation takes place in M steps, where M is the order of the initial functional program. During each step the order of the program is reduced by one, and the final outcome of the algorithm is an M-dimensional intensional program of order zero. As the resulting intensional code can be executed in a purely tagged-dataflow way, the proposed approach offers a promising new technique for the implementation of higher-order functional languages.

Branching-time logic programming: the language Cactus and its applications

Temporal programming languages provide a powerful means for the description and implementation of dynamic systems. However, most temporal languages are based on linear time, a fact that renders them unsuitable for certain types of applications (such as expressing properties of non-deterministic programs). In this paper we introduce the new temporal logic programming language Cactus, which is based on a branching notion of time. In Cactus, the truth value of a predicate depends on a hidden time parameter which varies over a tree-like structure. As a result, Cactus can be used to express in a natural way non-deterministic computations or generally algorithms that involve the manipulation of tree data structures. Moreover, Cactus appears to be appropriate as the target language for compilers or program transformers. Cactus programs can be executed using BSLD-resolution, a proof procedure based on the notion of canonical temporal atoms/clauses.

An infinite-game semantics for well-founded negation in logic programming☆

Abstract We present an infinite-game characterization of the well-founded semantics for function-free logic programs with negation. Our game is a simple generalization of the standard game for negation-less logic programs introduced by van Emden [M.H. van Emden, Quantitative deduction and its fixpoint theory, Journal of Logic Programming 3 (1) (1986) 37–53] in which two players, the Believer and the Doubter , compete by trying to prove (respectively disprove) a query. The standard game is equivalent to the minimum Herbrand model semantics of logic programming in the sense that a query succeeds in the minimum model semantics iff the Believer has a winning strategy for the game which begins with the Doubter doubting this query. The game for programs with negation that we propose follows the same rules as the standard one, except that the players swap roles every time the play “passes through” negation. We start our investigation by establishing the determinacy of the new game by using some classical tools from the theory of infinite-games. Our determinacy result immediately provides a novel and purely game-theoretic characterization of the semantics of negation in logic programming. We proceed to establish the connections of the game semantics to the existing semantic approaches for logic programming with negation. For this purpose, we first define a refined version of the game that uses degrees of winning and losing for the two players. We then demonstrate that this refined game corresponds exactly to the infinite-valued minimum model semantics of negation [P. Rondogiannis,W.W. Wadge, Minimum model semantics for logic programs with negation-as-failure, ACM Transactions on Computational Logic 6 (2) (2005) 441–467]. This immediately implies that the unrefined game is equivalent to the well-founded semantics (since the infinite-valued semantics is a refinement of the well-founded semantics).

On the number of spanning trees of multi-star related graphs

Abstract In this paper we compute the number of spanning trees of a specific family of graphs using techniques from linear algebra and matrix theory. More specifically, we consider the graphs that result from a complete graph K n after removing a set of edges that spans a multi-star graph K m ( a 1 , a 2 ,…, a m ). We derive closed formulas for the number of spanning trees in the cases of double-star ( m = 2), triple-star ( m = 3), and quadruple-star ( m = 4). Moreover for each case we prove that the graphs with the maximum number of spanning trees are exactly those that result when all the a i s are equal.

Petri-net-based deadlock analysis of process algebra programs

Abstract Recent research has been conducted on representing Process Algebra programs by safe Petri nets. We suggest that such a representation offers direct benefits: one can use methods that have been developed in the Petri net theory domain, to reason about Process Algebra programs. We propose for a subset of a specific Process Algebra (Milners Calculus of Communicating Systems or CCS), a deadlock detection algorithm which is based on Petri net reduction techniques. The Petri net model of a CCS program is transformed into a simpler one which contains a smaller number of states, without losing however any deadlock information. An implementation of the proposed technique confirms that net-based verification of processes is a promising area of research.

Minimum model semantics for extensional higher-order logic programming with negation

Extensional higher-order logic programming has been introduced as a generalization of classical logic programming. An important characteristic of this paradigm is that it preserves all the well-known properties of traditional logic programming. In this paper we consider the semantics of negation in the context of the new paradigm. Using some recent results from non-monotonic fixed-point theory, we demonstrate that every higher-order logic program with negation has a unique minimum infinite-valued model. In this way we obtain the first purely model-theoretic semantics for negation in extensional higher-order logic programming. Using our approach, we resolve an old paradox that was introduced by W. W. Wadge in order to demonstrate the semantic difficulties of higher-order logic programming.

A fixed point theorem for non-monotonic functions

We present a fixed point theorem for a class of (potentially) non-monotonic functions over specially structured complete lattices. The theorem has as a special case the Knaster-Tarski fixed point theorem when restricted to the case of monotonic functions and Kleenes theorem when the functions are additionally continuous. From the practical side, the theorem has direct applications in the semantics of negation in logic programming. In particular, it leads to a more direct and elegant proof of the least fixed point result of 12]. Moreover, the theorem appears to have potential for possible applications outside the logic programming domain.

Well-Founded semantics for boolean grammars

Boolean grammars [A. Okhotin, Information and Computation 194 (2004) 19-48] are a promising extension of context-free grammars that supports conjunction and negation. In this paper we give a novel semantics for boolean grammars which applies to all such grammars, independently of their syntax. The key idea of our proposal comes from the area of negation in logic programming, and in particular from the so-called well-founded semantics which is widely accepted in this area to be the “correct” approach to negation. We show that for every boolean grammar there exists a distinguished (three-valued) language which is a model of the grammar and at the same time the least fixed point of an operator associated with the grammar. Every boolean grammar can be transformed into an equivalent (under the new semantics) grammar in normal form. Based on this normal form, we propose an

Theorems on Pre-fixed Points of Non-Monotonic Functions with Applications in Logic Programming and Formal Grammars

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