Abstract: We present _tropical games_, a generalization of combinatorial min-max games based on tropical algebras. Our model breaks the traditional symmetry of rational zero-sum games where players have exactly opposed goals (min vs. max), is more widely applicable than min-max and also supports a form of pruning, despite it being less effective than alpha-beta. Actually, min-max games may be seen as particular cases where both the game and its dual are tropical: when the dual of a tropical game is also tropical, the power of alpha-beta is completely recovered. We formally develop the model and prove that the tropical pruning strategy is correct, then conclude by showing how the problem of approximated parsing can be modeled as a tropical game, profiting from pruning.

Abstract: A mathematical model describing the interactions between viral infection and acquired immunity is proposed. The model is formulated in terms of a system of partial integro-differential bilinear equations. Results of numerical experiments are presented.

Abstract: We propose a Case-Based Reasoning System as the most suitable and stable solution to response and manage the information concerning sequential terms of biological structures and the implementation of pattern recognition on secondary structures. We will concentrate on the theoretical approaches and the fundamental principles for the design of a machine-learning system.

Abstract: This paper presents a combination of several automated reasoning and proof presentation tools with the Mizar system for formalization of mathematics. The combination forms an online service called MizAR, similar to the SystemOnTPTP service for first-order automated reasoning. The main differences to SystemOnTPTP are the use of the Mizar language that is oriented towards human mathematicians (rather than the pure first-order logic used in SystemOnTPTP), and setting the service in the context of the large Mizar Mathematical Library of previous theorems, definitions, and proofs (rather than the isolated problems that are solved in SystemOnTPTP). These differences poses new challenges and new opportunities for automated reasoning and for proof presentation tools. This paper describes the overall structure of MizAR, and presents the automated reasoning systems and proof presentation tools that are combined to make MizAR a useful mathematical service.

Abstract: General purpose theorem provers provide sophisticated proof methods, but lack some of the advanced structuring mechanisms found in specification languages. This paper builds on previous work extending the theorem prover Isabelle with such mechanisms. A way to build the quotient type over a given base type and an equivalence relation on it, and a generalised notion of folding over quotiented types is given as a formalised high-level step called a design tactic. The core of this paper are four axiomatic theories capturing the design tactic. The applicability is demonstrated by derivations of implementations for finite multisets and finite sets from lists in Isabelle.

Abstract: Matrix interpretations generalize linear polynomial interpretations and have been proved useful in the implementation of tools for automatically proving termination of Term Rewriting Systems. In view of the successful use of rational coefficients in polynomial interpretations, we have recently generalized traditional matrix interpretations (using natural numbers in the matrix entries) to incorporate real numbers. However, existing results which formally prove that polynomials over the reals are more powerful than polynomials over the naturals for proving termination of rewrite systems failed to be extended to matrix interpretations. In this paper we get deeper into this problem. We show that, under some conditions, it is possible to transform a matrix interpretation over the rationals satisfying a set of symbolic constraints into a matrix interpretation over the naturals (using bigger matrices) which still satisfies the constraints.

Abstract: Krawtchouk polynomials appear in a variety of contexts, most notably as orthogonal polynomials and in coding theory via the Krawtchouk transform. We present an operator calculus formulation of the Krawtchouk transform that is suitable for computer implementation. A positivity result for the Krawtchouk transform is shown. Then our approach is compared with the use of the Krawtchouk transform in coding theory where it appears in MacWilliams' and Delsarte's theorems on weight enumerators.

Abstract: Given a regular set T in K[x], Lemaire et al. in ISSAC'08 give a nice algebraic property: the regular set T generates its saturated ideal if and only if it is primitive. We dirstly aim at giving a more direct proof of the above result, generalizing the concept of primitivity of polynomials and regular sets and presenting a new result which is equivalent to the above property. On the other hand, based upon correcting an error of the definition of U-set in AISC'06, we further develop some geometric properties of triangular sets. To a certain extent, the relation between the primitivity of T and its U-set is also revealed in this paper.

Abstract: Ordered resolution and superposition are the state-of-the-art proof procedures used in saturation-based theorem proving, for non equational and equational clause se ts respectively. In this paper, we present extensions of these calculi that permit one to reason about formulae built from terms with integer exponents (or I-terms), a schematisation language allowing one to denote infinite sequences of iterated terms. We prove that the ordered resolution calculus is still refutationally complete when applied on (non equational) clauses containing I-terms. In the equational case, we prove that the superposition calculus is not complete in the presence of I-terms and we devise a new inference rule, called H-superposition, that restores completeness.

Abstract: Many decision procedures for SMT problems rely more or less implicitly on an instantiation of the axioms defining the theories under consideration, and differ by making use of the additional properties of each theory, in order to increase efficiency. We present a new technique for devising complete instantiation schemes on SMT problems over a combination of linear arithmetic with another theory T . The method consists in first instantiating the arithmetic part of the formula, and then getting rid of the remaining variables in the problem by using an instantiation strategy which is complete for T . We provide examples evidencing that not only is this technique generic (in the sense that it applies to a wide range of theories) but it is also efficient, even compared to state-of-the-art instantiation schemes for specific theories.