this paper are (i) a construction by which, for a given bounded treewidth, a general MS graph property P is transformed to an MS binary tree property r(P), and a general labeled graph G with a suitable tree-decomposition is transformed to a labeled binary tree T(G) in time linear in the number of vertices of G and in such a way that P holds for G if and only if r(P) holds for T(G). This allows us, using techniques developed by Doner [20] and Thatcher and Wright [42], to compile a tree automaton which decides the MS-problem r(P) on the tree T(G) (and thus also P on the graph G) in linear time, and (ii) a procedure whereby such an automaton for a MS formula with free variables is modified to solve a related EMS problem involving counting

. We describe a model-theoretic approach to ordinal analysis via the finite combinatorial notion of an #-large set of natural numbers. In contrast to syntactic approaches that use cut elimination, this approach involves constructing finite sets of numbers with combinatorial properties that, in nonstandard instances, give rise to models of the theory being analyzed. This method is applied to obtain ordinal analyses of a number of interesting subsystems of first- and second-order arithmetic. x1. Introduction. Two of proof theory's defining goals are the justification of classical theories on constructive grounds, and the extraction of constructive information from classical proofs. Since Gentzen, ordinal analysis has been a major component in these pursuits, and the assignment of recursive ordinals to theories has proven to be an illuminating way of measuring their constructive strength. The traditional approach to ordinal analysis, which uses cut-elimination procedures to transfor...

A main goal of Formal Concept Analysis from its very beginning has been the support of rational communication. The source of this goal lies in our understanding of mathematics as a science which should encompass both its philosophical basis and its social consequences. This can be achieved by a process named 'restructuring'. This approach shall be extended to logic, which is based on the doctrines of concepts, judgments and conclusions. The program of restructuring logic is named Contextual Logic (CL). A main idea of CL is to combine Formal Concept Analysis and Concept Graphs (which are mathematical structures based on conceptual graphs). Concept graphs formulate judgments on the contained concepts, and conclusions can be drawn by inferring one concept graph from another. So we see that concept graphs can be understood as a crucial part of the mathematical implementation of CL, based on Formal Concept Analysis as the mathematization of the doctrine of concepts.

Abstract. We show that adding a total order to ACL2, via new axioms, allows for simpler and more elegant definitions of functions and libraries of theorems. We motivate the need for a total order with a simple example and explain how a total order can be used to simplify existing libraries of theorems (i.e., ACL2 books) on finite set theory and records. These ideas have been incorporated into ACL2 Version 2.6, which includes axioms positing a total order on the ACL2 universe. 1 Introduction ACL2 [7, 6, 8] is a logic of total functions. One particularly pleasant consequence is that many properties of functions can be stated as unconditional rewrite rules. For example, we can prove (equal ( * y ( * x z)) ( * x ( * y z))) without having to establish that x, y, and z are numbers. Such unconditional rewrite rules lead to simpler libraries of theorems, which in turn improve the ability of ACL2 to reduce large terms automatically and efficiently. Unfortunately, it is problematic to exploit fully the totality of functions in ACL2 Version 2.5. One is often forced to use rewrite rules with hypotheses because of the lack of a definable total order on the ACL2 universe.

This paper presents properties of relations between words that are realized by deterministic finite 2-tape automata. It has been made as complete as possible, and is structured by the systematic use of the matrix representation of automata. It is first shown that deterministic 2-tape automata are characterized as those which can be given a prefix matrix representation. Schutzenberger construct on representations, the one that gives semi-monomial representations for rational functions of words, is then applied to this prefix representation in order to obtain a new proof of the fact that the lexicographic selection of a deterministic rational relation on words is a rational function. R'esum'e Cet article donne une pr'esentation des propri'et'es des relations entre mots r'ealis'ees par des automates finis `a deux bandes d'eterministes, qu'on a voulu aussi compl`ete que possible. Elle est organis'ee autour de la notion de repr'esentation matricielle des automates. On montre d'abord que le...

. A method for building finite models is proposed. It combines enumeration of the set of interpretations on a finite domain with strategies in order to prune significantly the search space. The main new ideas underlying our method are to benefit from symmetries and from the information extracted from the structure of the problem and from failures of model verification tests. The algorithms formalizing the approach are given and the standard properties (termination, completeness, and soundness) are proven. The method can deal with first-order logic with equality. In contrast to existing ones, it does not require to transform the initial problem into a normal form and can be easily extended to other logics. Experimental results and comparisons with related works are reported. 1. Introduction The capital importance of the notion of "model" in Logic was naturally inherited by Automated Deduction, where, since the very beginning, the use of models has been recognized as an useful technique...

A string database is simply a collection of tables, the columns of which contain strings over some given alphabet. We address in this paper the issue of designing a simple, user friendly query language for string databases. We focus on the language FO(ffl), which is classical first order logic extended with a concatenation operator, and where quantifiers range over the set of all strings. We wish to capture all string queries, i.e., well-typed and computable mappings involving a notion of string genericity. Unfortunately, unrestricted quantification may allow some queries to have infinite output. This leads us to study the "safety" problem for FO(ffl), that is, how to build syntactic and/or semantic restrictions so as to obtain a language expressing only queries with finite output, hopefully all string queries. We introduce a family of such restrictions and study their expressivness and complexity. We prove that none of these languages express all string queries. We prov...

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We define a proof system for a language of finite domain (bounded arithmetic) formulas, closed under the propositional connectives. The system includes simple rules for propagating logical consequences and the Dilemma rule for assumption-based reasoning. Using the system, a saturation algorithm is given which checks the satisfiability of any formula in the language, and hence, it can also be used for verifying tautologies. The algorithm is a combination of two previous propagation algorithms, of which one is for propositional logic and one is for finite domain constraints. Contents 1

The aim of this note is to attract attention to the most important open problems and new directions of research in this exciting and promising area. 1 Distance spaces Recall that a metric space is a pair (\Delta; d), where \Delta is a nonempty set (of points) and d is a function from \Delta \Theta \Delta into the set R *0 (of non-negative real numbers) satisfying the following