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Disunification: a Survey
 Computational Logic: Essays in Honor of Alan
, 1991
"... Solving an equation in an algebra of terms is known as unification. Solving more complex formulas combining equations and involving in particular negation is called disunification. With such a broad definition, many works fall into the scope of disunification. The goal of this paper is to survey the ..."
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Solving an equation in an algebra of terms is known as unification. Solving more complex formulas combining equations and involving in particular negation is called disunification. With such a broad definition, many works fall into the scope of disunification. The goal of this paper is to survey these works and bring them together in a same framework. R'esum'e On appelle habituellement (algorithme d') unification un algorithme de r'esolution d'une 'equation dans une alg`ebre de termes. La r'esolution de formules plus complexes, comportant en particulier des n'egations, est appel'ee ici disunification. Avec une d'efinition aussi 'etendue, de nombreux travaux peuvent etre consid'er'es comme portant sur la disunification. L'objet de cet article de synth`ese est de rassembler tous ces travaux dans un meme formalisme. Laboratoire de Recherche en Informatique, Bat. 490, Universit'e de ParisSud, 91405 ORSAY cedex, France. Email: comon@lri.lri.fr i Contents 1 Syntax 5 1.1 Basic Defini...
A New Method for Undecidability Proofs of First Order Theories
 Journal of Symbolic Computation
, 1992
"... this paper is to define a framework for such reduction proofs. The method proposed is illustrated by proving the undecidability of the theory of a term algebra modulo the axioms of associativity and commutativity and of the theory of a partial lexicographic path ordering. 1. Introduction ..."
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this paper is to define a framework for such reduction proofs. The method proposed is illustrated by proving the undecidability of the theory of a term algebra modulo the axioms of associativity and commutativity and of the theory of a partial lexicographic path ordering. 1. Introduction
Checking amalgamability conditions for Casl architectural specifications
 Proc. Intl. Symp. on Mathematical Foundations of Computer Science, MFCS 2001. Springer LNCS 2136
, 2001
"... Introduction Architectural specifications are a mechanism to support formal program development [2, 11]. They have been introduced in the ..."
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Cited by 12 (9 self)
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Introduction Architectural specifications are a mechanism to support formal program development [2, 11]. They have been introduced in the
Comparative similarity, tree automata, and Diophantine equations
 In Proceedings of LPAR 2005, volume 3835 of LNAI
, 2005
"... Abstract. The notion of comparative similarity ‘X is more similar or closer to Y than to Z ’ has been investigated in both foundational and applied areas of knowledge representation and reasoning, e.g., in concept formation, similaritybased reasoning and areas of bioinformatics such as protein sequ ..."
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Cited by 12 (8 self)
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Abstract. The notion of comparative similarity ‘X is more similar or closer to Y than to Z ’ has been investigated in both foundational and applied areas of knowledge representation and reasoning, e.g., in concept formation, similaritybased reasoning and areas of bioinformatics such as protein sequence alignment. In this paper we analyse the computational behaviour of the ‘propositional ’ logic with the binary operator ‘closer to a set τ1 than to a set τ2 ’ and nominals interpreted over various classes of distance (or similarity) spaces. In particular, using a reduction to the emptiness problem for certain tree automata, we show that the satisfiability problem for this logic is ExpTimecomplete for the classes of all finite symmetric and all finite (possibly nonsymmetric) distance spaces. For finite subspaces of the real line (and higher dimensional Euclidean spaces) we prove the undecidability of satisfiability by a reduction of the solvability problem for Diophantine equations. As our ‘closer ’ operator has the same expressive power as the standard operator> of conditional logic, these results may have interesting implications for conditional logic as well. 1
Models of Lambda Calculi and Linear Logic: Structural, Equational and ProofTheoretic Characterisations
, 1994
"... Models of Lambda Calculi and Linear Logic: Structural, Equational and ProofTheoretic Characterisations Ralph Loader, of St. Hugh's College, Oxford. Thesis submitted for the Degree of D.Phil. Michaelmas term, 1994. T his thesis is an investigation into models of typed calculi and of linear logic. ..."
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Models of Lambda Calculi and Linear Logic: Structural, Equational and ProofTheoretic Characterisations Ralph Loader, of St. Hugh's College, Oxford. Thesis submitted for the Degree of D.Phil. Michaelmas term, 1994. T his thesis is an investigation into models of typed calculi and of linear logic. The models we investigate are denotational in nature; we construct various categories, in which types (or formulae) are interpreted by objects, and terms (proofs) by morphisms. The results we investigate compare particular properties of the syntax and the semantics of a calculus, by trying to use syntax to characterise features of a model, or vice versa. There are four chapters in the thesis, one each on linear logic and the simply typed calculus, and two on inductive datatypes. In chapter one, we look at some models of linear logic, and prove a full completeness result for multiplicative linear logic. We form a model, the linear logical predicates , by abstracting a little the structure ...
Combination Problems for Commutative/Monoidal Theories or How Algebra Can Help in Equational Unification
 J. Applicable Algebra in Engineering, Communication and Computing
, 1996
"... We study the class of theories for which solving unification problems is equivalent to solving systems of linear equations over a semiring. It encompasses important examples like the theories of Abelian monoids, idempotent Abelian monoids, and Abelian groups. This class has been introduced by the au ..."
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We study the class of theories for which solving unification problems is equivalent to solving systems of linear equations over a semiring. It encompasses important examples like the theories of Abelian monoids, idempotent Abelian monoids, and Abelian groups. This class has been introduced by the authors independently of each other as "commutative theories " (Baader) and "monoidal theories" (Nutt). We show that commutative theories and monoidal theories indeed define the same class (modulo a translation of the signature), and we prove that it is undecidable whether a given theory belongs to it. In the remainder of the paper we investigate combinations of commutative/monoidal theories with other theories. We show that finitary commutative/monoidal theories always satisfy the requirements for applying general methods developed for the combination of unification algorithms for disjoint equational theories. Then we study the adjunction of monoids of homomorphisms to commutative /monoidal t...
A Methodology for Constructing Predicate Transition Net Specifications
, 1991
"... this paper, a methodology for constructing hierarchical and structured predicate transition net specifications is developed, which includes new systematic notation extensions for supporting various transformation techniques upon predicate transition nets and several rules for applying such transform ..."
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Cited by 7 (0 self)
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this paper, a methodology for constructing hierarchical and structured predicate transition net specifications is developed, which includes new systematic notation extensions for supporting various transformation techniques upon predicate transition nets and several rules for applying such transformation techniques. The levelling technique in dataflow diagrams is adapted in the refinement and the abstraction techniques, and the state decomposition idea in statecharts is employed in designing various label formulation operators. The methodology is illustrated through the specification of a lift system. The methodology can significantly reduce the constructing complexity and enhance the comprehensibility of large predicate transition net specifications
Approximating the Shortest Path in Line Arrangements
 In Proc. 14th Canad. Conf. Computational Geometry
, 2001
"... Suppose one has a line arrangement and one wants to find a shortest path from one point lying on a line of the arrangement to another such point. The best known time bound for computing this is O(n 2 ). We develop an algorithm that finds a 1 + ffl approximation of the shortest path in time O(n log ..."
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Suppose one has a line arrangement and one wants to find a shortest path from one point lying on a line of the arrangement to another such point. The best known time bound for computing this is O(n 2 ). We develop an algorithm that finds a 1 + ffl approximation of the shortest path in time O(n log n + (minfn; 1 ffl 2 g) 1 ffl log 1 ffl ). 1
The Word Problem of ACDGround theories is Undecidable
"... We prove that there exists an ACDground theory  an equational theory defined by a set of ground equations plus the associativity and commutativity of two binary symbols and +, and the distributivity of over +  for which the word problem is undecidable. 1 Introduction Equations are ubiquit ..."
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We prove that there exists an ACDground theory  an equational theory defined by a set of ground equations plus the associativity and commutativity of two binary symbols and +, and the distributivity of over +  for which the word problem is undecidable. 1 Introduction Equations are ubiquitous in mathematics and the sciences. The word problem of a given a set of equations (that is the problem of deciding if an identity is a consequence of the equations), or equivalently of its equational theory, is undecidable in general. But there are known classes of equational theories which have a decidable word problem, in particular, ground equational theories. The most famous examples of theories with undecidable word problem are given by sets of ground equations over word algebras. Such theories can be considered as associativeground theories over a certain term algebra, whose signature contains only constants besides the binary (associative) symbol. Their word problem is known to be...
The number of certain integral polynomials and nonrecursive sets of integers, Part 1, this issue
"... We present some examples of mathematically natural nonrecursive sets of integers and relations on integers by combining results from Part 1, recursion theory, and from the negative solution to Hilbert’s 10th Problem ([3], [1], and [2]). 1 ..."
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We present some examples of mathematically natural nonrecursive sets of integers and relations on integers by combining results from Part 1, recursion theory, and from the negative solution to Hilbert’s 10th Problem ([3], [1], and [2]). 1