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Syntacticness, CycleSyntacticness and Shallow Theories
 INFORMATION AND COMPUTATION
, 1994
"... Solving equations in the free algebra T (F; X) (i.e. unification) uses the two rules: f(~s) = f( ~ t) ! ~s = ~ t (decomposition) and s[x] = x !? (occurcheck). These two rules are not correct in quotients of T (F; X) by a finitely generated congruence =E . Following C. Kirchner, we first define cl ..."
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Cited by 11 (0 self)
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Solving equations in the free algebra T (F; X) (i.e. unification) uses the two rules: f(~s) = f( ~ t) ! ~s = ~ t (decomposition) and s[x] = x !? (occurcheck). These two rules are not correct in quotients of T (F; X) by a finitely generated congruence =E . Following C. Kirchner, we first define classes of equational theories (called syntactic and cycle syntactic respectively) for which it is possible to derive some rules replacing the two above ones. Then, we show that these abstract classes are relevant: all shallow theories, i.e. theories which can be generated by equations in which variables occur at depth at most one, are both syntactic and cycle syntactic. Moreover, the new set of unification rules is terminating, which proves that unification is decidable and finitary in shallow theories. We give still further extensions. If the set of equivalence classes is infinite, a problem which turns out to be decidable in shallow theories, then shallow theories fulfill Colmerauer's indep...
Complete Axiomatizations of some Quotient Term Algebras
 In Proc. 18th Int. Coll. on Automata, Languages and Programming, Madrid, LNCS 510
, 1993
"... We show that T (F )= =E can be completely axiomatized when =E is a quasifree theory. Quasifree theories are a wider class of theories than permutative theories of [Mal71] for which Mal'cev gave decision results. As an example of application, we show that the first order theory of T (F )= =E ..."
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Cited by 10 (3 self)
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We show that T (F )= =E can be completely axiomatized when =E is a quasifree theory. Quasifree theories are a wider class of theories than permutative theories of [Mal71] for which Mal'cev gave decision results. As an example of application, we show that the first order theory of T (F )= =E is decidable when E is a set of ground equations. Besides, we prove that the \Sigma 1 fragment of the theory of T (F )= =E is decidable when E is a compact set of axioms. In particular, the existential fragment of the theory of associativecommutative function symbols is decidable. Introduction Mal'cev studied in the early sixties classes of locally free algebras that can be completely axiomatized [Mal71]. He proved in particular that what is today known as Clark's equality theory is decidable. He also studied some classes of permutative algebras in which, roughly, the axiom f(s 1 ; : : : ; s n ) = f(t 1 ; : : : ; t n ) ) s 1 = t 1 : : : s n = t n is replaced with f(s 1 ; : : : ; s n ) = f(t ...
Parallelizing Functional Programs by Generalization
 Journal of Functional Programming
, 1997
"... List homomorphisms are functions that are parallelizable using the divideandconquer paradigm. We study the problem of finding a homomorphic representation of a given function, based on the BirdMeertens theory of lists. A previous work proved that to each pair of leftward and rightward sequential ..."
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Cited by 9 (1 self)
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List homomorphisms are functions that are parallelizable using the divideandconquer paradigm. We study the problem of finding a homomorphic representation of a given function, based on the BirdMeertens theory of lists. A previous work proved that to each pair of leftward and rightward sequential representations of a function, based on cons and snoclists, respectively, there is also a representation as a homomorphism. Our contribution is a mechanizable method to extract the homomorphism representation from a pair of sequential representations. The method is decomposed to a generalization problem and an inductive claim, both solvable by term rewriting techniques. To solve the former we present a sound generalization procedure which yields the required representation, and terminates under reasonable assumptions. We illustrate the method and the procedure by the parallelization of the scanfunction (parallel prefix). The inductive claim is provable automatically.
Parallelizing Functional Programs by Term Rewriting
, 1997
"... List homomorphisms are functions that can be computed in parallel using the divideandconquer paradigm. We study the problem of finding a homomorphic representation of a given function, based on the BirdMeertens theory of lists. A previous work proved that to each pair of leftward and rightward se ..."
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Cited by 2 (2 self)
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List homomorphisms are functions that can be computed in parallel using the divideandconquer paradigm. We study the problem of finding a homomorphic representation of a given function, based on the BirdMeertens theory of lists. A previous work proved that to each pair of leftward and rightward sequential representations of a function, based on cons and snoclists, respectively, there is also a representation as a homomorphism. Our contribution is a mechanizable method to extract the homomorphism representation from a pair of sequential representations. The method is decomposed to a generalization problem and an inductive claim, both solvable by term rewriting techniques. To solve the former we present a sound generalization procedure which yields the required representation, and terminates under reasonable assumptions. We illustrate the method and the procedure by the parallelization of the scanfunction (parallel prefix). The inductive claim is provable automatically. Keywords: P...
Completion and Invariant Theory in Symbolic Computation and Artificial Intelligence
, 1993
"... . An outline for the study of invariant theoretic (as structural) and completion (as syntactical) concepts in symbolic computation and artificial intelligence is presented on a level of abstraction which permits a unifying viewpoint on problems in symbolic computation and artificial intelligence. We ..."
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. An outline for the study of invariant theoretic (as structural) and completion (as syntactical) concepts in symbolic computation and artificial intelligence is presented on a level of abstraction which permits a unifying viewpoint on problems in symbolic computation and artificial intelligence. We refer to applications in computational polynomial ideal theory and in general problemsolving in the sense of AI research. 1 Introduction In this article we discuss two methodological principles commonly used in mathematics and computer science. The first principle we refer to, integrates structural methods mainly used to provide for an external characterization of relational (algebraic) structures. The second envisioned principle embodies syntactical methods usually in force when dealing with an internal characterization of equational theories. Correspondingly, we will concentrate on two instances of these principles: 1. the search for invariants method, and 2. the completion method. It i...