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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 is de ..."
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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 ...
Formalization of the Development Process
, 1998
"... Introduction 14.1.1 What is development? Software development encompasses many phases including requirements engineering, specification, design, implementation, verification or testing, and maintenance. In this chapter we concentrate on the intermediate tasks: the transition from requirements spec ..."
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Introduction 14.1.1 What is development? Software development encompasses many phases including requirements engineering, specification, design, implementation, verification or testing, and maintenance. In this chapter we concentrate on the intermediate tasks: the transition from requirements specification to verified design and design optimization; in particular techniques for developing correct designs as opposed to ad hoc or a posteriori methods in which a postulated design is later verified or tested. We view software development as the composition of correctness preserving refinement steps. Hence, a development is meaningful relative to a notion of refinement and correctness. There are many such notions (e.g., Chapter 6), each defining a type of problem transformation or reduction. For example specification refinement from a specification SP to SP 0 through a refinement map ae might be defined as co
Unification algebras: an axiomatic approach to unification, equation solving and constraint solving
 Universitat Kaiserslautern
, 1988
"... Abstract. Traditionally unification is viewed as solving an equation in an algebra given an explicit construction method for terms and substitutions. We abstract from this explicit term construction methods and give a set of axioms describing unification algebras that consist of objects and mappings ..."
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Abstract. Traditionally unification is viewed as solving an equation in an algebra given an explicit construction method for terms and substitutions. We abstract from this explicit term construction methods and give a set of axioms describing unification algebras that consist of objects and mappings, where objects abstract terms and mappings abstract substitutions. A unification problem in a given unification algebra is the problem to find mappings for a system of equations 〈si = tii ∈ I〉, where si and ti are objects, such that si and ti are mapped onto the same object. Typical instances of unification algebras and unification problems are: Term unification with respect to equational theories and sorts, standard equation solving in mathematics, unification in the λcalculus, constraint solving, disunification, and unification of rational terms. Within this framework we give general purpose unification rules that can be used in every unification algorithm in unification algebras. Furthermore we demonstrate the use of this framework by investigating the analogue of syntactic unification and unification of rational terms.
Unification in permutative equational theories is undecidable
 J. SYMB. COMPUT
, 1989
"... An equational theory E is permutative if for all terms s, t: s =E t implies that the terms s and t contain the same symbols with the same number of occurrences. The class of permutative equational theories includes the theory of AC (associativity and commutativity). It is shown in this research note ..."
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An equational theory E is permutative if for all terms s, t: s =E t implies that the terms s and t contain the same symbols with the same number of occurrences. The class of permutative equational theories includes the theory of AC (associativity and commutativity). It is shown in this research note that there is no algorithm that decides Eunifiability of terms for all permutative theories. The proof technique is to provide for every Turing machine M, a permutative theory with a confluent termrewriting system such that narrowing on certain terms simulates the Turing machine M.