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 Paris-Sud, 91405 ORSAY cedex, France. E-mail: comon@lri.lri.fr i Contents 1 Syntax 5 1.1 Basic Defini...

In functional programming environments, one can use types as search keys in program libraries, if one disregards trivial differences in argument order or currying. A way to do this is to identify types that are isomorphic in every Cartesian closed category; simpler put, types should be identified if they are equal under an arithmetic interpretation, with Cartesian product as multiplication and function space as exponentiation. When the type system is polymorphic, one may also want to retrieve identifiers of types more general than the query. This paper describes a method to do both, that is, an algorithm for pattern matching modulo canonical CCC-isomorphism. The algorithm returns a finite complete set of matchers. An implementation shows that satisfactory speed can be achieved for library search. Contents 1 Introduction 2 2 Unification/Matching in Equational Theories 6 3 Comparison with Previous Work 9 4 An Algorithm for \Gamma-matching 14 5 Practical Experience of Library Search 25 6...

Introduction Interest in the theory and applications of rewriting has been growing rapidly, as evidenced in part by four conference proceedings #including this one# #15, 26, 41,66#; three workshop proceedings #33, 47, 77#; #ve special journal issues #5,88, 24, 40, 67#; more than ten surveys #2,7,27, 28, 44, 56,57,76, 82, 81#; one edited collection of papers #1#; four monographs #3, 12,55,65#; and seven books #four of them still in progress# #8,9, 35, 54, 60,75, 84#. To encourage and stimulate continued progress in this area, wehave collected #with the help of colleagues# a number of problems that appear to us to be of interest and regarding whichwe do not know the answer. Questions on rewriting and other equational paradigms have been included; manyhave not aged su#ciently to be accorded the appellation #open problem". Wehave limited ourselves to theoretical questions, though there are certainly many additional interesting questions relating to applications and implementation

We extend the ML-style type system with variable-arity procedures, supporting both optional arguments and arbitrarily long argument sequences. The language with variable-arity procedures is encoded in a core ML variant with infinitary tuples. We present an algebra of infinitary tuples and solve its unification problem. The resulting type discipline preserves principal typings and has a terminating type reconstruction algorithm. The expressive power of infinitary tuples is illustrated. 1 Introduction Most languages employing ML-style polymorphic type reconstruction do not support procedures with multiple arguments. Instead, multiple arguments are passed in an aggregate structure or via repeated application to a curried procedure. Extension of an ML-style type system to support higher, but fixed, arity procedures is straightforward. A variable-arity procedure accepts an indefinite number of arguments. Many languages provide variable-arity primitive procedures, and some allow creation of...

It is shown that for sets of Horn clauses saturated under basic paramodulation, the word and unifiability problems are in NP, and the number of minimal unifiers is simply exponential (i). For Horn sets saturated wrt. a special ordering under the more restrictive inference rule of basic superposition, the word and unifiability problems are still decidable and unification is finitary (ii). These two results are applied to the following languages. For shallow presentations (equations with variables at depth at most one) we show that the closure under paramodulation can be computed in polynomial time. Applying result (i), it follows that shallow unifiability is in NP, which is optimal since unifiability in ground theories is already NP-hard. The shallow word problem is even shown to be polynomial. Generalizing shallow theories to the Horn case, we obtain (two versions of) a language we call Catalog, a natural extension of Datalog to include functions and equality. The closure under paramo...

We show that unification in certain extensions of shallow equational theories is decidable. Our extensions generalize the known classes of shallow or standard equational theories. In order to prove decidability of unification in the extensions, a class of Horn clause sets called sorted shallow equational theories is introduced. This class is a natural extension of tree automata with equality constraints between brother subterms as well as shallow sort theories. We show that saturation under sorted superposition is effective on sorted shallow equational theories. So called semi-linear equational theories can be e ectively transformed into equivalent sorted shallow equational theories and generalize the classes of shallow and standard equational theories.

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 !? (occur-check). 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...

This paper considers the problem of E-unification for arbitrary equational theories E, and presents an inference rule approximating Paramodulation and leading to a complete E-unification procedure which generalizes Narrowing. This sheds some light on the boundary between arbitrary E-unification situations and E-unification under canonical E.

. We consider equational unification and matching problems where the equational theory contains a nilpotent function, i.e., a function f satisfying f(x;x) = 0 where 0 is a constant. Nilpotent matching and unification are shown to be NP-complete. In the presence of associativity and commutativity, the problems still remain NP-complete. But when 0 is also assumed to be the unity for the function f , the problems are solvable in polynomial time. We also show that the problem remains in P even when a homomorphism is added. Second-order matching modulo nilpotence is shown to be undecidable. Subject area: MECHANISMS: unification 1 Introduction Equational unification is an important computational problem in automated theorem proving. Its usefulness derives from the ability to `build in' many proof steps into the pattern matching algorithm, possibly shortening the search for a proof. As a new practical application, we define a class of set constraints and show that problems in this class ca...

) Robert Nieuwenhuis Technical University of Catalonia Pau Gargallo 5, 08028 Barcelona, Spain E-mail: roberto@lsi.upc.es. Abstract We prove that for sets of Horn clauses saturated under basic paramodulation, the word and unifiability problems are in NP, and the number of minimal unifiers is simply exponential (i). For Horn sets saturated wrt. a special ordering under the more restrictive inference rule of basic superposition, the word and unifiability problems are still decidable and unification is finitary (ii). We define standard theories, which include and significantly extend shallow theories. Standard presentations can be finitely closed under superposition and result (ii) applies. Generalizing shallow theories to the Horn case, we obtain (two versions of) a language we call Catalog, a natural extension of Datalog to include functions and equality. The closure under paramodulation is finite for Catalog sets, hence (i) applies. Since for shallow sets this closure is even polynom...