## Disunification: a Survey (1991)

Venue: | Computational Logic: Essays in Honor of Alan |

Citations: | 54 - 8 self |

### BibTeX

@INPROCEEDINGS{Comon91disunification:a,

author = {Hubert Comon},

title = {Disunification: a Survey},

booktitle = {Computational Logic: Essays in Honor of Alan},

year = {1991},

pages = {322--359},

publisher = {MIT Press}

}

### Years of Citing Articles

### OpenURL

### Abstract

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...

### Citations

812 | Semantics of constraint logic programs
- Jaffar, Maher, et al.
- 1998
(Show Context)
Citation Context ...rograms while keeping safe semantical foundations and a simple operational semantics. "Logic program" should be understood here in a broad sense: a logic program may be a (constrained) set o=-=f clauses [JL87]-=- or a (constrained) set of equations [KK89], or any other set of formulas in some logic. Enhancement of expressiveness is obvious, but some particular applications are shown in [KK89]. For example, it... |

470 | Termination of rewriting
- Dershowitz
- 1987
(Show Context)
Citation Context ... matching quasi-ordering in [KK89] and as a recursive path ordering 4 in [Com90a, JO91]. Each interpretation is motivated by some kind of application. 4 The recursive path ordering is defined in e.g. =-=[Der87]-=-. To each total ordering on F is associated a total recursive path ordering on T (F ). Such orderings are used in all implementations of Term Rewriting as well as in Ordered strategies in Automated De... |

241 |
Negation in logic programming
- Kunen
- 1987
(Show Context)
Citation Context ...)) the axioms were given in Figures 1 et 2: let EFT be this set of axioms. It turns out that EFT completely axiomatizes the algebra of finite trees over an infinite set of function symbols: Theorem 5 =-=[Kun87b]-=- EFT is complete when F is infinite. This result is also proved in a different way in [Mah88a]. EFT was also shown to be decidable in [Mal71] 8 . A locally free algebra is an F -algebra such that ever... |

228 |
Unification Revisited
- Lassez, Maher, et al.
- 1988
(Show Context)
Citation Context ... shows the completeness of the theory. A simple example for the first three steps is (syntactic) unification in finite trees: as shown e.g. in the survey on unification (this volume, chapter 8) or in =-=[LMM86]-=-, it simply consists in a strategy for the use of the axioms for syntactic equality recalled in Figures 1, 2. This paper aims at giving an overview of the main results on disunification as well as the... |

224 |
Mathematical Logic
- Shoenfield
- 1967
(Show Context)
Citation Context ...bly a single algebra A. Let T be a set of equational formulas (axioms). T is said to be correct with respect to K if K j= OE for every formula OE 2 T . The following definitions are derived from e.g. =-=[Sho67]. A s-=-entence is an equational formula without free variable. The first order theory of a set of sentences T is the set Th(T ) of all sentences OE such that T j= OE. Sometimes we will simply write "the... |

193 | Fast decision procedures based on congruence closure
- Nelson, Oppen
- 1980
(Show Context)
Citation Context ...dy the validity of an arbitrary formula in some restricted classes of algebras: the locally free algebras. Mal'cev also proposed some extensions to classes of permutative algebras. Nelson and Open in =-=[NO80]-=- show the decidability of satisfiability for the purely existential equational formulas. 2.3.8 Handling inequalities Considering general equational problems which may contain inequalities requires an ... |

144 |
Fundamental properties of infinite trees
- Courcelle
- 1983
(Show Context)
Citation Context ...orms are slightly different since an equation x = t[x] does have solutions. More generally, the following basic result shows the relationship between systems of equations and rational trees (see e.g. =-=[Cou81]-=-): Theorem 2 A system x 1 = t 1s: : :sxn = t n where x 1 ; : : : ; xn are variables, t 1 ; : : : ; t n 2 T (F; fx 1 ; : : : ; xn g) and, for every i, x i 6j t i , has a unique RT (F )-solution. Theref... |

137 |
Complete Axiomatization of the Algebras of Finite, Rational and Infinite Trees
- Maher
- 1988
(Show Context)
Citation Context ..., Mal'cev studied in his landmark paper [Mal71] classes of (term) algebras that are axiomatizable. Kirchner and Lescanne [KL87] followed by Comon and Lescanne [CL89], Kunen [Kun87b, Kun87a] and Maher =-=[Mah88a]-=- studied general equational formulas. 1.7 Extensions 1.7.1 Many-Sorted Signatures It is easy to extend the definitions to many-sorted signatures. All results presented here can be extended to this cas... |

117 |
The algebraic specification of abstract data types
- Guttag, Horning
- 1978
(Show Context)
Citation Context ...se rules. By "completely defined" we mean that every expression constructed on 0; s; + can effectively be reduced to a term that does not contain any occurrence of +. Introduced by Guttag an=-=d Horning [GH78], this pro-=-blem is called "sufficient completeness." It is related to correctness in algebraic specifications and to automated proofs by induction. For solving such problems, a simple idea is to look f... |

103 | Equational problems and disunification
- Comon, Lescanne
- 1989
(Show Context)
Citation Context ...s in algebras of trees. In this framework, Mal'cev studied in his landmark paper [Mal71] classes of (term) algebras that are axiomatizable. Kirchner and Lescanne [KL87] followed by Comon and Lescanne =-=[CL89]-=-, Kunen [Kun87b, Kun87a] and Maher [Mah88a] studied general equational formulas. 1.7 Extensions 1.7.1 Many-Sorted Signatures It is easy to extend the definitions to many-sorted signatures. All results... |

101 |
Equations and Inequations on Finite and Infinite Trees
- Colmerauer
- 1984
(Show Context)
Citation Context ..., the syntax may allow quantification [KL87, CL89, Mah88a] or not [Col84, Bur88]. The semantics may consider one particular algebra (e.g. the Herbrand Universe [LM87] or the algebra of rational trees =-=[Col84]-=- or irreducible trees [Com89]) or a class of algebras [Mal71, MSK90]. The definition of a solved form may also differ, depending on the application at hand. Indeed, we may be interested in deciding th... |

81 | Equational Logic as a Programming Language - O'Donnell - 1985 |

81 |
A Uni Algorithm for Associative-Commutative Functions
- Stickel
- 1981
(Show Context)
Citation Context ... in this volume). However, E-unification is decidable for some special theories such as AC-theories where E consists of a set of commutativity and associativity axioms for some functions symbols in F =-=[Sti81]-=- (see also the survey on unification in this volume). Since such cases are of practical interest, the problem arises to study equational problems in the algebra T (F )= =E (or T (F; X)==E ). Indeed, f... |

64 |
Decidable Theories
- Rabin
- 1977
(Show Context)
Citation Context ...Related Work Using a Similar Formalism Solving equational formulas is closely related to the study of decidable theories. However, we do not intend to survey the decidable theories (instead, see e.g. =-=[Rab77]-=-). Actually, we will only consider syntactic interpretations of our formulas, i.e. interpretations in algebras of trees. In this framework, Mal'cev studied in his landmark paper [Mal71] classes of (te... |

54 | A Decision Procedure for a Class of Set Constraints
- Heintze, Jaffar
- 1990
(Show Context)
Citation Context ... is decidable. (Treinen's result apply to any non-total precedence.) We do not know either whether the \Sigma 1 fragment is decidable in the case of a non-total pecedence. Heintze and Jaffar study in =-=[HJ90] set -=-constraints. Such constraints are formulas exp 1 ' exp 2 where exp 1 ; exp 2 are expressions involving usual operations on sets ("; [; :) and also function application (from a fixed alphabet F ) ... |

50 | Order-sorted equational computation - SMOLKA, Nu’rr, et al. - 1989 |

49 | Solving symbolic ordering constraints - Comon |

46 |
Axiomatizable classes of locally free algebras
- Malcev
- 1971
(Show Context)
Citation Context ...e symbol is equality (let's call it an equational formula) in an algebra A shows the decidability of the theory of A and provides a complete axiomatization of A. The work of logicians such as Mal'cev =-=[Mal71]-=- thus falls into the scope of disunification. There are also completely different approaches: solving certain kinds of equational formulas comes up when compiling functional programs, when dealing wit... |

40 |
Explicit representation of terms defined by counter examples
- Lassez, Marriott
- 1987
(Show Context)
Citation Context ...urns out that terms alone present a lack of expressiveness. In particular, the complement of a set of terms denoted by a term with variables cannot always be represented by a finite set of terms (see =-=[LM87]-=-). For example, it is not possible to find a finite set of terms whose instances are all terms that are not instances of f(x; x). This lack of expressiveness was encountered in many situations involvi... |

38 |
Proving refutational completeness of theorem proving strategies: The transfinite semantic tree method
- Hsiang, Rusinowitch
- 1991
(Show Context)
Citation Context ...ering on F is associated a total recursive path ordering on T (F ). Such orderings are used in all implementations of Term Rewriting as well as in Ordered strategies in Automated Deduction. (see e.g. =-=[HR87]-=-). 3 Solved Forms Solved forms are particular classes of formulas on which the problem to be solved is trivially decidable. As a first example, let us recall what can be used as a solved form in the u... |

37 |
A transformation approach to negation in logic programming
- Barbuti, Mancarella, et al.
- 1990
(Show Context)
Citation Context ...uch that every variable in U is mapped to an element of A. 2.3 Examples 2.3.1 A = T (F ) or A = T (F; X) This is the interpretation used for complement problems (for example [Sch87, Sch88b], [Thi84], =-=[BMPT90]-=-, [LM87], [Com86] and others). This interpretation is also considered in e.g [Ven87], [CL89], [Mah88a], [Bur88], [Kun87b, Kun87a, Nic87]. It appears that the main point for such interpretations is to ... |

35 |
Satisfiability of systems of ordinal notations with the subterm property is decidable
- Jouannaud, Okada
- 1991
(Show Context)
Citation Context ...existential fragment of the theory of lexicographic path ordering has been shown decidable [Com90c]. This result has been extended to other ordinal notations (in particular, recursive path orderings) =-=[JO91]-=-. Venkataraman's results on the theory of subterm ordering have been extended to rational trees [Tul91]. The set of transformation rules we gave in this survey for disunification in finite trees (with... |

31 |
Decidability of the purely existential fragment of the theory of term algebras
- Venkataraman
- 1987
(Show Context)
Citation Context ...F ) or A = T (F; X) This is the interpretation used for complement problems (for example [Sch87, Sch88b], [Thi84], [BMPT90], [LM87], [Com86] and others). This interpretation is also considered in e.g =-=[Ven87]-=-, [CL89], [Mah88a], [Bur88], [Kun87b, Kun87a, Nic87]. It appears that the main point for such interpretations is to know whether F is finite or infinite: the rules for solving equational formulas will... |

29 |
Is Appropriability a "Problem” for
- Davis
(Show Context)
Citation Context ...pleteness results when A is an arbitrary quotient of T (F ). This would indeed imply in particular the decidability of unification modulo any set of equational axioms which is known to be undecidable =-=[Dav77]-=-. It is not even possible to find a semi-decision procedure for the satisfiability of equational formulas in any quotient of T (F ) since the first order theory of T (F )= =E is complete. Hence, such ... |

29 | A new method for undecidability proofs of first order theories
- Treinen
- 1992
(Show Context)
Citation Context ...Mah88a, Com88b] [Mah88a] [Mah88a] [Com89, Com88b] Equational theory free quasi-free compact AC with decidable unification Equational Formulas: F finite [CL89, Mah88a] [Com88b] undecidable undecidable =-=[Tre90]-=- undecidable F infinite [Mah88a, Kun87b] [Com88b] \Delta \Delta undecidable Existential formulas: F finite \Delta \Delta [Com88b] [Com88b] undecidable [Fer91] F infinite [Col84] \Delta \Delta \Delta [... |

25 |
Answer sets and negation as failure
- Kunen
- 1987
(Show Context)
Citation Context ... / ODD(x) As shown, for example, in [BMPT90, Com88b] this automatic computation relies on the complement of the left hand sides defining EVEN. Another use of complement problems is developed by Kunen =-=[Kun87a]-=-. He uses them to represent infinite sets of answers to negative queries when dealing with negation as failure. For these reasons, we want to be able to compute a suitable representation of all the gr... |

24 | Compiling pattern matching by term decomposition
- Puel, Suhrez
- 1990
(Show Context)
Citation Context ... 1 + x 2 6= x + x expresses the complement of [[x + 0]] [ [[0 +x]] [ [[x + x]] [ [[s(s(0))]] (i.e., the left hand sides of the rules) into [[x 1 + x 2 ]]. Some complement problems are also studied in =-=[PS90]-=- with a slightly different definition. In their framework, a constant (ffl) plays a special role: its intended meaning is a non-terminating evaluation (or a non-evaluated term). Compiling (in an optim... |

23 |
Computing unification algorithms
- Kirchner
(Show Context)
Citation Context ...alent . Again, Unique Solution is a scheme of axioms that has infinitely many instances. But the set of instances is recursive. 4.4 Syntactic theories axioms C. Kirchner introduced syntactic theories =-=[Kir86]-=- which are a class of equational theories for which it is possible to systematically derive unification algorithms (see also [DJ90]). In such theories it is possible to give axioms such as those of Fi... |

22 |
R'esolution d"equations dans les langages d'ordre 1,2. . . !. Th`ese de doctorat d"etat, Universit'e de Paris VII
- Huet
- 1976
(Show Context)
Citation Context ...r a conjunction x 1 = t 1s: : :sxn = t n where x 1 ; : : : ; xn are distinct variables and, for every i, if t i is a variable, then x i occurs only once. The following result is given by e.g. G. Huet =-=[Hue76]: Th-=-eorem 3 For every equation s = t there is a system S in solved form such that s = t �� RT (F ) S. Note that we do not have a unicity result since the rational tree defined by x = g(x) is also defi... |

21 |
A constraint sequent calculus
- Lassez, McAloon
- 1990
(Show Context)
Citation Context ...n some further papers. Among them, let me cite [LMM86], [Com88b] where the lemma is extended to the case of some equational theories, [LM89] where the lemma is extended to systems of inequalities and =-=[LM90]-=- where this lemma is set up in a general framework. The independence of disequations lemma plays a central role not only for showing the existence of a solution but also in the transformation rules of... |

19 |
Constrained equational reasoning
- Kirchner, Kirchner
- 1989
(Show Context)
Citation Context ...ations and a simple operational semantics. "Logic program" should be understood here in a broad sense: a logic program may be a (constrained) set of clauses [JL87] or a (constrained) set of =-=equations [KK89]-=-, or any other set of formulas in some logic. Enhancement of expressiveness is obvious, but some particular applications are shown in [KK89]. For example, it is possible to deal with AC function symbo... |

17 | Inductive proofs by specification transformation - Comon - 1989 |

16 |
Sufficient completeness, term rewriting system and anti-unification
- Comon
- 1986
(Show Context)
Citation Context ...iable in U is mapped to an element of A. 2.3 Examples 2.3.1 A = T (F ) or A = T (F; X) This is the interpretation used for complement problems (for example [Sch87, Sch88b], [Thi84], [BMPT90], [LM87], =-=[Com86]-=- and others). This interpretation is also considered in e.g [Ven87], [CL89], [Mah88a], [Bur88], [Kun87b, Kun87a, Nic87]. It appears that the main point for such interpretations is to know whether F is... |

16 |
Equational formulas in order-sorted algebras
- Comon
- 1990
(Show Context)
Citation Context ... F infinite. More general sets of rules can also be provided, allowing to eliminate existential quantifiers as well as universal ones and dropping the requirement for the formula to be in prenex form =-=[Com90b]-=-. Other sets of rules (together with a different control) are given in [Mal71, Mah88a] together with their completeness and termination properties. 5.3 Completeness and termination results for rationa... |

13 |
Unification et disunification. Théories et applications. Thèse de Doctorat d’Université, Institut Polytechnique de Grenoble
- Comon
- 1988
(Show Context)
Citation Context ... [[t]] (i.e. without negation). In other words, the class of finite unions [[t 1 ]] [ : : : [ [[t n ]] is not closed under complement. That is the reason why constrained terms have been introduced in =-=[Com88b]-=-. A constrained term is a pair t & d, where t is a term and d is a conjunction of non-trivial disequations whose variables are those of t. Then [[t &d]] is the set of all ground instances of t which s... |

13 | Extending Equation Solving and Constraint Handling in Logic Programming - Dincbas, Simonis, et al. - 1987 |

12 |
Refined compilation of pattern-matching for functionnal languages
- Schnoebelen
- 1988
(Show Context)
Citation Context ...plied. Hence, we have some knowledge about the ground terms on which the second rule should be tried : they are in the complement of the first left hand side. This is effectively used in, for example =-=[Sch88a]-=-. Another kind of term complements is used in [Lav87, Lav88, PS90] in order to produce a non-ambiguous set of patterns . ffl There are basically two kinds of applications of complement problems in log... |

11 |
Solving disequations in equational theories
- Burckert
- 1988
(Show Context)
Citation Context ... the interpretation used for complement problems (for example [Sch87, Sch88b], [Thi84], [BMPT90], [LM87], [Com86] and others). This interpretation is also considered in e.g [Ven87], [CL89], [Mah88a], =-=[Bur88]-=-, [Kun87b, Kun87a, Nic87]. It appears that the main point for such interpretations is to know whether F is finite or infinite: the rules for solving equational formulas will be different in the two ca... |

11 |
Solving inequations in term algebras
- Comon
- 1990
(Show Context)
Citation Context ... sentence is undecidable. By contrast, he shows that the existential fragment is decidable. Interpreting ? as a recursive path ordering leads to similar results: the existential fragment is decidable =-=[Com90c]-=- and the first order theory of embedding 10 is undecidable [Tre90]. It is still open whether the first order theory of a recursive path ordering extending a total precedence on F is decidable (althoug... |

10 | Complete Axiomatizations of Some Quotient Term Algebras, Theoretical Computer Science 118
- Comon
- 1993
(Show Context)
Citation Context ... axioms fulfill this condition but not associativity. Also, equational theories which are defined by ground equations do have a conservative extension which is generated by a quasi-free set of axioms =-=[Com91a]. There ar-=-e however "few" equational theories satisfying these requirements. Theorem 8 EST is a complete axiomatization of T (F )= =E when F is infinite and E is quasi-free. EST [ fDCAg is a complete ... |

8 | On solving equations and disequations - Buntine, Bürckert - 1994 |

8 |
A general complete E-unification procedure
- Gallier, Snyder
- 1987
(Show Context)
Citation Context ...eneralization of syntactic unification: syntactic unification corresponds to the case E = ;. Of course, general E-unification is undecidable. Some complete strategies for this problem where proposed (=-=[GS87]-=- for example. See also the survey on unification in this volume). However, E-unification is decidable for some special theories such as AC-theories where E consists of a set of commutativity and assoc... |

8 |
Decidability of the existential theory of infinite terms with subterm relation
- Tulipani
- 1994
(Show Context)
Citation Context ...his result has been extended to other ordinal notations (in particular, recursive path orderings) [JO91]. Venkataraman's results on the theory of subterm ordering have been extended to rational trees =-=[Tul91]. Th-=-e set of transformation rules we gave in this survey for disunification in finite trees (with slight modifications) has been shown confluent [Del90], showing that �� T (F ) can be decided by rewri... |

7 | Order-sorted algebra I: Partial and overloaded operators, errors and inheritance - Goguen, Meseguer - 1987 |

7 | Lazy pattern matching in the ML language - Laville - 1987 |

6 | An effective method for handling initial algebras - Comon - 1988 |

6 |
Independence of negative constraints
- Lassez, McAloon
- 1989
(Show Context)
Citation Context ...a proposed by Colmerauer in [Col84]. This lemma has been studied in some further papers. Among them, let me cite [LMM86], [Com88b] where the lemma is extended to the case of some equational theories, =-=[LM89]-=- where the lemma is extended to systems of inequalities and [LM90] where this lemma is set up in a general framework. The independence of disequations lemma plays a central role not only for showing t... |

6 |
A Deduction Procedure for First Order Programs
- Lugiez
- 1989
(Show Context)
Citation Context ...rived only for a particular subclass of formulas. Nevertheless, this is sufficient for deriving a complete deduction procedure for logic programs where negative goals and general programs are allowed =-=[Lug89]-=-. It is also shown in [Lug89] how to use these results for the transformation of logic programs. 1.8 Constraint Languages A constraints system is a logic which enjoys two particular properties: ffl On... |

5 |
Solving disequations
- Kirchner, Lescanne
- 1987
(Show Context)
Citation Context ...s of our formulas, i.e. interpretations in algebras of trees. In this framework, Mal'cev studied in his landmark paper [Mal71] classes of (term) algebras that are axiomatizable. Kirchner and Lescanne =-=[KL87]-=- followed by Comon and Lescanne [CL89], Kunen [Kun87b, Kun87a] and Maher [Mah88a] studied general equational formulas. 1.7 Extensions 1.7.1 Many-Sorted Signatures It is easy to extend the definitions ... |

5 |
Bon preordres sur les arbres associes a des ensembles inevitables et preuves de terminaison de systemes de reecriture. These d'Etat
- Puel
- 1987
(Show Context)
Citation Context .... Simplifications of such complement problems as well as properties of the corresponding constrained terms are studied in [PS90]. Classical complement problems are also studied and used by L. Puel in =-=[Pue87]-=-. Here, the purpose is to decompose terms into patterns of a given (unavoidable) set in order to compare them using a pattern path ordering. 1.6 Related Work Using a Similar Formalism Solving equation... |