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## Unification and Matching modulo Nilpotence (1996)

Venue: | In Proc. CADE-13, volume 1104 of LNCS |

Citations: | 11 - 1 self |

### Citations

13826 |
Computers and Intractability, A Guide to the Theory of NP-completeness (Freeman
- Garey, Johnson
- 1968
(Show Context)
Citation Context ...g. NP -hardness is shown by reduction from THREE SATISFIABILITY [22]. ACN-unification, where the function is also associative and commutative, is shown to be NP -hard by reduction from 3-COLORABILITY =-=[9]-=-. 2 Unification modulo nilpotence The unifiability problem whose complexity we consider is: Definition1. The problem of elementary NILPOTENT UNIFIABILITY is defined as follows. Instance: A set S of pa... |

815 | Rewrite Systems
- Dershowitz, Jouannaud
- 1990
(Show Context)
Citation Context ...Definitions of some of the basic terminology of rewrite rule theory that we later use are given below. For a more extensive list of definitions and a survey of basic results the reader is referred to =-=[7]-=-. A term rewriting system R is a finite set of oriented equations fl i ! r i j i 2 Ng where l i and r i are terms, and Var(r i ) ` Var(l i ). These oriented equations are commonly called rewrite rules... |

562 |
The complexity of satisfiability problems
- Schaefer
- 1978
(Show Context)
Citation Context ...omplete problem, despite the extreme simplicity of the theory. Membership in NP is easy to show, especially since E is optimally reducing. NP -hardness is shown by reduction from THREE SATISFIABILITY =-=[22]-=-. ACN-unification, where the function is also associative and commutative, is shown to be NP -hard by reduction from 3-COLORABILITY [9]. 2 Unification modulo nilpotence The unifiability problem whose ... |

370 | An efficient unification algorithm
- Martelli, Montanari
- 1982
(Show Context)
Citation Context ...e can assume without loss of generality that there are no free variables in s i ae and t i ae where 1sisn. These terms are also first-order terms. We recall that the term decomposition transformation =-=[18]-=- for first-order unification replaces an equation of the form f(u 1 ; : : : ; um ) = f(v 1 ; : : : ; vm ) where f is a m-place constant symbol by the equations u i = v i where 1sism. Repeatedly apply ... |

293 | Unification theory.
- Baader, Snyder
- 2001
(Show Context)
Citation Context ...has been work on unification algorithms that build-in these theories. There are many other properties that can be treated similarly. Several equational theories have been considered in the literature =-=[5, 13]-=-. Decidability and complexity results have been obtained for many of them [15, 23]. ? Some of the results reported here are in partial fulfillment of Qing Guo's Ph.D. requirements, and will form part ... |

177 |
Solving equations in abstract algebras: a rule-based survey of unification
- Jouannaud, Kirchner
- 1990
(Show Context)
Citation Context ...has been work on unification algorithms that build-in these theories. There are many other properties that can be treated similarly. Several equational theories have been considered in the literature =-=[5, 13]-=-. Decidability and complexity results have been obtained for many of them [15, 23]. ? Some of the results reported here are in partial fulfillment of Qing Guo's Ph.D. requirements, and will form part ... |

174 |
Canonical Forms and Unification
- Hullot
- 1980
(Show Context)
Citation Context ...iable if and only if f(t 1 ; 0); : : : ; (t n ; 0)g is E-unifiable. The theory N can easily be seen to be finitary (i.e., complete sets of unifiers are always finite) by Proposition 2 (page 327) from =-=[12]-=-. 3 ACN-unification and matching Let + be a nilpotent function. We consider the case where + additionally satisfies associativity and commutativity. In other words, the equational theory ACN is (x + y... |

119 |
Building-in Equational Theories
- Plotkin
- 1972
(Show Context)
Citation Context ...ing equational theories in unification algorithms has been guided by the properties of the functions that we often encounter in mathematical theories. For instance, it was observed by Plotkin in 1972 =-=[21]-=- that properties such as associativity and commutativity can be built into unification algorithms. This resulted in extensive research into associative-commutative (AC) unification and rewriting. Unit... |

117 |
Set-based analysis of ML programs
- Heintze
- 1994
(Show Context)
Citation Context ...n whether this problem can be solved in P. 5.1 Application to Set Constraints Set constraints have been considered as a simple, accurate and intuitive formalism in program analysis and type inference =-=[1, 11]-=-. Generally, solving a system of set constraints is very expensive. The satisfiability problem of set constraints is NEXPT IME-complete in general. Even for systems of set constraints that only contai... |

109 |
Unification in the union of disjoint equational theories: Combining decision procedures
- Baader, Schulz
- 1996
(Show Context)
Citation Context ...olynomial time. However general unification and matching modulo ACUN are NP -complete. Theorem 9. General ACUN-UNIFIABILITY is NP -complete. Proof. Membership in NP can be shown using techniques from =-=[4]-=-. We omit the details. We reduce the NP -complete problem 3SAT to general ACUN-UNIFIABILITY. The reduction is similar to the one used in [16] to show NP -completeness of set-matching. Let a set of cla... |

73 | Set constraints: Results, applications, and future directions
- Aiken
- 1994
(Show Context)
Citation Context ...n whether this problem can be solved in P. 5.1 Application to Set Constraints Set constraints have been considered as a simple, accurate and intuitive formalism in program analysis and type inference =-=[1, 11]-=-. Generally, solving a system of set constraints is very expensive. The satisfiability problem of set constraints is NEXPT IME-complete in general. Even for systems of set constraints that only contai... |

73 | The complexity of set constraints
- Aiken, Kozen, et al.
- 1993
(Show Context)
Citation Context ... NEXPT IME-complete in general. Even for systems of set constraints that only contain nullary set constructors (constants) over set operations, the satisfiability problem turns out to be NP -complete =-=[2]-=-. Therefore identifying tractable subclasses of the set constraints problem is important. We define a subcase of the set constraints problem which can be solved using our algorithm for ACUNh-unificati... |

48 |
Np-completeness of the set unification and matching problems
- Kapur, Narendran
- 1986
(Show Context)
Citation Context ... Proof. Membership in NP can be shown using techniques from [4]. We omit the details. We reduce the NP -complete problem 3SAT to general ACUN-UNIFIABILITY. The reduction is similar to the one used in =-=[16]-=- to show NP -completeness of set-matching. Let a set of clauses C = fc 1 ; c 2 ; : : : ; c m g be any instance of 3SAT , over a set of Boolean variables X = fx 1 ; x 2 ; : : : ; xng. Each c i is of th... |

34 | Fast parallel computation of Hermite and Smith forms of polynomials matrices
- Kaltofen, Krishnamoorthy, et al.
- 1987
(Show Context)
Citation Context ...Delta \Delta + x c l n c l Theorem10. S c1 ; S c2 ; : : : ; S c l have solutions iff S is ACUNh-unifiable and `(s i ) =ACUNh `(t i ), 1sisk. Since solvability of linear equations over Z 2 [h] is in P =-=[14]-=-, we get Corollary11. Elementary ACUNh-unifiability problem can be solved in polynomial time. General ACUNh-unifiability is again NP-complete. The above method of solving the ACUNh-unification problem... |

29 |
P.: Complexity of unification problems with associativecommutative operators
- Kapur, Narendran
- 2002
(Show Context)
Citation Context ...any other properties that can be treated similarly. Several equational theories have been considered in the literature [5, 13]. Decidability and complexity results have been obtained for many of them =-=[15, 23]-=-. ? Some of the results reported here are in partial fulfillment of Qing Guo's Ph.D. requirements, and will form part of his dissertation. Paliath Narendran was partially supported by NSF grant CCR-94... |

23 |
Computing unification algorithms
- Kirchner
(Show Context)
Citation Context ... a function is the exclusive-or function in Boolean algebra. Nilpotent unification is a (first-order) E-unification problem in the presence of the theory E = ff(x; x) = 0g. This is a syntactic theory =-=[17]-=-: it is finite, and resolvant, which means that if s = E t holds, then there is an equational derivation of it which uses at most one application of f(x; x) = 0 at the top level. It is also a shallow ... |

20 |
The Clausal Theory of Types, volume 21 of Cambridge Tracts
- Wolfram
- 1993
(Show Context)
Citation Context ...in the proof to represent numerical constants, and disagreement pairs are constructed to represent the equations of any instance of Hilbert's Tenth Problem. The encoding used below is a reformulation =-=[24]-=- in simply-typed -calculus of Goldfarb's [10]. If C denotes a natural number cs0, it is represented by the term cA, where c is the curried typed Church numeral (ty: t(\Delta \Delta \Delta t --- --zsc ... |

19 | On the unification problem for Cartesian closed categories
- Narendran, Pfenning, et al.
- 1993
(Show Context)
Citation Context ...titution `, if `(s) is irreducible for every proper subterm s of l, then `(r) is irreducible too. A term rewriting system R is optimally reducing if and only if every rule in it is optimally reducing =-=[19]-=-. A function f is nilpotent if it satisfies the identity f(x; x) = 0. As mentioned earlier, an example of such a function is the exclusive-or function in Boolean algebra. Nilpotent unification is a (f... |

17 | Unification in Commutative Theories, Hilbert’s Basis Theorem, and Gröbner Bases - Baader - 1993 |

17 |
Unification in monoidal theories
- Nutt
- 1990
(Show Context)
Citation Context ...by solving equations over Z 2 [h]. The connection between elementary unification problems of theories with homomorphisms and solving linear equations over algebraic domains was first observed by Nutt =-=[20]-=-. For example, consider the pair ! h(x)+x; h(h(a))+a ?, where a is a constant. Equation (h + 1)x = (h 2 + 1)a is generated, which is solvable on Z 2 [h]: (h + 1)x = (h 2 + 1)a (h + 1)x = (h + 1) 2 a x... |

14 | Decidable Matching for Convergent Systems
- Dershowitz, Mitra, et al.
- 1992
(Show Context)
Citation Context ...f(x; x) ! 0 is convergent, and the right-hand side of the rule is an irreducible ground term. Thus the system is optimally reducing [19] and the unifiability check is in NP . (See also Theorem 5.1 in =-=[8]-=-.) To show NP -hardness, the NP -complete problem THREE SATISFIABILITYswill be polynomially reduced to NILPOTENT UNIFIABILITY. Let U = fx 1 ; : : : ; xng and B = fc 1 ; : : : ; c m g be any instance o... |

12 |
The Undecidability of the Second-Order Unification Problem. Theoretical Computer Science
- Goldfarb
- 1981
(Show Context)
Citation Context ...e satisfiability problem of a system of equality constraints over M can be solved in polynomial time. 6 Second-order matching modulo nilpotence 6.1 Undecidability of Second-Order Unification Goldfarb =-=[10]-=- showed that the unification of second-order terms is recursively undecidable by a reduction from Hilbert's Tenth Problem, the general solution of a finite set of Diophantine equations. Any instance o... |

11 |
Decidable problems in shallow equational theories (extended abstract
- Comon, Haberstrau, et al.
- 1992
(Show Context)
Citation Context ... finite, and resolvant, which means that if s = E t holds, then there is an equational derivation of it which uses at most one application of f(x; x) = 0 at the top level. It is also a shallow theory =-=[6]-=-, i.e., the variables in every equation (in the presentation) do not appear below level 1. Furthermore, it is not hard to see that the term rewriting system ff(x; x) ! 0g is convergent and optimally r... |

3 |
An algorithm for distributive unification
- Schmidt-Schauss
(Show Context)
Citation Context ...any other properties that can be treated similarly. Several equational theories have been considered in the literature [5, 13]. Decidability and complexity results have been obtained for many of them =-=[15, 23]-=-. ? Some of the results reported here are in partial fulfillment of Qing Guo's Ph.D. requirements, and will form part of his dissertation. Paliath Narendran was partially supported by NSF grant CCR-94... |