Comon, H. and Lescanne, P. Equational problems and disuni cation, J. of Symbolic Computation 7, 1989, 371-425.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....only if the equational problem P (9 x) 6 j=1 (b 1 ; b k j )2(f j ) 1) x i j1 = b 1 x i jk j = b k j ) 7 5 over the Herbrand universe H = fa 1 ; aK g is satis able. A good overview of the wide range of applications of equational problems can be found in [CL89] In many of these applications, testing the satis ability of an equational problem is even more important than actually computing the solutions. In this section, we present a survey of complexity results for this satis ability problem, where we consider several restrictions on the equational ....

....is no di erence between free variables and existentially quanti ed ones. In particular, 9 w 8 y P ( w; x; y ) is satis able, if and only if 9 x 9 w 8 y P ( w; x; y ) is. Without loss of generality we therefore only consider equational problems without free variables. In analogy with [CL89] universally quanti ed variables will be referred to as parameters. In order to distinguish between syntactical identity and the equivalence of two equational problems, we use the notation and , respectively. We shall thus write P Q to denote that the two equational problems P and Q ....

[Article contains additional citation context not shown here]

H. Comon and P. Lescanne. Equational problems and disuni cation. Journal of Symbolic Computation, 7(2 & 3):371-425, 1989.

....by comparing Proposition 34 and Proposition 38. Compared to [22, Page 70] we nd that the termination of our procedure is more evident and the extension to the term power algebra in Section 6 easier. Our base formulas somewhat resemble formulas arising in other quanti er elimination procedures [31, 11, 30]. Our terminology also borrows from congruence closure graphs like those of [39, 38] although we are not primarily concerned with eciency of the algorithm described. Term algebra is an example of a theory of pairing functions, and [15] shows that non empty family of theories of pairing functions ....

....A semi base formula is a base formula i the graph associated with is acyclic. A semi base formula whose associated graph is cyclic is unsatis able in the term algebra of nite terms. Checking the cyclicity of a base formula corresponds to occur check in uni cation algorithms (see e.g. [29, 11]) De nition 22 By height H(u) of a node u in the acyclic graph we mean the length of the longest path starting from u. A node u is sink i H(u) 0. De nition 23 We say that an internal variable u l is a source variable of a base formula i u l is represented by a node that is source in ....

[Article contains additional citation context not shown here]

Hubert Comon and Pierre Lescanne. Equational problems and disuni cation. Journal of Symbolic Computation, 7(3):371, 1989.

....equivalence of CIs are for free, the computation of lfp(W P ) must take more than polynomial time in general (under the assumptions of noncollapsing complexity classes) Consider now deciding whether a constrained atom is true in a CI. It is known that equational reasoning on H is decidable [3, 14]. Hence, it is decidable whether a given a constraint atom A (not necessarily ground) is true in a given nite CI CI. However, the complexity is tremendous; recent results on equational reasoning immediately imply that it is nonelementary [ For the function free case, we have much lower ....

H. Comon, P. Lescanne. Equational Problems and Disuni cation, J. Symbolic Computation, 7:371-425, 1989.

....by 2 and 1, respectively. Actually, A is equivalent to x 1 6=f(y 3 ; g(x 2 ) x 3 6=g(x 2 ) y 2 6= g(a) y 1 6=g(x 2 ) by uni cation. Moreover, it can be shown that all disequations of the form y 6= t, where y is a universally quanti ed variable, may be deleted (cf. the U 2 rule from [6]) Hence, we are left with the disjunction x 1 6= f(y 3 ; g(x 2 ) x 3 6= g(x 2 ) Our goal is to transform the disequations into the simple form x i 6= x j , where x i and x j are two distinct, existentially quanti ed variables. To this end, we make use of the explosion rule from [6] as ....

....from [6] Hence, we are left with the disjunction x 1 6= f(y 3 ; g(x 2 ) x 3 6= g(x 2 ) Our goal is to transform the disequations into the simple form x i 6= x j , where x i and x j are two distinct, existentially quanti ed variables. To this end, we make use of the explosion rule from [6] as follows. Observe that the disjunction (9 z ) t = a t = g(z 1 ) t = f(z 2 ; z 3 ) holds over our signature = fa; f; gg for any term t 2 H. In particular, we may apply the explosion rule to the variable x 1 . Hence, P is equivalent to (9 x ) 8 y ) x 1 = a x 1 = g(x 4 ) x 1 = f(x 5 ; x 6 ....

[Article contains additional citation context not shown here]

H. Comon and P. Lescanne. Equational problems and disuni cation. In Journal of Symbolic Computation, 7(3/4):371-425, 1989.

....S P ) since it contains no existential quanti er. Indeed, P (x) D P ] V P (t i ) R i P (x) 8y i :x 6= t i :R i ] S P . Let F 0 P = 8x:P (x) D P . We have: F 0 P :P (x) W P (t i ) R i 9y i :x = t i :R i ] Now, we apply the equational formula transformation rules of [11] on :F 0 P in order to eliminate existential quanti ers. For the sake of completeness, we give a specialized version of the quanti er elimination process below. For any conjunction of literals c we denote by size(c) the total number of position p in terms in c such that c jp contains an ....

H. Comon and P. Lescanne. Equational problems and disunication. Journal of Symbolic Computation, 7:371-475, 1989.

....a simple example adapted from [15] Example 1. Let the constraint domain be the algebra of nite trees and consider the following conjunction of open function equations: a; 1) X (X; 1) b(Y ) We will use some simple rules for rewriting equations between terms which can be found in e.g. [5]: true; a; 1) X (X; 1) b(Y ) 7 FS (X 6= a 1 6= 1 X = b(Y ) a; 1) X (X; 1) b(Y ) 7 (X 6= a X = b(Y ) a; 1) X (X; 1) b(Y ) 7 Replace (X 6= a a = b(Y ) a; 1) X (X; 1) b(Y ) 7 Clash X 6= a; a; 1) X (X; 1) b(Y ) Theorem 1. Let R be a ....

....select. This result shows that we can reduce the problem of nding a solution to a constraint with open functions to solving a constraint without open functions. A set of rules for the domains of nite, rational and in nite trees which satisfy the conditions of the above theorem can be found in [5]. 3.2 Finite Domain Constraints Given a constraint c with n occurrences of open function symbols then the rule N can be applied at most n times and consequently the rule FS can be applied n times. In many practical CLP applications it is preferable to have a constraint solver with smaller ....

H. Comon and P. Lescanne. Equational problems and disunication. Journal of Symbolic Computation, 7:371-425, 1989.

....SP ) since it contains no existential quanti er. Indeed, P (x) DP ] V P (t i ) R i P (x) 8y i :x 6= t i :R i ] SP . Let F 0 P = 8x:P (x) DP . We have: F 0 P :P (x) W P (t i ) R i 9y i :x = t i :R i ] Now, we apply the equational formula transformation rules of [7] on :F 0 P in order to eliminate existential quanti ers. For the sake of completeness, we give a specialized version of the quanti er elimination process below. For any conjunction of literals c we denote by size(c) the total number of position p in terms in c such that c jp contains an ....

....transformation process, x = t can be simply removed from the formula, since y does not occur in the rest of the conjunction. Therefore, the only equational literals are introduced by the case 1. b, hence can easily be eliminated afterward using the so called universality of parameters rule from [7]: 8x: x 6= t R) Rfx tg (if x does not occur in t) Lemma 4. Let be an ordering, and S be a set of clauses satisfying (C ) The maximal depth of terms in M (S) is equal to the maximal depth of the clauses in S. 9 Proof. sketch) First, we remark that all the equational literals ....

H. Comon and P. Lescanne. Equational problems and disunication. Journal of Symbolic Computation, 7:371-475, 1989.

....of complement problems. Since higher order disunication contains higher order unication, our goal of checking automatically the completeness of denitions seems to have little chance of success. Indeed, we prove in this paper that higher order disunication, contrary to rst order disuni cation [Comon and Lescanne, 1989], is even not semi decidable and that second order complement problems are undecidable (by encoding Minsky machines) However, we are able to prove the decidability of such formulae when some conditions are set on secondorder variables and bound variables, but not on rst order variables. Moreover, ....

....ffl Basic rules and new rules for eliminating universal variables are used to get problems free of universal variables. ffl The resulting problems are simplied further to get constrained substitutions which are our solved forms. These solved forms are similar to the rst order solved forms of [Comon and Lescanne, 1989], extended with dependence constraints. Contrary to second order complement problems, universal variables can occur in each side of an equation or disequation, but the restriction that each term is a pattern allows to devise new rules like the occurrence test rule or the compatibility rules. The ....

Comon and Lescanne, 1989 Comon, H. and Lescanne, P. (1989). Equational problems and disunication. Journal of Symbolic Computation, 7:371425.

No context found.

Comon, H. and Lescanne, P. Equational problems and disuni cation, J. of Symbolic Computation 7, 1989, 371-425.

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC