J-L. Lassez, K. Marriott, "Explicit representation of terms defined by counter examples". Journal of Automated Reasoning 3, pp 301-317, 1987.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....is not necessary for completeness of, e.g. resolution procedures, H subsumption is a necessary redundancy elimination technique for automated model building which cannot be dispensed with. Moreover, the strengthened redundancy elimination techniques have various applications in machine learning [25] , logic programming [16; 24] functional programming [24] and as an extension of many familiar resolution and paramodulation based inference systems without destroying the refutational completeness [32] For all of the three redundancy elimination techniques mentioned above, practically ....

Lassez, J.-L. and K. Marriott: 1987, `Explicit Representation of Terms Defined by Counter Examples'. Journal of Automated Reasoning 3(3), 301--317.

....a wish to infer disjunctive concepts. We show that the class of sets definable by disjunctive term expressions is not closed under complementation, so that, in general, a rational term expression has no equivalent purely disjunctive form. However we present an algorithm, similar to an algorithm of [20], which produces an equivalent disjunctive expression, if one exists, and otherwise halts with failure. Unfortunately the decision problem is co NP complete, even for relatively simple rational term expressions. A second important decision problem is to determine whether a rational term expression ....

....problem is decidable follows from the solution to the previous problem. However, we show that this problem is NP complete. Here the technical difficulty lies in showing that the problem lies in NP. Corresponding problems in other term languages have previously been addressed. Lassez and Marriott [20] gave a similar analysis (similar in the technical results, but not discussing complexity) for a universe of finite trees, and a description language based on finite terms. Indeed, our work was inspired by [20] This work and its range of applications is surveyed in [22] That work was extended ....

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J-L. Lassez & K.G. Marriott, Explicit Representation of Terms Defined by Counter Examples, Journal of Automated Reasoning, 3, 301--317, 1987.

....of negative constraints for example, 5] and [18] are not applicable. However there are still several constraint domains known to have this property, including the algebras of finite, rational and infinite trees with equational constraints, when there are infinitely many function symbols [17, 25], feature trees with infinitely many sorts and features [34] linear arithmetic equations over the rational or real numbers, infinite Boolean algebras with positive constraints [8] 19] has some other examples. The following result extends this list. A Horn theory is a collection of closed Horn ....

J-L. Lassez & K.G. Marriott, Explicit Representation of Terms Defined by Counter Examples, Journal of Automated Reasoning, 3, 301--317, 1987.

....answers possibly containing disequations and universal quantifiers. We choose anti subsumption constraints [14] and, with a slight abuse of notation, consider (dis)equations over atoms instead of terms. The anti subsumption constraints provide a compact representation of counter examples [16] or exceptions [3] Definition 2.1. Let A and B be atoms, and be a ground substitution. Then A = B is true under if A and B are identical ground atoms. The standard definitions of truth, validity, and (un)satisfiability can be extended to equations over atoms. Definition 2.2. ....

....in B. The variables in A are called free variables, and those in B are called bound variables. We write A62B as an abbreviation of 8X:A 6= B. Intuitively, an AS constraint of the form A62B means that A cannot be an instance of B (if B is viewed as the set of its ground instances) As shown in [16], 5 constraints in general cannot be converted into a finite disjunction of equations (or substitutions) Definition 2.3. A constrained atom, A, is a pair of the form (A; OE) where A is an atom and OE is a conjunction of AS constraints and OE is satisfiable. Let X be all the free variables in ....

J.-L. Lassez and K.G. Marriott. Explicit representation of terms defined by counter examples. Journal of Automated Reasoning, 3(3):1--17, September 1987.

....is aimed directly at allowing negative goals to return computed answers. A positive derivation for G returns fl such that comp(P ) j= 8Gfl, this approach attempts to let a negative derivation for G return such that comp(P ) j= 8:G . is called a fail substitution. Unfortunately [Lassez and Marriott(1987)] have shown that in general answers to negative queries cannot be represented by finite positive information alone, thus this approach must sometimes return an infinite number of fail substitutions. Drabent(1993) proposes a similar approach to [Ma luszy nski and N aslund(1989) but allows ....

....operator in Section 6. In Sections 7 and 8 we give the soundness and completeness results. Finally in Section 9 we discuss which structures are particularly suited for constructive negation. 7 2 A Brief Review of Constraint Logic Programming A constraint logic programming [Jaffar and Lassez(1987)] language, CLP(A) exists in the context of a particular structure A which determines the meaning of the function and (constraint) relation symbols of some language L. Constraints in the structure are relations upon terms of the structure. A primitive constraint takes the form r(t 1 ; t n ) ....

Lassez, J--L. and Marriott, K.(1987), Explicit Representation of Terms Defined by Counter Examples, Journal of Automated Reasoning, 3.

....of the negative constraints is inconsistent with the positive constraints. Independence of negative constraints has been investigated in greater generality in [163] The property has been shown to hold for several classes of constraints including equations on finite, rational and infinite trees [161, 160, 174], linear real arithmetic constraints (where only equations may be negated) 162] sort and feature constraints on feature trees [12] and infinite Boolean algebras with positive constraints [106] among others [163] We consider a restricted form of independence of negative constraints [177] ....

....of the other results cited above, at least not in their full generality. However there are still several useful constraint domains known to have this property, including the algebras of finite, rational and infinite trees with equational constraints, when there are infinitely many function symbols [161, 174], feature trees with infinitely many sorts and features [12] linear arithmetic equations over the rational or real numbers, and infinite Boolean algebras with positive constraints [106] Example 2.9. In the Herbrand constraint domain FT with only two function symbols, a constant a and a unary ....

J-L. Lassez & K.G. Marriott, Explicit Representation of Terms Defined by Counter Examples, Journal of Automated Reasoning, 3, 301--317, 1987.

....to analyze communicating processes expressed by infinite transition systems, as well as in machine learning and inductive theorem proving. In logic programming Kunen [Kun87] used them to represent infinite set of answer to negative queries. An other approach is discussed in 8. Lassez et Marriot [LM87] proposed the seminal uncover algorithm for computing relative complements and introduced the now familiar restriction to linear terms. We here quote the definition of the Not algorithm for the (singleton) complement problem given in [BMPT90] as an gentler introduction to our Definition 61. ....

J.-L. Lassez and K. Marriot. Explicit representation of terms defined by counter examples. Journal of Automated Reasoning, 3(3):301--318, September 1987.

....structured lattice (P typ ; The computation of the most specific generalisation and most general specialisation of type expressions with respect to the subtype relations are lattice operations u; t. We implemented these operations by special antiunification and subsumption algorithms (cf. [Plo70, Rey71, Ver75, LM87, Bun88]) 2.2 Formulas In our extended language we admit horn clauses and formulas of the form: 8T : p(T ) typ p 1 (T ) typ pr (T ) j q 1 (T ; q p (T ; 9T : p(T ) typ p 1 (T ) typ pr (T ) j q 1 (T ; q p (T ; 8T : p(T ) hyp ( typ p 1 (T ) typ ....

J.L. Lassez and K. Marriot. Explicit representation of terms defined by counter examples. J. of Automated Reasoning, 3:301--317, 1987.

....sets of answers to negative queries. Our main motivation has been the explicit synthesis of the negation of higherorder logic programs, as discussed in Section 8. 1 This notion of linearity should not be confused with the eponymous concept in linear logic and calculus. Lassez and Marriot [7] proposed the seminal uncover algorithm for computing relative complements and introduced the now familiar restriction to linear terms. We quote the definition of the Not algorithm for the (singleton) complement problem given in [1] which we generalize in Definition 9. Given a finite signature ....

J.-L. Lassez and K. Marriot. Explicit representation of terms defined by counter examples. Journal of Automated Reasoning, 3(3):301--318, Sept. 1987.

....harsh. However, if we add the rule q(f(a) to the program above then we are faced with somehow trying to represent the success set for p(x) as x may be anything except f(a) a concept that requires a broader notion of answer substitution. Some results in this direction have been presented in [12], and some more recent work has described a process called constructive negation in which negative subgoals are used to generate negative bindings [4, 20] Whether such methods will be useful in practice, or whether ground negation is sufficient for most purposes remains to be seen. Restricting ....

J. L. Lassez and K. Marriott. Explicit representation of terms defined by counter examples. Journal of Automated Reasoning, 3:301--317, 1987.

....Continuing Example 2.1, consider t = f list(v) v (x u ) The set of ground well sorters of t is the solution set of flist(v) ug, and the set of ground ill sorters is the solution set of flist(v) 6= ug. But there is no finite equation set which represents the solution set of flist(v) 6= ug, see [4]. Hence there does not exist a finite number of equation sets to represent the set of ground ill sorters in this case. For a ground substitution Theta to be a well sorter or ill sorter of a term is equivalent to Theta being a solution of a certain disjunctive system, as illustrated by the ....

Lassez, J.-L. and K. Marriott, "Explicit Representation of Terms Defined by Counter Examples", Journal of Automated Reasoning, 3(1987) 301-317.

....unify s and t means to find such a u or, equivalently, to solve the equation s = t. However, it turns 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 involving symbolic manipulations on terms. In order to overcome this difficulty, researchers had to ....

....differs from the unification case. Actually, depending on the application at hand, syntaxes as well as semantics may differ. For example, 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 the validity (then solved forms are just true and ....

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J.-L. Lassez and K. G. Marriott. Explicit representation of terms defined by counter examples. J. Automated Reasoning, 3(3):1--17, September 1987.

.... Given a first order formula with equality as the only predicate symbol, can negation be effectively eliminated from an arbitrary formula OE when OE is equivalent to a positive formula Equivalently, if OE has a finite complete set of unifiers, can they be computed Special cases were solved in [20, 64]. Problem 43. Design a framework for combining constraint solving algorithms. Problem 44 (H. Comon) Syntactic theories enjoy the property that a (semi ) unification algorithm can be derived from the axioms [42, 53] This algorithm terminates for some particular cases (for instance, if all ....

J. Lassez and K. G. Marriott. Explicit representation of terms defined by counter examples. J. Automated Reasoning, 3(3):1--17, Sep. 1987.

....L with axioms from Ax(P ) Fax(P ) we get P proofs of a finite covering of L, with axioms from Ax(P ) Nax(P ) i.e. we will not use DCA. Since the P system is regular, Ax(P ) Nax(P ) allows to search for answers of open negative goals, basically using the computation model of SLD resolution. In [22] and later in [5] more targeted to logic programs) algorithms to compute the (relative) complement of first order terms are described (see also [11] Those algorithms can be adapted to compute regular splitting, as informally described in the examples below. Example 40 Let us consider the ....

....we consider x = 0. Being even(0) already ground, we can build by contraposition the corresponding negative axiom or regular instance: Nax(even(0) true :even(0) The terms not covered by the above regular instance are kvk n k0k = ks(W )k where n represents set difference among terms as in [22]. Since even(s(W ) is a s.c. instance w.r.t. Fax(even) we may obtain the regular instance: Nax(even(s(W ) even(W ) even(s(W ) Now all terms are covered. Hence the regular splitting contains Nax(even(0) and Nax(even(s(W ) Example 41 Consider the program SUM (2.2) Its failure axiom ....

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J-L. Lassez and K. Marriot. Explicit Representation of Terms Defined by Counter Examples. Journal of Automated Reasoning, 3, 301--317, 1987.

....specialisation in Classical Logic representations should be mentioned. Vere [137] developed methods to deal with counterexamples during induction. These are termed counterfactuals and they define a set of conditions which must be false if a generalisation is to be satisfied. Lassez and Marriott [49] have dealt with the problem of generalisation given a set of counter examples using a technique of representing explicit exceptions. As discussed previously, work in Machine Learning such as that by Michalski [63] covers handling of counter examples in a representation based on classical logic. ....

J.-L. Lassez and K. Marriott. Explicit Representation of Terms Defined by Counter Examples. Journal of Automated Reasoning, 3:301--317, 1987.

.... formula with equality as the only predicate symbol, can negation be effectively eliminated from an arbitrary formula OE when OE is equivalent to a positive formula Equivalently, if OE has a finite complete set of unifiers, can they be computed Special cases were solved in [Comon, 1988; Lassez and Marriott, 1987] A positive solution is given in [ Tajine, 1993 ] Problem 43. Design a framework for combining constraint solving algorithms. Some particular cases have been attacked: In [ Baader and Schulz, 1992 ] it was shown how decision procedures for solvability of unification problems can be ....

J.-L. Lassez and K. G. Marriott. Explicit representation of terms defined by counter examples. J. Automated Reasoning, 3(3):1--17, September 1987.

....learning and inductive theorem proving. In logic programming Kunen [8] used term complement to represent infinite sets of answers to negative queries. Our main motivation has been the explicit synthesis of the negation of higher order logic programs, as discussed in Section 8. Lassez and Marriot [9] proposed the seminal uncover algorithm for computing relative complements and introduced the now familiar restriction to linear terms. We quote the definition of the : algorithm for the (singleton) complement problem given in [2] which we generalize in Definition 11. Given a finite signature ....

J.-L. Lassez and K. Marriot. Explicit representation of terms defined by counter examples. Journal of Automated Reasoning, 3(3):301--318, September 1987.

....convincingly the operations of generalisation (also called antiunification or disunification) and specialisation (unification) on such first order terms. This trend of ideas was followed by more work done in the direction of Machine Learning, where disjunction and negation were studied in [9] and [22]. These ideas were studied in an unsorted world (corresponding to PROLOG terms for instance) althought these results usually extend to many sorted signatures. Nevertheless adding sorts is a reasonable step when wanting to describe the real world : any given term will have a meaning, whereas in ....

....will study in this paper the negation of a term and of a disjunction of terms. For instance not (on top (red, ball, W) empty, on top (blue, W, W) on top (green, W, W) on top (W, cube, W) In the case of first order terms, and also using the closed world assumption, it is proved (in [9] or [22]) that the explicit (positive) negation of a term may not be finite. The negation of a term f(x, x) is indeed infinite as soon as the set of all terms is so. A first way to make sure that negation introduces finite sets, is to consider constrained terms [9] where constraints concern the ....

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J-L. Lassez, K. Marriott, "Explicit representation of terms defined by counter examples". Journal of Automated Reasoning 3, pp 301-317, 1987.

....finite set of terms T 2 C there exists a finite set of terms T 0 2 C , such that H(T 0 ) T c holds. The set T 0 is called a finite complement representation. For first order terms, Lassez and Marriott proved that finite sets of linear terms always have a finite complement representation [LM87]. On the other hand, they showed that this is not true for arbitrary finite sets of first order terms. Since schematizations were introduced to increase the expressive power of first order terms, we might expect to be able to represent the complements of non linear terms by a finite set of primal ....

J.-L. Lassez and K. Marriott. Explicit representation of terms defined by counter examples. J. Automated Reasoning, 3(3):301--317, 1987.

....Conclusion Note that our concepts of negative bindings and negative constraints are somewhat related to the concepts of sets of term inequalities in [13] However, the constraints in [13] are used to specify an infinite set of substitutions. Another similar kind of constraints has been used in [11] to define a generalisation operator for machine learning which respects a set of counter examples. Our constraint programming language also resembles the CLP(FT ) language with dif constraints in [17, 18] An adaptation of our technique might yield a useful abstraction operator for CLP(FT ) In ....

J.-L. Lassez and K. Marriott. Explicit representation of terms defined by counter examples. Journal of Automated Reasoning, 3:301--317, 1987.

....[Wri89a] T PL6 has finite elasticity. Hence, the conclusion immediately follows from Theorem 8. The class (T PL6 ) k is compact with respect to containment if for any L 2 T PL6 and any union L 1 [ 1 1 1 [ L k 2 (T PL6 ) k , L L 1 [ L k ) L L i for some 1 i k: Lassez et al.[LM86] showed that if 6 is an infinite alphabet, then the class T PL6 of any finite unions of tree pattern languages over 6 is compact with respect to containment. Now, we refine the result of Lassez et al. for a finite 6. For a set S, we denote by ]S the number of elements of S. Proposition 12. ....

J-L. Lassez and K. Marriott. Explicit representation of terms defined by counter examples. Journal of Automated Reasoning, 3:301--317, 1986.

....with quantification, and the use of the constraint generalization operation to achieve successful inductive learning in these models. Section 2 of the paper reviews ordinary generalization, which has been studied by Plotkin [1970; 1971] Reynolds [1970] Lassez, Maher and Marriott [1988] and Lassez and Marriott [1987]. Ordinary generalization is a purely syntactic operation that operates solely on the basis of expression structure. Section 3 defines the sorted and constraint generalization problems. Sorted generalization operates on atoms whose variables are constrained by associated sorts, or monadic ....

J-L. Lassez and K. Marriott. Explicit representation of terms defined by counter examples. JAR, 3:301--317, 1987.

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J-L. Lassez, K. Marriott, "Explicit representation of terms defined by counter examples". Journal of Automated Reasoning 3, pp 301-317, 1987.

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J.-L. Lassez and K. Marriott. Explicit Representation of Terms Defined by Counter Examples. Journal of Automated Reasoning, 1987.

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Lassez, J-L. and Marriott, K. (1986) Explicit representation of terms defined by counter examples. Journal of Automated Reasoning, 3:301-- 317. 22 A Generalization of the Least General Generalization

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