J.-L. Lassez and K. Marriot. Explicit representation of terms dened by counter examples. Journal of Automated Reasoning, 3(3):301-318, Sept. 1987.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....i.e. we avoid any extension to the (meta) language. Traditionally, this is achieved by bringing the negation of the completion [7] of a Horn program into negation normal form and then by extracting positive information from negated atoms via complementing terms, a problem rst addressed in [13] for rst order terms. A nal issue, which we do not tackle here, is dealing with local variables, which, during the transformation, become (extensionally) universally quanti ed [1] Unfortunately, this approach does not scale immediately to logical frameworks such as HHF, for three main ....

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

....theorem proving. In logic programming, Kunen [1987] used term complement to represent in nite sets of answers to negative queries. Our main motivation has been the explicit synthesis of the negation of higher order logic programs [Momigliano 2000a; 2000b] as discussed brie y in Section 9. Lassez and Marriot [1987] proposed the seminal uncover algorithm for computing rst order relative complements and introduced the now familiar restriction to linear terms. We quote the de nition of the Not algorithm for the (singleton) complement problem given in [Barbuti et al. 1990] which we generalize in De nition ....

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

....can generalize this and prove that if all patterns in the set are linear, a nite complement representation of this set can be constructed. However, one can prove that the set S = ff(x; x)g does not have a nite complement representation. The exhaustive analysis of the situation has been given in [LM87]. The main result can be stated as follows. Theorem 1 ( LM87] A set of patterns S has a nite complement representation i there exists a set of linear patterns S lin such that Inst(S) Inst(S lin ) Moreover, if such a set S lin exists, it can be obtained by instantiating the non linear ....

....are linear, a nite complement representation of this set can be constructed. However, one can prove that the set S = ff(x; x)g does not have a nite complement representation. The exhaustive analysis of the situation has been given in [LM87] The main result can be stated as follows. Theorem 1 ([LM87]) A set of patterns S has a nite complement representation i there exists a set of linear patterns S lin such that Inst(S) Inst(S lin ) Moreover, if such a set S lin exists, it can be obtained by instantiating the non linear variables in the patterns of S by terms, the property of ....

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

....variable are useless but in other applications, we may be interested in the values of the original variables such that the formula is true, therefore free variables are required. A particular class of equational problems deserves a denition because of its great importance in Computer Science, see [Lassez and Marriot, 1987] for a complete study of the rst order case. Definition 4. A complement problem is an equational problem of the form 9 X8 Y : t 6= jfi t 1 : t 6= jfi t n where X = FV (t) and Y = i=1; n FV (t i ) The next section gives examples of applications of such problems. 1.3. What are ....

Lassez and Marriot, 1987 Lassez, J.-L. and Marriot, K. (1987). Explicit representation of terms dened by counter examples. Journal of Automated Reasoning, 3(3):117.

....nd another set c(S) with the set of instances consisting exactly of those ground terms which are not instances of terms of S. c(S) is naturally called the complement representation of S. Probably the rst study of how to compute a complement representation has been done by Lassez and Marriot [LM87] in the context of learning a set speci ed by counter examples. It can be easily observed that if terms of S contain only linear terms (i.e. terms without repetitive occurrences of variables) then the nite complement representation always exists. Lassez and Marriot proved that S has a nite ....

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

....0 . This has the neat e ect that negation and its problems are eliminated, i.e. we avoid any extension to the (meta) language. Technically, we can achieve this by transforming a Horn program into negation normal form and then by negating atoms via complementing terms, a problem rst addressed in [10] for rstorder terms. A nal issue, which we do not tackle here, is dealing with local variables, which, during the transformation, become (extensionally) universally quanti ed [1] Unfortunately, this approach does not scale immediately to logical frameworks such as HHF, for three main reasons: ....

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

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

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

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