Schorlemmer, W. M. and Agust'i, J. (1995). Theorem proving with transitive relations from a practical point of view. Research Report IIIA 95/12, Institut d'Investigaci'o en Intel\Deltalig`encia Artificial (CSIC).  Home/Search   Document Details and Download   Summary   Related Articles   Check

....( AEGL92] and, what is more important here, they have an automated deduction system. The deduction with transitive relations like inclusions and equalities cannot be handled by resolution, like ordinary relations. There is need of specialized deduction procedures. This has been studied in [LA93, SA95]. The application of this automated theorem prover to help the construction of GraSp diagrams and is posterior translation into Prolog is still under development. For instance, the deduction system can help to see that the refinement step from figure 2 to figure 4 is a particular one among others. ....

W. Marco Schorlemmer and Jaume Agust'i. Theorem Proving with Transitive Relations from a Practical Point of View. Technical Report IIIA 95/12, Institut d'Investigaci'o en Intel\Deltalig`encia Artificial, 1995.

....calculus We have seen that calculi based on bi rewriting, like ordered chaining, are suitable as proof calculi for rewriting logic, since ordering restriction on terms and atoms significantly prune the search space of the prover. But these calculi are still highly prolific in the general case [Schorlemmer and Agust i, 1995]. Inferences require unification on variable positions, although only when they appear repeated in the same term (see Definition 3.4) and, if the operators are monotonic with respect to the transitive relation (e.g. the rewrite relation ) in rewrite theories) functional reflexive axioms are ....

.... s(x y) Such rewrite systems correspond, for example, to the semantics of Maude s functional modules [Meseguer, 1993] Overlaps on variable positions and the functional reflexive axioms are not needed: All those overlaps are convergent, because rewrite rules appear in both rewrite systems (see [Schorlemmer and Agust i, 1995]) If the set of equations E is Church Rosser (in the traditional sense of equational rewrite systems, for instance see [Dershowitz and Jouannaud, 1990] the birewrite system hR) R( i obtained from set of rules R in which E is mapped to is also ChurchRosser (in the sense of Theorem 3.5) as ....

Schorlemmer, W. M. and Agust'i, J. (1995). Theorem proving with transitive relations from a practical point of view. Research Report IIIA 95/12, Institut d'Investigaci'o en Intel\Deltalig`encia Artificial (CSIC).

....studied the unification of linear second order terms and proposes a semi decision procedure for it. The decidability of the unification of linear second order terms remains an open problem. For a more extended survey on the state of the art of theorem proving with transitive relations we refer to [19]. 5 Towards Declarative Programming with Inclusions We have seen in the previous sections, that reasoning with transitive relations, and hence with inclusions, is well captured by term rewriting. The use of ordering restrictions on chaining based inferences has contributed to the improvement of ....

W. M. Schorlemmer and J. Agust'i. Theorem proving with transitive relations from a practical point of view. Technical Report IIIA 95/12, Institut d'Investigaci'o en Intel\Deltalig`encia Artificial (CSIC), 1995.

....avoided completely [3] 14 J. Agust i, J. Puigsegur W. M. Schorlemmer parent parent grandparent parent bob john mary parent tom sally ann parent bob charly ann Figure 4: A Visual Program For a more extended survey on the state of the art of theorem proving with transitive relations we refer to [23]. 3.3 Visually solving queries As mentioned in section 2.1, our visual notation is based on using graphical set inclusion between diagrams instead of implication. The transitivity of the inclusion relation, which is also implicit in the chaining inference rule is trivially captured in our ....

W. M. Schorlemmer and J. Agust'i. Theorem proving with transitive relations from a practical point of view. Research Report IIIA 95/12, Institut d'Investigaci'o en Intel\Deltalig`encia Artificial (CSIC), 1995.

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