Jurgen Stuber. Superposition theorem proving for abelian groups represented as integer modules. Theoretical Computer Science, 208(1-2):149-177, 1998.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....under equational ordered paramodulation w.r.t. a west ordering extending a give reduction ordering r , and let R denote the (unique) canonical TRS for E and r . Then E 0 R. 4 Building in abelian groups Paramodulation with built in abelian groups (AG) has been investigated by many authors [Che86,Mar94,Mar96,GW96,Wal98,Wal99,Stu98]. This is not surprising since abelian groups are of course ubiquitous in many applications of (semi )automated reasoning. But building in AG is also attractive for at least two more reasons. On the one hand, due to the fact that diophantine equation solving is easier in the integers than in the ....

....into variables can be avoided, but abstraction remains necessary [Wal98,Wal99] These results have in turn been extended in [Wal01] to superposition and chaining for totally ordered divisible abelian groups. In Stuber s work on paramodulation for abelian groups represented as integer modules [Stu98], symmetrization is again crucial, but AG unification is not applied. Instead, AC unification is used, and hence paramodulation inferences with the AG axioms on the remaining clauses are needed. For example, refuting a clause like f( Gammab x a) 6 f(0) requires inferences with the AG axioms, ....

Jurgen Stuber. Superposition theorem proving for abelian groups represented as integer modules. Theoretical Computer Science, 208(1--2):149--177, 28 November 1998.

....algebra system Weyl. Other work in theorem proving in this domain concentrates mainly on the equational reasoning aspect in abstract algebra. As examples we refer to term rewrite systems for nite groups as presented for instance by [3] and to the specialized superposition calculi for groups of [32] and for monoids of [12] Work on exploration and automated discovery in nite algebra is reported in [11, 21, 30, 34] where model generation techniques are used to tackle quasi group existence problems. In particular, systems such as finder ( 29] and sato ( 33] were successfully employed to ....

J. Stuber. Superposition theorem proving for abelian groups represented as integer modules. Theoretical Computer Science, 208(1-2):149-177, 1998.

.... recognized that automated theorem provers have problems proving theorems in theories with permutative axioms like associativity, commutativity, distributivity and the inverse law which are common in algebra, and there have been various approaches to the integration of these axioms into provers [17, 11, 16, 5, 2, 8, 13, 14, 15]. A similar argument holds for the use of equivalences on the level of logical formulas. State of the art resolution theorem provers such as SPASS do a clause normal form transformation once at the beginning, which completely destroys in particular the equivalences. With some e ort it is possible ....

Jurgen Stuber. Superposition theorem proving for abelian groups represented as integer modules. Theoretical Computer Science, 208(1-2):149-177, 1998.

....and Rubio 1997] and in Section 6. 2 (see also [Rubio 1996] for builtin semigroups, i.e. associative theories) This approach is considered as well for arbitrary regular theories in [Vigneron 1996] Recent research concerns algebraic structures richer than abelian semigroups, like abelian groups [Stuber 1998a, Godoy and Nieuwenhuis 2000], cancellative abelian monoids [Ganzinger and Waldmann 1996] commutative rings [Stuber 1998b] or divisible torsion free abelian groups [Waldmann 1998] 52 R. Nieuwenhuis and A. Rubio 6.1. E compatible reduction orderings Many results in the literature on ordered paramodulation and ....

Stuber J. [1998a], `Superposition theorem proving for abelian groups represented as integer modules', Theoretical Computer Science 208(1--2), 149--177.

....specialized techniques to work efficiently with standard algebraic theories, since a nave handling of some axioms (like associativity and commutativity, AC) leads to an explosion of the search space. Paramodulation with built in abelian groups (AG) has been investigated by many authors [6, 23, 11, 12, 10, 21, 22, 18]. This is not surprising since abelian groups are of course ubiquitous in many applications of (semi )automated reasoning. But building in AG is also attractive for at least two more reasons. On the one hand, due to the fact that diophantine equation solving is easier in the integers than in the ....

....increases the number of possible inferences on f . By specialising to torsionfree divisible abelian groups, AC unification and inferences into variables can be avoided, but abstraction remains necessary [21, 22] In Stuber s work on paramodulation for abelian groups represented as integer modules [18], symmetrisation is again crucial, but AG unification is not applied. Instead, AC unification is used, and hence paramodulation inferences with the AG axioms on the remaining clauses are needed. For example, refuting a clause like f( Gammab x a) 6 f(0) requires inferences with the AG axioms, ....

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J. Stuber. Superposition theorem proving for abelian groups represented as integer modules. Theoretical Computer Science, 208(1--2):149--177, 28 Nov. 1998.

....specialized techniques to work efficiently with standard algebraic theories, since a nave handling of some axioms (like associativity and commutativity, AC) leads to an explosion of the search space. Paramodulation with built in abelian groups (AG) has been investigated by many authors [6, 23, 11, 12, 10, 21, 22, 18]. This is not surprising since abelian groups are of course ubiquitous in many applications of (semi )automated reasoning. But building in AG is also attractive for at least two more reasons. On the one hand, due to the fact that diophantine equation solving is easier in the integers than in the ....

....increases the number of possible inferences on f . By specialising to torsionfree divisible abelian groups, AC unification and inferences into variables can be avoided, but abstraction remains necessary [21, 22] In Stuber s work on paramodulation for abelian groups represented as integer modules [18], symmetrisation is again crucial, but AG unification is not applied. Instead, AC unification is used, and hence paramodulation inferences with the AG axioms on the remaining clauses are needed. For example, refuting a clause like f( Gammab x a) 6 f(0) requires inferences with the AG axioms, ....

[Article contains additional citation context not shown here]

J. Stuber. Superposition theorem proving for abelian groups represented as integer modules. Theoretical Computer Science, 208(1--2):149--177, 28 Nov. 1998.

....the present approach is a considerable improvement over Hsiang, Rusinowitch, and Sakai s extension of ordered paramodulation [17, 27] to handle cancellation laws. Our approach is related to normalized rewriting modulo the group axioms (March e [20] and superposition for integer modules (Stuber [28]) Both handle only the stronger case of groups. As will become apparent, working in non groups makes some aspects of equational theorem proving significantly more difficult while others are simplified. For instance, there will sometimes be a need for abstraction 2 ; in other situations, the ....

Jurgen Stuber. Superposition theorem proving for abelian groups represented as integer modules. To appear in Proc. RTA'96, 1996.

.... recognized that automated theorem provers have problems proving theorems in theories with permutative axioms like associativity, commutativity, distributivity and the inverse law which are common in algebra, and there have been various approaches to the integration of these axioms into provers [17, 11, 16, 5, 2, 8, 13, 14, 15]. A similar argument holds for the use of equivalences on the level of logical formulas. State of the art resolution theorem provers such as SPASS do a clause normal form transformation once at the beginning, which destroys in particular the equivalences. With some e ort it is possible to ....

Jurgen Stuber. Superposition theorem proving for abelian groups represented as integer modules. Theoretical Computer Science, 208(1-2):149-177, 1998.

....he calls symmetrization) but he explicitly adds extended rules to obtain convergence with the theory. He doesn t consider special critical pair criteria for overlaps of symmetrizations. Note that the construction in this paper is needed for rings but not for abelian groups or integer modules (Stuber 1996). 2 Preliminaries We assume that the reader is familiar with term rewriting (Dershowitz and Jouannaud 1990) We write s t if s t and t is irreducible; that is, t is a normal form of s. We write T (s) for the normal form of s with respect to a convergent term rewriting system T . We use ....

Stuber, J. (1996). Superposition theorem proving for abelian groups represented as integer modules. In H. Ganzinger (ed.), Proc. 7th Int. Conf. on Rewriting Techniques and Applications, New Brunswick, NJ, USA, LNCS 1103, pp. 33--47. Springer.

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Stuber, J. (1998a). Superposition theorem proving for abelian groups represented as integer modules. Theoretical Computer Science 208(1-2): 149-177.

.... 1997) The special case of divisible torsion free abelian groups allows us to eliminate unshielded variables, which avoids the most proli c inferences (Waldmann 1997, Waldmann 1998) Previously we have shown that our approach is compatible with constraints for the special case of integer modules (Stuber 1996, Stuber 1998a) We have also carried it out for commutative rings in the ground case (Stuber 1998b) In term rewriting March e (1996) builds a range of theories from AC to commutative rings into equational completion. He explicitly adds symmetrizations and is not concerned with redundancy ....

Stuber, J. (1996). Superposition theorem proving for abelian groups represented as integer modules. In H. Ganzinger (ed.), Proc. 7th Int. Conf. on Rewriting Techniques and Applications, New Brunswick, NJ, USA, LNCS 1103, pp. 33-47. Springer.

....technique. Bundgen (1996) has used a similar method for the special case of polynomial rings, in order to formalize the relation between Knuth Bendix completion and Grobner base computation. Normalized rewriting is due to March e (1996) We have previously presented a calculus for integer modules (Stuber 1996). 2 Preliminaries We assume that the reader is familiar with term rewriting (Dershowitz and Jouannaud 1990) We write s t if s t and t is irreducible; that is, t is a normal form of s. We write T (s) for the normal form of s with respect to a convergent term rewriting system T . We ....

Stuber, J. (1996). Superposition theorem proving for abelian groups represented as integer modules. In H. Ganzinger (ed.), Proc. 7th Int. Conf. on Rewriting Techniques and Applications, New Brunswick, NJ, USA, LNCS 1103, pp. 33--47. Springer.

No context found.

Stuber, J. (1998a). Superposition theorem proving for abelian groups represented as integer modules. Theoretical Computer Science 208(1--2): 149--177.

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Jurgen Stuber. Superposition theorem proving for abelian groups represented as integer modules. Theoretical Computer Science, 208(1--2):149--177, 28 November 1998.

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