J. Hsiang and M. Rusinowitch, Proving Refutational Completeness of Theorem-Proving Strategies: The Transfinite Semantic Tree Method. J. of the ACM 38(3):559--58, July 1991.  Home/Search   Document Not in Database   Summary   ACM   TOC   Related Articles   Check

.... most successfull methods for automated deduction with equality [BG98,NR01] For many years all known completeness results for Knuth Bendix completion and ordered paramodulation required the term ordering to be well founded, monotonic and total (or extendable to a total ordering) on ground terms [HR91,BDH86,BD94,BG94], until in [BGNR99] the monotonicity requirement was dropped and well foundedness and the subterm property were shown to be sucient (note that any such ordering can be totalized without loosing these two properties) After this, the fundamental question arises whether more requirements can be ....

J. Hsiang and M Rusinowitch. Proving refutational completeness of theorem proving strategies: the trans nite semantic tree method. Journal of the ACM, 38(3):559-587, July 1991.

.... most successful methods for automated deduction with equality [BG98,NR01] For many years all known completeness results for Knuth Bendix completion and ordered paramodulation required the term ordering to be well founded, monotonic and total (or extendable to a total ordering) on ground terms [HR91,BDH86,BD94,BG94], until in [BGNR99] the monotonicity requirement was dropped and well foundedness and the subterm property were shown to be sufficient (note that any such ordering can be totalized without loosing these two properties) After this, the fundamental question arises whether more requirements can be ....

J. Hsiang and M Rusinowitch. Proving refutational completeness of theorem proving strategies: the transfinite semantic tree method. Journal of the ACM, 38(3):559--587, July 1991.

....termination of the resolution paramodulation process (see Section 3) The class of restricted g clauses is very similar to the PVD and PVD = g classes de ned in [3, 4] A rough comparison is given in the appendix. From now, we assume given a Complete Simpli cation Ordering t (see for example [7] for the de nition of CSO) t may be extended to a partial ordering on g literals using the following relation and to g clauses using the multiset extension of the ordering on g literals: 9x 1 ; x n )L t (9x 0 1 ; x 0 m )L 0 i n m or if n = m and L t L 0 . Remark ....

J. Hsiang and M. Rusinowitch. Proving refutational completeness of theorem proving strategies: The transnite semantic tree method. Journal of the ACM, 38(3), July 1991.

....During the last two decades this field has importantly progressed through new Knuth Bendix like completion techniques and their extensions to ordered paramodulation for first order clauses. These techniques have lead to important results on theorem proving in first order logic with equality [HR91,BDH86,BD94,BG94] (that have been applied to stateof the art theorem provers like Spass [Wei97] results on logic based complexity and decidability analysis [BG96,Nie98] on deduction with constrained clauses [KKR90,NR95] and on many other applications like inductive theorem proving, symbolic constraint ....

J. Hsiang and M Rusinowitch. Proving refutational completeness of theorem proving strategies: the transfinite semantic tree method. Journal of the ACM, 38(3):559--587, jul 1991.

....During the last two decades this field has importantly progressed through new Knuth Bendix like completion techniques and their extensions to ordered paramodulation for first order clauses. These techniques have lead to important results on deduction in firstorder logic with equality, like [HR91,BDH86,BD94,BG94,BG98,NR01], results that have been applied to state of the art theorem provers like Spass [Wei97] and Vampire [RV01] These techniques have also led to results on logic based complexity and decidability analysis [BG01,Nie98] on deduction with constrained clauses [KKR90,NR95] on inductive theorem proving ....

....equations, the paramodulation steps involve only maximal terms of maximal equations of both premises. In this case it is called superposition [BG94] Ordered paramodulation is the slightly less restricted version of superposition where inferences also take place on non maximal sides of equations [HR91]. 2.1 Paramodulation with constrained clauses By expressing the ordering and unification restrictions as inherited constraints, paramodulation remains complete [NR95] Then it becomes: C s t j T D j T 0 C D[t] p j T T 0 s = Dj p OC where the part OC of the constraint represents the ....

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J. Hsiang and M Rusinowitch. Proving refutational completeness of theorem proving strategies: the transfinite semantic tree method. Journal of the ACM, 38(3):559--587, July 1991.

....(Rusinowitch 1991, Zhang 1988, Bachmair and Ganzinger 1990, Pais and Peterson 1991) and the basic strategy has actually first been developed for first order clauses. Associativity and commutativity have been built into ordered paramodulation (Paul 1992, Rusinowitch and Vigneron 1991) a calculus (Hsiang and Rusinowitch 1991) that does not generalize completion, but includes similar rewrite techniques; and Wertz (1992) designed an associative commutative superposition calculus. Unfortunately, the completeness proofs proposed for these calculi are technically involved and quite complicated. The calculus described in ....

J. Hsiang and M. Rusinowitch, 1991. Proving refutational completeness of theorem proving strategies: The transfinite semantic Tree method. Journal of the ACM, Vol. 38, No. 3, pp. 559--587.

....equations s t, but only under similar restrictions as for equality factoring. Ordered paramodulation does not impose the ordering constraints (ii) and (vi) and hence permits replacement of subterms in the smaller side of equations. Equality factoring is not needed for this variant. THEOREM 5. (Hsiang and Rusinowitch, 1991). Ordered paramodulation is refutationally complete. The original proof of this theorem in (Hsiang and Rusinowitch, 1991) employs a proof technique based on transfinite semantic trees, and covers only specific selection strategies such as the positive strategy or the positive unit strategy in the ....

....ordering constraints (ii) and (vi) and hence permits replacement of subterms in the smaller side of equations. Equality factoring is not needed for this variant. THEOREM 5. Hsiang and Rusinowitch, 1991) Ordered paramodulation is refutationally complete. The original proof of this theorem in (Hsiang and Rusinowitch, 1991) employs a proof technique based on transfinite semantic trees, and covers only specific selection strategies such as the positive strategy or the positive unit strategy in the case of Horn clauses. 6.1. Candidate Models and Counterexamples Candidate models and reduction of counterexamples are ....

Hsiang, J. and M. Rusinowitch: 1991, `Proving refutational completeness of theorem proving strategies: The transfinite semantic tree method'. J. Association for Computing Machinery 38(3), 559--587.

....terms. We are giving only the basic results and references on the literature as well as more detailed account of results could be found in the survey paper [CT94] As this will be detailed in other lectures of this school, this kind of ordering constraints is extremely useful in theorem proving [HR91, BGLS95] as well as in programming language design since they allow to keep into account simpli cation as well as deletion strategies. This section is devoted to ordering constraints where the ordering predicate is interpreted as a recursive path ordering, but several other possibilities have ....

J. Hsiang and M. Rusinowitch. Proving refutational completeness of theorem proving strategies: The transnite semantic tree method. Journal of the ACM, 38(3):559-587, July 1991.

....and functional reflexivity. It is also based on a notion of replacement. Over time, several refinements have been added to this rule. Brand [6] has shown that only the reflexivity axiom x x is needed. Peterson [18] has shown that paramodulations into variables are useless. Hsiang and Rusinowitch [10] have introduced ordering restrictions to the application of these rules, and have proved the completeness of the following ordering strategy : each inference step has to be applied between maximal (w.r.t. an ordering) literals in clauses, and in each paramodulation step, a term cannot be replaced ....

....the term f(a; a) is E equal to a, and for any ground term t, f(a; t) E t. Indeed, since f and a are the only symbols, any ground term (a, f(a; a) f(f(a; a) a) is E equal to a. We prove the Theorem of Completeness by the transfinite semantic tree method of Hsiang and Rusinowitch [10], extended to deduction modulo an equational theory in [26, 29] Let us give a sketch of this proof, as it is rather similar to the proof for the particular case of associative and commutative theories [26] see [29, 30] for the detailed proofs) Proof. Let E be a theory satisfying (P1) or (P2) ....

[Article contains additional citation context not shown here]

J. Hsiang and M. Rusinowitch. Proving Refutational Completeness of Theorem Proving Strategies: The Transfinite Semantic Tree Method. Journal of the ACM, 38(3):559--587, July 1991.

....based on a notion of replacement. Over the time, several refinements have been added to this rule. Brand has shown that only the reflexivity axiom (x x) is needed. Peterson has shown that paramodulations into variables are useless. But the crucial step has been achieved by Hsiang and Rusinowitch [2]. They have introduced ordering restrictions to the application of these rules. Moreover, they have proved the completeness of the ordering strategy defined as follows: each inference step has to be applied between maximal (w.r.t. an ordering) literals in clauses, and in each paramodulation step, ....

J. Hsiang and M. Rusinowitch. Proving Refutational Completeness of Theorem Proving Strategies: The Transfinite Semantic Tree Method. Journal of the ACM, 38(3):559--587, July 1991.

....is refutationally complete in AC theories. Theorem 17 Let S be an AC unsatisfiable set of clauses with empty constraints. Then, every fair derivation from S generates the empty clause. The method we use for proving this theorem is based on the notion of transfinite semantic trees introduced in [Rus89, HR91], which was adapted to AC theories in [RV93] for the paramodulation strategy and in [Vig94b] for the superposition strategy. We present here only a sketch of the proof, but it is detailed in the Appendix A. Sketch of Proof : In the empty theory, the proof is done by defining the transfinite tree ....

J. Hsiang and M. Rusinowitch. Proving refutational completeness of theorem-proving strategies : the transfinite semantic tree method. Journal of the ACM, 38(3):559--587, July 1991. 14

....seems rather elegant, it has the disadvantage that the counter model constructed depends directly on the inference operator employed. 2 Notions and Results We assume familiarity with the concept of semantic trees, especially transfinite E semantic trees, and maximal consistent trees mct(C) c.f. [8, 6, 9]. To simplify notation, we assume a fixed set of input clauses C. We write I(C) to denote the closure of all inference I 2 I. We assume that tautologies are deleted wrt. I and that the redundancy criteria embodied in I is a semantic, contrary to a proof theoretic one 2 . Either P eqf or P mep ....

....factoring, superfluous. 98 should only be applied to a maximal atom: S3) u[s]v) L for all literals L in D 1 or even to the maximal term thereof (S4) u[s] v. Paramodulation calculi which fulfill all four conditions (S1) S4) are called superposition calculi. The ordered literal strategy [6] imposes only restrictions (S1) S2.a) and (S3) M. Rusinowitch established refutational completeness for a paramodulation rule satisfying (S1) S3) and (S4) called weak superposition. The proof method used is an extension of the ordinary completeness proof by semantic trees to transfinite ....

J. Hsiang and M. Rusinowitch. Proving refutational completeness of theorem-proving strategies: The transfinite semantic tree method. Journal of the ACM, 383:559--587, 1991.

....attractive, and indeed, we show in [CP93b] that this cooperation improves the effectiveness of semantic hyper linking on a number of problems. However, the definition of rough resolution seems arbitrary; the logical way to eliminate large literals is to use ordered resolution, as described in [BG90, HR91]. Also, the manner in which the semantic tree is constructed and searched seems to have an arbitrary element to it; this also makes this part of the method harder to describe formally. In addition to semantic hyper linking and rough resolution, UR (unit resulting) resolution is a component of the ....

....in a similar way. In fact, both strategies are somewhat similar to model elimination in this respect [Lov69] this similarity may be more apparent for ordered semantic hyper linking than for semantic hyper linking. However, instead of a semantic tree, we have a transfinite semantic tree, as in [HR91]. Also, in ordered semantic hyper linking we specify more precisely how this tree is constructed. That is, the interpretations are examined in a sequence I 0 ; I 1 ; I 2 : consistent with the total ordering; the first interpretation I 0 to be considered is the one that is least in this ....

J. Hsiang and M Rusinowitch. Proving refutational completeness of theorem-proving strategies: the transfinite semantic tree method. J. Assoc. Comput. Mach., 38(3):559--587, July 1991.

....e 1 ; e 2 ; e n of equations and inequations where each e i is either an element of q or is a critical pair between two previous equations in the sequence. A critical pair refutation is a critical pair proof e 1 ; e n where for some term t, e n is of the form t 6= t. It is known [15, 2] that if q is an Eq unsatisfiable set of equations and inequations, then there is a critical pair refutation from q in which each critical pair is with respect to . Theorem 3.6 Suppose that q is a set of equations and inequations from S. Then if q is Eq unsatisfiable, q has a minimal ....

....T of S and a substitution fi such that T (T fi oe false) in this system. Proof. Suppose S is Eq unsatisfiable. Then there is a (non rigid) resolutionparamodulation proof of the empty clause from S [ fx = xg, since resolution and ordered paramodulation are complete relative to equality [15, 3]. It follows that there is a (rigid) resolution paramodulation proof tree Pi of the empty clause from S. In particular, we can choose Pi so that all of the labels of the leaves have disjoint variables. We show that there is a substitution fi such that A Pi (A Pi fi oe false) that is, we can ....

J. Hsiang and M Rusinowitch. Proving refutational completeness of theoremproving strategies: the transfinite semantic tree method. J. Assoc. Comput. Mach., 38(3):559--587, July 1991.

....or restricted equality clauses [Nieuwenhuis and Orejas 1990] The generalization of this kind of completion procedure to full first order clauses with equality required the development of more powerful proof techniques for establishing completeness. Using the transfinite semantic tree method Hsiang and Rusinowitch [1991] proved the refutation completeness of ordered paramodulation, while Bachmair [1989] applied an extension of the so called proof ordering technique for obtaining similar results. By means of their model generation proof method, similar to other forcing techniques developed by Zhang [1988] and ....

Hsiang J. and Rusinowitch M. [1991], `Proving refutational completeness of theorem proving strategies: the transfinite semantic tree method', Journal of the ACM 38(3), 559--587.

....is, an ordering that will produce polynomial behavior for A ordering. Later we show that this is not always possible, but give some special cases where such a good ordering always exists. 5. 5 Implications for term rewriting Note that the first order strategies based on term rewriting techniques [HR91, BG90] generally reduce to A ordering methods on clauses without equality. This shows that these methods also sometimes suffer from exponential search inefficiency and often lack goal sensitivity. Nor do they have smaller search depth than all negative resolution. However, there is some advantage for ....

J. Hsiang and M Rusinowitch. Proving refutational completeness of theorem-proving strategies: the transfinite semantic tree method. J. Assoc. Comput. Mach., 38(3):559--587, July 1991.

....transitive frames the standard embedding methods lead outside the guarded fragment. 3 The Superposition Calculus For the decision procedure to be described below we only need a rather weak form of the superposition calculus of Bachmair Ganzinger (1990) called ordered paramodulation, for which Hsiang Rusinowitch (1991) have also given a completeness proof. Here (ordered) paramodulation into the larger side of an equation is permitted. We use the symbol # to denote formal equality and do not distinguish between equations s # t and t # s. Disequations (s # t ) will also be written as s ## t . The ....

Hsiang, J. & Rusinowitch, M. (1991), `Proving refutational completeness of theorem proving strategies: The transfinite semantic tree method', J. Association for Computing Machinery 38(3), 559--587.

....of clauses (though, often, the basic argument can easily be generalized to the countable case) Methods to overcome this limitation have been proposed too. For instance, in order to prove the completeness of equality based resolution calculi, the notion of trans nite semantic tree is exploited in [HR91]. Consequently, a generalization of the Herbrand Theorem is proposed in order to deal with the lifting of derivations of (potentially) trans nite length obtained from (potentially) trans nite sets of clauses. Approaches that explicitly avoid any assumption on the cardinality of the objects they ....

J. Hsiang and M. Rusinowitch. Proving Refutational Completeness of Theorem Proving Strategies: The Transnite Semantic Tree Method. Journal of the ACM, 38(3):559-587, 1991.

....equations, the paramodulation steps involve only maximal terms of maximal equations of both premises. In this case it is called superposition [BG94] Ordered paramodulation is the slightly less restricted version of superposition where inferences also take place on non maximal sides of equations [HR91]. 1 Here Djp denotes the subterm of the clause D at position p, and D[t]p denotes the result of replacing in D that subterm by t. If Djp is in the term u of a (positive or negative) equation u v in D, then we say that the paramodulation step involves the terms s and u; more precisely, it takes ....

J. Hsiang and M Rusinowitch. Proving refutational completeness of theorem proving strategies: the transfinite semantic tree method. Journal of the ACM, 38(3):559--587, jul 1991.

.... [Tam91] Tam92] and [Win82] To enlargement of known decideable classes, enlargement of symbolic model representation techniques, respectively: BGW93a] BGW93b] DG79] FS93] FLTZ93] GH95] Lei95] Pel95] and [Sal91] Equational Calculi: BG91] BG94] BGLS92] Hue80] HKLR92] [HR91], NR94] and [Pla93] Implementation of superposition calculi: BG94] NR92] NN93] and [Sal91] 4 Aims of the Project The general aim consists in working out a firm mathematical basis for model building with equality. From this basis efficient algorithms should be developed REFERENCES 7 and ....

J. Hsiang and M. Rusinowitch. Proving refutational completeness of theorem-proving strategies: The transfinite semantic tree method. Journal of the ACM, 383:559--587, 1991.

....the number of potential deduction steps, the paramodulation rule has been restricted by an ordering, to guarantee it replaces big terms by smaller ones. This notion of ordered paramodulation has been applied to the KnuthBendix completion procedure [16] for avoiding failure in some situations (see [14] and [2] A lot of work has been devoted to putting more restrictions on paramodulation in order to limit combinatorial explosion [23] In particular paramodulation is often inecient with axioms such as associativity and commutativity since these axioms allow for many successful uni cations ....

J. Hsiang and M. Rusinowitch. Proving Refutational Completeness of TheoremProving Strategies : the Transnite Semantic Tree Method. JACM, 38(3):559-587, July 1991.

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J. Hsiang and M. Rusinowitch, Proving Refutational Completeness of Theorem-Proving Strategies: The Transfinite Semantic Tree Method. J. of the ACM 38(3):559--58, July 1991.

No context found.

J. Hsiang and M. Rusinowitch, Proving Refutational Completeness of Theorem-Proving Strategies: The Transfinite Semantic Tree Method. J. of the ACM 38(3):559--58, July 1991.

No context found.

J. Hsiang and M Rusinowitch. Proving refutational completeness of theorem proving strategies: the transfinite semantic tree method. Journal of the ACM, 38(3):559--587, July 1991.

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J. Hsiang and M. Rusinowitch. Proving Refutational Completeness of Theorem Proving Strategies: The Transnite Semantic Tree Method. Journal of the ACM 38 (1991) pp. 559587.

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