R.E. Shostak, Refutation graphs, Artif. Intell. 7 (1976) 51-64.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....starting from T i . It is clear that tableaux produced in SOL deductions are connected. Notice that Skip operation is essentially to blindly assume a literal to be false. Hence Skip can be viewed as a variant of the folding up operation [12] while Merge and Factoring correspond to C reduction [18] or E tableau reduction [12] with a (negated) skipped literal. The regularity condition for s literals in SOL tableaux is a properly improved point to the original SOL resolution. The original version only considers complementary pairs between s literals and subgoals. In SOL tableau calculus, ....

....omit the detail here. The unit axiom matching is a mandatory rule. Identical Reduction and C reduction: The identical reduction is a reduction with empty substitution. This reduction is also well known in model eliminationlike calculi [19] The identical C reduction is a restricted C reduction [18] with empty substitution. Both identical reductions are clearly mandatory rules. 5.3 Subsumption A lot of work on subsumption has been carried out to avoid logical redundancies in connection tableaux or model elimination like calculi [2, 12] Almost all of them are compatible with SOL tableau ....

R.E. Shostak, Refutation graphs, Artif. Intell. 7 (1976) 51-64.

....lemmas to be added to the input set. Linear resolution with Selection function (SL resolution) Kowalski Kuehner, 1971] was the second major chain format system presented. As well as the three core deduction operations, SL resolution incorporates factoring. The Graph Construction (GC) procedure [Shostak, 1976] introduced the use of C literals in chain format systems. There have been many implementations of chain format systems, e.g. Fleisig, Loveland, Smiley and Yarmush, 1974; Stickel, 1986; Letz, Schumann, Bayerl and Bibel 1992; Astrachan Stickel, 1992; Sutcliffe, 1992b] A more detailed survey of ....

....operations. For example, the lemma r l(a) produced in the example could also be used at step 6. This detrimental effect has been noted in various places, the principal problem being cited that lemmas tend to be highly redundant they are often subsumed by other lemmas and input chains [Shostak, 1976, p. 63] As the lemma mechanism is not necessary for the completeness of the host deduction system, restrictions can be used to reduce this problem. This approach has been taken by Fleisig, et al. 1974] whose implementation of the lemma mechanism includes a simplified subsumption test, and by ....

Shostak R.E. (1976), Refutation Graphs, In Artificial Intelligence 7, Elsevier Science, Amsterdam, The Netherlands, 51-64.

....are replaced by merges in proof ordering. In a manner similar to MESON [21] ancestor merges correspond to the insertion of merge paths in build ordering. These ideas are extended in Sections 3.3 and 3. 4 to the selected literal SL procedure [16] and Shostak s graph construction GC procedure [29] respectively. Using clause trees, it is easily seen that GC is a restriction of a variant of SL, and that SL is a restriction of ME. Section 4 develops the new concepts of visibility and support, which are relations between nodes of clause trees. The results of this paper rest on these relations. ....

....Hence useful information may be lost. Procedures using clause trees do not depend on truncation; they use the partial order on nodes to define which later resolutions are legal. Some of this information can be used by the procedure in the next section. 3. 4 Left paths and the GC Procedure Shostak [29] developed the graph construction (GC) procedure. It uses C literals in addition to the A literals (ancestors) and B literals (open siblings of ancestors) of ME and SL. An example from his paper (p. 60) is used to illustrate the GC procedure. The clauses are: N, T , M, Q, N , L, M , L, Q , ....

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R. E. Shostak, Refutation graphs, Artif. Intell. 7(1) (1976) 51--64.

....link sequence 1, 2, 3 is a cycle. The example shows that cyclic paths represent derivation chains biting their own tail : in order to prove P , one has to prove R; to prove R, one has to prove Q; to prove Q, one has to prove P ; Refutation graphs were first examined by Robert E. Shostak [Shostak, 1976]. He showed that a clause set is unsatisfiable if and only if for a sufficient number of copies of these clauses there exists a refutation graph, that is a nonempty, noncyclic clause graph in which every literal node is incident with exactly one R link, so that some global substitution unifies ....

Shostak, R. E. (1976). Refutation graphs. Artificial Intelligence, 7(1):51--64.

....is always executed if a new clause has been added to the input set in the iteration. In the HPDS these features are affected not only by clauses created locally, but also by clauses received from the other components. The GLD Component GLD is based on Shostak s Graph Construction (GC) procedure [Shostak,#1976], but also incorporates features from other chain format linear deduction systems. In particular, GLD has a combined lemma C literal mechanism, which improves on the original separate mechanisms in Loveland s Model Elimination (ME) procedure [Loveland,#1969] and the GC procedure. The lemmas ....

Shostak R.E. (1976), Refutation Graphs, In Artificial Intelligence 7, Elsevier Science, Amsterdam, The Netherlands, 51-64.

....these tautologies admit of short proofs using other refinements of resolution, namely the Davis Putnam procedure, the connection method is not the computationally least complex theorem proving method. Other theorem proving techniques such as Kowalski s connection graph resolution [Kowalski 1975, Shostak 1976] and Prawitz s matrix reduction procedure [Prawitz 1970] have been compared to the connection method [Bibel 1982] These comparisons suggest that simulation results similar to the one proved in this chapter can also be proved for these methods. 77 CHAPTER 7 Hard Examples For Tree Resolution ....

.... SL resolution simulate s linear resolution Further research could be directed at answering similar questions concerning the relative complexity of other restrictions of resolution such as Lock resolution [Boyer 1971] and other kinds of theorem proving methods such as connection graph resolution [Shostak 1976]. It would undoubtedly 96 be useful, both for deepening our understanding of these systems and for the practical requirements of automated theorem proving, if it were possible to characterize the Achilles heel that causes all these methods to suffer combinatorial explosions. Examples that are ....

Shostak, R. E. (1976). "Refutation Graphs" Artificial Intelligence 7, pp.51-64.

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