Boyer, R. S. (1971). Locking: a restriction of resolution. PhD thesis, Mathematics Department, University of Texas, Austin.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....concerning equality, such as demodulation and paramodulation [27] Finally, another weak point of ACT P that needs further work is due to the fact that ACT P offers no direct constructs for attaching information to literals and terms. Thus, ACT P can incorporate strategies like Lock resolution [3] and Connection graphs [14] only with great difficulty, if at all. Acknowledgements We would like to express our thanks to Frank van Harmelen for extensive comments on an earlier stage of this work. Most of this work was done when both authors were 24 at University of Nottingham, UK. The first ....

R. S. Boyer, Locking: a restriction of resolution, PhD Thesis, University of Texas at Austin, Austin, TX, 1971.

....the models employed, i.e. there are no clauses that present evaluation problems, such as that in example 2.2, in any of the models. Brown s method is complete for renamable sets. 10 2. 6 Hereditary Lock Resolution Sandford [Sandford 80] combines Semantic resolution with Boyer s Lockresolution [Boyer 71] and Luckham s Model Strategy [Luckham 70] into a system he calls Hereditary Lock resolution (HLR) This is complete for both Horn and non Horn sets. The strategy builds up False Substitution Lists (FSLs) for each clause. A FSL is a list of literals that appear in the set. These literals may ....

R. Boyer. Locking: A Restriction of Resolution. Unpublished PhD thesis, University of Texas at Austin, 1971.

.... the same consequences as # in L(l(i, j) i.e. if we add those clauses to A j we get a conservative extension of A j ) Resolution strategies that can be readily used in RES MP, while preserving completeness, include linear resolution [23] directional resolution [25] and lock resolution [7] . Strategies such as set of support and semantic resolution can be used with somewhat different treatments. 4 Minimizing Node Coupling Using Polarity FMP and RES MP use the communication language to determine relevant inference steps between formulae in connected partitions. This section ....

R. S. Boyer. Locking: a restriction of resolution. PhD thesis, Mathematics Department, University of Texas, Austin, 1971.

.... i has the same consequences as in L(l(i; j) i.e. if we add those clauses to A j we get a conservative extension of A j ) Resolution strategies that can be readily used in RES MP, while preserving completeness, include linear resolution [23] directional resolution [25] and lock resolution [7] . Strategies such as set of support and semantic resolution can be used with somewhat different treatments. 4 Minimizing Node Coupling Using Polarity FMP and RES MP use the communication language to determine relevant inference steps between formulae in connected partitions. This section ....

R. S. Boyer. Locking: a restriction of resolution. PhD thesis, Mathematics Department, University of Texas, Austin, 1971.

....substitutions . In literal ordered resolution, the orderings must be liftable to maintain completeness. Since liftable orderings are often are not total (but see [3] in these cases the restriction cannot choose a unique maximal literal, leading to fan out in the search space. Lock resolution [1], incidentally, is closely related. In lock resolution, as in rank activity, the ranks are assigned to literal occurrences, chosen in any order, not according to an overall literal ordering. Like literal ordered resolution, lock resolution uses only the maximal case of the resolutions in Figure 4. ....

R. S. Boyer. Locking: A Restriction of Resolution. PhD thesis, University of Texas at Austin, 1971.

....generated. They also discuss how to automatically extend an ordering to the new predicate symbols in such a way that symbols that represent small formulas are preferred in ordered inferences. 7.4. Lock Resolution Extension results in interesting variations of resolution, such as lock resolution [Boyer 1971], which can essentially be encoded by positive hyper resolution. Lock resolution is applied to standard clauses in which each occurrence of a literal has been assigned a positive integer, called a lock index. For example, in the following set N 0 of four clauses, 8 A 7 B; 6 :A 5 :B; 4 B ....

....literal occurrence has been assigned a unique index, but in general di erent literal occurrences may be assigned the same index. The lock restriction states that only literals with a maximal index must be resolved. 8 More formally, we have the 8 We have departed from the original de nition in [Boyer 1971], which restricts resolution to minimal literals, so as to avoid confusion with ordered resolution, which resolves maximal literals. 54 L. Bachmair and H. Ganzinger Lock resolution C i A j :A D C D where no literal in C has a larger index than i, and no literal in D has a larger index ....

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Boyer R. S. [1971], Locking: A restriction of resolution, PhD thesis, University of Texas at Austin, Austin, TX.

....order) but it can be given over all atoms if the order satisfies some natural properties (see [72] A ordering is refutation complete. It is sometimes combined with other strategies (e.g. linear resolution, semantic resolution, etc. still preserving refutation completeness. Lock resolution [16] is another resolution strategy for first order theories. It generalizes both directional resolution and A ordering. In a theory A every literal instance is given an index. The same literal appearing in two di#erent clauses may receive a di#erent index for each instance. Resolution is allowed ....

Robert S. Boyer. Locking: a restriction of resolution. PhD thesis, Mathematics Department, University of Texas, Austin, 1971.

....added motivation for showing that FLH is complete for FOL, because it will then follow that the above version of IMPLY is also complete. The ground completeness of first literal hyper resolution has been an open question for about ten years. This technique also bears some similarities to Locking [3] and Indexing [5] in that one can impose an arbitrary order on the literals of a clause by means of indices. However, it differs from these in two important respects: 1) no literal ever appears more than once in a clause, and (2) the order of the positive literals in any nucleus are preserved ....

Boyer, Robert S. Locking: a Restriction of Resolution. Ph. D. Thesis, University of Texas at Austin, Texas, 1971, 74 pp.

....Here we show a few proofs using the PC prover. See Section 3.2 for the formal rules for this prover. Our currently implemented PC prover has automatically proved all of process 5 Each literal L in Lock Resolution consists of two parts, an ordinary literal, lit(L) and its index, i(L) See [8, 22] 8 Examples 1 4,4L below, and many others, using a proof plan for each. Examples 1,1 0 ,2,2 0 , though trivial, are given only to help explain the details of the procedure, using the rules of Section 3.2. Examples 4 and 4L are much more difficult. 6 Before presenting these examples we make ....

....main strength lies in its proof of theorems in elementary set theory, or where most of the predicates are from set theory. 6 The GCR and GCLR Examples We now discuss two substantial examples: GCR, the ground completeness of Resolution [31, 2] and GCLR, the ground completeness of Lock Resolution [8]. Lock Resolution is similar to Loveland s Index Ordering, which is even more general than Locking. See [22] pp. 129134. The statements of these theorems and their proof plans are given in Sections 6.1 and 6.2 below. See Section 3.2 for the formal rules. PLAN GCR is given by the user and is ....

Boyer, R. S., Locking: A Restriction of Resolution, PhD Dissertation, University of Texas at Austin, (1971).

....hand, to develop strategies is a crucial work in theorem proving. Indeed, one challenge is to give strategies which are not only efficient but also complete. We know that the basic ideas of many famous strategies for resolution, such as the linear, set of support, semantic and lock resolutions (Boyer 1971; Loveland 1969; Slagle 1967; Wos et al. 1965) do not depend upon resolution itself. For example, up to overlaps, we can view the N strategy (as well as the odd strategy) and the Grobner basis method as the strategies utilized in the semantic (or more strictly, negative) resolution and the lock ....

Boyer, R. S. (1971): Locking: A Restriction of Resolution. Ph.D thesis, University of Texas at Austin, USA.

....extend the ordering to the new P LffiL 0 atoms in a way such that predicates that abbreviate small formulas are preferred for ordered inferences. 8.4 Lock resolution Extension allows us to obtain some interesting variations of resolution inference systems. For instance, lock resolution (Boyer 1971) can essentially be encoded by positive hyper resolution. Lock resolution is applied to clauses in which each occurrence of a literal has been assigned an integer, called a lock index. For example, in the following matrix N 0 of four clauses, 1 A 2 B; 3 :A 4 :B; 5 B 6 :A; 7 :B 8 A each ....

Boyer, R. S. (1971), Locking: A restriction of resolution, PhD thesis, University of Texas at Austin, Austin, TX.

....is that the number of literals in clauses never grows; it suffers from the disadvantage of being a bottom up method. Ordered resolution, in which the literals of each clause are arranged in a linear order and only the largest literal may serve as a resolvent, is also complete for Horn clauses [ Boyer, 1971 ] The purpose here is to design Horn clause strategies that make more comprehensive use of orderings in controlling inference. A conditional equation is a universally quantified Horn clause in which the only predicate symbol is equality ( Conditional equations are important for specifying ....

Robert S. Boyer. Locking: A restriction of resolution. PhD thesis, University of Texas at Austin, Austin, TX, 1971.

.... him; many of them went on to become first rate scientists: John Wade Ulrich, 1968, computer sciences; Stephen Charles Darden, 1969, computer sciences; Charles Edward Wilks, 1969, mathematics; James Bertram Morris, 1969, computer sciences; Robert Brockett Anderson, 1970, mathematics; Robert Stephen Boyer, 1971, mathematics; Dallas Sylvester Lankford III, 1972, mathematics; Vesko Genov Marinov, 1973, computer sciences; Mark Steven Moriconi, 1977, computer sciences; John Threecivelous Minor, 1979, computer sciences; Peter Leonard Bruell, 1979, computer sciences; William Mabry Tyson, 1981, computer ....

....on completeness proofs. Along with his student Robert Anderson, Bledsoe invented a powerful restriction (Anderson and Bledsoe 1970) They introduced (Anderson and Bledsoe 1970) the excess literal method, a simple but general approach for establishing completeness of resolution restrictions. Boyer (1971) discovered a resolution restriction called locking and proved its completeness also using the excess literal method. Although these restricted resolution algorithms tend to generate fewer clauses on each round, it often takes more rounds to find a proof if one exists. Thus, what is saved on ....

Boyer, R. S. 1971. Locking: A Restriction of Resolution. Ph. D. diss., Department of Mathematics, University of Texas at Austin.

....of Ground Lock Resolution Next we want to prove the completeness of lock resolution. Lock resolution is a refinement of binary resolution where an index referring to an ordering is assigned to each literal in a clause and only resolving on minimal literals respecting the ordering is allowed (cf. [4, 18]) 2 Of course it would be very useful to have one single general method for the application of arbitrary definitions or lemmata. It is, however, a non trivial problem to consider all possibilities, in particular, when applying an ND rule for quantified formulae, a special treatment for the ....

R. S. Boyer. Locking: A Restriction of Resolution. PhD thesis, U. Texas at Austin, USA, 1971.

....Lock Resolution Next we want to prove the completeness of lock resolution. Lock resolution is a refinement of binary resolution where an index referring to an ordering is assigned to each literal in a clause and only resolving on maximal literals with respect to the ordering is allowed (cf. [4, 13]) The completeness proof also uses the k parameter technique and is almost the same as in the case of binary resolution. The only exception is that one uses a maximal literal to partition the clause set S. Therefore an additional predicate Q(S; L) has to be added to the lemmata 1, 2, and 2A ....

R. S. Boyer. Locking: A Restriction of Resolution. PhD thesis, U. Texas at Austin, USA, 1971.

....the resolution and factoring steps are performed separately. Ground case: It is proved that the variant is complete for the ground case, that is, for the Herbrand expansion of the set of clauses. Instead of using semantic trees we will use the completeness proof for ground lock resolution (see [Boyer 71] Chang, Lee 73] where an induction on the k parameter is used (k parameter is the difference between the sum of lengths of clauses and the number of clauses) Lifting: Let A and B be two non ground clauses: A = A 1 ; A n ) and B = B 1 ; Bm ) Let A 0 and B 0 be any ....

....a single literal (assuming we treat factorization as a separate rule) the fact (L 6 L) for all literals L follows from condition (A) As demonstrated by the following example, the property (A) is not enough for the general completeness of refinement for ground clauses. Example (taken from [Boyer 71] Consider the following unsatisfiable set of ground clauses: 1: P; R) 2: R; P ) 3: R; P ) 4: P; R) Let us allow to resolve only upon the first literal of each clause. At the first level we can thus get only the following two clauses, which both are tautologies: 1,4 give 5: R; R) ....

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Boyer, R.S. Locking: a restriction of resolution. The University of Texas at Austin, Ph.D. dissertation, 1971.

....strategy prohibits the generation of resolvents from two hypothesis clauses. Somewhat more general, one can distinguish any satisfiable subset of the initial clause set, and prohibit resolution between members of this distinguished subset; only the other clauses are supported . Lock resolution [Boyer, 1971] is a restriction strategy using a concept similar to that of ordering 18 Note that input resolution is by definition a special form of linear resolution. Therefore, input resolution is also known as linear input resolution. 19 It can also be seen as a special form of negative ....

Boyer, R. S. (1971). Locking: A Restriction of Resolution. PhD thesis, University of Texas, Austin, TX.

.... improved tableau (iii) can analytic tableau simulate the improved analytic tableau (iv) can 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 ....

Boyer, R. S. (1971) "Locking: a Restriction of Resolution" Ph.D. Thesis, University of Texas at Austin.

....of literals in the set of clauses and the number of clauses in the set. This method has been used successfully by many people, including one of the authors of this note (RSB) in his Ph. D. thesis under Bledsoe, which introduced the locking restriction of resolution and proved its completeness [48, 51]. Other Ph. D. students of Bledsoe who made serious contributions to the resolution literature include Robert Anderson [1, 2] Dallas Lankford [55] James B. Morris [56, 57] and T. C. Wang [46] One of the first extensions Bledsoe made to his proof checker was a resolution theorem prover coded by ....

R. S. Boyer (1971): Locking: A Restriction of Resolution. Ph. D. Dissertation, University of Texas at Austin.

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Boyer, R. S. (1971). Locking: a restriction of resolution. PhD thesis, Mathematics Department, University of Texas, Austin.

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