Murray, N.V., and Rosenthal, E. Dissolution: Making paths vanish. J.ACM 40,3 (July 1993), 504--535.  Home/Search   Document Not in Database   Summary   ACM   TOC   Related Articles   Check

....clauses are omitted) which involves only restricted clauses and cubes in the representation and, in addition, prohibits that a single literal is both an implicant and an implicate of the same subformula. Other representations for nnf formulas have been proposed in the literature: For example, in [7] graphs are used, however, their representation does not di#er substantially from the standard one, and it is only useful for the search of links which is the main part of the dissolution method they propose. Another widespread alternative representation are the BDDs and its variants [2, 10] but ....

N.V. Murray and E. Rosenthal. Dissolution: Making paths vanish. Journal of the ACM, 40(3):504--535, 1993.

....[Rob65] who collapses the two steps into a unique and uniform inference rule (i.e. resolution) However resolution is inherently a local inference rule. The major drawback of this fact is that even trivial theorems require several applications of the rule. The latest generation of procedures [And81, Bib82, MR] have rediscovered the separation between the two steps of instance generation and propositional analysis (which adopting Andrews terminology are called amplification and search over the set of vertical paths respectively) The key characteristic of FOLTAUT is a neat separation between such ....

N.V. Murray and E. Rosenthal. Dissolution: Making Paths Vanish. Journal of the ACM. To appear.

....also some ideas based on the data structure of BDDs have been # Partially supported by CICYT project number TIC97 0579 C02 02. 1 TAS stands for Transformaciones de Arboles Sintacticos, Spanish translation of Syntactic Trees Transformations. 1 used in this context. Recently, path dissolution [6] has been introduced as a generalisation of the analytic tableaux, allowing tableaux deductions to be substantially speeded up. The central point for e#ciency of any satisfiability tester is the control over the branching, and our approach is focussed on the previous reduction of the formula to be ....

N.V. Murray and E. Rosenthal. Dissolution: Making paths vanish. Journal of the ACM, 40(3):504--535, 1993.

....is not compatible with applying substitutions and the strong normal form property is lost [120] 3.2. 4 Other Calculi There is a deduction method which, like non clausal resolution, avoids to compute any normal form altogether: Murray Rosenthal s dissolution rule is available both for classical [109] and finite valued logics [108, 111] Many valued dissolution operates on formulas in signed negation normal form (NNF) i.e. formulas built up from , and signed literals. The dissolution rule selects in a signed NNF formula an implicitly conjunctively connected pair of literals S p, S 0 p and ....

N. V. Murray and E. Rosenthal. Dissolution: Making paths vanish. Journal of the ACM, 3(40):504--535, 1993.

....paths (more precisely: disjunctive paths) through a formula. In the case of a CNF formula the set of its clauses is identical to the set of its disjunctive paths. Appropriate tools for a formal treatment of the notions required here were developed by Andrews [1] Bibel [6] and Murray Rosenthal [13]. Here we use a notation which is close to that of the latter paper (but this can be mainly considered as a matter of taste the other formalisms could be used just as well) Definition 7 An NNF formula is defined recursively as follows: 1. A (possibly non ground) literal is an NNF formula. 2. ....

....that p(x)oe is not maximal in Coe. In the completeness proof for A ordered ground clause tableau we used that a clause is satisfied by an interpretation if one of the clause s literals is satisfied by the interpretation. This result can be generalized to NNF formulas. Proposition 1 (see, e.g. [13]) A ground NNF formula OE is satisfied by an interpretation I iff at least one literal in every d path of OE is satisfied by I. 3.2 Ordered Links The basis of each refutation procedure is the detection of complementary pairs of literal occurrences. A pair of literal occurrences that might ....

[Article contains additional citation context not shown here]

N. V. Murray and E. Rosenthal. Dissolution: Making paths vanish. Journal of the ACM, 3(40):504--535, 1993.

....paths (more precisely: disjunctivepaths) through a formula. In the case of a CNF formula the set of its clauses is identical to the set of its disjunctive paths. Appropriate tools for a formal treatment of the notions required here were developed by Andrews [1] Bibel [5] and Murray Rosenthal [10]. Here we use a notation which is close to that of the latter paper (but this can be mainly considered as a matter of taste the other formalisms could be used just as well) Definition 7 An NNF formula is defined recursively as follows: 1. A (possibly non ground) literal is an NNF formula. 2. ....

....that p(x)oe is not maximal in Coe. In the completeness proof for A ordered ground clause tableau we used that a clause is satisfied by an interpretation if one of the clause s literals is satisfied by the interpretation. This result can be generalized to NNF formulas. Proposition 1 (see, e.g. [10]) A ground NNF formula OE is satisfied by an interpretation I iff at least one literal in every d path of OE is satisfied by I. The basis of each refutation procedure is the detection of complementary pairs of literal occurrences. A pair of literal occurrences that might become complementary ....

[Article contains additional citation context not shown here]

N. V. Murray and E. Rosenthal. Dissolution: Making paths vanish. Journal of the ACM, 3(40):504--535, 1993.

....it. Paramodulation inference can be implemented in clause trees using paths to justify equality substitutions analogously to the way in which merge paths justify factoring. Another open question is how to extend clause trees to handle proofs that use formulas other than just clauses, as done in [24]. Many propositional satisfiability problems can be solved only with an exponential number of resolution steps [4] 5] Hence these problems admit only exponentially large closed clause trees. However, such problems can sometimes be solved in polynomial time [4] 7] 39] One method that allows the ....

N. V. Murray and E. Rosenthal, Dissolution: making paths vanish, J. ACM 40 (3) (1993) 504--535.

....with unbounded nesting depth without having to compute a bounded depth circuit first. This can greatly reduce the size required for intermediate representations. We stress that the anti link technique is not intended to replace existing and successful techniques such as BDDs or dissolution [4] (it is not even a complete inference rule for propositional logic) rather, the latter can be augmented and improved by our analysis. In this paper, we define an anti link operation on a generic language for expressing many valued logic formulas called signed NNF and we show that all interesting ....

....In this paper, we define an anti link operation on a generic language for expressing many valued logic formulas called signed NNF and we show that all interesting properties of two valued anti links generalize to the many valued setting, although in a non trivial way. Our notation follows [4]; but instead of employing the special concept of semantic graphs for the representation of NNF formulas, we use terminology that solely relies on well known notions like formulas, subformulas, etc. Before giving the technical details, in the remainder of this section we briefly outline our ....

[Article contains additional citation context not shown here]

Neil V. Murray and Erik Rosenthal. Dissolution: Making Paths Vanish. Journal of the ACM, 3(40):504--535, 1993.

No context found.

Murray, N.V., and Rosenthal, E. Dissolution: Making paths vanish. J.ACM 40,3 (July 1993), 504--535.

No context found.

Murray, N.V., and Rosenthal, E. Dissolution: Making paths vanish. J. ACM 40, 3, 504--535, July 1993.

....do contain A. In Figure 1, CC(A, X) D V E; CPE(A, X) A. It is intuitively clear that the paths through (X I0 that do not contain the link are those through (CPE(A, X) CC(A, D) plus those through (CC(A, X) CPE( D) plus those through (CC(A, X) CC( D) The reader is referred to [5] and [7] for the formal definitions of CC and of CPE and for the proofs of the lemmas below; there, they are stated in full generality for arbitrary subgraphs H. In this paper, we need these structures only with respect to single nodes, and the lemmas are restricted accordingly. Lemma 2. The c paths of ....

....full block M except those of CPE(A, X) CPE(A, Y) i.e. except those through the link. Theorem 1. Let H be a link in semantic graph G, and let M be the smallest full block containing H. Then M and DV(H, M) are equivalent. A proof of Theorem 1 (in its full generality) can be found in [5] and in [7]. We may therefore select an arbitrary link H in G and replace the smallest full block containing H by its dissolvent, producing (in the ground case) an equivalent graph. We call the resulting graph the dissolvent of G with res19ect to H and denote it Diss(G,H) Since the paths of the new graph ....

Murray, N.V., and Rosenthal, E. Dissolution: Making paths vanish. J. ACM 40,3 (July 1993), 504-535.

....single fault diagnoses are generated. This approach does not rely on a clause form representation (although it is applicable to systems represented in clause form) nor does it require generating the set of minimal conflicts. The approach is based on dissolution, an inference rule described in [10] for formulas in negation normal form (NNF) Dissolution has been implemented in Dissolver, a theorem prover for NNF formulas, and included in PI, a prime implicant implicate generating system [13,14] We have obtained experimental results on the performance of our algorithm on commonly used ....

....of atom, literal, and formula from classical logic. We consider only formulas in negation normal form (NNF) The only connectives used are conjunction and disjunction, and all negations are at the atomic level. In this section, we review a number of technical terms and definitions taken from [9,10]. They make the paper somewhat self contained even for readers not familiar with dissolution. Semantic Graphs A semantic graph G is a triple (N,C,D) of nodes, c arcs, and d arcs, respectively, where a node is a literal occurrence, a c arc is a conjunction of two semantic graphs, and a d arc is a ....

[Article contains additional citation context not shown here]

Murray, N.V., and Rosenthal, E. Dissolution: Making paths vanish. J. ACM 40,3 (July 1993), 504-535.

....and minimal diagnosis algorithms. 4 The software is also available through anonymous ftp at ftp: ftp.cs.albany.edu nvm diag. 6 Chapter 2 Preliminaries This chapter contains a brief review of propositional logic and path dissolution. Path dissolution was developed by Murray and Rosenthal [40] as an inference rule for propositional two valued logic. It was then extended to multiple valued logics using the notion of signed formulas. The definitions and notation used in this section are taken from [55, 37, 40] In this thesis we deal with formulas in propositional logic only. 2.1 The ....

....logic and path dissolution. Path dissolution was developed by Murray and Rosenthal [40] as an inference rule for propositional two valued logic. It was then extended to multiple valued logics using the notion of signed formulas. The definitions and notation used in this section are taken from [55, 37, 40]. In this thesis we deal with formulas in propositional logic only. 2.1 The Language of Propositional Logic A language consists of a set of symbols, and a set of rules describing the set of strings which are legal sentences. The set of symbols and the set of rules form the syntax of the ....

[Article contains additional citation context not shown here]

Neil V. Murray and Erik Rosenthal. Dissolution: Making paths vanish. Journal of the ACM, 48(3):504--535, July 1993.

....More than one author 1 tried to prove strong completeness of cg resolution until Eisinger s famous example [7] dispelled that idea. He showed that even a quite restrictive notion of fairness cannot prevent cyclic derivations. An example of a strongly complete calculus is path dissolution [12], which operates by deleting all paths (and thus the link itself) through a given link, decreasing the number of paths in the formula. Since there are nitely many paths, the process terminates in a linkless formula. That formula is empty if the original formula was unsatis able; otherwise the ....

....Not surprisingly, reconciling completeness issues with the destructive nature of the calculus has proved to be considerably more elusive for cg resolution than for the more standard resolution systems. Other examples of destructive calculi are free variable tableaux [2] and dissolution [12]. Level saturation is an instance of what is often called a fair selection rule. In cg resolution a fair rule guarantees that each link in each derivation vanishes within a nite amount of time (called a coveringthree lter by Siekmann and Stephan [16] It can easily be implemented locally by an ....

Murray, N.V. and Rosenthal, E. Dissolution: Making paths vanish. Journal of the ACM, 40(3):504-535, 1993.

....normal form (DNF) The algorithm of [9] requires the input to be a conjunction of DNF formulas. In [11] we proposed a set of techniques for finding the prime implicates of formulas in negation normal form (defined below) The techniques are based on dissolution, an inference rule described in [8], and on an algorithm called PI. We have discovered classes of formulas for which these techniques are polynomial but for which This work was supported in part by the National Science Foundation under grants CCR 9101208 and CCR 9202013. any CNF DNF based technique must be exponential in the size ....

....in part by the National Science Foundation under grants CCR 9101208 and CCR 9202013. any CNF DNF based technique must be exponential in the size of the input. Ngair [9] has also introduced similar examples. In the next section, a brief discussion of background material is presented; see [8] for details. In Section 3, the new inference rule semi resolution is introduced. The spanning property and its preservation are discussed in Section 4, and completeness is proved in Section 5. Some proofs are omitted for lack of space. 2 Background In this section we briefly review concepts ....

[Article contains additional citation context not shown here]

Murray, N.V., and Rosenthal, E. Dissolution: Making paths vanish. J.ACM 40,3 (1993), 504-535.

No context found.

Murray, N.V., and Rosenthal, E. Dissolution: Making paths vanish. J.ACM 40,3 (July 1993), 504-535.

....in some respects to the one presented here, but it requires a non trivial double induction and relies on the existence of refutations that must conform to an excessively rigid structure. Link deletion is sometimes possible with inference rules that do not rely on clause form. Path dissolution [8] and the tableau method implicitly delete links. Link deletion is also possible with semi resolution [9] but proving that the spanning property is preserved is highly non trivial. Some insight into link deletion in negation normal form is provided in Section 5.3. In Section 2, a variation of the ....

....the c paths of SP will be a subset of the c paths of S, SP will also be spanned. It turns out that a good choice for SP is the c path complement of P in S, denoted CC(P ; S) We use a definition of CC(P ; S) which is tailored to the special case required in this paper; for the general case, see [8]. Definition. Let H be a subformula of the formula G. Then the c path complement of H with respect to G, denoted CC(H;G) is the subformula that results from replacing H by false and making obvious simplifications. Alternatively, CC(H;G) can be simply characterized with the CE operator defined as ....

[Article contains additional citation context not shown here]

Murray, N.V., and Rosenthal, E. Dissolution: Making paths vanish. J.ACM 40(3), 1993, 504-- 535.

No context found.

Neil V. Murray and Eric Rosenthal. Dissolution: Making paths vanish. Journal of the ACM, 48(3):504--535, July 1993.

....single fault diagnoses are generated. This approach does not rely on a clause form representation (although it is applicable to systems represented in clause form) nor does it require generating the set of minimal conflicts. The approach is based on dissolution, an inference rule described in [7] for formulas in negation normal form (NNF) We have obtained experimental results on the performance of our techniques on commonly used benchmark problems [10] A system is a pair (SD; Comp) where SD, the system description, is a propositional formula, and Comp, the set of system components, is a ....

....formula F(SD;Obs;D) SD Obs 0 ab(c)2D ab(c) 1 A 0 ab(c)2(AbV nD) ab(c) 1 A is consistent. A diagnosis D is minimal if no proper subset of D is a diagnosis. A diagnosis D is a single fault diagnosis if D is singleton set, otherwise it is a multiple fault diagnosis. 1 Dissolution [7] is an inference rule that preserves equivalence and that terminates naturally in the propositional case. If we dissolve in formula F until it is linkless, the resulting formula is called the full dissolvent of F ; we denote it by FD(F ) The set of conjunctive paths 2 (c paths) in FD(F ) is ....

N. Murray and E. Rosenthal. Dissolution: Making paths vanish. Journal of the ACM, 40(3):504--535, 1993.

No context found.

Neil V. Murray and Eric Rosenthal. Dissolution: Making paths vanish. Journal of the ACM, 48(3):504--535, July 1993.

....can be converted to NNF in polynomial time. We deal only with propositional logic in this paper, although some of the following results like the Path Dissolution Rule are completely general. In this section, we introduce a number of technical terms and definitions that are treated in detail in [9]. They are required for the development of the anti link operations defined in Section 3, and they make the paper self contained even for readers not familiar with dissolution. 2.1. Semantic Graphs A semantic graph G is a triple (N,C,D) of nodes, c arcs, and d arcs, respectively, where a node is ....

....Note that c arcs and d arcs are indicated by the usual symbols for conjunction and disjunction. Essentially, the only difference between a semantic graph and a formula in NNF is the point of view, and we will use either term depending upon the desired emphasis. For a more detailed exposition, see [9]. ################## (Ramesh and Murray) and by Deutsche Forschungsgemeinschaft within the Schwerpunktprogramm Deduktion (H . ahnle and Beckert) 1 Anti links and some associated operators were first proposed by Beckert and H . ahnle personal communication. The first motivation for ....

[Article contains additional citation context not shown here]

Murray, N.V., and Rosenthal, E. Dissolution: Making paths vanish. J.ACM 40,3 (July 1993), 504-535.

....are collapsed into one. All other disjunctive paths remain unchanged, and the resulting formula is therefore logically equivalent to the formula F . Note, that the DADV operator does not in general produce a formula in clause form (CNF or DNF) The operation is closely related to Path Dissolution [6]; DADV is similar to the operator used there to remove unsatisfiable (or tautological, in the dual case) paths. Path Dissolution works by selecting a link (i.e. occurrences of a literal A and its negation :A) and restructuring the formula so that all paths through the link are eliminated. One ....

N. Murray and E. Rosenthal. Dissolution: Making paths vanish. Journal of the ACM, 48(3):504--535, July 1993.

No context found.

Neil V. Murray and Erik Rosenthal. Dissolution: Making paths vanish. Journal of the ACM, 40(3):504-535, July 1993.

No context found.

Neil V. Murray and Erik Rosenthal. Dissolution: Making paths vanish. Journal of the ACM, 3(40):504-- 535, 1993.

No context found.

Neil V. Murray and Erik Rosenthal. Dissolution: Making Paths Vanish. Journal of the ACM, 3(40):504--535, 1993.

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC