Cantone, D., E. G. Omodeo and A. Policriti, The automation of syllogistic. II. Optimization and complexity issues, J. Automated Reasoning 6 (1990), pp. 173-- 187.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....in the sentence. Coincidentally, the number of distinct EMLS conjuncts is 2 2n 3 4n 2 , whose exponent is of the same order of magnitude. Policriti [Pol87] originally proves the NP completeness of MLS. The problem of finding a satisfying assignment for sentences in MLS2 is NP complete [COP90]. In this paper, in addition to proving the NP average completeness of MLS (and related languages) we prove the stronger result that deciding a subset of EMLS, that is, conjuncts of elementary positive literals, is both NP complete and NP average complete. 2.2 Presburger arithmetic and the ....

....it follows from the definition that f witnesses the NP average completeness of L 2 . 2 5 Average case complexity of correct program technology 5. 1 The average case complexity of EMLS and fragments of set theory With respect to proving the following theorem, we recall reduction f of SAT to MLS2 [COP90]. Let OE j OE 1 Delta Delta Delta OE n where each of the OE i is a disjunction of literals (i.e. OE is in conjunctive normal form) Let v i , i = 1; m be the distinct propositional variables occurring in OE. For each propositional variable we include a corresponding set variable v i . ....

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Domenico Cantone, Eugenio Omodeo, and Alberto Policriti. The automation of syllogistic. II. Optimization and complexity issues. Journal of Automated Reasoning, 6(2):173--188, June 1990. 19

....to true or false. Alternatively, an algorithm can be provided that, in case 9 V i is satisfiable, find a substitution for the variables of V i . It can be obtained by first reducing 9 V i to a disjunction of cardinality constraints by lemmas (10) and (11) An algorithm proposed in [4] to decide validity of a subclass of the language of set theory, can be instantiated to our case to solve cardinality constraints. Once found such solutions, we have, for each variable X of V i , the set of variables SX of the term to which X must be mapped to for V i to be satisfied. At ....

D. Cantone, E. G. Omodeo, and A. Policriti. The Automation of Syllogistic II. Optimization and Complexity Issues. Journal of Automated Reasoning, 6(2):173-- 187, 1990.

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Cantone, D., E. G. Omodeo and A. Policriti, The automation of syllogistic. II. Optimization and complexity issues, J. Automated Reasoning 6 (1990), pp. 173-- 187.

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Domenico Cantone, Eugenio G. Omodeo, and Alberto Policriti. The automation of syllogistic. II. Optimization and complexity issues. Journal of Automated Reasoning, 6(2):173--187, 1990.

.... Schwartz in 1977 [Sch77, DS77, Sch78] as an enhancement of the Floyd Hoare approach [Flo67, Hoa69] Subsequently, the most successful directly related developments have been a thought piece on transformational programming [Dea77, Dea80] and theoretical work on decidable sublanguages of set theory [FOS80, CFOS87, CFO89, COP90, Pol87a, Pol87b, PS92, PT92]. We first define factors to consider in reviewing a given approach towards building verification tools, and then we briefly consider three quite distinct approaches to building such tools. 2.1 Specification We define the notion of a function, then we say what it means to specify a function, and ....

Domenico Cantone, Eugenio Omodeo, and Alberto Policriti. The automation of syllogistic. II. Optimization and complexity issues. Journal of Automated Reasoning, 6(2):173--188, June 1990.

....in Lemma 6 allows us to conclude that the satis ability problem for normalized UGRA conjunctions is in NP . In view of Lemma 5, the above discussion yields that the satis ability problem for the theory UGRA is in NP . On the other hand, by using a reduction similar to the one described in [COP90], it can easily be seen that the decision problem SAT can be reduced in polynomial time to the satis ability problem for UGRA, thus showing the NP hardness of the latter decision problem. Hence we have: Theorem 1. The satis ability problem for the theory UGRA is NP complete. ut 3 The directed ....

D. Cantone, E.G. Omodeo, and A. Policriti. The automation of syllogistic. II. Optimization and complexity issues. J. Automated Reasoning 6:173-187, 1990.

....r 9 of the given r under the axioms. The third possibility, r 9 neither provable nor refutable, is ruled out by the syntactic peculiarities of r. The method to be proposed inherits from [BFOS81] cf. also [CFO89] Chapter 7) but is closer to a much improved decision algorithm given in [COP90]; both of those ancestors referred uniquely to sets, though. Notice that the correctness of the methods crucially depends on the extensonality axiom (E) actually, in the case of sets the axioms that enter into play are (E) A) N) and (W) in the case of hypersets they are (H) and (6 R) Let ....

....a flex G, which also has g places i 1 ; i g . We introduce an arc [ between two principal nodes ; of G if and only if 2 ; also, we introduce an arc from each i h to the corresponding j h . Showing that all this characterizes an implicant of r requires well established techniques (cf. [BFOS81, CFO89, COP90]) 2 Remark 3. Optimizations of the search method implicit in this theorem are at hand. The size of the search space can in fact be cut down by requiring that the sought G has: ffl excess size g smaller than the number of variables that occur on the right of 2 in restricted quantifiers of r ....

D. Cantone, E. G. Omodeo, and A. Policriti. The Automation of Syllogistic II. Optimization and Complexity Issues. Journal of Automated Reasoning, 6(2):173-- 187, 1990.

....y) by letting e be the result of identifying in equivalent variables, then e is injectively satis able. 3.1 Complexity issues In this subsection we prove that the satis ability problem for (extended) normalized conjunctions of MLSS is NP complete. NP hardness is immediate (cfr. [COP90]) Concerning the NP completeness, notice that, in view of Corollary 3.1, the size of the directed acyclic graph G = N; b 2) considered in the statement of Theorem 3.1 is O(j j 4 ) Also, given G = N; b 2) the construction of the U realization R U as well as the veri cation that R U ....

D. Cantone, E.G. Omodeo, and A. Policriti. The automation of syllogistic. II. Optimization and complexity issues. J. Automated Reasoning 6:173-187, 1990.

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D. Cantone, E. G. Omodeo, and A. Policriti. The Automtion of Syllogistic. II. Optimization and Complexity Issues. Journal of Automated Reasoning, 6: 173--187, 1990.

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