Galil, Z.: On the complexity of regular resolution and the Davis-Putnam procedure. Theoretical Computer Science, 4 (1977) 23-46  Home/Search   Document Not in Database   Summary   Related Articles   Check

....obtaining bounds for the length of propositional resolution refutations. The first super polynomial lower bound for resolution (satisfying a restriction called regularity) was obtained by Tseitin [32] Subsequent work simplified Tseitin s proof and improved the lower bounds for regular resolution [14], 33] However great di#culty was experienced in extending Tseitin s arguments to unrestricted resolution ( dag resolution) In [16] Haken managed to give a super polynomial lower bound for the pigeon hole principle for dag resolution. Later this result was improved considerably by Ajtai [1] 2] ....

Galil, Z.: On the complexity of regular resolution and the Davis-Putnam procedure. Theoretical Computer Science, 4 (1977) 23-46

....structured instances characterized by a low induced width, DP may perform better than DLL implementations (without backjumping and learning techniques) However, most of the time, DP produces successive resolvents whose size grows in a prohibitive way. Galil first proved the intractability of DP [9], using a problem for which any proof by directed resolution is exponentially long. Ten years later, Haken [10] and Urquhart [21] extended this result to all resolution based procedures. These theoretical results suggest that resolution based procedures like DP and DLL are not appropriate to solve ....

....based procedures like DP and DLL are not appropriate to solve such instances. But they rely on the implicit assumption that resolution steps are performed one by one. We here show that taking advantage of ZBDD structures for representing sets of clauses may lead to reconsider this point. In [9], Galil describes the DP procedure as: I. Choose a propositional variable x of f . II. Replace all the clauses of f containing x (or :x) by those which can be obtained by resolution on x. III. a. If the empty clause is present, then the original set is unsatisfiable. b. If the current set is ....

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Zvi Galil. On the complexity of regular resolution and the Davis-Putnam procedure. Theorical Computer Science, 4:23--46, 1977.

....picture is very di erent with DP, the original Davis Putnam algorithm [3] DP is able to determine if a propositional formula f , expressed under conjunctive normal form (CNF) is satis able or not. Assuming the reader is familiar with propositional logic, DP may be roughly described as follows [9]: I. Choose a propositional variable x of f . II. Replace all the clauses which contain the literal x (or :x) by all binary resolvents (on x) of these clauses (cut elimination of x) and remove all subsumed clauses. III. a. If the new set of clauses is reduced to the empty clause, then the ....

....thanks to ecient data structures for representing very large sets of clauses. Several authors have pointed out that resolution based provers (like DP and DLL) are intrinsically limited, since they have found instances that require an exponential number of resolution steps to be solved (e.g. [9, 10, 14]) This is the case for the Pigeon Hole [10] and for the Urquhart problem [14] They suggest that more powerful proof systems have to be used practically to solve such problems eciently. However, all these results are based on the implicit hypothesis that successive resolutions in DP and DLL are ....

Zvi Galil. On the complexity of regular resolution and the davis-putnam procedure. Theorical Computer Science, 4:23-46, 1977.

....of the tree inward towards x, assign values to the edges attached to vertices other than x so that Clauses(y) is satisfied. The resulting assignment must be x critical since Clauses(G) is contradictory. # The graphs used in the lower bound for resolution are the expander graphs used by Galil [31] to prove an exponential lower bound for regular resolution, with a small modification to simplify the proof. The expander graph Hm is a simple bipartite graph in which each vertex has degree at most 5 and each side contains m 2 vertices (for brevity we write n = m 2 ) The particular family ....

Zvi Galil, On the complexity of regular resolution and the Davis-Putnam procedure, Theoretical Computer Science, vol. 4 (1977), pp. 23--46.

....of the tree inward towards x, assign values to the edges attached to vertices other than x so that Clauses(y) is satisfied. The resulting assignment must be x critical since Clauses(G) is contradictory. a The graphs used in the lower bound for resolution are the expander graphs used by Galil [31] to prove an exponential lower bound for regular resolution, with a small modification to simplify the proof. The expander graph H m is a simple bipartite graph in which each vertex has degree at most 5 and each side contains m 2 vertices (for brevity we write n = m 2 ) The particular family ....

Zvi Galil, On the complexity of regular resolution and the Davis-Putnam procedure, Theoretical Computer Science, vol. 4 (1977), pp. 23--46.

....test material for developers of SAT procedures (see Selman et al. 1992, for example) The SAT procedure we used for our tests is the Davis Putnam procedure, which we describe below. We believe this was a good choice for two reasons: First, it is basically a variant of resolution (Vellino 1989; Galil 1977), the most widely used general reasoning method in AI; second, almost all empirical work on SAT testing has used one or another refinement of this method, which facilitates comparison. We suspect that our results on hard and easy areas generalize to all SAT procedures. The rest of the paper is ....

Galil, Z. (1977). On the complexity of regular resolution and the Davis-Putnam procedure. Theoretical Computer Science, 4, 1977, 23--46.

....of Nullstellensatz refutations to Thue systems. First, we introduce and study (see section 1) boolean multiplicative Thue systems (basically, they consist of binomials necessarily containing among them the polynomials X 2 i Gamma 1, 1 i n) They extend slightly Tseitin s tautologies [16] [9], 17] 18] We exploit the construction of the Tseitin s tautologies ( 9] 17] 18] based on expanders ( 1] 11] 12] and give a somewhat simpler proof of a linear lower bound for the case of used in section 1 notion of refutations (lemma 4) Relying on it, we first prove a linear lower ....

.... study (see section 1) boolean multiplicative Thue systems (basically, they consist of binomials necessarily containing among them the polynomials X 2 i Gamma 1, 1 i n) They extend slightly Tseitin s tautologies [16] 9] 17] 18] We exploit the construction of the Tseitin s tautologies ([9], 17] 18] based on expanders ( 1] 11] 12] and give a somewhat simpler proof of a linear lower bound for the case of used in section 1 notion of refutations (lemma 4) Relying on it, we first prove a linear lower degree bound for Nullstellensatz refutations for the systems which include ....

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Z.Galil. On the complexity of regular resolution and the Davis-Putnam procedure. Theor. Comput. Sci., 1977, 4, p.23--46.

....of resolution performed systematically along some ordering of the variables in a propositional theory is sufficient for deciding satisfiability. This algorithm, in its original form, has received limited attention, and analyses of its performance have emphasized its worst case exponential behavior [12, 14], while neglecting the algorithm s virtues. This happened, in our view, because the algorithm was immediately overshadowed by a competitor with nearly the same name: the Davis Putnam Procedure. This competing algorithm, proposed in 1962 by Davis, Logemann, and Loveland [5] searches through the ....

....First, we show that, in addition to determining satisfiability, the algorithm generates an equivalent theory that facilitates model generation and query processing. Consequently, it may be viewed as a knowledge compilation algorithm. Second, we offset the known worst case exponential complexities [12, 14] by showing the tractability of DP elimination for many known tractable classes of satisfiability and constraint satisfaction problems (e.g. 2 cnfs, causal theories, and theories having a bounded induced width [7, 8] Third, we introduce a new parameter, called diversity, that gives rise to new ....

Galil, Z., On the Complexity of Regular Resolution and the Davis-Putnam Procedure, Theoretical Computer Science, 4:23-46 (1977).

....that a restricted amount of resolution performed along some ordering of the propositions in a propositional theory is sufficient for deciding satisfiability. However, this algorithm has received limited attention and analyses of its performance have emphasized its worst case exponential behavior [35, 39], while overlooking its virtues. It was quickly overshadowed by the Davis Putnam Procedure, introduced in 1962 by Davis, Logemann, and Loveland [11] They proposed a minor syntactic modification of the original algorithm: the resolution rule was replaced by a splitting rule in order to avoid an ....

....Even small problems having 20 variables already demonstrate the exponential behavior of DR (see Figure 20a) On larger problems DR often ran out of memory. We did not proceed with more extensive experiments in this case, since the exponential behavior of DR on uniform 3 cnfs is already well known [35, 39]. However, the behavior of the algorithms on chain problems was completely different. DR was by far more efficient than DP, as can be seen from Table 1 and from Figure 20b, summarizing the results on 3 cnf chain problems that contain 25 subtheories, each having 5 variables and 9 to 23 clauses (24 ....

Z. Galil. On the complexity of regular resolution and the Davis-Putnam procedure. Theoretical Computer Science, 4:23--46, 1977.

....of resolution performed systematically along some ordering of the variables in a propositional theory is sufficient for deciding satisfiability. This algorithm, in its original form, has received limited attention, and analyses of its performance have emphasized its worst case exponential behavior [12, 14], while neglecting the algorithm s virtues. This happened, in our view, because the algorithm was immediately overshadowed by a competitor with nearly the same name: The Davis Putnam Procedure. This competing algorithm, proposed in 1962 by Davis, Logemann, and Loveland [5] searches through the ....

....we show that, in addition to determining satisfiability, the algorithm generates an equivalent theory that facilitates model generation and query processing. Consequently, it may be better viewed as a knowledge compilation algorithm. Second, we offset the known worst case exponential complexities [12, 14] by showing the tractability of DP elimination for many known tractable classes of satisfiability and constraint satisfaction problems (e.g. 2 cnfs, Horn clauses, causal theories and theories having a bounded induced width [7, 8] Third, we introduce a new parameter, called diversity, that gives ....

Galil, Z., On the Complexity of Regular Resolution and the Davis-Putnam Procedure, Theoretical Computer Science 4:23-46 (1977).

....one value during execution of A 3 but literals may be assigned the values true and false at different points during execution of DPP ; that is, DPP contains a backtracking component whereas A 3 does not. Thus, A 3 is a polynomial time algorithm whereas DPP requires exponential time on some inputs [9]. However, A 3 is not guaranteed to find a truth assignment (implicitly) which satisfies a given instance of SAT if one exists. But, if A 3 does not give up then the truth assignment found implicitly by A 3 satisfies the instance input to A 3 . Another difference between A 3 and DPP is that A 3 ....

Galil, Z., " On the complexity of regular resolution and the Davis-Putnam Procedure," Theoretical Computer Science 4 (1977), pp. 23--46.

....bound for the length of symmetric resolution refutations by constructing a family of graphs with no symmetries for which the corresponding set of clauses Clauses(G 0 ) require long resolution refutations. The graphs used in the lower bound for resolution are the expander graphs used by Galil [8] to prove an exponential lower bound for regular resolution, with a small modification to simplify the proof. The expander graph Hm is a simple bipartite graph in which each vertex has degree at most 5 and each side contains m 2 vertices (for brevity we write n = m 2 ) The particular family ....

Zvi Galil. On the complexity of regular resolution and the Davis-Putnam procedure. Theoretical Computer Science, 4:23--46, 1977.

....belongs to the class of backtracking search algorithms (e.g. dynamic backtracking) It implements a form of dynamic variable ordering by performing unit resolution until quiesience at each step. Analyses of the original Davis Putnam algorithm have emphasized its worst case exponential behavior [Galil, 1977, Goerdt, 1992] while neglecting its virtues. Consequently, it was overshadowed by its competitor, DP backtracking, and most work on the Davis Putnam procedure, from then on, quotes the backtracking version [Goldberg et al. 1982, Selman, 1992] wrongly suggesting that this is the algorithm ....

Z. Galil, "On the complexity of regular resolution and the Davis-Putnam procedure", Theoretical Computer Science 4, 1977, 23-46.

....restricted amount of resolution, if performed systematically along some order of the atomic formulas, is sufficient for deciding satisfiability. This algorithm, in its original form, has received limited attention, and analyses of its performance have emphasized its worst case exponential behavior [Galil, 1977, Goerdt, 1992] while neglecting its virtues. This happened, in our view, because the algorithm was immediately overshadowed by a competitor with nearly the same name: The Davis Putnam Procedure. This 3 This work was partially supported by NSF grant IRI9157636, by Air Force Office of Scientific ....

....we show that, in addition to determining satisfiability, the algorithm generates an equivalent theory that facilitates model generation and query processing. Consequently, it may be better viewed as a knowledge compilation algorithm. Second, we offset the known worst case exponential complexities [Galil, 1977, Goerdt, 1992] by showing that the algorithm is tractable for many of the known tractable classes for satisfiability (e.g. 2 cnfs and Horn clauses) and for constraint satisfaction problems [Dechter and Pearl, 1987, Dechter and Pearl, 1991] e.g. causal theories and theories having a bounded ....

Z. Galil, "On the complexity of regular resolution and the Davis-Putnam procedure", Theoretical Computer Science 4, 1977, 23-46.

....of our complexity bound (even the relative sizes of the bounds so far established) is not accidental and can if at all only be improved by using reductions and tests that cannot be simulated polynomially by resolution. Regular resolution resolution 1966 p n; p k; p [T] 1977 n; k; [Ga] 1985 p n; 3 p k; 3 p [H] 1987 n; k; U] Table 2: Lower bounds on 1 c log 2 #fclausesg in resolution proofs 3 Results 3.1 Analysis of backtracking algorithms In Section 5 we present a general method for analyzing the complexity of backtracking algorithms (which may use an ....

Galil, Z.: On the complexity of regular resolution and the Davis-Putnam-procedure. Theoretical Computer Science 4 (1977), 23--46.

.... a large class of strategies used so far in (complete) TAUT(SAT) algorithms can be simulated polynomially by resolution (see also [32] Therefore the exponential lower bounds found by Urquhart ( 55] for the number of (different) clauses in resolution proofs (continuing work of Tseitin [54] Galil [14] and Haken [24] for further developments see [56, 1] apply to these algorithms. 1) Hence the lower bounds in [55] yield lower bounds 1 ff n (p) ff k (p) ff (p) 0 (depending (only) on p 3) for power coefficients for p DNF with respect to n, k or , when in the definition of power ....

Z. Galil. On the complexity of regular resolution and the Davis-Putnam-procedure. Theoretical Computer Science, 4:23--46, 1977.

....Any unsatisfiable formula has a resolution proof. Haken, solving a question that had been open for many years, proved that there are formulas requiring exponential size proofs [Ha85] His work followed that of Tseitin [Ts62] who had solved regular resolution, a further restricted form (see also [Ga77]) Extended resolution, a generalized form, remains open (see [Co75] Kozen [Ko77] considered other proof systems and established lower bounds for them. Proof systems are related to the class NP. If all unsatisfiable formulas had polynomial length resolution proofs then it would easily follow ....

Z. Galil, On the complexity of regular resolution and the Davis-Putnam procedure, Theoretical Computer Science 4, 23--46, 1977.

....than ordinary resolution, because only a single branch of the search tree (of at most depth n) is stored in memory at any one time. None the less, Haken s result implies that showing unsatisfiability using DP requires in the worst case exponential time. A direct proof of this result is also by Galil (1977). Note that this result holds no matter what variable choice heuristics are added to DP. In fact, the proofs of almost all random, unsat 3SAT formulas grow exponentially with the number of variable n (Chvatal and Szemeredi 1988) This may explain the difficulty in getting DP to handle hard, ....

Galil, Z., On the complexity of regular resolution and the Davis-Putnam procedure. Theoretical Computer Science, 4 (1977) 23--46.

....algorithms [37] Figure 1: Some typical algorithms for the SAT problem. and regular resolution [27, 59, 63, 70, 84, 86, 92] independent set al..gorithm [49] and matrix inequality system [83] have been proposed. Other specific algorithms using these principles include simplified DP algorithms [24, 29, 68], and a simplified DP algorithm with strict ordering of variables [47] The DP algorithm improved in certain aspects over Gilmore s proof method [28] Analyses of SAT algorithms often concentrates on algorithms that are simple because it is difficult to do a correct analysis of the best ....

Z. Galil. On the complexity of regular resolution and the Davis-Putnam procedure. Theoretical Computer Science, pages 23--46, 1977.

....by G.S. Tseitin [36] who showed that a slightly restricted method, which he called regular resolution, has resolution complexity at least 2 Omega Gamma p n) A. Haken [20] proved that resolution complexity is at least 2 Omega Gamma 3 p n) A. Urquhart [37] using an idea of [36] and [15], showed that resolution complexity is at least 2 Omega Gamma n) The constant in [15] for regular resolution was not given and later estimated to be around 1 50 0 000 . Makowsky calculated a constant to be around 1 387 , cf. 29] and for the details [31] This also applies to Urquhart s ....

....regular resolution, has resolution complexity at least 2 Omega Gamma p n) A. Haken [20] proved that resolution complexity is at least 2 Omega Gamma 3 p n) A. Urquhart [37] using an idea of [36] and [15] showed that resolution complexity is at least 2 Omega Gamma n) The constant in [15] for regular resolution was not given and later estimated to be around 1 50 0 000 . Makowsky calculated a constant to be around 1 387 , cf. 29] and for the details [31] This also applies to Urquhart s result. In [9] a bound of 1 129 is obtained, and the (anonymous) referee informed us, ....

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Z. Galil. On the complexity of regular resolution and the Davis--Putnam procedure. Theoretical Computer Science, 4:23--46, 1977.

....[94] Binary Decision Diagrams [2, 8] chip and conquer [29] independent set al..gorithm [64] and resolution and regular resolution [30, 74, 79, 88, 103, 106] have been proposed. Specific versions of resolution principle include Davis Putnam algorithm [18, 17] simplified Davis Putnam algorithms [27, 31, 85], Davis Putnam algorithm in Loveland s form (DPL) 74, 74] Davis Putnam Loveland procedure plus heuristic (DPLH) 65] and a simplified Davis Putnam algorithm with strict ordering of variables [61] A number of special SAT problems, such as 2 satisfiability and Horn clauses, are solvable in ....

Z. Galil. On the complexity of regular resolution and the Davis-Putnam procedure. Theoretical Computer Science, pages 23--46, 1977.

....proof systems. The computation of either DP60 or DPLL on an unsatisfiable input instance gives rise to a regular resolution proof for that instance whose number of lines is bounded by the number of steps in the computation. Hence the exponential lower bound on regular resolution proofs [Tse70, Gal77, BA80] gives rise to worst case exponential lower bounds on the time required for both DP60 and DPLL, and the exponential lower bounds for general resolution [Hak85, Urq87] show that no smarter use of resolution will help. From our point of view, the strongest lower bound for proof systems is due ....

Zvi Galil. On the complexity of regular resolution and the Davis-Putnam procedure. Theoretical Computer Science, 4:23--46, 1977.

....material for developers of SAT procedures (see Selman et al. 1992, for example) The SAT procedure we used for our tests is the DavisPutnam procedure, which we describe below. We believe this was a good choice for two reasons: First, it has been shown to be a variant of resolution (Vellino 1989, Galil 1977), the most widely used general reasoning method in AI; second, almost all empirical work on SAT testing has used one or another refinement of this method, which facilitates comparison. We suspect that our results on hard and easy areas generalize to all SAT procedures, but this remains to be seen. ....

Galil, Zvi (1977). On the complexity of regular resolution and the Davis-Putnam procedure. Theoretical Computer Science, 4, 1977, 23--46.

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