## Deciding propositional tautologies: Algorithms and their complexity (1997)

Citations: | 36 - 8 self |

### BibTeX

@TECHREPORT{Kullmann97decidingpropositional,

author = {O. Kullmann and H. Luckhardt},

title = {Deciding propositional tautologies: Algorithms and their complexity},

institution = {},

year = {1997}

}

### OpenURL

### Abstract

We investigate polynomial reductions and efficient branching rules for algorithms deciding propositional tautologies for DNF and coNP--complete subclasses. Upper bounds on the time complexity are given with exponential part 2 ff\Delta(F ) where (F ) is one of the measures n(F ) = #f variables g, `(F ) = #f literal occurrences g and k(F ) = #f clauses g. We start with a discussion of variants of the algorithms from [Monien/Speckenmeyer85] and [Luckhardt84] with the known upper bound 2 0:695\Deltan for 3-DNF and (roughly) (2 \Delta (1 \Gamma 2 \Gammap )) n for p-DNF, p 3, where p is the maximal clause length, giving now an uniform treatment for all p-DNF including the easy decidable case p 2. Recently for 3-DNF the bound has been lowered to 2 0:5892\Deltan ([K2]; see also [Sch2], [K3]). In this article further improvements are achieved by studying two additional characteristic groups of parameters. The first group differentiates according to the maximal numbers (a; b) of occ...

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Citation Context ...to parameter q. 3. The process of differentiating complexity of TAUT/SAT should be combined with experiences and heuristics gained by experiments emphasizing the worst case aspect (see [GW], and also =-=[BKB]-=-, [MSL] and [VGT]). Since there is special interest in 3-SAT/TAUT, we give also the best known combined power coefficients for 3-DNF. Note that `=ks3 holds and thus q = k=ns1=3 \Delta `=ns1=3. 3:5 1=3... |

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Citation Context ...ccidental 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 meth... |

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Citation Context ...n in Table 3.s! C # n k ` all :; ;s- formulas 1 --------- 1/4 [VG] DNF 1 1=3 [MS2] 0:1279 [VG] 1=8 [K1] 1=9 Theorem 5 p-DNF /(p) [L] [MS3] 3-DNF 0:8232 [D] 0:6943 8 ? ? ! ? ? : [MS1] [L] [MS3] 0:6583 =-=[Sch1]-=- 0:5893 [K2] 0:5814 1) DNF-(1; 1) 0.5284 [L] DNF-(1; 2) 1=3 [MS2] 0:1762 Theorem 4 0.0588 Corol. 9.11 3-DNF-(1; 2) 0.1762 Corol. 9.12 Table 1: Power coefficients for all boolean formulas and for coNP-... |

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Citation Context ... DNF's in which every variable appears in one sign at most once and in the other sign at most s times. p-DNF-(1; s) := p-DNF " DNF-(1; s). For a given Boolean formula Tseitin, Luckhardt and Cook =-=(see [L]-=-) give a method for constructing in polynomial time a formula in 3-DNF which is TAUT-equivalent to the original one. Thus all p-DNF for ps3 are representative in this sense. In [L] this is extended to... |

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Citation Context ...nts which are given in the literature and in this paper (bold-faced). Another type of results is given in Table 3.s! C # n k ` all :; ;s- formulas 1 --------- 1/4 [VG] DNF 1 1=3 [MS2] 0:1279 [VG] 1=8 =-=[K1]-=- 1=9 Theorem 5 p-DNF /(p) [L] [MS3] 3-DNF 0:8232 [D] 0:6943 8 ? ? ! ? ? : [MS1] [L] [MS3] 0:6583 [Sch1] 0:5893 [K2] 0:5814 1) DNF-(1; 1) 0.5284 [L] DNF-(1; 2) 1=3 [MS2] 0:1762 Theorem 4 0.0588 Corol. ... |

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Citation Context ...er (bold-faced). Another type of results is given in Table 3.s! C # n k ` all :; ;s- formulas 1 --------- 1/4 [VG] DNF 1 1=3 [MS2] 0:1279 [VG] 1=8 [K1] 1=9 Theorem 5 p-DNF /(p) [L] [MS3] 3-DNF 0:8232 =-=[D]-=- 0:6943 8 ? ? ! ? ? : [MS1] [L] [MS3] 0:6583 [Sch1] 0:5893 [K2] 0:5814 1) DNF-(1; 1) 0.5284 [L] DNF-(1; 2) 1=3 [MS2] 0:1762 Theorem 4 0.0588 Corol. 9.11 3-DNF-(1; 2) 0.1762 Corol. 9.12 Table 1: Power ... |

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Citation Context ...blished) 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 pres... |

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Citation Context ...ade in between DP l -applications or only at the end, but elimination of subsumed clauses as soon as possibly is more efficient (for unrestricted applications exponentially many clauses can be saved: =-=[C], [S]). For generali-=-zing the use of "Subsumption" in this context by "K-Extended Subsumption" see Definition 7.10. 5. We will see in Lemma 7.4 part 2 that the result of DP l 1 ;:::;l M does not depend... |

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Citation Context ...jections: f1; : : : ; mg ! f1; : : : ; mg we have: r S ffi DP l 1 ;:::;l m = r S ffi DP l (1) ;:::;l (m) : 6) but in general not with respect to restriction R (see Remark 5); furthermore, as shown in =-=[Go]-=- at least for the unrestricted case and for DP-resolution proof systems, the amount of work needed for computing DP l 1 ;:::;l m by successive computations of DP l i may depend heavily on the order of... |

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Citation Context ...formulas. In Subsection 3.12 we show diagrams of simplified combined graduated power coefficients for DNF and 3-DNF. For applications of corresponding "combined" algorithms we refer to "=-=;Multi-Sat" in [Gu]-=-, [GuPu], [Gu2]. 3.10.2 Linear combinations of measures and improved algorithms The efficiency of backtracking algorithms we gauge by the totality of differences \Delta labelling the edges of the test... |

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Citation Context ...ubsection 3.12 we show diagrams of simplified combined graduated power coefficients for DNF and 3-DNF. For applications of corresponding "combined" algorithms we refer to "Multi-Sat&quo=-=t; in [Gu], [GuPu], [Gu2]-=-. 3.10.2 Linear combinations of measures and improved algorithms The efficiency of backtracking algorithms we gauge by the totality of differences \Delta labelling the edges of the test tree for an ap... |

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Citation Context ...type of results is given in Table 3.s! C # n k ` all :; ;s- formulas 1 --------- 1/4 [VG] DNF 1 1=3 [MS2] 0:1279 [VG] 1=8 [K1] 1=9 Theorem 5 p-DNF /(p) [L] [MS3] 3-DNF 0:8232 [D] 0:6943 8 ? ? ! ? ? : =-=[MS1]-=- [L] [MS3] 0:6583 [Sch1] 0:5893 [K2] 0:5814 1) DNF-(1; 1) 0.5284 [L] DNF-(1; 2) 1=3 [MS2] 0:1762 Theorem 4 0.0588 Corol. 9.11 3-DNF-(1; 2) 0.1762 Corol. 9.12 Table 1: Power coefficients for all boolea... |

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Citation Context ...n between DP l -applications or only at the end, but elimination of subsumed clauses as soon as possibly is more efficient (for unrestricted applications exponentially many clauses can be saved: [C], =-=[S]). For generalizing -=-the use of "Subsumption" in this context by "K-Extended Subsumption" see Definition 7.10. 5. We will see in Lemma 7.4 part 2 that the result of DP l 1 ;:::;l M does not depend on t... |

1 |
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Citation Context ...as. In Subsection 3.12 we show diagrams of simplified combined graduated power coefficients for DNF and 3-DNF. For applications of corresponding "combined" algorithms we refer to "Multi=-=-Sat" in [Gu], [GuPu]-=-, [Gu2]. 3.10.2 Linear combinations of measures and improved algorithms The efficiency of backtracking algorithms we gauge by the totality of differences \Delta labelling the edges of the test tree fo... |

1 | On a note by Q.G - Guggenheimer - 1964 |

1 |
Personal communication. (According to a diplom thesis by Prof
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(Show Context)
Citation Context ... Lemma 5.2 as follows: (1; : : : ; p \Gamma 1)s(1 + 1; 1 + 2; : : : ; 1 + (p \Gamma 1); 2) = (2; : : : ; p; 2) ? (2; : : : ; p + 1). However, in this way we have lost the quick decision of 2-DNF (see =-=[Le]-=- where an exponential lower bound on 2-DNF is given). To repair this one has to extend the assignments ' i by all immediatly following 1-clause eliminations. 8.3.3 Local Autarkness and new clauses All... |