Showing papers on "Disjunctive normal form published in 1984"

The intractability of resolution (complexity)

Abstract: We prove that for infinitely many disjunctive normal form (DNF) propositional calculus tautologies (theta), the length of the shortest resolution proof of (theta) cannot be bounded by any polynomial in the length of (theta). The conjunctions of a DNF propositional formula are called clauses. We denote logical "and" by juxtaposition, and "not x" by x'. Call x a positive literal and x' a negative literal. Cook and Reckhow have given propositional tautologies that encode the pigeonhole principle: For n > 2, let PF(,n) be the disjunction of the clauses in the following set: {y(,1)'(,,j)y(,2)'(,,j)...y(,n)'(,,j) (VBAR) 1 (LESSTHEQ) j (LESSTHEQ) n + 1} union with {y(,i, j)y(,i,k) (VBAR) 1 (LESSTHEQ) i (LESSTHEQ) n and 1 (LESSTHEQ) j 0, so that for n large and n divisible by 100, any resolution proof of PF(,n) must generate at least 2('cn) different clauses. For notational simplicity the clauses arising from PF(,n) are represented by arrays with n rows and n + 1 columns. Let a resolution proof of PF(,n) be given. For every set S of n/4 variables such that no two variables from S are in the same row or column in the array representation, we define (gamma)(,S), a clause in the resolution proof. We show that there must be n/4 + 1 columns K in the array representation such that (gamma)(,S) contains a positive literal from column K or at least n/2 negative literals from column K. Furthermore (gamma)(,S) does not contain the negative literal for any variable in S, nor two positive literals in any column. A combinatorial counting argument shows that a large number of the clauses (gamma)(,S) must be different. Extended resolution can prove PF(,n) by generating only on the order of n('4) different clauses. Therefore resolution is seen to be much more complex on certain problems than is extended resolution.

Minimal cut-set methodology for artificial intelligence applications

Abstract: This paper suggests that given the considerable (and growing) literature of expert systems for diagnostics and maintenance, consideration of the theory of minimal cut sets should be most beneficial. The minimal cut-set approach uses disjunctive normal form in Boolean algebra and various Boolean operators to simplify very complicated tree structures composed of AND/OR gates. The simplification reduces the tree to an equivalent diagram displaying the smallest combinations of independent component failures which could result in the fault symbolized by the root of the tree and called the top event. This paper reviews minimal cut-set theory and illustrates its application with an example. Using this approach, expert diagnostic systems would have a tool in which, with minimum search, the description of fault causes is made clear and explicit, contributor sequences to a top event fault are easily quantified and ranked, and the probability of the top event is easily computed. Finally, the application of minimal cut sets to planning and problem solving is developed.