3 Definitions Satisfiability Product of Sums n m K An instance of the problem is defined by a Boolean expression written using only AND, OR, NOT, variables, and parentheses. The question is: given the expression, is there some assignment of TRUE and FALSE values to the variables that will make the entire expression true?*Product of SumsA way of arranging a boolean expression such that the final output is a result of ‘AND’ing some ‘OR’ clauses where each term within an ‘OR’ clause may or may not be ‘NOT’ed.nThe number of boolean variables found in the expression (with or without a NOT)mThe number of ‘OR’ clauses ‘AND’ed together in the boolean expressionKThe maximum number of variables found within each OR clause* Definition from
4 K-SatisfiabilityK is the maximum number of terms allowed per ‘OR’ clause.For any K > 2, the problem lies in NP-CompleteSolution of order 2nSpecial K’s3SAT3 Colorable Graph problemStill NP-Complete2SATPolynomial time solutionK = 5(A + B’ + C)(B + C’ + E’)(A’ + B + C’+ D’ + E)(B + D)K = 3 (3SAT)(X’ + Y)(X + Y’ + Z)(X’ + Z’)(Y + Z’)K = 2 (2SAT)(X1 + X2)(X2’ + X3)(X1’ + X2’)(X3 + X4)(X3’ + X5)(X4’ + X5’)(X3’ + X4)
5 3 – Colorable Transformation Procedure Transform(G) for each vertex vi in G output(ri + yi + bi) for each edge (vi,vj) in G output(ri’ + rj ’)(yi’ + yj’)(bi’ + bj’) output : boolean expression with K = 3
8 Davis - Putnam Recursive but still 2n Prunes off branches which will not yield a result.Does not examine the entire expression at each guessProcedure split (E) 1. if E has an empty clause, then return 2. if E has no clauses, then exit with current partial assignment 3. select next unassigned variable, xi, in E 4. split ( E ( xi = 0 ) ) 5. split ( E ( xi = 1 ) )
10 (X1 + X2)(X2’ + X3)(X1’ + X2’)(X3 + X4)(X3’ + X5)(X4’ + X5’)(X3’ + X4) 2- Satisfiability(X1 + X2)(X2’ + X3)(X1’ + X2’)(X3 + X4)(X3’ + X5)(X4’ + X5’)(X3’ + X4)Pick an assignment Drop terms that become true Make assignments based on initial assignment. Repeat….
11 SummaryFor K-Satisfiability where K > 2 it is an NP-Complete Problem2-Satisfiability is solvable in polynomial timeDavis-Putnam improves efficiency but does not decrease order of complexity3-Colorable problem is synonymous to 3-Satisfiability
12 References http://condor.stcloudstate.edu/~pkjha/CSCI504/2SAT.pdf "The New Turing Omnibus", by A.K. Dewdney A.W.H. Freeman/Owl Book, 2001 New York