SAT Solver Math Foundations of Computer Science Boolean

SAT Solver Math Foundations of Computer Science

Boolean Expressions A Boolean expression is a Boolean function Any Boolean function can be written as a Boolean expression s x 0 x 1 f 0 0 Disjunctive normal form (sums of products) For each row in the truth table where the output is true, write a product such that the corresponding input is the only input combination that is true Not unique E. G. (multiplexor function) 0 0 1 0 0 1 1 0 0 0 1 1 0 0 1 1 2

Conjunctive Normal Form s x 0 x 1 f 0 0 1 0 0 1 1 0 0 0 1 1 0 0 1 1 3

Tautology Checker A program can be written to check to see if a Boolean expression is a tautology. Simply generate all possible truth assignments for the variables occurring in the expression and evaluate the expression with its variables set to each of these assignments. If the evaluated expressions are always true, then the given Boolean expression is a tautology. A similar program can be written to check if any two Boolean expressions E 1 and E 2 are equivalent, i. e. if E 1 E 2. Such a program has been provided. 4

Satisfiability A formula is satisfiable if there is an assignment to the variables that make the formula true A formula is unsatisfiable if all assignments to variables eval to false A formula is falsifiable if there is an assignment to the variables that make the formula false A formula is valid if all assignments to variables eval to true (a valid formula is a theorem or tautology)

Satisfiability Checking to see if a formula f is satisfiable can be done by searching a truth table for a true entry Exponential in the number of variables Does not appear to be a polynomial time algorithm (satisfiability is NP-complete) There are efficient satisfiability checkers that work well on many practical problems Checking whether f is satisfiable can be done by checking if f is a tautology An assignment that evaluates to false provides a counter example to validity

Propositional Logic in ACL 2 In beginner mode and above ACL 2 S B !>QUERY (thm (implies (and (booleanp p) (booleanp q)) (iff (implies p q) (or (not p) q)))) << Starting proof tree logging >> Q. E. D. Summary Form: ( THM. . . ) Rules: NIL Time: 0. 00 seconds (prove: 0. 00, print: 0. 00, proof tree: 0. 00, other: 0. 00) Proof succeeded.

Propositional Logic in ACL 2 >QUERY (thm (implies (and (booleanp p) (booleanp q)) (iff (xor p q) (or p q)))) … **Summary of testing** We tested 500 examples across 1 subgoals, of which 1 (1 unique) satisfied the hypotheses, and found 1 counterexamples and 0 witnesses. We falsified the conjecture. Here are counterexamples: [found in : "Goal''"] (IMPLIES (AND (BOOLEANP P) (BOOLEANP Q) P) (NOT Q)) -- (P T) and (Q T)

SAT Solvers Input expected in CNF Using DIMACS format One clause per line delimited by 0 Variables encoded by integers, not variable encoded by negating integer We will use Mini. SAT (minisat. se)

Mini. SAT Example (x 1 | -x 5 | x 4) & (-x 1 | x 5 | x 3 | x 4) & (-x 3 | x 4). DIMACS format (c = comment, “p cnf” = SAT problem in CNF) c SAT problem in CNF with 5 variables and 3 clauses p cnf 5 3 1 -5 4 0 -1 5 3 4 0 -3 -4 0

Mini. SAT Example (x 1 | -x 5 | x 4) & (-x 1 | x 5 | x 3 | x 4) & (-x 3 | x 4). This is Mini. Sat 2. 0 beta ==============[ Problem Statistics ]========= | | | Number of variables: 5 | | Number of clauses: 3 | | Parsing time: 0. 00 s | …. SATISFIABLE v -1 -2 -3 -4 -5 0

Avionics Application Aircraft controlled by (real time) software applications (navigation, control, obstacle detection, obstacle avoidance …) Applications run on computers in different cabinets 500 apps 20 cabinets Apps 1, 2 and 3 must run in separate cabinets Problem: Find assignment of apps to cabinets that satisfies constraints

Corresponding SAT problem

Constaints in CNF

DIMACS Format

Avionics Example

Avionics Example p cnf 50 25 c clauses for valid map forall a exists c AC^c_a 1 2 3 4 5 0 6 7 8 9 10 0 11 12 13 14 15 0 16 17 18 19 20 0 21 22 23 24 25 0 26 27 28 29 30 0 31 32 33 34 35 0 36 37 38 39 40 0 41 42 43 44 45 0 46 47 48 49 50 0

Avionics Example c constaints ~AC^c_1 + ~AC^c_2 and ~AC^c_1 + ~AC^c_3 -1 -6 0 -1 -11 0 -2 -7 0 -2 -12 0 -3 -8 0 -3 -13 0 -4 -9 0 -4 -14 0 -5 -10 0 -5 -15 0 c constraint ~AC^c_2 + ~AC^c_3 -6 -11 0 -7 -12 0 -8 -13 0 -9 -14 0 -15 0

Avionics Assignment SAT -1 -2 3 -4 -5 -6 7 -8 -9 -10 11 -12 -13 -14 -15 16 -17 -18 -19 -20 21 -22 -23 -24 -25 26 -27 -28 -29 -30 31 -32 -33 -34 -35 36 -37 -38 -39 -40 41 -42 -43 -44 -45 46 -47 -48 -49 -50 0 True indicator variables: 3 = 5*0 + 3 => AC(1, 3) 7 = 5*1 + 2 => AC(2, 2) 11 = 5*2 + 1 => AC(3, 1) 16 = 5*3+1 => AC(4, 1) 21 = 5*4+1 => AC(5, 1) 26 = 5*5=1 => AC(6, 1) 31 = 5*6+1 => AC(7, 1) 36 = 5*7+1 => AC(8, 1) 41 = 5*8 + 1 => AC(9, 1) 46 = 5*9+1 => AC(10, 1)