HW #1. Due Mar 22 Midnight 1

1. Verify the following program using SAT solver 1. Translate the program into a SSA form 2. Create a Boolean formula from the SSA representation 3. Completely translate arithmetic and equality into propositional formula 4. Translate the formula A into CNF form 5. Run Minisat on A_CNF and capture the result (see the class homepage to find relevant information) and interpret the result Intro. to Logic CS402 2 /* Assume that x and y are 2 bit unsigned integers */ /* Also assume that x+y <= 3 */ void f(unsigned int y) { unsigned int x=1; x=x+y; if (x==2) x+=1; else x=2; assert(x ==2); }

2. 2. A Latin square is an n × n table filled with n different symbols in such a way that each symbol occurs exactly once in each row and exactly once in each column. Here is an example for the 3x3 table. - - Encode a 3x3 Latin square problem in a SAT formula using the following variable names and explain your encoding scheme - - Make a Latin square solver like Sudoku solver using minisat2 - - Translate your SAT encoding into DIMACS CNF formula - - For example, if TA adds the 3 clauses “x111 /\ x212 /\ x313” (i.e., (1, 2, 3) as the first row) into your formula, minisat2 should make a model - - “x122/\ x223 /\ x321/\ x133/\ x231 /\x332” - - Or “x132/\ x233 /\ x331/\ x123/\ x221 /\x322” Intro. to Logic CS402 3 x111 x211 x311 x121 x221 x321 x131 x231 x331 x112 x212 x312 x113 x213 x313 x322 x332

Intro. to Logic CS Show validity and satisfiability of the following propositional formulas using both - Truth table - Semantic tableau

4. Show that semantic tableau method is sound and complete. I.e. prove that A is satisfiable if and only if a semantic tableau tree of A is open. In other words, prove Theorem 2.49 (read pg carefully) Intro. to Logic CS402 5