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lecture1 - Satisfiability Modulo Theories Summer School on...

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Unformatted text preview: Satisfiability Modulo Theories Summer School on Formal Methods Menlo College, 2011 Bruno Dutertre and Leonardo de Moura bruno@csl.sri.com, leonardo@microsoft.com SRI International, Microsoft Research SAT/SMT p.1/50 Whats Satisfiability Modulo Theory Satisfiability is the problem of determining whether a formula has a model If is propositional , a model is a truth assignemt to Boolean variables If is a first-order formula , a model assigns values to variables and interpretations to the function and predicate symbols SAT Solvers: check satisfiability of propositional formulas SMT Solvers: check satisfiability of formulas in a decidable first-order theory (e.g., linear arithmetic, uninterpreted functions, array theory, bitvectors) SAT/SMT p.2/50 Example b + 2 = c f ( read ( write ( a, b, 3) , c 2)) negationslash = f ( c b + 1) SAT/SMT p.3/50 Example b + 2 = c f ( read ( write ( a, b , 3 ) , c 2 )) negationslash = f ( c b + 1 ) Arithmetic SAT/SMT p.3/50 Example b + 2 = c f ( read ( write ( a , b, 3) , c 2)) negationslash = f ( c b + 1) Array theory SAT/SMT p.3/50 Example b + 2 = c f ( read ( write ( a, b, 3) , c 2)) negationslash = f ( c b + 1) Uninterpreted function SAT/SMT p.3/50 Example b + 2 = c f ( read ( write ( a, b, 3) , c 2)) negationslash = f ( c b + 1) SAT/SMT p.3/50 Example b + 2 = c f ( read ( write ( a, b, 3) , c 2)) negationslash = f ( c b + 1) By arithmetic , this is equivalent to b + 2 = c f ( read ( write ( a, b, 3) , b )) negationslash = f (3) SAT/SMT p.3/50 Example b + 2 = c f ( read ( write ( a, b, 3) , c 2)) negationslash = f ( c b + 1) By arithmetic , this is equivalent to b + 2 = c f ( read ( write ( a, b, 3) , b )) negationslash = f (3) then, by the array theory axiom : read ( write ( v, i, x ) , i ) = x b + 2 = c f (3) negationslash = f (3) SAT/SMT p.3/50 Example b + 2 = c f ( read ( write ( a, b, 3) , c 2)) negationslash = f ( c b + 1) By arithmetic , this is equivalent to b + 2 = c f ( read ( write ( a, b, 3) , b )) negationslash = f (3) then, by the array theory axiom : read ( write ( v, i, x ) , i ) = x b + 2 = c f (3) negationslash = f (3) then, the formula is unsatisfiable SAT/SMT p.3/50 Example 2 x f ( x ) y f ( y ) x negationslash = y SAT/SMT p.4/50 Example 2 x f ( x ) y f ( y ) x negationslash = y This formula is satisfiable SAT/SMT p.4/50 Example 2 x f ( x ) y f ( y ) x negationslash = y This formula is satisfiable : Example model: x 1 y 2 f (1) f (2) 1 f ( . . . ) SAT/SMT p.4/50 SMT Solving Input a first-order formula Output the status of : satisfiable or unsatisfiable optionally, if is satisfiable, a model of also optionally, if is unsatisfiable, a proof of unsatisfiability Main issues Formula size (e.g., thousands of atoms or more) Formulas with complex Boolean structure Combinations of theories SAT/SMT p.5/50 Overview of SMT Solving...
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This note was uploaded on 02/07/2012 for the course CS 4322 taught by Professor Martinrinard during the Spring '11 term at MIT.

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lecture1 - Satisfiability Modulo Theories Summer School on...

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