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An Introduction to Z3 Huixing Fang National Trusted Embedded Software Engineering Technology Research Center April 12, 2017 . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . .. . . . . . Outline SMT 1 Z3 2 . .

  1. An Introduction to Z3 Huixing Fang National Trusted Embedded Software Engineering Technology Research Center April 12, 2017 . . . . . .. . . . . . . .. . . . . . . .. . . . . . . .. . . .. . . . . .

  2. Outline SMT 1 Z3 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Huixing Fang (ECNU) An Introduction to Z3 April 12, 2017 2 / 24

  3. Outline SMT 1 Z3 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Huixing Fang (ECNU) An Introduction to Z3 April 12, 2017 3 / 24

  4. SMT: Satisfiability Modulo Theory 1 Satisfiability is the problem of determining whether a formula φ has a model If φ is propositional, a model is a truth assignemt to Boolean variables 1 If φ is a first-order formula, a model assigns values to variables and 2 interpretations to the function and predicate symbols 2 SAT Solvers: check satisfiability of propositional formulas 3 SMT Solvers: check satisfiability of formulas in a decidable first-order theory (e.g., linear arithmetic, array theory, bitvectors) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Huixing Fang (ECNU) An Introduction to Z3 April 12, 2017 4 / 24

  5. Outline SMT 1 Z3 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Huixing Fang (ECNU) An Introduction to Z3 April 12, 2017 5 / 24

  6. Z3 Context . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Figure: Context View Huixing Fang (ECNU) An Introduction to Z3 April 12, 2017 6 / 24

  7. Z3 Architecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Huixing Fang (ECNU) An Introduction to Z3 April 12, 2017 7 / 24

  8. Z3 Architecture 1 Simplifier : Input formulas are first processed using an incomplete, but efficient simplification p ∧ true �− → p x = 4 ∧ q ( x ) �− → x = 4 ∧ q ( 4 ) 2 Compiler: The simplified abstract syntax tree representation of the formula is converted into a different data-structure comprising of a set of clauses and congruence-closure nodes 3 Congruence Closure Core: Handles equalities and uninterpreted functions. Receives the assignment from the SAT solver, and processed using E-matching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Huixing Fang (ECNU) An Introduction to Z3 April 12, 2017 8 / 24

  9. Basic Commands 1 displays a message: (echo "starting Z3...") . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Huixing Fang (ECNU) An Introduction to Z3 April 12, 2017 9 / 24

  10. Basic Commands 1 displays a message: (echo "starting Z3...") 2 declares a constant of a given type: (declare-const a Int) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Huixing Fang (ECNU) An Introduction to Z3 April 12, 2017 9 / 24

  11. Basic Commands 1 displays a message: (echo "starting Z3...") 2 declares a constant of a given type: (declare-const a Int) 3 declares a function : (declare-fun f (Int Bool) Int) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Huixing Fang (ECNU) An Introduction to Z3 April 12, 2017 9 / 24

  12. Basic Commands 1 displays a message: (echo "starting Z3...") 2 declares a constant of a given type: (declare-const a Int) 3 declares a function : (declare-fun f (Int Bool) Int) 4 asserts a formula : (assert (> a 10)) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Huixing Fang (ECNU) An Introduction to Z3 April 12, 2017 9 / 24

  13. Basic Commands 1 displays a message: (echo "starting Z3...") 2 declares a constant of a given type: (declare-const a Int) 3 declares a function : (declare-fun f (Int Bool) Int) 4 asserts a formula : (assert (> a 10)) 5 defines a function: (define-fun a () Int 100) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Huixing Fang (ECNU) An Introduction to Z3 April 12, 2017 9 / 24

  14. Basic Commands 1 displays a message: (echo "starting Z3...") 2 declares a constant of a given type: (declare-const a Int) 3 declares a function : (declare-fun f (Int Bool) Int) 4 asserts a formula : (assert (> a 10)) 5 defines a function: (define-fun a () Int 100) 6 defines a function: (define-fun f ((x Int) (y Bool)) Int (ite (and (= x 11) (= y true)) 0 1)) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Huixing Fang (ECNU) An Introduction to Z3 April 12, 2017 9 / 24

  15. Basic Commands 1 displays a message: (echo "starting Z3...") 2 declares a constant of a given type: (declare-const a Int) 3 declares a function : (declare-fun f (Int Bool) Int) 4 asserts a formula : (assert (> a 10)) 5 defines a function: (define-fun a () Int 100) 6 defines a function: (define-fun f ((x Int) (y Bool)) Int (ite (and (= x 11) (= y true)) 0 1)) 7 check: (check-sat) determines whether the current formulas on the Z3 stack are satisfiable or not . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Huixing Fang (ECNU) An Introduction to Z3 April 12, 2017 9 / 24

  16. Basic Commands 1 displays a message: (echo "starting Z3...") 2 declares a constant of a given type: (declare-const a Int) 3 declares a function : (declare-fun f (Int Bool) Int) 4 asserts a formula : (assert (> a 10)) 5 defines a function: (define-fun a () Int 100) 6 defines a function: (define-fun f ((x Int) (y Bool)) Int (ite (and (= x 11) (= y true)) 0 1)) 7 check: (check-sat) determines whether the current formulas on the Z3 stack are satisfiable or not 8 model: (get-model) is used to retrieve an interpretation that makes all formulas on the Z3 internal stack true . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Huixing Fang (ECNU) An Introduction to Z3 April 12, 2017 9 / 24

  17. Basic Commands Basic Building Blocks 1 The basic building blocks of SMT formulas are constants and functions. Constants are just functions that take no arguments. So everything is really just a function. 2 An uninterpreted function or function symbol is one that has no other property than its name and n-ary form. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Huixing Fang (ECNU) An Introduction to Z3 April 12, 2017 10 / 24

  18. Basic Commands Basic Building Blocks 1 The basic building blocks of SMT formulas are constants and functions. Constants are just functions that take no arguments. So everything is really just a function. 2 An uninterpreted function or function symbol is one that has no other property than its name and n-ary form. Valid and Satisfiable 1 A formula F is valid if F always evaluates to true for any assignment of appropriate values to its uninterpreted function and constant symbols. 2 A formula F is satisfiable if there is some assignment of appropriate values to its uninterpreted function and constant symbols under which F evaluates to true. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Huixing Fang (ECNU) An Introduction to Z3 April 12, 2017 10 / 24

  19. Propositional Logic The pre-defined sort Bool is the sort (type) of all Boolean propositional expressions. Z3 supports the usual Boolean operators and, or, xor, not, => (implication), ite (if-then-else). Example Formulas: ( declare − const p Bool ) ( declare − const q Bool ) ( declare − const r Bool ) ( define − fun c o n j e c t u r e () Bool (=> ( and (=> p q) (=> q r )) (=> p r ) ) ) ( a s s e r t ( not c o n j e c t u r e )) ( check − sat ) Result: unsat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Huixing Fang (ECNU) An Introduction to Z3 April 12, 2017 11 / 24

  20. Uninterpreted functions and constants The basic building blocks of SMT formulas are constants and functions. Constants are just functions that take no arguments. So everything is really just a function. Example Formulas: ( declare − fun f ( I n t ) I n t ) ( declare − fun a () I n t ) ; a i s a constant ( declare − const b I n t ) ; syntax sugar f o r ( declare − fun b () I n t ) ( a s s e r t (> a 20)) ( a s s e r t (> b a )) ( a s s e r t (= ( f 10) 1)) ( check − sat ) ( get − model ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Huixing Fang (ECNU) An Introduction to Z3 April 12, 2017 12 / 24

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