Formalizing Simplex within Isabelle/HOL

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1 Formalizing Simplex within Isabelle/HOL Mirko Spasić Filip Marić {mirko Department of Computer Science Faculty of Mathematics University of Belgrade Formal and Automated Theorem Proving and Applications, 2011

2 Outline 1 Background Information

3 Outline 1 Background Information 2

4 Outline 1 Background Information 2 3

5 Background Information Outline 1 Background Information 2 3

6 Background Information Linear Arithmetic 1 Linear arithmetic over reals (LRA) 2 Linear arithmetic over integers (LIA) Formula Quantifier-free linear arithmetic formula a 1 x a n x n b, a 1,..., a n, b Q, x 1,..., x n Q(N)

7 Background Information Decision Procedure Linear arithmetic is decidable Decision procedure, returning sat if and only if an input linear arithmetic formula is satisfiable, and returning unsat, otherwise Two Methods 1 Fourier-Motzkin procedure 2 Simplex method

8 Background Information Simplex Method Originally constructed to solve linear programming optimization problem Decision procedure for linear arithmetic does not have to maximize anything, but have to find a single feasible solution of input constraints One Variant of Simplex Method Dual Simplex algorithm

9 Background Information DPLL(T ) SMT solvers, in the DPLL(T ) framework, use Simplex-based linear arithmetic solver Procedure is capable of deciding the satisfiability of conjunctions of linear constraints

10 Outline 1 Background Information 2 3

11 Reliability SMT solvers are quite complex software Result of SMT solving is sat or unsat Main Question Can we be so sure that solver give us right answer? If the answer is sat, we simply can check if the calculated model is realy model for input formula, otherwise, there is no simple checking Real problems require a high level of reliability

12 Outline 1 Background Information 2 3

13 Isabelle Our Goal Prove correctness for Simplex method, within the Isabelle/HOL theorem proving system Isar (Intelligible semiautomated reasoning) language Primitive recursion Isabelle s built-in theory of Lists, theory of Sets, of Functions, Mappings, Rationals Effective executable code (without bugs) in functional programming languages can be generated

14 Outline 1 Background Information 2 3

15 linear_poly Theory of multivariate linear polynomials Vector space Associativity, commutativity, neutral element,... Effectively executable operations (addition, subtraction, multiplication by constant, valuation)

16 Outline 1 Background Information 2 3

17 Small Example Find solution for conjunctions of linear constraints: [x 1, y 1, x 2, y + 2x + z 4, x + z = 5]

18 Small Example Tableau s 1 = y + 2x + z s 2 = x + z Constraints [x 1, y 1, x 2, s 1 4, s 2 = 5] Bound and Valuation Variable LBound Valuation UBound x 0 y 0 z 0 s 1 0 s 2 0

19 Small Example Tableau s 1 = y + 2x + z s 2 = x + z Constraints [x 1,y 1, x 2, s 1 4, s 2 = 5] Bound and Valuation Variable LBound Valuation UBound x 1 0 y 0 z 0 s 1 0 s 2 0

20 Small Example Tableau s 1 = y + 2x + z s 2 = x + z Constraints [x 1,y 1, x 2, s 1 4, s 2 = 5] Bound and Valuation Variable LBound Valuation UBound x 1 1 y 0 z 0 s 1 2 s 2 1

21 Small Example Tableau s 1 = y + 2x + z s 2 = x + z Constraints [x 1, y 1,x 2, s 1 4, s 2 = 5] Bound and Valuation Variable LBound Valuation UBound x 1 1 y 0 1 z 0 s 1 2 s 2 1

22 Small Example Tableau s 1 = y + 2x + z s 2 = x + z Constraints [x 1, y 1,x 2, s 1 4, s 2 = 5] Bound and Valuation Variable LBound Valuation UBound x 1 1 y 1 1 z 0 s 1 1 s 2 1

23 Small Example Tableau s 1 = y + 2x + z s 2 = x + z Constraints [x 1, y 1, x 2,s 1 4, s 2 = 5] Bound and Valuation Variable LBound Valuation UBound x y 1 1 z 0 s 1 1 s 2 1

24 Small Example Tableau s 1 = y + 2x + z s 2 = x + z Constraints [x 1, y 1, x 2, s 1 4,s 2 = 5] Bound and Valuation Variable LBound Valuation UBound x y 1 1 z 0 s s 2 1

25 Small Example Tableau x = 0.5s 1 0.5y 0.5z s 2 = 0.5s 1 0.5y +0.5z Constraints [x 1, y 1, x 2, s 1 4,s 2 = 5] Bound and Valuation Variable LBound Valuation UBound x y 1 1 z 0 s s 2 1

26 Small Example Tableau x = 0.5s 1 0.5y 0.5z s 2 = 0.5s 1 0.5y +0.5z Constraints [x 1, y 1, x 2, s 1 4,s 2 = 5] Bound and Valuation Variable LBound Valuation UBound x y 1 1 z 0 s s 2 2.5

27 Small Example Tableau z = s 1 y 2x s 2 = s 1 y x Constraints [x 1, y 1, x 2, s 1 4,s 2 = 5] Bound and Valuation Variable LBound Valuation UBound x y 1 1 z 0 s s 2 2.5

28 Small Example Tableau z = s 1 y 2x s 2 = s 1 y x Constraints [x 1, y 1, x 2, s 1 4,s 2 = 5] Bound and Valuation Variable LBound Valuation UBound x y 1 1 z 1 s s 2 3

29 Small Example Tableau z = s 1 y 2x s 2 = s 1 y x Constraints [x 1, y 1, x 2, s 1 4, s 2 = 5] Bound and Valuation Variable LBound Valuation UBound x y 1 1 z 1 s s

30 Small Example Tableau z = s 2 x s 1 = s 2 + y + x Constraints [x 1, y 1, x 2, s 1 4, s 2 = 5] Bound and Valuation Variable LBound Valuation UBound x y 1 1 z 1 s s

31 Small Example Tableau z = s 2 x s 1 = s 2 + y + x Constraints [x 1, y 1, x 2, s 1 4, s 2 = 5] Bound and Valuation Variable LBound Valuation UBound x y 1 1 z 3 s s

32 Small Example Tableau z = s 2 x s 1 = s 2 + y + x Constraints [x 1, y 1, x 2, s 1 4, s 2 = 5] Bound and Valuation Variable LBound Valuation UBound x y 1 1 z 3 s s

33 Step of Input list of constraints = List of basic constraints l init1 (l) List of basic constraints = state * init1 (l) init2 (init1 (l)) state = state s check (assert_next (s)) *state (tableau constraints LBound Valuation UBound)

34 Outline 1 Background Information 2 3

35 Main Lemmas and Theorem Lemmas satisfiable(l) satisfiable(init1(l)) satisfiable(init1(l)) satisfiable(init2(init1(l))) satisfiable(s) satisfiable(check(assert_next(s))) Theorem satisfiable(l) simplex(l)

36 Doubtless Perhaps Finallize this proof Formalize whole SMT solver for linear arithmetic, using formalized SAT solver

37 The End Thanks for your attention!

SMT BASICS WS 2017/2018 ( ) LOGIC SATISFIABILITY MODULO THEORIES. Institute for Formal Models and Verification Johannes Kepler Universität Linz

SMT BASICS WS 2017/2018 ( ) LOGIC SATISFIABILITY MODULO THEORIES. Institute for Formal Models and Verification Johannes Kepler Universität Linz LOGIC SATISFIABILITY MODULO THEORIES SMT BASICS WS 2017/2018 (342.208) Armin Biere Martina Seidl biere@jku.at martina.seidl@jku.at Institute for Formal Models and Verification Johannes Kepler Universität