Combining Decision Procedures: The Nelson-Oppen approach

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1 Combining Decision Procedures: The Nelson-Oppen approach Albert Oliveras and Enric Rodríguez-Carbonell Deduction and Verification Techniques Session 4 Fall 2009, Barcelona Combining Decision Procedures:The Nelson-Oppen approach p. 1

2 Need for combination In software verification, formulas like the following one arise: a=b+2 A=write(B,a+1,4) (read(a,b+3)=2 f(a 1) = f(b+1)) Here reasoning is needed over The theory of lineararithmetic(t LA ) The theory of arrays (T A ) The theory of uninterpreted functions (T UF ) Remember that T-solvers only deal with conjunctions of lits. Given T-solvers for the three individual theories, canwecombine them to obtainone for (T LA T A T UF )? Under certain conditions the Nelson-Oppen combination method gives a positive answer Combining Decision Procedures:The Nelson-Oppen approach p. 2

3 Motivating example - Convex case Consider the following set of literals: f(f(x) f(y)) = a f(0) = a +2 x = y There are twotheories involved: T LA(R) and T UF FIRST STEP:purify eachliteralsothatitbelongs toasingletheory f(f(x) f(y)) = a = f(e 1 ) = a = f(e 1 ) = a e 1 = f(x) f(y) e 1 = e 2 e 3 e 2 = f(x) e 3 = f(y) Combining Decision Procedures:The Nelson-Oppen approach p. 3

4 Motivating example - Convex case Consider the following set of literals: f(f(x) f(y)) = a f(0) = a +2 x = y There are twotheories involved: T LA(R) and T UF FIRST STEP:purify eachliteralsothatitbelongs toasingletheory f(0) = a +2 = f(e 4 ) = a +2 = f(e 4 ) = e 5 e 4 = 0 e 4 = 0 e 5 = a +2 Combining Decision Procedures:The Nelson-Oppen approach p. 3

5 Motivating example - Convex case (2) EUF Arithmetic f(e 1 ) = a e 2 e 3 = e 1 f(x) = e 2 e 4 = 0 f(y) = e 3 e 5 = a +2 f(e 4 ) = e 5 x = y The two solversonlyshare constants: e 1,e 2,e 3,e 4,e 5,a Tomergethe twomodelsintoasingleone, the solvershaveto agree on equalities between shared constants (interface equalities) This can be done by exchanging entailed interface equalities Combining Decision Procedures:The Nelson-Oppen approach p. 4

6 Motivating example - Convex case (2) EUF Arithmetic f(e 1 ) = a e 2 e 3 = e 1 f(x) = e 2 e 4 = 0 f(y) = e 3 e 5 = a +2 f(e 4 ) = e 5 e 2 = e 3 x = y The two solversonlyshare constants: e 1,e 2,e 3,e 4,e 5,a EUF-Solver says SAT Ari-Solver says SAT EUF = e 2 =e 3 Combining Decision Procedures:The Nelson-Oppen approach p. 5

7 Motivating example - Convex case (2) EUF Arithmetic f(e 1 ) = a e 2 e 3 = e 1 f(x) = e 2 e 4 = 0 f(y) = e 3 e 5 = a +2 f(e 4 ) = e 5 e 2 = e 3 x = y e 1 = e 4 The two solversonlyshare constants: e 1,e 2,e 3,e 4,e 5,a EUF-Solver says SAT Ari-Solver says SAT Ari = e 1 =e 4 Combining Decision Procedures:The Nelson-Oppen approach p. 6

8 Motivating example - Convex case (2) EUF Arithmetic f(e 1 ) = a e 2 e 3 = e 1 f(x) = e 2 e 4 = 0 f(y) = e 3 e 5 = a +2 f(e 4 ) = e 5 e 2 = e 3 x = y a = e 5 e 1 = e 4 The two solversonlyshare constants: e 1,e 2,e 3,e 4,e 5,a EUF-Solver says SAT Ari-Solver says SAT EUF = a=e 5 Combining Decision Procedures:The Nelson-Oppen approach p. 7

9 Motivating example - Convex case (2) EUF Arithmetic f(e 1 ) = a e 2 e 3 = e 1 f(x) = e 2 e 4 = 0 f(y) = e 3 e 5 = a +2 f(e 4 ) = e 5 e 2 = e 3 x = y a = e 5 e 1 = e 4 The two solversonlyshare constants: e 1,e 2,e 3,e 4,e 5,a EUF-Solver says SAT Ari-Solver says UNSAT Hence the original setof litswasunsat Combining Decision Procedures:The Nelson-Oppen approach p. 8

10 Nelson-Oppen The convex case A theory T is stably-infinite iff every T-satisfiable quantifier-free formula has an infinite model A theory T isconvex iff S = T a 1 =b 1... a n =b n = S = a i =b i for some i Deterministic Nelson-Oppen: Giventwostably-infiniteandconvex theories T 1 and T 2 Givenasetof literals S over the signatureof T 1 T 2 The (T 1 T 2 )-satisfiabilityof S canbe checked withthe algorithm Combining Decision Procedures:The Nelson-Oppen approach p. 9

11 Nelson-Oppen The convex case (2) Deterministic Nelson-Oppen 1. Purify S andsplititinto S 1 S 2. Let E the setof interface equalitiesbetween S 1 and S 2 2. If S 1 is T 1 -unsatisfiablethen UNSAT 3. If S 2 is T 2 -unsatisfiablethen UNSAT 4. If S 1 = T1 x=y with x=y E \S 2 then S 2 := S 2 {x=y} and goto 3 5. If S 2 = T2 x=y with x=y E \S 1 then S 1 := S 1 {x=y} and goto 2 6. Report SAT Combining Decision Procedures:The Nelson-Oppen approach p. 10

12 Motivating example Non-convex case Consider the following UNSATISFIABLE set of literals: 1 x 2 f(1) = a f(x) = b a = b +2 f(2) = f(1) +3 There are twotheories involved: T LA(Z) and T UF FIRST STEP:purify eachliteralsothatitbelongs toasingletheory f(1) = a = f(e 1 ) = a e 1 = 1 Combining Decision Procedures:The Nelson-Oppen approach p. 11

13 Motivating example Non-convex case Consider the following UNSATISFIABLE set of literals: 1 x 2 f(1) = a f(x) = b a = b +2 f(2) = f(1) +3 There are twotheories involved: T LA(Z) and T UF FIRST STEP:purify eachliteralsothatitbelongs toasingletheory f(2) = f(1) +3 = e 2 = 2 f(e 2 ) = e 3 f(e 1 ) = e 4 e 3 = e 4 +3 Combining Decision Procedures:The Nelson-Oppen approach p. 11

14 Motivating example Non-convex case(2) Arithmetic EUF 1 x f(e 1 ) = a x 2 f(x) = b e 1 = 1 f(e 2 ) = e 3 a = b +2 f(e 1 ) = e 4 e 2 = 2 e 3 = e 4 +3 a = e 4 The two solversonlyshare constants: x,e 1,a,b,e 2,e 3,e 4 Ari-Solver says SAT EUF-Solver says SAT EUF = a=e 4 Combining Decision Procedures:The Nelson-Oppen approach p. 12

15 Motivating example Non-convex case(2) Arithmetic EUF 1 x f(e 1 ) = a x 2 f(x) = b e 1 = 1 f(e 2 ) = e 3 a = b +2 f(e 1 ) = e 4 e 2 = 2 e 3 = e 4 +3 a = e 4 The two solversonlyshare constants: x,e 1,a,b,e 2,e 3,e 4 Ari-Solver says SAT EUF-Solver says SAT No theory entails any other interface equality, but... Combining Decision Procedures:The Nelson-Oppen approach p. 13

16 Motivating example Non-convex case(2) Arithmetic EUF 1 x f(e 1 ) = a x 2 f(x) = b e 1 = 1 f(e 2 ) = e 3 a = b +2 f(e 1 ) = e 4 e 2 = 2 e 3 = e 4 +3 a = e 4 The two solversonlyshare constants: x,e 1,a,b,e 2,e 3,e 4 Ari-Solver says SAT EUF-Solver says SAT Ari = T x = e 1 x = e 2. Let sconsiderboth cases. Combining Decision Procedures:The Nelson-Oppen approach p. 14

17 Motivating example Non-convex case(2) Ari-Solver says SAT Arithmetic EUF 1 x f(e 1 ) = a x 2 f(x) = b e 1 = 1 f(e 2 ) = e 3 a = b +2 f(e 1 ) = e 4 e 2 = 2 x = e 1 e 3 = e 4 +3 a = e 4 x = e 1 EUF-Solver says SAT EUF = T a=b, that whensentto Ari makesitunsat Combining Decision Procedures:The Nelson-Oppen approach p. 15

18 Motivating example Non-convex case(2) Let s trynow with x=e 2 Arithmetic EUF 1 x f(e 1 ) = a x 2 f(x) = b e 1 = 1 f(e 2 ) = e 3 a = b +2 f(e 1 ) = e 4 e 2 = 2 e 3 = e 4 +3 a = e 4 Combining Decision Procedures:The Nelson-Oppen approach p. 16

19 Motivating example Non-convex case(2) Ari-Solver says SAT Arithmetic EUF 1 x f(e 1 ) = a x 2 f(x) = b e 1 = 1 f(e 2 ) = e 3 a = b +2 f(e 1 ) = e 4 e 2 = 2 x = e 2 e 3 = e 4 +3 a = e 4 x = e 2 EUF-Solver says SAT EUF = T b=e 3,that whensentto Ari makesitunsat Combining Decision Procedures:The Nelson-Oppen approach p. 17

20 Motivating example Non-convex case(2) Arithmetic EUF 1 x f(e 1 ) = a x 2 f(x) = b e 1 = 1 f(e 2 ) = e 3 a = b +2 f(e 1 ) = e 4 e 2 = 2 x = e 2 e 3 = e 4 +3 a = e 4 x = e 2 Since both x=e 1 and x = e 2 are UNSAT, the setof literalsisunsat Combining Decision Procedures:The Nelson-Oppen approach p. 18

21 Nelson-Oppen - The non-convex case In the previous example Deterministic NO does not work Thiswasbecause T LA(Z) isnot convex: S LA(Z) = TLA(Z) x=e 1 x=e 2, but S LA(Z) = TLA(Z) x=e 1 and S LA(Z) = TLA(Z) x=e 2 However, there is a version of NO for non-convex theories Givenasetconstants C, anarrangement A over C is: A set of equalities and disequalites between constants in C For each x,y C either x=y A or x = y A Combining Decision Procedures:The Nelson-Oppen approach p. 19

22 Nelson-Oppen The non-convex case (2) Non-deterministic Nelson-Oppen: Giventwostably-infinitetheories T 1 and T 2 Givenasetof literals S over the signature T 1 T 2 The (T 1 T 2 )-satisfiabilityof S canbe checked via: 1. Purify S andsplititinto S 1 S 2 Let C be the setofsharedconstants 2. For everyarrangement A over C do If (S 1 A) is T 1 -satisfiableand (S 2 A) is T 2 -satisfiable report SAT 3. Report UNSAT Combining Decision Procedures:The Nelson-Oppen approach p. 20

Theory Combination. Clark Barrett. New York University. CS357, Stanford University, Nov 2, p. 1/24

Theory Combination. Clark Barrett. New York University. CS357, Stanford University, Nov 2, p. 1/24 CS357, Stanford University, Nov 2, 2015. p. 1/24 Theory Combination Clark Barrett barrett@cs.nyu.edu New York University CS357, Stanford University, Nov 2, 2015. p. 2/24 Combining Theory Solvers Given