Satisfiability Modulo Theories

1. Satisfiability Modulo Theories Sinan Hanay

2. Boolean Satisfiability (SAT) Is there an assignment to the p1, p2, …, pn variables such that  evaluates to 1? Slide taken from [Barret09]

3. Satisfiability Modulo Theories (SMT) Is there an assignment to the x,y,z,w variables s.t.  evaluates to 1? Slide taken from [Barret09]

4. SAT vs SMT • SMT extends SAT solving by adding extensions • An SMT solver can solve a SAT problem, but not vice-versa. • SMT Applications • Analog Circuit Verification • RTL Verification • Software Model Checking

5. Overview • Introduction • SMT Theories • Example: Difference Logic • Combining Theories • SMT Solvers and SMT Libraries. • Conclusion

6. SMT Theories • Real or Integer Arithmetic • Equality and Uninterpreted Functions • Example: If x1 = x2, thenf(x1) = f(x2) else f(x1) ≠ f(x2) • Bitvectors and Arrays • Properties: • Decidable: An effective procedure exists to check if a formula is a member of a theory T. • Often Quantifier-free: Free from quantifiers such as (∃, ∀)

7. SMT Theories • Core Theory • Type: Boolean • Constants: {TRUE, FALSE} • Functions: {AND, OR, XOR} • Functions: Implication (=>) • Integer Theory (Ints) • Type: Int • All numerals are Int constants • Functions: { + , - , x, mod, div, abs}

8. SMT Theories • Reals Theory • Type: Real • Functions: { +, -, x, / } • Functions: { <, > } • Arrays with Extentionality Theory (ArraysEx) • Type: type of index and type of values • Functions: {select, store}

9. Overview • Introduction • SMT Theories • Case Study: Difference Logic Theory • SMT Solvers • SMT-LIB • Conclusion

10. SMT Example I– Difference Logic • Can solve problems such as: • Is there a solution {x,y} satisfying x-y < 20 and x -y > 4 • x,y can be integers or reals • If x,y are integers (QF_IDL: Integer Difference Logic) • If x,y are reals (QF_RDL : Real Difference Logic) • QF: Quantifier-free

11. SMT Theories– Difference Logic • In difference logic [NO05], we are interested in the satisfiability of a conjunction of arithmetic atoms. • Each atom is of the form x − y OP c, where x and y are variables, c is a numeric constant, and OP ∈ {=,<,≤,>,≥}. Examples: x-y > 10, y-x < 12 • The variables can range over either the integers(QF_IDL) or the reals(QF_RDL). Slide taken from [Barret09]

12. Difference Logic • The first step is to rewrite everything in terms of ≤: • x − y = c ⇒ x − y ≤ c∧x − y ≥ c • x − y ≥ c ⇒ y − x ≤ −c • x − y > c ⇒ y − x < −c • x − y < c ⇒ x − y ≤ c − 1 (integers) • x − y < c ⇒ x − y ≤ c − δ (reals) Slide adopted from [Barret09]

13. x cy Difference Logic • Now we have a conjunction of literals, all of the form x − y ≤ c. • From these literals, we form a weighted directed graph with a vertex for each variable. • For each literal x − y ≤ c, create an edge • The set of literals is satisfiable iff there is no cycle for which the sum of the weights on the edges is negative. • There are a number of efficient algorithms for detecting negative cycles in graphs [CG96]. Slide adopted from [Barret09]

14. x− y = 5 z − y ≥ 2 z − x > 2 w − x = 2 z − w < 0 x − y ≤ 5 ∧ y − x ≤ −5 y − z ≤ −2 x − z ≤ −3 w − x ≤ 2 ∧ x − w ≤ −2 z − w ≤ −1 Transform to a-b ≤ c Difference Logic x−y = 5 ∧ z −y ≥ 2 ∧ z −x > 2 ∧ w −x = 2 ∧ z −w < 0 Slide adopted from [Barret09]

15. Difference Logic Is there a negative cycle? Satisfiable if there is not any. Slide taken from [Barret09]

16. Combining Theories • QF_UFLIA • How to Combine Theory Solvers? 1 ≤ x ∧ x ≤ 2 ∧ f(x) ≠ f(1) ∧ f(x) ≠ f(2) Linear Integer Arithmetic (LIA) Uninterpreted Functions(UF)

17. Combining Theory Solvers • Theory solvers become much more useful if they can be used together. mux_sel = 0 → mux_out = select(regfile, addr) mux_sel = 1 → mux_out = ALU(alu0, alu1) • For such formulas, we are interested in satisfiability with respect to a combination of theories. • Fortunately, there exist methods for combining theory solvers. • The standard technique for this is the Nelson-Oppen method [NO79, TH96]. Slide taken from [Barret09]

18. The Nelson-Oppen Method • Suppose that T1 and T2 are theories and that Sat 1 is a theory solver for T1-satisfiability and Sat 2 for T2-satisfiability. • We wish to determine if φ is T1∪T2-satisfiable. • Convert φ to its separate form φ1 ∧ φ2. • Let S be the set of variables shared between φ1 and φ2. • For each arrangement D of S: • Run Sat 1 on φ1 ∪ D . • Run Sat 2 on φ2 ∪ D. Slide taken from [Barret09]

19. Combining Theories • QF_UFLIA φ =1 ≤ x ∧ x ≤ 2 ∧ f(x) ≠ f(1) ∧ f(x) ≠ f(2) • We first convert φ to a separate form: • φUF = f(x) ≠ f(y) ∧ f(x) ≠ f(z) • φLIA = 1 ≤ x ∧ x ≤ 2 ∧ y = 1 ∧ z = 2 Slide taken from [Barret09]

20. Φ IS UNSAT Combining Theories • φUF = f(x) ≠ f(y) ∧ f(x) ≠ f(z) • φLIA = 1 ≤ x ∧ x ≤ 2 ∧ y = 1 ∧ z = 2 • {x, y, z} can have 5 possible arrangements based on equivalence classes of x, y, and z • Assume All Variables Equal: • {x = y, x = z, y = z}inconsistent with φUF • Assume Two Variables Equal, One Different • {x = y, x ≠ z, y ≠ z}inconsistent with φUF • {x ≠ y, x = z, y ≠ z}inconsistent with φUF • {x ≠ y, x ≠ z, y = z}inconsistent with φLIA • Assume All Variables Different: • {x ≠ y, x ≠ z, y ≠ z}inconsistent with φLIA Slide adopted from [Barret09]

21. Overview • Introduction • SMT Theories • Case Study: Difference Logic Theory • SMT Solvers and Libraries • Summary

22. SMT-LIB • SMT Library • Provides standard rigorous descriptions of background theories • Common input and output languages for SMT solvers • Provides a library of benchmarks Ref: The SMT-LIB Standard

23. SMT Solvers • Proprietary • Z3, Yices, Barcelogic, MathSAT • Open Source • Open-SMT, CVC3, Boolector • Some SMT-LIB Compatibility Solvers (Even partially) • CVC3, Open-SMT, MathSAT5, Sonolar

24. UNINTERPRETED FUNCTIONS UNSATISFIABLE SMT-LIB Example • Check if (p AND p’) is satisfiable? Ref: SMT-LIB Tutorial by David R. Cok and GrammaTech Inc.

25. x=8, y= 6 SMT-LIB Example Is there a solution to x+2y = 20 and x-y = 2 LINEAR INTEGER ARITHMETIC SATISFIABLE

26. SUMMARY • SMT problems include a wider range of problems than SAT. • SMT-LIB initiative to bring standards to solvers. • SMT Applications Include: • Analog, Mixed-Signal Circuit Checker [Walter07] • Software Testing • RTL Verification • Nelson-Oppen Method for Combining Theory Solvers

27. Trivia • SMT Competition (SMT-COMP) • SMT Solvers Competition • Since 2005 • 2010 Winners: CVC3, OpenSMT, MathSAT 5, test_pmathsat, MiniSmt, simplifyingSTP. • First International SAT/SMT Solver Summer School 2011 • June 12- 17 at MIT. • Free for students.

28. References • [Barret09] Clark Barrett, Sanjit A. Seshia, ICCAD Tutorial 2009 • [NO79] Greg Nelson and Derek C. Oppen. Simplification by cooperating decision procedures. ACM Trans. on Programming Languages and Systems, 1(2):245–257, October 1979 • [Walter07] David Walter, Scott Little, Chris Meyers, “Bounded model checking of analog and mixed-signal circuits using an SMT solver”, Proceeding ATVA'07.

29. Questions • Thank you.

30. Equivalence Checking of Programs int fun1(int y) { int x, z; z = y; y = x; x = z; return x*x; } SMT formula  Satisfiable iff programs non-equivalent ( z = y ∧ y1 = x ∧ x1 = z ∧ ret1 = x1*x1) ∧ ( ret2 = y*y ) ∧ ( ret1  ret2 ) Using SAT to check equivalence (w/ Minisat) 32 bits for y: Did not finish in over 5 hours 16 bits for y: 37 sec. 8 bits for y: 0.5 sec. SMT: Using EUF solver: 0.01 sec What if we use SAT to check equivalence? int fun2(int y) { return y*y; } Slide adopted from [Barret09]