1 / 30

Satisfiability Modulo Theories

Satisfiability Modulo Theories. Sinan Hanay. Boolean Satisfiability (SAT). Is there an assignment to the p 1 , p 2 , …, p n variables such that  evaluates to 1?. Slide taken from [Barret09]. Satisfiability Modulo Theories (SMT).

helki
Download Presentation

Satisfiability Modulo Theories

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  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]

More Related