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]
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, then f(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, / }
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. 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].
x c y
Slide adopted from [Barret09]

14. Difference Logic
x−y = 5 ∧ z −y ≥ 2 ∧ z −x > 2 ∧ w −x = 2 ∧ z −w < 0
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
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. 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 Φ IS
UNSAT
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 Open Source
Z3, Yices, Barcelogic, MathSAT
Open Source
Open-SMT, CVC3, Boolector
Some SMT-LIB Compatibility Solvers (Even partially)
CVC3, Open-SMT, MathSAT5, Sonolar

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

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