Satisfiability Modulo Theories

Applications

• Static Checking
• Test-Case Generation
• Bounded Model Checking
• Symbolic Simulation
• Planning and Scheduling
• Program Synthesis

Characteristics

• SAT Generalization
• First-Order Logic
• Decidable/Undecidable

SAT aka Boolean Satisfiability

SAT

• Decidable
• NP-hard
• Hundreds of variables

SMT

• Multisorted first-order logic
• Linear arithmetic
• Non-linear arithmetic
• Arrays
• Bit-vectors
• Quantifiers
• ...

SMT

(declare-const x Int)
(declare-const y Int)
(assert (= (+ x y) 10))
(assert (= (+ x (* 2 y)) 20))
(check-sat)
(get-model)

Logic Basics

Language

Interpretation

Proof

Propositional calculus

• A, B, C, ...
• not A, (A implies B)
• And that's all folks

Interpretation

• {0, 1}
• (A and B) = ([A] * [B])
• And so on

Truth tables

Proofs

• MP: (A=>B), A |- B
• Couple of axioms
• Soundness and Completeness
• Excluded middle (classical logic)

Predicate calculus

• The same plus
• Atoms (constants): a, b, c, ...
• Variables: x, y, z, ...
• (Uninterpreted) functions: f(x, b), ...
• Predicates: P(x, g(x, b), c), ...
• Quantifiers:  ∀ and ∃

Interpretaton

• Set theory
• Domain theory
• Category theory
• ...

Predicate calculus

• Still Sound and Complete
• But semi-decidable (Turing-complete)
• Proofs are much more complicated

SMT-LIB Basics

SMT-LIB

• International collaboration
• Formal documentation
• De-facto standard
• http://smtlib.cs.uiowa.edu

Elements of Syntax

• Lisp-like (s-expressions)
• Commands to the solver
• Sets options
• Sets theory
• Sets assertions

Example

(set-logic QF_LIA)
(declare-const x Int)
(declare-const y Int)
(assert (= (- x y) (+ x (- y) 1)))
(check-sat)
; unsat
(exit)

Example

(set-logic QF_LIA)
(declare-const x Int)
(declare-const y Int)
(assert (= (+ x (* 2 y)) 20))
(assert (= (- x y) 2))
(check-sat)
; sat
(get-model)
; ((define-fun x () Int 8)
;  (define-fun y () Int 6)
; )
(exit)

Example

; Modeling sequential code in SSA form
;; Buggy swap
; int x, y;
; int t = x;
; y = t;
; x = y;

(set-logic QF_UFLIA)
(declare-fun x (Int) Int)
(declare-fun y (Int) Int)
(declare-fun t (Int) Int)
(assert (= (t 0) (x 0)))
(assert (= (y 1) (t 0)))
(assert (= (x 1) (y 1)))

(assert (not 
  (and (= (x 1) (y 0)) 
       (= (y 1) (x 0)))))

(check-sat)
(get-model)
; possible returned model:
; (
;  (define-fun x ((_ufmt_1 Int)) Int (- 1))
;  (define-fun y ((_ufmt_1 Int)) Int (ite (= _ufmt_1 1) (- 1) 2))
;  (define-fun t ((_ufmt_1 Int)) Int (- 1))
; )
(exit)

Z3

Features

• C/C++ lib with bindings
• Optimized extended DPLL
• Is used for everything
• http://rise4fun.com/z3

Exercise

  CHOO
+ CHOO
 -----
 TRAIN

Stack ADT

TYPES
• INT_STACK

FUNCTIONS
• put: INT_STACK x INT -> INT_STACK
• remove: INT_STACK -/> INT_STACK
• item: INT_STACK -/> INT
• empty: INT_STACK -> BOOLEAN
• new: INT_STACK

AXIOMS
For any x:INT, s:INT_STACK
• A1 - item(put(s,x)) = x
• A2 - remove(put(s,x)) = s
• A3 - empty(new)
• A4 - not empty(put(s,x))

PRECONDITIONS
• remove(s:INT_STACK) require not empty(s)
• item(s:INT_STACK) require not empty(s)