1. Decision Procedures in First Order Logic
Decision Procedures for Equality Logic
Daniel Kroening and Ofer Strichman

2. Decision Procedures An algorithmic point of view
Outline
Introduction
Definition, complexity
Reducing Uninterpreted Functions to Equality Logic
Using Uninterpreted Functions in proofs
Simplifications
Introduction to the decision procedures
The framework: assumptions and Normal Forms
General terms and notions
Solving a conjunction of equalities     
Decision Procedures An algorithmic point of view

3. Basic assumptions and notations
Input formulas are in NNF
Input formulas are checked for satisfiability
Formula with Uninterpreted Functions: UF
Equality formula: E
Decision Procedures An algorithmic point of view

4. First: conjunction of equalities
Input: A conjunction of equalities and disequalities
Define an equivalence class for each variable. For each equality x = y unite the equivalence classes of x and y. Repeat until convergence.
For each disequality u  v if u is in the same equivalence class as v return 'UNSAT'.
Return 'SAT'.
student’s project: implement it with set-union. See page 448 in the algorithms book (CLR)
Decision Procedures An algorithmic point of view

5. Decision Procedures An algorithmic point of view
Example
x1 = x2 Æ x2 = x3 Æ x4=x5 Æ x5  x1
x1,x2,x3
x4,x5
This is SAT – the only disequality is between different classes
Equivalence class
Equivalence class
Is there a disequality between members of the same class ?
Decision Procedures An algorithmic point of view

6. Next: add Uninterpreted Functions
x1 = x2 Æ x2 = x3 Æ x4=x5 Æ x5  x1 Æ F(x1) F(x2)
F(x1)
x4,x5
x1,x2,x3
Equivalence class
Yes!
F(x2)
Equivalence class
Equivalence class
Equivalence class
Decision Procedures An algorithmic point of view

7. Next: Compute the Congruence Closure
x1 = x2 Æ x2 = x3 Æ x4=x5 Æ x5  x1 Æ F(x1) F(x2)
x1,x2,x3
F(x1),F(x2)
x4,x5
Yes!
Equivalence class
Equivalence class
Now - is there a disequality between members of the same class ?
This is called the Congruence Closure
Decision Procedures An algorithmic point of view

8. And now: consider a Boolean structure
x1 = x2 Ç (x2 = x3 Æ x4=x5 Æ x5  x1 Æ F(x1)  F(x2))
x1,x2
x2,x3
x4,x5
F(x1)
F(x2)
Equivalence class
Equivalence classes
Yes!
case 1
case 2
Syntactic case splitting: this is what we want to avoid!
Decision Procedures An algorithmic point of view

9. Deciding Equality Logic with UFs
Input: Equality Logic formula UF
Convert UF to DNF
For each clause:
Define an equivalence class for each variable and each function instance.
For each equality x = y unite the equivalence classes of x and y. For each function symbol F, unite the classes of F(x) and F(y). Repeat until convergence.
If all disequalities are between terms from different equivalence classes, return 'SAT'.
Return 'UNSAT'.
Note that we ignore constants here for the time being.
if there are constants then the presence of two constants in the same class makes it unsat
Decision Procedures An algorithmic point of view