The document describes two methods for computing the minimal unsatisfiable core of a conjunctive formula: 1. Dropping one atom at a time and checking satisfiability, requiring up to n queries where there are n atoms. 2. Using a binary search approach by partitioning atoms into two sets and recursively minimizing, requiring only log(n) queries. This approach avoids modifying the underlying theory solver.

1. Computing Minimal Unsat Core
How can we compute minimal unsat core of conjunctive T
formula without modifying theory solver?
Let φ be original unsatisfiable conjunct
Drop one atom from φ, call this φ
If φ is still unsat, φ := φ
Repeat this for every atom in φ
Clearly, resulting φ is minimal unsat core of original formula
I¸ıl Dillig,
s
CS780: Automated Logical Reasoning
Lecture 17: SMT Solvers and the DPPL(T ) Framework
24/44

2. Example
Let’s compute minimal unsat core of
φ : x = y ∧ f (x ) + z = 5 ∧ f (x ) = f (y) ∧ y ≤ 3
Drop x = y from φ. Is result unsat? no, so keep x = y
Drop f (x ) + z = 5. Is result unsat? yes, so drop f (x ) + z = 5
New formula: φ : x = y ∧ f (x ) = f (y) ∧ y ≤ 3
Drop f (x ) = f (y). Is result unsat? no, keep f (x ) = f (y)
Finally, drop y ≤ 3. Is result unsat? yes, drop y ≤ 3
What is minimal unsat core? x = y ∧ f (x ) = f (y)
I¸ıl Dillig,
s
CS780: Automated Logical Reasoning
Lecture 17: SMT Solvers and the DPPL(T ) Framework
25/44

3. How Many Sat Queries?
To compute minimal unsat core this way, how many
satisfiability queries do we need?
We make one sat query per literal, so if there are n literals, we
make n sat queries
However, we can do much better!
Idea: Instead of dropping one literal at a time, drop half the
literals, i.e., do binary search
I¸ıl Dillig,
s
CS780: Automated Logical Reasoning
Lecture 17: SMT Solvers and the DPPL(T ) Framework
26/44

4. Better Way to Find Minimal Unsat Core
Given conjunction of atoms a1 , . . . , an , partition them into
two sets of equal cardinality A and B
If conjunction of literals in A is UNSAT, recursively minimize
A and return result
Otherwise, if conjunction of literals in B is UNSAT,
recursively minimize B and return result
Interesting case: What if A and B are both sat on their own?
Idea: Minimize A assuming unsat core includes B
I¸ıl Dillig,
s
CS780: Automated Logical Reasoning
Lecture 17: SMT Solvers and the DPPL(T ) Framework
27/44

5. Better Way to Find Minimal Unsat Core, cont.
How do we minimize A assuming unsat core includes B ?
Every time we issue sat query for a subset of A, also conjoin
B because we assume B is part of unsat core
Let A∗ be the result of minimizing A under this assumption
Now, we know A∗ ∧ B is UNSAT, but not yet minimal. Why?
Because not everything in B is necessary to derive ⊥
Now, minimize B assuming A∗ is in unsat core
Let B ∗ be result of minimizing B
Claim: A∗ ∧ B ∗ is a minimal unsat core
I¸ıl Dillig,
s
CS780: Automated Logical Reasoning
Lecture 17: SMT Solvers and the DPPL(T ) Framework
28/44

6. Example
Let’s find minimal unsat core using binary search strategy:
φ : x = y ∧ f (x ) + z = 5 ∧ f (x ) = f (y) ∧ y ≤ 3
Partition into two sets:
A : x = y ∧ f (x ) + z = 5 B : f (x ) = f (y) ∧ y ≤ 3
Are A and B are satisfiable on their own? yes
Let’s minimize A assuming B is in unsat core
Partition A into sets A1 and A2 :
A1 : x = y
I¸ıl Dillig,
s
CS780: Automated Logical Reasoning
A2 : f (x ) + z = 5
Lecture 17: SMT Solvers and the DPPL(T ) Framework
29/44

7. Example, cont
A1 : x = y
A2 : f (x ) + z = 5 B : f (x ) = f (y) ∧ y ≤ 3
Is A1 unsat assuming B is in unsat core? Yes, A1 ∧ B unsat
Now, minimize A1 further assuming B is in unsat core
⇒ cannot be minimized further; A∗ : x = y
Next, minimize B assuming A∗ is in unsat core
Partition B into two sets:
B1 : f (x ) = f (y) B2 : y ≤ 3 A∗ : x = y
Is B1 unsat assuming A∗ is in unsat core? Yes
B1 cannot be minimized further, thus B ∗ : f (x ) = f (y)
Thus, minimal unsat core is A∗ ∧ B ∗ : x = y ∧ f (x ) = f (y)
I¸ıl Dillig,
s
CS780: Automated Logical Reasoning
Lecture 17: SMT Solvers and the DPPL(T ) Framework
30/44

8. Minimal Unsat Core Summary
Using binary search strategy, can find minimal unsat core with
log(n) sat queries to theory solver
Alternative approach: Rather than using theory solver as
“blackbox” augment it to provide minimal unsat core
,
Advantages of blackbox strategy:
1. Oblivious to what theory we use; works for any theory
2. Doesn’t require modifying decision procedure for theory
However, other strategy is potentially much more efficient
⇒ can take theory-specific knowledge into account
I¸ıl Dillig,
s
CS780: Automated Logical Reasoning
Lecture 17: SMT Solvers and the DPPL(T ) Framework
31/44