1. Minimizing Unsatisfiable Formulas
SAT/SMT seminar
07/05/2017
Minimizing Unsatisfiable Formulas
Alexander Ivrii
IBM

2. Disclaimers:
All the experts in the room are welcome to speak at any time!
This is NOT a comprehensive overview
This is probably highly subjective
Many slides are borrowed from other presentations

3. Motivation
Given an unsatisfiable CNF, understand why it is unsatisfiable

14. Many variants of the problem
Given an unsatisfiable CNF formula, we can look for:
A smaller unsatisfiable core
A minimal unsatisfiable core (MUC or MUS)
A minimum-sized core
A lexicographically/anti-lexicographically preferred minimal unsatisfiable core
Find MUS which contains/avoids clauses (Clauses are ordered, subsets of clauses are ordered lexicographically) …
And we can also look for more than one core
Enumerate all MUSes

15. Many useful generalizations
MUSes – minimal unsatisfiable subformulas
Goal: Find a subset-minimal set of clauses that is unsatisfiable
GMUS / HLMUC – group/high-level MUSes
Clauses are partitioned into disjoint groups (sets) of clauses
There is also a special “don-t care” group-0 consisting of “non-interesting” clauses
Goal: Find a subset-minimal set of interesting groups that (together with group-0) is unsatisfiable
VMUS – variable MUSes
Goal: Find a subset-minimal set of variables so that the set of all clauses containing these variables is unsatisfiable
LMUS – labeled MUSes
Each clause has an associated set of labels (possibly empty)
Goal: find a subset-minimal set of labels so that the set of all clauses with at least one label from this set is unsatisfiable
Similar to GMUSes, but the groups do no need to be disjoint
Simple & useful & drove a lot of research
Contains all others as special cases, allows preprocessing

16. Example

17. Example MUSes: {C1, C2, C3} {C1, C2, C4, C6}
GMUSes – assume G0 = {C1, C2}, G1 = {C3, C4}, G2 = {C5, C6}
{G1}
VMUSes:
{p, q}
LMUSes – assume L(C1) = {}, L(C2) = {}, L(C3) = {1}, L(C4) = {1}, L(C5) = {2}, L(C6) = {2}
{1}
LMUSes – assume L(C1) = {1}, L(C2) = {2}, L(C3) = {1,2}, L(C4) = {1,3}, L(C5) = {1,2}, L(C6) = {2,3}
{1,2}

18. Application: Abstraction Refinement
Given a model checking problem (Init, Tr, P) and a bound k, a bounded model checking problem (with bound k) checks if
Init(X0)  Tr(X0, X1)  …  Tr(Xk-1, Xk)  (P0  …  Pk)
is satisfiable.
When unsatisfiable, may want to construct a latch-level or gate-level abstraction
Can be naturally formulated as a GMUS (or a VMUS) problem
References:
Alexander Nadel: “Boosting minimal unsatisfiable core extraction”, FMCAD 2010

19. Application: Minimal Equivalent Subformulas
Problem Statement:
Given a satisfiable CNF formula F, find a subset-minimal set of clauses with the same set of satisfying assignments
Idea:
Given SF, S has the same set of assignments as F iff S  F is unsatisfiable
Reduction to GMUS:
Given F = {c1, …, cn}, define:
Don’t-care group G0 = CNF(F)
For each clause ciF, define a group Gi = {ci}
Claim: E is a MES of F if and only if { Gi | ciE } is a GMUS of G0  G1  …  Gn
References:
Anton Belov, Mikolás Janota, Inês Lynce, João Marques-Silva: “On Computing Minimal Equivalent Subformulas”, CP 2012

20. Application: Minimal Independent Support
An independent support of a Boolean formula F is a subset of variables whose values uniquely determine the values of the remaining variables in any satisfying assignment to the formula
Example: F = (c  a)  (c  b)  (c  a  b).
Then {a, b} is an independent support of F
Problem Statement:
Given a CNF formula F, find its minimal independent support
References:
Alexander Ivrii, Sharad Malik, Kuldeep Meel, and Moshe Vardi: “On computing Minimal Independent Support and its applications to sampling and counting”, Constraints 21(1) 2016

21. Application: Minimal Independent Support
Equivalently, S Vars(F) is an independent support of F if
F(x1, …, xn)  F(y1, …, yn)  xiS(xi = yi)  xiVars(F)(xi = yi)
Reduction to GMUS:
Given F(x1, …, xn), define:
Don’t-care group G0 = F(x1, …, xn)  F(y1, …, yn)  (xiVars(F) (xi  yi))
For each variable viVars(F), define a group Gi = {xi = yi}
Claim: S is a MIS of F if and only if { Gi | xiS } is a GMUS of G0  G1  …  Gn

22. Algorithms for computing MUSes
Basic Deletion-based algorithm
Basic Insertion-based algorithm
Many optimizations exist
Hybrid algorithm
Some features of both insertion-based and deletion-based algorithms
Follow-up research
Various algorithms based on the MUS – MCS duality
Most of the algorithms and optimizations extend to compute LMUSes

23. Backup slide: MUS-MCS duality
A correction subset S of an unsatisfiable CNF formula F is such that F \ S is satisfiable
A minimal correction subset (MCS) is a subset-minimal such subset
Easy to see that every MUS of F and every MCS of F must intersect
The MUS-MCS duality states that M is a MUS of F iff M is a minimal hitting set for AllMCSes(F)
Dually, M is an MCS of F iff M is a minimal hitting set for AllMUSes(F)
Mostly used to compute multiple MUSes

56. A note on MUSers and competitions
Three state-of-the-art approaches to extract a MUS:
Resolution-based using resolution graphs (HaifaMUC)
Assumption-based using selector variables (Muser2)
Assumption-based with assumptions organized in a form of a partial resolution graph (MinisatAbb)
The only competition in this area was associated with SAT’11
Features both a MUS track and a GMUS track
Problems were selected from easy unsatisfiable instances used in SAT competitions

57. Clause-Set Trimming Motivation
Reduce the formula before running the MUS extraction algorithm
Iteratively compute smaller unsatisfiable clause-sets
Run SAT-solver on F0 = F
The query should be unsatisfiable
Extract core F1
Run SAT-solver on F1
Extract core F2 …
Stop based on some criteria
Number of iterations, number/percentage of clauses removed, etc.

59. Model Rotation for MUS problem
Motivation
Cheaply produce more necessary clauses on each SAT outcome
Idea
Suppose that F is unsatisfiable formula
Further suppose that  is an assignment satisfying all clauses except for c (cF)
Then c is necessary (belongs to every MUS of F)
Flip values of certain variables  in the hope to produce satisfying assignments to F\cj for other clauses cj
Variants:
Model Rotation
Recursive Model Rotation
Extended Recursive Model Rotation
Can be extended to rotation for LMUS formulas

70. Redundancy Removal Motivation Make SAT queries easier Idea
Suppose that F is unsatisfiable formula
Checking whether F\c is satisfiable is equivalent to checking whether (F\c)(c) is satisfiable
As c consists of unit assumptions, the problem usually becomes simpler to solve
Extensions:
Path Strengthening (Alexander Nadel, Vadim Ryvchin, Ofer Strichman: “Efficient MUS extraction with resolution, FMCAD 2013)
Backbone Literals (Alexander Ivrii, Vadim Ryvchin, Ofer Strichman: “Mining Backbone Literals in Incremental SAT - A New Kind of Incremental Data”, SAT 2015)
Drawback: clause-set refinement (removing clauses after an UNSAT result) becomes tricky

71. Preprocessing Motivation
Use standard preprocessing techniques (BCP, subsumption, self-subsumption, variable elimination, blocked clause elimination) to make MUS extraction easier
Problem: Simply preprocessing the original formula does not work!
Example:
F = (x  p)  (x  p  q)  (p)  (x  q)  (x)
F’ = (x  p)  ( p  q)  (p)  (x  q)  (x)
Here (p  q) was obtained by self-subsumption of (x  p  q) with (x  p)
M’ = { (p  q), (p), (x  q), (x)} is a MUS of F’
As (p  q) is derived from (x  p  q) with (x  p), we are tempted to say that M = {(x  p), (x  p  q), (p), (x  q), (x)} is a MUS of F
But M is NOT a MUS of F!
The situation is even worse for GMUS computations!

73. Preprocessing and LCNF
All standard preprocessing techniques can be easily extended to work with labeled clauses
Example: self-subsumption reduces (x  p){1}  (x  p  q){2} to (x  p){1}  (p  q){1,2}
And become sound for LMUS extraction!

75. Cool new idea! In my experiments with Valeriy Balabanov:
VE on LCNF formula coming from GMUS problems had a highly beneficial effect
Further improved by allowing VE to eliminate a variable even if this increases the CNF size (cl-lim=200, grow=40)
Reason: aggressive-VE significantly helps rotation and does not hurt solving
But no improvement on formulas coming from MUS problems
Reason: aggressive-VE significantly helps rotations but hurts solving

76. Cool new idea!
Idea:
Certain preprocessing techniques make rotation simpler, while certain preprocessing techniques make solving simpler
Let solving and rotation operate on different (yet synchronized) formulas
For example:
the CNF formula for rotation can be obtained from the original formula by the variable elimination, subsumption, self-subsumption, blocked clause elimination
The CNF formula for solving can be obtained from the original formula by blocked clause addition

77. Thank you!