Maximum Satisfiability Ruben Martins - - PowerPoint PPT Presentation
Maximum Satisfiability Ruben Martins http://www.cs.cmu.edu/~mheule/15816-f19/ Automated Reasoning and Satisfiability, October 1, 2019 1/41 What is Boolean Satisfiability? Fundamental problem in Computer Science The first problem to be
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◮ Fundamental problem in Computer Science
◮ The first problem to be proven NP-Complete ◮ Has a wide range of applications
◮ Formula:
◮ ϕ = (¬x2 ∨ ¬x1) ∧ (x2 ∨ ¬x3) ∧ (x1) ∧ (x3)
◮ Boolean Satisfiability (SAT):
◮ Is there an assignment of true or false values to variables
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◮ Set of packages we want to install: {p1, p2, p3, p4} ◮ Each package pi has a set of dependencies:
◮ Packages that must be installed for pi to be installed
◮ Each package pi has a set of conflicts:
◮ Packages that cannot be installed for pi to be installed
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◮ Giving up?
◮ The problem is NP-hard, so let’s develop heuristics or
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◮ Giving up?
◮ The problem is NP-hard, so let’s develop heuristics or
◮ No! Current tools can find solutions for very large
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◮ Encoding dependencies:
◮ p1 ⇒ (p2 ∨ p3) ≡ (¬p1 ∨ p2 ∨ p3) ◮ p2 ⇒ p3 ≡ (¬p2 ∨ p3) ◮ p3 ⇒ p2 ≡ (¬p3 ∨ p2) ◮ p4 ⇒ (p2 ∧ p3) ≡ (¬p4 ∨ p2) ∧ (¬p4 ∨ p3)
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◮ Encoding conflicts:
◮ p1 ⇒ ¬p4 ≡ (¬p1 ∨ ¬p4) ◮ p3 ⇒ ¬p4 ≡ (¬p3 ∨ ¬p4)
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◮ Encoding installing all packages:
◮ (p1) ∧ (p2) ∧ (p3) ∧ (p4)
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◮ ϕ = (¬p1 ∨p2 ∨p3)∧(¬p2 ∨p3)∧(¬p3 ∨p2)∧(¬p4 ∨p2)∧
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◮ Formula is unsatisfiable ◮ Can you find an unsatisfiable subformula?
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◮ Formula is unsatisfiable ◮ We cannot install all packages ◮ How many packages can we install?
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◮ Maximum Satisfiability (MaxSAT):
◮ Clauses in the formula are either soft or hard ◮ Hard clauses: must be satisfied
◮ Soft clauses: desirable to be satisfied
◮ Goal: Maximize number of satisfied soft clauses
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◮ What are the hard constraints?
◮ (Hint) Dependencies, conflicts or installation packages?
◮ What are the soft constraints?
◮ (Hint) Dependencies, conflicts or installation packages?
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◮ What are the hard constraints?
◮ Dependencies and conflicts
◮ What are the soft constraints?
◮ Installation of packages
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◮ Dependencies and conflicts are encoded as hard clauses ◮ Installation of packages are encoded as soft clauses ◮ Goal: maximize the number of installed packages
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◮ Dependencies and conflicts are encoded as hard clauses ◮ Installation of packages are encoded as soft clauses ◮ Optimal solution (3 out 4 packages are installed)
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◮ Many real-world applications can be encoded to MaxSAT:
◮ Software package upgradeability ◮ Error localization in C code ◮ Haplotyping with pedigrees ◮ . . .
◮ MaxSAT algorithms are very effective for solving
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200 400 600 800 1000 1200 1400 1600 1800 200 400 600 800 1000 1200 seconds instances Open-WBO (2015) MaxHS (2016) MSCG (2015) Eva (2014) Open-WBO (2014) Z3 (Microsoft 2016) QMaxSAT (2013) WPM2 (2013) PM2 (2010) QMaxSAT (2011-12) QMaxSAT (2010) CPLEX (IBM 2013) SAT4J (2009-10) IncWMaxSatz (2008)
◮ Best solver can solve 3× more benchmarks than in 2008! ◮ Better than tools like CPLEX (IBM) and Z3 (Microsoft)!
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200 400 600 800 1000 1200 1400 1600 1800 100 200 300 400 500 600 700 800 seconds instances MaxHS (2016) LHMS (2015-16) MSCG (2015) MaxHS (2013) Eva (2014) QMaxSAT (2014) Z3 (Microsoft) CPLEX (IBM) WPM2 (2013) WPM1 (2011-12) WBO (2010) IncWMaxSatz (2008) SAT4J (2009-10)
◮ Best solver can solve 2.5× more benchmarks than in
◮ Better than tools like CPLEX (IBM) and Z3 (Microsoft)!
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◮ MaxSAT Algorithms:
◮ Upper bound search on the number of unsatisfied soft
◮ Lower bound search on the number of unsatisfied soft
◮ Partitioning in MaxSAT:
◮ Use the structure of the problem to guide the search
◮ Using MaxSAT solvers
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◮ Satisfying assignment:
◮ Assignment to the variables that evaluates the formula
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◮ Satisfying assignment:
◮ Assignment to the variables that evaluates the formula
◮ µ = {x1 = 1, x2 = 1, x3 = 0}
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◮ Formula is unsatisfiable
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◮ Formula is unsatisfiable ◮ Unsatisfiable subformula (core):
◮ ϕ′ ⊆ ϕ, such that ϕ′ is unsatisfiable
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◮ MaxSAT algorithms build on SAT solver technology ◮ MaxSAT algorithms use constraints not defined in causal
◮ AtMost1 constraints, n
j=1 xj ≤ 1
◮ General cardinality constraints, n
j=1 xj ≤ k
◮ Pseudo-Boolean constraints, n
j=1 ajxj ≤ k ◮ Efficient encodings to CNF
◮ Sinz, Totalizer, . . .
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◮ Relax all soft clauses ◮ Relaxation variables:
◮ VR = {r1, r2, r3, r4} ◮ If a soft clause ωi is unsatisfied, then ri = 1 ◮ If a soft clause ωi is satisfied, then ri = 0
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◮ Formula is satisfiable
◮ ν = {x1 = 1, x2 = 0, x3 = 0, r1 = 0, r2 = 1, r3 = 1, r4 = 0}
◮ Goal: Minimize number of relaxation variables assigned
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◮ r2 and r3 were assigned truth value 1:
◮ Current solution unsatisfies 2 soft clauses
◮ Can less than 2 soft clauses be unsatisfied?
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ri∈VR ri ≤ 1)
◮ Add cardinality constraint that excludes solutions that
◮ CNF(r1 + r2 + r3 + r4 ≤ 1)
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ri∈VR ri ≤ 1)
◮ Formula is unsatisfiable:
◮ There are no solutions that unsatisfy 1 or less soft
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◮ Optimal solution: given by the last model and
◮ ν = {x1 = 1, x2 = 0, x3 = 0}
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◮ We have just seen a search on the upper bound ◮ What other kind of search can we do to find an optimal
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◮ We have just seen a search on the upper bound ◮ What other kind of search can we do to find an optimal
◮ What if we start searching from the lower bound?
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◮ Relax all soft clauses ◮ Relaxation variables:
◮ VR = {r1, r2, r3, r4} ◮ If a soft clause ωi is unsatisfied, then ri = 1 ◮ If a soft clause ωi is satisfied, then ri = 0
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ri∈VR ri ≤ 0)
◮ Add cardinality constraint that excludes solutions that
◮ CNF(r1 + r2 + r3 + r4 ≤ 0)
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ri∈VR ri ≤ 0)
◮ Formula is unsatisfiable:
◮ There are no solutions that unsatisfy 0 or less soft
◮ Add cardinality constraint that excludes solutions that
◮ CNF(r1 + r2 + r3 + r4 ≤ 1)
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ri∈VR ri ≤ 1)
◮ Formula is unsatisfiable:
◮ There are no solutions that unsatisfy 1 or less soft
◮ Add cardinality constraint that excludes solutions that
◮ CNF(r1 + r2 + r3 + r4 ≤ 2)
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ri∈VR ri ≤ 2)
◮ Formula is satisfiable:
◮ µ = {x1 = 1, x2 = 0, x3 = 0, r1 = 0, r2 = 1, r3 = 1, r4 =
◮ Optimal solution unsatisfies 2 soft clauses
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◮ What are the problems of this algorithm?
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◮ What are the problems of this algorithm?
◮ We relax all soft clauses! ◮ The cardinality constraint contain as many literals as we
◮ Can we do better?
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◮ Formula is unsatisfiable
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◮ Formula is unsatisfiable ◮ Identify an unsatisfiable core
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◮ Relax non-relaxed soft clauses in unsatisfiable core:
◮ Add cardinality constraint that excludes solutions that
◮ CNF(r1 + r2 ≤ 1) ◮ Relaxation on demand instead of relaxing all soft clauses
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◮ Formula is unsatisfiable
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◮ Formula is unsatisfiable ◮ Identify an unsatisfiable core
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◮ Relax non-relaxed soft clauses in unsatisfiable core:
◮ Add cardinality constraint that excludes solutions that
◮ CNF(r1 + r2 + r3 + r4 ≤ 2) ◮ Relaxation on demand instead of relaxing all soft clauses
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◮ Formula is satisfiable:
◮ µ = {x1 = 1, x2 = 0, x3 = 0, r1 = 0, r2 = 1, r3 = 1, r4 =
◮ Optimal solution unsatisfies 2 soft clauses
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◮ What are the problems of this algorithm?
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◮ What are the problems of this algorithm?
◮ We must translate cardinality constraints into CNF! ◮ If the number of literals is large than we may generate a
◮ Can we do better?
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◮ Formula is unsatisfiable
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◮ Formula is unsatisfiable ◮ Identify an unsatisfiable core
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◮ Relax unsatisfiable core:
◮ Add relaxation variables ◮ Add AtMost1 constraint
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◮ Formula is unsatisfiable
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◮ Formula is unsatisfiable ◮ Identify an unsatisfiable core
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◮ Relax unsatisfiable core:
◮ Add relaxation variables ◮ Add AtMost1 constraint
◮ Soft clauses may be relaxed multiple times
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◮ Formula is satisfiable ◮ An optimal solution would be:
◮ ν = {x1 = 1, x2 = 0, x3 = 0}
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◮ Formula is satisfiable ◮ An optimal solution would be:
◮ ν = {x1 = 1, x2 = 0, x3 = 0}
◮ This assignment unsatisfies 2 soft clauses
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◮ Unsatisfiable cores found by the SAT solver are not
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◮ Unsatisfiable cores found by the SAT solver are not
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◮ Unsatisfiable cores found by the SAT solver are not
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◮ Unsatisfiable cores found by the SAT solver are not
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◮ Unsatisfiable cores found by the SAT solver are not
◮ Minimizing unsatisfiable cores is computationally hard
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◮ Partitioning in MaxSAT:
◮ Partition the soft clauses into disjoint sets ◮ Iteratively increase the size of the MaxSAT formula
◮ Advantages:
◮ Easier formulas for the SAT solver ◮ Smaller unsatisfiable cores at each iteration
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SAT Solver Satisfying assignment Unsatisfiable subformula Refinement Optimal Solution SAT UNSAT
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SAT Solver Satisfying assignment Unsatisfiable subformula Refinement Optimal Solution SAT UNSAT
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SAT Solver Satisfying assignment Unsatisfiable subformula Refinement Optimal Solution SAT UNSAT
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SAT Solver Satisfying assignment Unsatisfiable subformula Refinement Optimal Solution SAT UNSAT
1
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SAT Solver Satisfying assignment Unsatisfiable subformula Refinement Optimal Solution SAT UNSAT
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SAT Solver Satisfying assignment Unsatisfiable subformula Refinement Optimal Solution Solution may not be optimal! SAT UNSAT
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SAT Solver Satisfying assignment Unsatisfiable subformula Refinement
Partitions?
Optimal Solution SAT UNSAT No Yes
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SAT Solver Satisfying assignment Unsatisfiable subformula Refinement
Partitions?
Optimal Solution SAT UNSAT No Yes
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SAT Solver Satisfying assignment Unsatisfiable subformula Refinement
Partitions?
Optimal Solution SAT UNSAT
1 ∪ ϕ2 ϕ′′ 1 ∪ ϕ′ 2 No Yes
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SAT Solver Satisfying assignment Unsatisfiable subformula Refinement
Partitions?
Optimal Solution SAT UNSAT No Yes
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SAT Solver Satisfying assignment Unsatisfiable subformula Refinement
Partitions?
Optimal Solution SAT UNSAT No Yes
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SAT Solver Satisfying assignment Unsatisfiable subformula Refinement
Partitions?
Optimal Solution SAT UNSAT No Yes
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SAT Solver Satisfying assignment Unsatisfiable subformula Refinement
Partitions?
Optimal Solution SAT UNSAT
1 ∪ ϕ′ 2 ∪ ϕ3 ϕ′′′ 1 ∪ ϕ′′ 2 ∪ ϕ′ 3
No Yes
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SAT Solver Satisfying assignment Unsatisfiable subformula Refinement
Partitions?
Optimal Solution SAT UNSAT No Yes
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SAT Solver Satisfying assignment Unsatisfiable subformula Refinement
Partitions?
Optimal Solution SAT UNSAT No Yes
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SAT Solver Satisfying assignment Unsatisfiable subformula Refinement
Partitions?
Optimal Solution SAT UNSAT No Yes
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◮ Graph representation of the MaxSAT formula:
◮ Vertices: Variables ◮ Edges: Between variables that appear in the same clause
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◮ Graph representation of the MaxSAT formula:
◮ Vertices: Variables ◮ Edges: Between variables that appear in the same clause
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◮ There are many ways to represent MaxSAT as a graph:
◮ Clause-Variable Incidence Graph (CVIG) ◮ Variable Incidence Graph (VIG) ◮ Hypergraph ◮ Resolution Graph ◮ . . .
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◮ There are many ways to represent MaxSAT as a graph:
◮ Clause-Variable Incidence Graph (CVIG) ◮ Variable Incidence Graph (VIG) ◮ Hypergraph ◮ Resolution Graph ◮ . . .
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◮ MaxSAT solvers rely on the identification of
◮ How can we capture sets of clauses that are closely
◮ Represent MaxSAT formulas as resolution graphs! ◮ Resolution graphs are based on the resolution rule
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◮ MaxSAT solvers rely on the identification of
◮ How can we capture sets of clauses that are closely
◮ Represent MaxSAT formulas as resolution graphs! ◮ Resolution graphs are based on the resolution rule
◮ Example of the resolution rule:
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◮ MaxSAT solvers rely on the identification of
◮ How can we capture sets of clauses that are closely
◮ Represent MaxSAT formulas as resolution graphs! ◮ Resolution graphs are based on the resolution rule
◮ Example of the resolution rule:
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◮ Vertices: Represent each clause in the graph ◮ Edges: There is an edge between two vertices if you can
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◮ Vertices: Represent each clause in the graph ◮ Edges: There is an edge between two vertices if you can
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◮ Vertices: Represent each clause in the graph ◮ Edges: There is an edge between two vertices if you can
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◮ Vertices: Represent each clause in the graph ◮ Edges: There is an edge between two vertices if you can
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200 400 600 800 1000 1200 1400 1600 1800 200 400 600 800 1000 1200 CPU Time (in seconds) Number of problems solved OPEN-WBO’14 Partitioning
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200 400 600 800 1000 1200 1400 1600 1800 200 400 600 800 1000 1200 CPU Time (in seconds) Number of problems solved OPEN-WBO’14 Partitioning
◮ The techniques in Open-WBO have been adopted by
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◮ Java:
◮ SAT4J ◮ http://www.sat4j.org/
◮ Python:
◮ RC2 ◮ Best solver in 2018 and 2019! ◮ SAT solvers written in C++ ◮ https://pysathq.github.io
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◮ C++:
◮ MaxHS ◮ One of the best solvers for weighted problems! ◮ Combines SAT and MIP solvers ◮ http://www.maxhs.org/ ◮ Open-WBO ◮ Winner of multiples tracks in the MaxSAT Competition
◮ https://github.com/sat-group/open-wbo
◮ Annual competition:
◮ http://maxsat-evaluations.github.io ◮ Modify a solver today and enter this year competition!
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◮ Variables indexed from 1 to n ◮ Negation: -
◮ -3 stands for ¬x3
◮ 0: special end-of-line character ◮ One special header “p”-line:
◮ #vars: number of variables ◮ #clauses: number of clauses ◮ top: “weight” of hard clauses
◮ Clauses represented as lists of integers
◮ Weight is the first number ◮ (¬x3 ∨ x1 ∨ ¬x45), weight 2:
◮ Clause is hard if weight is equal to top
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p wcnf 17997976 23364255 9223372036854775807 142 -11393180 12091478 0 200 -12496389 -1068725 13170751 0 209 -8854604 -8854942 -8854943 -8253894 9864153 0 174 -9406753 -8105076 11844088 0 200 -10403325 -8104972 12524177 0 142 -11987544 12096893 0 37 -10981341 -10980973 10838652 0 209 -9578314 -9579250 -9579251 -8254733 9578317 0 209 -8868994 -8870298 -8870299 -8254157 8868997 0 209 -9387012 -9387508 -9387509 -8253943 9387015 0 174 -9834074 -8106628 12074710 0 200 -10726788 -8105074 12909526 0 ... 9223372036854775807 -13181184 0 9223372036854775807 -13181215 0 ... truncated 763 MB
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c Open-WBO: a Modular MaxSAT Solver c Version: MaxSAT Evaluation 2016 c Authors: Ruben Martins, Vasco Manquinho, Ines Lynce c Contributors: Miguel Neves, Saurabh Joshi, Mikolas Janota ... c Problem Type: Weighted c Number of variables: 17,997,976 c Number of hard clauses: 8,237,870 c Number of soft clauses: 15,126,385 c Parse time: 5.60 s ...
s OPTIMUM FOUND c Total time: 361.26 s v 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15... ...17997976
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On Solving the Partial MAX-SAT Problem. SAT 2006: 252-265.
straints for MaxSAT. CP 2014: 531-548
SAT 2014: 438-445
Community-Based Partitioning for MaxSAT Solving. SAT 2013: 182-191
Resolution-Based Representations for MaxSAT Solving. SAT 2015: 272-286 Jessica Davies, Fahiem Bacchus: Postponing Optimization to Speed Up MAXSAT Solving. CP 2013: 247-262 Alexey Ignatiev, Antnio Morgado, Joao Marques-Silva: PySAT: A Python Toolkit for Prototyping with SAT Oracles. SAT 2018: 428-437
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Boolean Constraints into CNF Such That Generalized Arc Consistency Is
straints into CNF. SAT 2015: 9-16 http://tools.computational-logic.
aldez-Cru, Jordi Levy. The Community Structure of SAT
MaxSAT Evaluation: http://www.maxsat.udl.cat/ Open-WBO: http://sat.inesc-id.pt/open-wbo/ SAT4J: http://www.sat4j.org/ RC2: https://pysathq.github.io MaxHS: http://www.maxhs.org/ SATGraf: https://bitbucket.org/znewsham/satgraf