Equisatisfiable SAT Encodings of Arithmetical Operations

73 Pages·2015·0.32 MB·English

Equisatisfiable SAT Encodings
of Arithmetical Operations
M
zur Erlangung des akademischen Grades
Master of Science
im Studiengang Medieninformatik
der Fakultät Informatik, Mathematik und Naturwissenschaften
der Hochschule für Technik, Wirtschaft und Kultur (HTWK) Leipzig
vorgelegt von
Martin Finke
Leipzig, den 18. November 2015
Erstgutachter: Prof. Dr. rer. nat. Johannes Waldmann
Zweitgutachter: Alexander Bau
Eidesstattliche Versicherung
Ich erkläre hiermit, dass ich diese Masterarbeit selbstständig ohne Hilfe Dritter
und ohne Benutzung anderer als der angegebenen Quellen und Hilfsmittel ver-
fasst habe. Alle den benutzten Quellen wörtlich oder sinngemäß entnommenen
Stellen sind als solche einzeln kenntlich gemacht.
Diese Arbeit ist bislang keiner anderen Prüfungsbehörde vorgelegt und auch
nicht veröffentlicht worden.
Ich bin mir bewusst, dass eine falsche Erklärung rechtliche Folgen haben wird.
Leipzig, den 18. November 2015
Martin Finke
Abstract
In Constraint Programming, problems containing both Boolean and arithmeti-
cal constraints are common in areas such as scheduling and software verifica-
tion. To solve them using a SAT solver, the arithmetical constraints must be
encoded as propositional formulas in conjunctive normal form (CNF). One way
is to Tseitin-transform logic circuits, adding an extra variable for each gate.
This thesis shows two CNF encoding methods that add only a small number
of extra variables. One replaces sub-formulas by extra variables (similar to
the Tseitin transform), the other is syntax-independent and does not constrain
the semantics of the extra variables. The methods are used to encode the
operations +,·,> and ≥ on natural numbers with up to 7 bits. The resulting
CNFs are up to 86% smaller than a Tseitin encoding and up to 84% smaller
than a minimal CNF without extra variables. However, using them to encode
an example problem results in an increased SAT solver time.
Table of Contents
Abstract
1 Introduction 1
2 Background 4
2.1 Boolean Satisfiability (SAT) . . . . . . . . . . . . . . . . . . . . . 4
2.1.1 Formula Syntax . . . . . . . . . . . . . . . . . . . . . . . . 4
2.1.2 Formula Semantics . . . . . . . . . . . . . . . . . . . . . . 8
2.2 Satisfiability Modulo Theories (SMT) . . . . . . . . . . . . . . . . 12
2.3 Bit Blasting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.3.1 Addition and Multiplication Circuits . . . . . . . . . . . . 14
2.3.2 Handling Integer Overflow . . . . . . . . . . . . . . . . . . 16
2.3.3 Binary Relations . . . . . . . . . . . . . . . . . . . . . . . 18
3 Requirements 19
3.1 Functionality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.1.1 Encoding Arithmetical Operations . . . . . . . . . . . . . 19
3.1.2 Minimizing the Size of Formulas . . . . . . . . . . . . . . . 20
3.1.3 Integrating Results for Evaluation . . . . . . . . . . . . . 21
3.2 Runtime Performance . . . . . . . . . . . . . . . . . . . . . . . . 21
3.3 Quality Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
4 Existing CNF Encoding Methods 23
4.1 Conversion to the Equivalent Canonical CNF . . . . . . . . . . . 23
4.1.1 Specification . . . . . . . . . . . . . . . . . . . . . . . . . . 23
4.1.2 Implementation . . . . . . . . . . . . . . . . . . . . . . . . 24
4.2 Conversion to an Equisatisfiable CNF . . . . . . . . . . . . . . . 25
4.2.1 Specification . . . . . . . . . . . . . . . . . . . . . . . . . . 27
4.2.2 Implementation: The Tseitin Transform . . . . . . . . . . 27
4.2.3 Improvements on the Tseitin Transform . . . . . . . . . . 29
4.3 CNF Minimization . . . . . . . . . . . . . . . . . . . . . . . . . . 30
4.3.1 Specification . . . . . . . . . . . . . . . . . . . . . . . . . . 30
4.3.2 Implementation . . . . . . . . . . . . . . . . . . . . . . . . 30
4.3.2.1 Finding the Set of Primes . . . . . . . . . . . . . 31
4.3.2.2 Finding a Minimum Subset of Primes . . . . . . 33
4.3.3 Heuristic Methods . . . . . . . . . . . . . . . . . . . . . . 34
4.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
5 Syntactic Method 35
5.1 Specification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
5.2 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
5.2.1 Selecting Sub-Formulas . . . . . . . . . . . . . . . . . . . . 39
5.2.2 Replacing Sub-Formulas . . . . . . . . . . . . . . . . . . . 39
5.2.3 Conversion to a CNF and Minimization . . . . . . . . . . 41
5.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
6 Semantic Method 43
6.1 Specification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
6.2 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
6.3 Removing Redundant Clauses . . . . . . . . . . . . . . . . . . . . 47
6.3.1 Illegal Clauses . . . . . . . . . . . . . . . . . . . . . . . . . 47
6.3.2 Non-minimal Clauses . . . . . . . . . . . . . . . . . . . . . 48
6.3.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
6.4 Complexity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
6.5 Heuristic Methods . . . . . . . . . . . . . . . . . . . . . . . . . . 52
7 Results 53
7.1 Formula Size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
7.2 SAT Solver Time . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
8 Conclusions and Future Work 57
Software Overview 59
List of Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
Installation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
Usage Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
Glossary 63
References 65
1 Introduction
Constraint programming is a type of declarative programming where a problem
is expressed as a constraint program, which is a formula in predicate or proposi-
tional logic. The Boolean Satisfiability Problem (SAT) is the problem of finding
an assignment to the variables in a Boolean formula such that the formula is
satisfied. A SAT solver is a program that solvesthis problem for a givenformula.
Many modern SAT solvers require the formula to be in conjunctive normal form
(CNF).1
Inthelasttwodecades, SATsolverperformancehasincreasedsignificantly[3, 4].
To use a SAT solver to solve a problem with arithmetical constraints, they have
to be encoded in propositional logic. This thesis examines different methods
to translate logic circuits for +,·,> and ≥ to a CNF. Since SAT solver time
depends on formula size, the aim is to find CNFs with a minimum number of
literals. To this end, extra variables may be introduced, as long as the CNF is
equisatisfiable2 to the circuit.
Converting a formula to a CNF without adding extra variables incurs an expo-
nential growth in the worst case.
Example 1.1 (CNF Without Extra Variables):
Consider the formula F = v ⊕v ⊕v ⊕v . The ⊕ operator is the exclusive
0 1 2 3
or (XOR).
A minimal equivalent CNF is: (¬v ∨¬v ∨¬v ∨¬v )∧(¬v ∨¬v ∨v ∨
0 1 2 3 0 1 2
v )∧(¬v ∨v ∨v ∨¬v )∧(v ∨¬v ∨¬v ∨v )∧(v ∨v ∨¬v ∨¬v )∧
3 0 1 2 3 0 1 2 3 0 1 2 3
(¬v ∨ v ∨ ¬v ∨ v ) ∧ (v ∨ ¬v ∨ v ∨ ¬v ) ∧ (v ∨ v ∨ v ∨ v ). It has 8
0 1 2 3 0 1 2 3 0 1 2 3
1This includes all solvers based on the Davis–Putnam–Logemann–Loveland (DPLL) algorithm
[1, 2].
2Equisatisfiability is formally introduced in Def. 4.9.
1
clauses and 32 literals.3
The Tseitin transform (described in Subsec. 4.2.2) introduces an extra variable
for every binary operator and accomplishes a linear increase in size.
Example 1.2 (CNF from the Tseitin Transform):
Converting the formula F from Ex. 1.1 to an equisatisfiable CNF using the
Tseitin transform introduces 3 extra variables (v ,v ,v ) and results in the
4 5 6
CNF: v ∧(v ∨v ∨¬v )∧(v ∨¬v ∨v )∧(¬v ∨v ∨v )∧(¬v ∨¬v ∨
6 2 3 4 2 3 4 2 3 4 2 3
¬v )∧(v ∨v ∨¬v )∧(v ∨¬v ∨v )∧(¬v ∨v ∨v )∧(¬v ∨¬v ∨¬v )∧
4 1 4 5 1 4 5 1 4 5 1 4 5
(v ∨v ∨¬v )∧(v ∨¬v ∨v )∧(¬v ∨v ∨v )∧(¬v ∨¬v ∨¬v ). It has 13
0 5 6 0 5 6 0 5 6 0 5 6
clauses and 37 literals. F is small, so in this case the linear growth is higher
than the exponential one.
Adding a small number of extra variables can result in a smaller CNF than not
adding any extra variables or performing the Tseitin transform. The following
example shows this.
Example 1.3 (CNF with One Extra Variable):
Introducing one extra variable t with the semantics t ⇔ (v ⊕v ) results in
0 3
the CNF: (¬v ∨¬v ∨t)∧(¬v ∨v ∨¬t)∧(¬v ∨¬v ∨¬t)∧(¬v ∨v ∨
1 2 1 2 0 3 0 3
t)∧(v ∨¬v ∨t)∧(v ∨v ∨¬t)∧(v ∨¬v ∨¬t)∧(v ∨v ∨t). It has 8
0 3 0 3 1 2 1 2
clauses and 24 literals.
The research questions are: Can the size of CNF encodings for arithmetical
operations be reduced by introducing extra variables, compared to no extra
variables or using the Tseitin transform? And if the encodings are smaller, does
SAT solver time decrease when they are used?
This thesis shows two algorithms to find small CNF encodings with a given
number of extra variables. One replaces selected sub-formulas with extra vari-
ables, the other is based on the Boolean function realized by the formula. If
the smaller CNFs indeed decrease SAT solver times, they can be hardcoded and
3Moregenerally,forn>0,aminimalequivalentCNFencodingofanXORovernvariableshas
2n−1 clauses,andeachclausecontainsnliterals. Theintuitionisthecheckerboardpatternon
the Karnaugh map: for each zero, one clause is needed to cover it, and not cover anything
else.
2
reused in constraint solvers such as Satchmo [5].
Ch. 2 formally introduces propositional logic and the Boolean Satisfiability
Problem (SAT). Furthermore, it describes how arithmetical operations can be
encoded as propositional formulas to be used in a SAT encoding.
Ch. 3 specifies the requirements for the developed software and how the quality
of the resulting formulas is evaluated.
Ch. 4 shows two existing methods for conversion of propositional formulas to
conjunctive normal form. One (Sec. 4.1) is based on the truth table realized by
a formula (similar to Ex. 1.1), the other (Sec. 4.2) is the Tseitin transform (as
in Ex. 1.2). Furthermore, CNF minimization is covered.
The following two chapters show two methods to find a small CNF for an input
formula F and a number k of allowed extra variables. The syntactic method
(Ch. 5) is to replace k sub-formulas of F by extra variables, choosing the set
of sub-formulas that results in the smallest CNF. The semantic method (Ch. 6)
expresses the problem as a series of constraint satisfaction problems and uses a
SAT solver to find a solution. Its output is a minimal CNF independent of the
syntax of F.
In Ch. 7, the results are evaluated with regards to the requirements from Ch. 3.
The formula sizes are compared and SAT solver time of a termination problem
is measured, comparing different encodings.
Ch. 8 concludes the text by pointing out possible improvements and areas for
future work.
3
2 Background
2.1 Boolean Satisfiability (SAT)
This section introduces the syntax and semantics of propositional logic used in
this thesis, and the Boolean Satisfiability Problem (SAT).
2.1.1 F S
The fundamental components in propositional logic are Booleans and variables.
Definition 2.1 (Set of Booleans):
The set of Booleans is B = {True,False}.
Definition 2.2 (Set of Variables):
The set of variables is denoted as V = {v ,v ,v ,...}.
0 1 2
These components can be combined using logical operators.
Definition 2.3 (Logical Operators):
The set of operators is {¬,∧,∨,⊕,→,⇔}. They have the following mean-
ings:
• ¬ is the logical complement (negation, not)
• ∧ is the logical conjunction (and)
• ∨ is the logical disjunction (or)
• ⊕ is the exclusive or (xor)
• → is the logical implication
• ⇔ is the logical equivalence
4
The complement (¬) is unary. The logical implication (→) is binary: the
left side is the premise, the right side is the conclusion. The other operators
(∧,∨,⊕ and ⇔) are applied to a finite set of operands (which may be empty).
In this text, a formula is a formula in propositional logic. Formulas are created
by combining Booleans, variables and operators as follows.
Definition 2.4 (Formula):
A formula is an expression that has one of the following forms:
• v, where v ∈ V
• b, where b ∈ B
• ¬F, where F is a formula
• F → F , where F and F are formulas
prem conc prem conc
∧ ∨ ⊕ ⇔
• Fs or Fs or Fs or Fs, where Fs is a finite set of formulas.
For n ≥ 2, a formula ∧{F ,...,F } can be written as F ∧ ... ∧ F .
1 n 1 n
The same applies analogously to the operators ∨,⊕ and ⇔.
The set of formulas is denoted as F.
Example 2.5 (Formula):
The expression v ∨¬¬(¬False ⊕v ) is a formula.
1 4
(3+5)∨v is not a formula. Neither is (∀v ∈ V : ¬v).
0
Formulascanbeparenthesizedtoavoidambiguity. Toeasereading,theformulas
∧{F},∨{F} and ⊕{F} can be simplified to F.
For a given formula, the set of all variables in it can be determined as follows.
Definition 2.6 (Variable Set of a Formula):
For a formula F, the variable set V is defined as:
F

∅{F,}if, Fif F∈ B∈ V
V = V , if F has the form ¬F′
F ∪VF{′ VF∪′ |VF′ ∈,Fisf}F, ihfaFs hthaes ftohremfoFrm ∧→FsF,∨Fs,⊕Fs, or ⇔Fs
Fprem Fconc prem conc
5