SAT and SMT solvers
Ayrat Khalimov
(based on Georg Hofferek‘s slides)
AKDV 2014
Motivation
• SAT solvers:
- They rocketed the model checking
• First-Order Theories
- Very expressive
- Efficient SMT Solvers
But:
• What are they?
• How do solvers work?
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Outline
• Propositional SAT solver
- DPLL algorithm
• Predicate Logic (aka. First-Order Logic)
- Syntax
- Semantics
• First Order Logic
• First-Order Theories
• SMT solver
- Eager Encoding
- Lazy Encoding
- DPLL(T)
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Scope of Solvers
theory of equality
linear integer arithmetic
propositional logic
SAT solvers
difference logic
theory of arrays
…
SMT solvers
first order logic
Theorem provers
Notation
• propositional variables
- e.g., a, b, c, d, …
• literal is a variable or its negation
- e.g., a, b, …
• partial assignment A is a conjunction of literals
- e.g., A = a  d
• clause is a disjunction of literals
- e.g., c = a   b
•  is a CNF formula (i.e. conjunction of clauses):
- e.g.,  = (a  b  d)  c
• [A] is  with all variables set according to A
- e.g., [A] = (FALSE  b   TRUE)  c = b  c
SAT Solver
Formula in CNF
SAT
Solver
Satisfiable
(+ model)
Unsatisfiable
(+ refutation
proof)
DPLL Algorithm
• Due to Davis, Putnam, Loveland, Logemann
- two papers: 1960, 1962
• Basis for all modern SAT solvers
CNF as a Set of Clauses
• Formula:
𝜑=
𝑎 ∨ ¬𝑏 ∨ 𝑐 ∧
¬𝑎 ∨ ¬𝑑 ∧
(¬𝑐)
• Set Representation
{
𝑎, ¬𝑏, 𝑐 ,
¬𝑎, ¬𝑑 ,
¬𝑐
}
Idea of DPLL-based SAT Solvers
• Recursively search an A:
- [A] is TRUE
• Proves  satisfiable
• “A” is a satisfying model
• No such A exists
-  is unsatisfiable
Setting Literals
• Compute [l], for a literal l:
- Remove all clauses that contain l:
• They are true
- Remove all literals l:
• They are false (i.e., 𝑎 ∨ ¬𝑙 becomes a, ¬𝑙 becomes empty)
- An empty clause is false
- An empty set of clauses is true
Truth Value of a CNF
𝒄𝒍𝒂𝒖𝒔𝒆 ∧ 𝒄𝒍𝒂𝒖𝒔𝒆 ∧ 𝒄𝒍𝒂𝒖𝒔𝒆 ∧ ⋯ ∧ 𝒄𝒍𝒂𝒖𝒔𝒆
{
…
∧
…
∧
…
∧ … ∧
• At least one clause is empty:
- FALSE
• Clause set empty:
- TRUE
• Otherwise:
- Unassigned Literals left
…
}
DPLL Algorithm
// sat(, A)=TRUE iff [A] is satisfiable
// sat(, true)=TRUE iff  is satisfiable
sat(, A){
if([A] = true) return TRUE;
if([A] = false) return FALSE;
// Some unassigned variables left
l = pick unassigned variable;
AT = A  l;
if(sat(, AT)) return TRUE;
AF = A  l;
if(sat(, AF)) return TRUE;
return FALSE;
}
DPLL Example
• Formula to check: (a  b)  (b  c)  (c  a)
1. sat((a  b)  (b  c)  (c  a), true)
2. sat( (a  b)  (b  c)  (c  a), a)
3. sat( (a  b)  (b  c)  (c  a), ab)
4. sat( (a  b)  (b  c)  (c  a), abc) unsat
5. sat( (a  b)  (b  c)  (c  a), abc) unsat
6. sat( (a  b)  (b  c)  (c  a), ab) unsat
7. sat( (a  b)  (b  c)  (c  a), a)
8. sat((a  b)  (b  c)  (c  a), ab)
9. sat((a  b)  (b  c)  (c  a), abc) sat
Boolean Constraint Propagation (BCP)
• Unit clause:
- a clause with a single unassigned literal
- Examples:
• (a)
• (b)
• Unit Clause exists  set its literal
- Very simple but very important heuristic!
DPLL with BCP
sat(, A){
while(unit
// l is
// unit
A = A 
}
clause occurs){
only unassigned literal in
clause;
l;
if([A] = true) return TRUE;
if([A] = false) return FALSE;
l = pick unassigned variable;
AT = A  l;
if(sat(, AT)) return TRUE;
AF = A  l;
if(sat(, AF)) return TRUE;
return FALSE;
}
Example
• Formula to check: (a  b)  (b  c)  (c  a)
1. sat((a  b)  (b  c)  (c  a), true)
2. sat( (a  b)  (b  c)  (c  a), a)
3. [BCP]: sat( (a  b)  (b  c)  (c  a), ab)
4. [BCP]: sat( (a  b)  (b  c)  (c  a), abc) unsat
5. sat( (a  b)  (b  c)  (c  a), a)
6. sat( (a  b)  (b  c)  (c  a), ab)
7. sat((a  b)  (b  c)  (c  a), abc) sat
Can we do better?
sat(, A){
while(unit clause occurs){
// l is only unassigned literal in
// unit clause;
A = A  l;
}
if([A] = true) return TRUE;
if([A] = false) return FALSE;
l = pick unassigned variable;
AT = A  l;
if(sat(, AT)) return TRUE;
AF = A  l;
if(sat(, AF)) return TRUE;
return FALSE;
}
Pure Literals
• Pure literal:
- Literal for unassigned variable
- The variable appears in one phase only
• Pure literals  true them
DPLL with BCP and Pure Literals
sat(, A){
while(unit clause occurs){ // BCP
let l be only unassigned literal in c;
A = A  l;
}
while(pure literal l exists){ // Pure literals
A = A  l;
}
if([A] = true) return TRUE;
if([A] = false) return FALSE;
l = pick a literal that does not occur in A;
AT = A  l;
if(sat(, AT)) return TRUE;
AL = A  l;
if(sat(, AL)) return TRUE;
return FALSE;
}
Example
• Formula to check: (a  b)  (b  c)  (c  a)
1. sat((a  b)  (b  c)  (c  a), true) [a pure]
2. sat( (a  b)  (b  c)  (c  a), a) [b pure]
3. sat( (a  b)  (b  c)  (c  a), ab) sat
Can we do better?
sat(, A){
while(unit clause l occurs) A = A  l;
while(pure literal l exists) A = A  l;
if([A] = true) return TRUE;
if([A] = false) return FALSE;
l = pick a literal that does not occur in A;
AT = A  l;
if(sat(, AT)) return TRUE;
AL = A  l;
if(sat(, AL)) return TRUE;
return FALSE;
}
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Learning: informal
• Whenever we get the conflict
- analyze it
• add clauses to avoid in future
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Learning
1.
2.
3.
4.
5.
6.
7.
(a  c)
(b  c)
(a  b  c)
(a  b)
a
(a  b)
(a  b)
UNSAT
(a  b)
c
Learning
1.
2.
3.
4.
5.
6.
7.
(a  c)
(b  c)
(a  b  c)
(a  b)
a
(a  b)
(a  b)
UNSAT
(a  b)
Without learning
c
a
a
UNSAT UNSAT
a
UNSAT
The problem is with a: no need to set c=true!
Learning
1.
2.
3.
4.
5.
6.
7.
(a  c)
(b  c)
c
(a  b  c)
(a  b)
a
(a  b)
(a  b)
UNSAT
(a  b)
a
We learn: a
6 b
false
7
Learning & Backtracking
1.
2.
3.
4.
5.
6.
7.
8.
(a  c)
(b  c)
c
(a  b  c)
(a  b)
a
(a  b)
(a  b)
UNSAT
(a  b)
a
LEVEL 0
LEVEL 1
a
6 b
false
7
LEVEL 2
We learn: a
Jump back to level 0 is smart
Learning & Backtracking
1.
2.
3.
4.
5.
6.
7.
8.
(a  c)
(b  c)
c
(a  b  c)
(a  b)
a
(a  b)
(a  b)
UNSAT
(a  b)
a
a
LEVEL 0
LEVEL 1
LEVEL 2
Jump back to level 0 is smart
Learning & Backtracking
1.
2.
3.
4.
5.
6.
7.
8.
(a  c)
(b  c)
c
(a  b  c)
(a  b)
a
(a  b)
(a  b)
UNSAT
(a  b)
a
a
4 b
false
5
LEVEL 0
LEVEL 1
LEVEL 2
Learning & Backtracking
1.
2.
3.
4.
5.
6.
7.
8.
(a  c)
(b  c)
c
(a  b  c)
(a  b)
a
(a  b)
(a  b)
UNSAT
(a  b)
a
UNSAT
a
4 b
false
5
LEVEL 0
LEVEL 1
LEVEL 2
We learn: UNSAT, because
no decision was necessary
Backtrack Level
• Three important possibilities
1. Backtrack as usual
2. Restart for every learned clause
3. Go to the earliest level in which the conflict
clause is a unit clause
• Option 3 often performs better
Can we do better?
(learning is not shown)
sat(, A){
while(unit clause l occurs) A = A  l;
while(pure literal l exists) A = A  l;
if([A] = true) return TRUE;
if([A] = false) return FALSE;
l = pick a literal that does not occur in A;
AT = A  l;
if(sat(, AT)) return TRUE;
how to pick literals?
AF = A  l;
if(sat(, AF)) return TRUE;
return FALSE;
}
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Effect of picking heuristics on SAT solver performance
Source: Armin Biere’s slides: http://fmv.jku.at/rerise14/rerise14-sat-slides.pdf
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Can we do better? -- Special cases
• Horn clauses can be solved in polynomial time
• Cut width algorithm
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source: http://gauss.ececs.uc.edu/SAT/
Syntax of Predicate Logic
• Two sorts:
- Objects
•
•
•
•
“Terms”
Numbers
Strings
Elements of sets
…
- Truth values
• IsEven(42)
“Formulas”
From Terms to Formulas
Formula
𝑃 𝑥, 𝑓 𝑦
Predicate Term
Term
FOL formulae: informal definition
∀𝒙∀𝒛∃𝒚 𝑷 𝒙, 𝒛 ∧ 𝑭 𝒚 →
𝑷(𝒕 𝒙, 𝒚 , 𝒛)
quantifiers over variables
unary
functions
predicates: binary , etc.
• can FO formulae quantify over functions/predicates?
• can FO formulae have free (non-quantified) variables?
• * can FO formulae have ‘uninterpreted’ functions?
• * can FO formula has infinite number of atoms?
Syntax of Predicate Logic
• Variables 𝕍
- x, y, z, …
• Functions 𝔽
- f, g, h, … (arity > 0)
- constants (arity = 0)
• Predicates ℙ
- P, Q, R, … (with arity > 0)
• Terms 𝕋 and Formulae defined next
Terms 𝕋
• Variable is a term
• Constant is a term
• If 𝒕𝟏 , 𝒕𝟐 , … , 𝒕𝒏 are terms, 𝒇 is 𝑛-ary function
then
𝒇(𝒕𝟏 , 𝒕𝟐 , … , 𝒕𝒏 ) is a term
Formulae
Preconditions:
 𝑷(𝒕𝟏 , 𝒕𝟐 , … , 𝒕𝒏 )
• Terms 𝒕𝟏 , 𝒕𝟐 , … , 𝒕𝒏
 ¬𝝓
• 𝑛-ary predicate
symbol 𝑷
 𝝓 ∧ 𝝍, 𝝓 ∨ 𝝍, 𝝓 → 𝝍
• formulae 𝝓, 𝝍
 ∀𝒙 𝝓
• Variable 𝒙
 ∃𝒙 𝝓
True and False FO formulae
• Functions and predicates in FO formulae are
‘uninterpreted’
- they can be any
• Variables in FO formulae have no domains
- what can x, y be?
• What does it mean that this formula is true? or false?
∀𝒙∃𝒚: 𝑷 𝒙, 𝒚 → 𝑷 𝒇 𝒙, 𝒚 , 𝒚
• Depends..
Model 𝑀 for (𝔽, ℙ,𝕍)
• Non-empty set 𝐷
- Domain for variables
- Possibly infinite
- Non-empty
• For constansts 𝑐 ∈ 𝔽: concrete element 𝑐 𝑀 ∈ 𝐷
• For functions 𝑓 ∈ 𝔽: concrete function 𝑓 𝑀 : 𝐷𝑛 → 𝐷
• For predicates 𝑃 ∈ ℙ: subset 𝑃𝑀 ⊆ 𝐷𝑛 (of arity n)
- i.e., set of tuples on which 𝑃 is true
Semantics of Predicate Logic
• Formula 𝜙
- Over 𝔽, ℙ, 𝕍
• Model 𝑀
- For 𝔽, ℙ, 𝕍
•𝑴⊨𝝓?
- (𝜙 has no free variables)
 Inductive
Definition
Semantics of Predicate Logic
• For 𝜙 of the form ∀𝑥 𝜓
- 𝑀 ⊨ ∀𝑥 𝜓 iff 𝑀 ⊨ 𝜓[𝑥↦𝑎] , for all 𝑎 ∈ 𝐷
• For 𝜙 of the form ∃𝑥 𝜓
- 𝑀 ⊨ ∃𝑥 𝜓 iff 𝑀 ⊨ 𝜓[𝑥↦𝑎] , for at least one 𝑎 ∈ 𝐷
• For 𝜙 of the form ¬𝜓, 𝜓1 ∧ 𝜓2 , 𝜓1 ∨ 𝜓2
- Like in propositional logic
• No free variables => any predicate
𝑃 𝑡1 , 𝑡2 , … , 𝑡𝑛 has concrete arguments
Examples
• Let model M be:
- D = {1,2}
- 𝑃 1,1 = P 1,2 = T, others gives F
- f(1, ..)=1, f(2, 1)=1, f(2,2)=2
• 𝝓 = ∀𝒙∃𝒚: 𝑷 𝒙, 𝒚 → 𝑷 𝒇 𝒙, 𝒚 , 𝒚
Does 𝑴 ⊨ 𝝓?
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Satisfiable FO formulae
∀𝒙∃𝒚: 𝑷 𝒙, 𝒚 → 𝑷 𝒇 𝒙, 𝒚 , 𝒚 is sat
means there is a model:
• there is a non-empty domain D for x, y
- for example, D={1,2}
• there is predicate P, function 𝑓:
- for example, 𝑃 = 1,2 , i.e. P(1,2)=true, P(2,.)=false
- for example, 𝑓 = { 1,2 ↦ 1, . . }, i.e. 𝑓(1,2) = 1
such that
𝑴 ⊨ ∀𝒙 ∃𝒚: 𝑷 𝒙, 𝒚 → 𝑷 𝒇 𝒙, 𝒚 , 𝒚
Valid FO formulae
∀𝒙 𝑷 𝒙 → ∀𝒙 𝑷 𝒇 𝒙 is valid
iff it is satisfied by any model
Let us check for example the model:
• D={1,2}
• P={1,2}
- i.e., P(1)=P(2)=T
• function 𝜏 is any from {1,2} to {1,2}
Some facts about our world
• Gödel proved that
- every valid FO formula has a finite proof.
• Church-Turing proved that
- no algorithm exists that can decide if FO formula is invalid
proof
FO formula
deduction
algorithm
if invalid
may never terminate
Notion of “Theory”
Application
Domain
Structures &
Objects
Arithmetic
Numbers
(Integers,
Rationals, Reals)
Computer
Programs
Arrays, Bitvectors
Predicates &
Functions
Array-Read,
Array-Write, …
Definition of a Theory
First-Order Theory 𝓣:
1. Signature Σ
-
Constants
Predicates
Functions
𝚺-formula:
(non-logic) symbols
from Σ only
2. Set of Axioms 𝒜
-
Sentences (=Formulas without free variables) with
symbols from Σ only
𝚺, 𝓐: possibly infinite
Example: Theory of Equality 𝒯𝐸
• Signature Σ𝐸 = =, 𝑎0 , 𝑏0 , 𝑐0 , 𝑑0 , …
- Binary equality predicate =
- Arbitrary constant symbols
- (no function/predicate symbols!)
• Axioms 𝒜𝐸 :
1. ∀𝑥. 𝑥 = 𝑥
(reflexivity)
2. ∀𝑥. ∀𝑦. 𝑥 = 𝑦 → 𝑦 = 𝑥
(symmetry)
3. ∀𝑥. ∀𝑦. ∀𝑧. 𝑥 = 𝑦 ∧ 𝑦 = 𝑧 → 𝑥 = 𝑧
(transitivity)
Model View
All possible Models
Models satisfying
all axioms
• We check satisfiability and validity only wrt models
that satisfy axioms
-  “Satisfiability modulo (=‘with respect to’) theories”
𝒯-Satisfiability
• Green: Models Satisfying all Axioms
• Violet: Models Satisfying Formula in Question
𝓣-Satisfiable
𝓣-Satisfiable
Not 𝓣-Satisfiable
𝒯-Validity
• Green: Models Satisfying all Axioms
• Violet: Models Satisfying Formula in Question
𝓣-Valid
𝓣-Valid
Not 𝓣-Valid
Theory Formulas vs. FO Formulas
Theory
Formula
𝝓𝑻
𝒜→𝝓
𝒜∧𝝓
Fragment of a Theory
• Syntactically restricted subset
- Quantifier-free fragment
- Conjunctive fragment
• e.g.: 𝑎 = 𝑏 ∧ 𝑏 = 𝑐 ∧ 𝑏 ≠ 𝑑
Scope of Solvers
theory of equality
linear integer arithmetic
propositional logic
SAT solvers
difference logic
theory of arrays
…
SMT solvers
first order logic
Theorem provers
Deciding Satisfiability (quantifier free theory):
main methods
1. Eager Encoding
- Equisatisfiable
2. Lazy Encoding
 Theory Solver
 Conjunctive Fragment
propositional formula
 Blocking Clauses
- one fat SAT call
 numerous SAT calls
3. DPLL (T)
Example: Theory of Uninterpreted
Functions and Equality 𝒯𝑈𝐸
• Signature Σ𝑈𝐸 = =, 𝑎, 𝑏, 𝑐, 𝑑, …
- Binary equality predicate =
- Arbitrary constant- and function-symbols
• Axioms 𝒜𝑈𝐸 :
1.-3. same as in 𝒜𝐸
4.
∀𝑥. ∀ 𝑦. (
𝑖 𝑥𝑖
(reflexivity), (symmetry), (transitivity)
= 𝑦𝑖 ) → 𝑓 𝑥 = 𝑓 𝑦
(function congruence)
Axiom Schema: Template for (infinite number of) axioms
Two-Stage Eager Encoding
(quant.-free)
𝓣𝑼𝑬 formula
Ackermann’s
Reduction
equisatisfiable
𝓣𝑬 formula
Graph-based
Reduction
equisatisfiable
propositional formula
SAT Solver
Ackermann’s Reduction (from 𝓣𝑼𝑬 to 𝓣𝑬 )
• Fresh Variables
- 𝑓 𝑥 ⇝ 𝑓𝑥 , 𝑔 𝑥 ⇝ 𝑔𝑥 , ...
• Functional Constraints
-
𝑥 = 𝑦 → 𝑓𝑥 = 𝑓𝑦
• 𝓣𝑬 formula: 𝜙𝐸 = 𝜙𝐹𝐶 ∧ 𝜙𝑈𝐸
𝝓𝑼𝑬
Perform Ackermann’s Reduction for
≔𝒇 𝒈 𝒙 =𝒇 𝒚 ∨ 𝒛=𝒈 𝒚 ∧𝒛≠𝒇 𝒛
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Graph-Based Reduction (from 𝓣𝐄 to propositional)
• Non-Polar Equality Graph
f
- Node per variable
e
d
- Edge per (dis)equality
• Make it chordal
- No chord-free cycles (size > 3)
c
a
b
g
Graph-Based Reduction (from 𝓣𝐄 to propositional)
• Fresh Propositional Variables
- 𝑎 = 𝑏 ⇝ 𝑒𝑎=𝑏
- Order!
𝑏 = 𝑎 ⇝ 𝑒𝑎=𝑏
𝒂
• Triangle (𝑎, 𝑏, 𝑐):
- Transitivity Constraints
- 𝑒𝑐=𝑏 ∧ 𝑒𝑏=𝑎 → 𝑒𝑐=𝑎 ∧
𝑒𝑐=𝑏 ∧ 𝑒𝑐=𝑎 → 𝑒𝑏=𝑎 ∧
𝑒𝑐=𝑎 ∧ 𝑒𝑏=𝑎 → 𝑒𝑐=𝑏
- 𝜙𝑝𝑟𝑜𝑝 = 𝜙 𝑇𝐶 ∧ 𝜙𝐸
𝒄
𝒃
 SAT Solver
Perform Graph-Based Reduction for
𝝓𝑬 ≔ 𝒂 = 𝒃 ∧ 𝒃 = 𝒄 ∧ 𝒄 = 𝒅 ∧ 𝒅 ≠ 𝒂
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Summary: Eager Encoding
(quant.-free)
𝓣𝑼𝑬 formula
Ackermann’s
Reduction
equisatisfiable
𝓣𝑬 formula
𝝓𝑬 = 𝝓𝑭𝑪 ∧ 𝝓𝑼𝑬
Graph-based
Reduction
equisatisfiable
propositional formula
𝝓𝒑𝒓𝒐𝒑 = 𝝓𝑻𝑪 ∧ 𝝓𝑬
SAT Solver
Lazy Encoding
𝒔𝒌𝒆𝒍(𝝓)
SAT
Solver
UNSAT
Assignment of Literals
Blocking Clause
Theory
Solver
SAT
Conjunctive (quant-free) Fragment of 𝒯𝑈𝐸
• Conjunction of theory literals, where literals
are:
- (𝑡1 = 𝑡2 ), 𝑓(𝑡2 ) = 𝑡1 , 𝑡1 ≠ 𝑡2 , 𝑡1 ≠ 𝑓(𝑡2 ) , … .
Congruence-Closure Algorithm
• Equivalence Classes
- introduce class for each term 𝑡1 , 𝑡2 , 𝑓 𝑡1 … .
- 𝑡1 = 𝑡2 :
merge classes of 𝑡1 𝑎𝑛𝑑 𝑡2 into one
larger class
- two classes shared terms -- merge classes! (repeat)
- 𝑡𝑖 , 𝑡𝑗 from same class:
Merge classes of 𝑓 𝑡𝑖 , 𝑓 𝑡𝑗 (repeat)
• Check Disequalities 𝑡𝑘 ≠ 𝑡𝑙
- 𝑡𝑘 , 𝑡𝑙 in same class: UNSAT!
- Otherwise: SAT!
Perform Congruence Closure for
𝝓𝑼𝑬 ≔ 𝒂 = 𝒃 ∧ 𝒃 = 𝒄 ∧ 𝒅 = 𝒆 ∧ 𝒆 ≠ 𝒂 ∧ 𝒇 𝒂 ≠ 𝒇(𝒄)
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Lazy Encoding
𝒔𝒌𝒆𝒍(𝝓)
SAT
Solver
UNSAT
Assignment of Literals
Blocking Clause
Theory
Solver
SAT
DPLL(T)
Decide
partial assignment
SAT
Learn &
Backtrack
Start
partial
assignment
full assignment
BCP/PL
conflict
Analyze
Conflict
partial
assignment
Theory Solver
theory propagation
/ conflict
Add Clauses
UNSAT
Scope of Solvers
theory of equality
linear integer arithmetic
propositional logic
SAT solvers
difference logic
theory of arrays
…
SMT solvers
first order logic
Theorem provers
Summary
• Propositional SAT Problem
- DPLL
• First-Order Theories
- Examples: 𝒯𝐸 , 𝒯𝑈𝐸
• Satisfiability modulo theories
- Eager Encoding
- Lazy Encoding
- DPLL(T)
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Self-check: learning targets
• Explain Satisfiability Modulo Theories
• Describe Theory of Uninterpreted Functions
and Equality
• Explain and use
-
Ackermann’s Reduction
Graph-based Reduction
Congruence Closure
DPLL
DPLL(T)
Institute for Applied Information Processing and Communications
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some reading
• History of satisfiability: http://gauss.ececs.uc.edu/SAT/articles/FAIA1850003.pdf
• SAT basics: http://gauss.ececs.uc.edu/SAT/articles/sat.pdf
• Conflict Driven Clause Learning:
http://gauss.ececs.uc.edu/SAT/articles/FAIA185-0131.pdf
• Armin Biere’s slides: http://fmv.jku.at/rerise14/rerise14-sat-slides.pdf
• SAT game http://www.cril.univartois.fr/~roussel/satgame/satgame.php?level=1&lang=eng
• Logic and Computability classes by Georg
http://www.iaik.tugraz.at/content/teaching/bachelor_courses/logik_und_ber
echenbarkeit/
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