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CS389L: Automated Logical Reasoning
Lecture 1: Introduction and
Review of Basics
Işıl Dillig
Işıl Dillig,
CS389L: Automated Logical Reasoning
Lecture 1: Introduction and Review of Basics
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Instructor: Işil Dillig
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E-mail: [email protected]
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Office hours: Tuesday-Thursday after class in GDC 5.726
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TA: Pengxiang Cheng ([email protected])
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Office hours: Mon, Wed 2:30-3:30pm
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What is this Course About?
CS389L: Automated Logical Reasoning
Lecture 1: Introduction and Review of Basics
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Why Should You Care?
Logic is a fundamental part of computer science:
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This course is about computational logic and its applications.
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Explore logical theories widely used in computer science.
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Learn about decision procedures that allow us to automatically
decide satisfiability and validity of logical formulas.
Işıl Dillig,
CS389L: Automated Logical Reasoning
Lecture 1: Introduction and Review of Basics
Işıl Dillig,
Artificial intelligence: planning, automated game playing, ...
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Programming languages: Static analysis, software
verification, program synthesis, ...
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Software engineering: automated test generation, automated
program repair, ...
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Overview of the Course
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CS389L: Automated Logical Reasoning
Lecture 1: Introduction and Review of Basics
Overview, cont
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Part I: Propositional logic
Part II: First-order theorem proving
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SAT solvers
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Semantics of FOL and theoretical properties
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Applications and variations (e.g., MaxSAT)
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Basics of first-order theorem proving
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Binary Decision Diagrams
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Decidable fragments of FOL
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Lecture 1: Introduction and Review of Basics
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Lecture 1: Introduction and Review of Basics
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Overview, cont.
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Overview, cont.
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Part III: SMT Solving
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Decision procedures for commonly used theories (e.g., equality,
linear arithmetic)
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Combining theories, Nelson-Oppen method
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DPLL(T) and practical SMT solvers
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CS389L: Automated Logical Reasoning
Lecture 1: Introduction and Review of Basics
Part IV: Applications in formal methods
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Program verification (Hoare logic)
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Invariant generation
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Bounded model checking
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Program synthesis
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Logistics
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Lecture 1: Introduction and Review of Basics
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Reference #1
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Classroom will move to GDC 1.304 after today’s lecture!!
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All class material (slides, relevant reading etc.) posted on the
course website:
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The Calculus of Computation
by Aaron Bradley and Zohar Manna
http://www.cs.utexas.edu/˜ idillig/cs389L
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Also have a Piazza page:
piazza.com/utexas/spring2017/cs389l
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Please post all non-personal questions on Piazza instead of
emailing us!
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No required books, but following textbooks may be useful
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CS389L: Automated Logical Reasoning
Lecture 1: Introduction and Review of Basics
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Reference #2
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Işıl Dillig,
CS389L: Automated Logical Reasoning
Lecture 1: Introduction and Review of Basics
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Workload and Grading
Decision Procedures: An Algorithmic Point of View
by Daniel Kroening and Ofer Strichman
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Lecture 1: Introduction and Review of Basics
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Işıl Dillig,
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My philosophy: Maximize your knowledge without taking too
much time
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No programming assignments or big projects
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Expect problem set once every 2 weeks (35% grade)
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2 in-class, closed-book exams (30% of grade each)
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Class attendance and participation (5% of grade)
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Lecture 1: Introduction and Review of Basics
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Exams
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Exam dates: March 2, May 2 – put these dates on your
calendar! (free during finals week)
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All exams closed-book, closed-notes, closed-laptop,
closed-phone etc, but can bring 2 cheat sheets
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Let’s get started!
Today: Review of basic propositional logic
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Should already know this stuff – quick refresher!
Please introduce yourself!
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CS389L: Automated Logical Reasoning
Lecture 1: Introduction and Review of Basics
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Işıl Dillig,
Review of Propositional Logic: PL Syntax
Atom
truth symbols > (“true”) and ⊥ (“false”)
propositional variables p, q, r , p1 , q1 , r1 , · · ·
Literal
atom α or its negation ¬α
Formula
literal or application of a
logical connective to formulae F , F1 , F2
¬F
F1 ∧ F2
F1 ∨ F2
F1 → F2
F1 ↔ F2
Işıl Dillig,
“not”
“and”
“or”
“implies”
“if and only if”
CS389L: Automated Logical Reasoning
Base Cases:
I |= >
I |= p
I 6|= p
iff
iff
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(negation)
(conjunction)
(disjunction)
(implication)
(iff)
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Interpretation I : mapping from each propositional variables
in F to exactly one truth value
Formula F + Interpretation I = Truth value
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We write I |= F if F evaluates to > under I (satisfying
interpretation)
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Similarly, I 6|= F if F evaluates to ⊥ under I (falsifying
interpretation).
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Simple Example
iff I 6|= F
iff I |= F1 and I |= F2
iff I |= F1 or I |= F2
iff, I 6|= F1 or I |= F2
iff, I |= F1 and I |= F2
or I 6|= F1 and I 6|= F2
Lecture 1: Introduction and Review of Basics
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I [p] = >
I [p] = ⊥
CS389L: Automated Logical Reasoning
Lecture 1: Introduction and Review of Basics
I : {p 7→ >, q 7→ ⊥, · · · }
I 6|= ⊥
Inductive Cases:
I |= ¬F
I |= F1 ∧ F2
I |= F1 ∨ F2
I |= F1 → F2
I |= F1 ↔ F2
CS389L: Automated Logical Reasoning
PL Semantics
Inductive Definition of PL Semantics
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Consider formula F1 : (p ∧ q) → (p ∨ ¬q)
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What is its truth value under interpretation
I1 : {p 7→ >, q 7→ ⊥} ?
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What about formula F2 : (p ↔ ¬q) → (q → ¬r ) and
interpretation I2 = {p 7→ ⊥, q 7→ >, r 7→ >}?
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Lecture 1: Introduction and Review of Basics
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Satisfiability and Validity
Examples
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F is satisfiable iff there exists an interpretation I such that
I |= F .
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F valid iff for all interpretations I , I |= F .
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F is contingent if it is satisfiable but not valid.
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Duality between satisfiability and validity:
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Sat, unsat, or valid?
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(p ∧ q) → ¬p
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(p → q) → (¬(p ∧ ¬q))
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(p → (q → r )) ∧ ¬((p ∧ q) → r )
F is valid iff ¬F is unsatisfiable
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Thus, if we have a procedure for checking satisfiability, this
also allows us to decide validity
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CS389L: Automated Logical Reasoning
Lecture 1: Introduction and Review of Basics
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Deciding Satisfiability and Validity
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Before we talk about practical algorithms for deciding
satisfiability, let’s review some simple techniques
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Two very simple techniques:
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Truth table method: essentially a search-based technique
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Semantic argument method: deductive way of deciding
satisfiability
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Lecture 1: Introduction and Review of Basics
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Method 1: Truth Tables
F : (p ∧ q) → (p ∨ ¬q)
Example
p
0
0
1
1
q
0
1
0
1
p ∧ q
0
0
0
1
¬q
1
0
1
0
p ∨ ¬q
1
0
1
1
F
1
1
1
1
Thus F is valid.
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Modern SAT solvers combine search and deduction!
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Lecture 1: Introduction and Review of Basics
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Another Example
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Lecture 1: Introduction and Review of Basics
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Bad Idea!
F : (p ∨ q) → (p ∧ q)
p
0
0
1
1
q
0
1
0
1
p ∨ q
0
1
1
1
p ∧ q
0
0
0
1
F
1
0
0
1
← satisfying I
← falsifying I
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Truth tables are completely brute-force, impractical ⇒ must
list all 2n interpretations!
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Does not work for any other logic where domain is not finite
(e.g., first-order logic)
Thus F is satisfiable, but invalid.
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Lecture 1: Introduction and Review of Basics
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Lecture 1: Introduction and Review of Basics
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Method 2: Semantic Argument
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Semantic argument method is essentially a proof by
contradiction, and is also applicable for theories with
non-finite domain.
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Main idea: Assume F is not valid ⇒ there exists some
falsifying interpretation I such that I 6|= F
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Apply proof rules.
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If we derive a contradiction in every branch of the proof, then
F is valid.
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If there is some branch where we cannot derive ⊥ (after
applying all proof rules), then F is not valid.
The Proof Rules (I)
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I |= ¬F
I 6|= F
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Lecture 1: Introduction and Review of Basics
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Işıl Dillig,
CS389L: Automated Logical Reasoning
According to semantics of conjunction, from I |= F ∧ G, we
can deduce:
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According to semantics of disjunction, from I |= F ∨ G, we
can deduce:
Similarly, from I 6|= F ∧ G, we can deduce:
Similarly, from I 6|= F ∨ G, we can deduce:
I |6 = F ∨ G
I |6 = F
I |6 = G
I 6|= F ∧ G
I 6|= F | I 6|= G
The second deduction results in a branch in the proof, so each
case has to be examined separately!
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Lecture 1: Introduction and Review of Basics
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The Proof Rules (IV)
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CS389L: Automated Logical Reasoning
According to semantics of implication:
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And:
Lecture 1: Introduction and Review of Basics
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The Proof Rules (V)
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According to semantics of iff:
I |= F → G
I 6|= F | I |= G
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I |= F ∨ G
I |= F | I |= G
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Lecture 1: Introduction and Review of Basics
The Proof Rules (III)
I |= F ∧ G
I |= F
←and
I |= G
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Similarly, from I 6|= ¬F , we can deduce:
I |6 = ¬F
I |= F
The Proof Rules (II)
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According to semantics of negation, from I |= ¬F , we can
deduce I 6|= F :
I |= F ↔ G
I |= F ∧ G | I |= ¬F ∧ ¬G
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I |6 = F → G
I |= F
I 6|= G
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Işıl Dillig,
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And:
I 6|= F ↔ G
I |= F ∧ ¬G | I |= ¬F ∧ G
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Lecture 1: Introduction and Review of Basics
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The Proof Rules (Contradiction)
An Example
F : (p ∧ q) → (p ∨ ¬q)
Prove
I
is valid.
Let’s assume that F is not valid and that I is a falsifying
interpretation.
Finally, we derive a contradiction, when I both entails F and
does not entail F :
1.
2.
3.
4.
5.
6.
7.
8.
I |= F
I 6|= F
I |= ⊥
I
I
I
I
I
I
I
I
6|=
|=
6|=
|=
|=
6|=
6|=
|=
(p ∧ q) → (p ∨ ¬q)
p ∧ q
p ∨ ¬q
p
q
p
¬q
⊥
assumption
1 and →
1 and →
2 and ∧
2 and ∧
3 and ∨
3 and ∨
4 and 6 are contradictory
⇒ Thus F is valid.
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Lecture 1: Introduction and Review of Basics
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Another Example
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Lecture 1: Introduction and Review of Basics
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Equivalence
Prove that the following formula is valid using semantic
argument method:
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F : ((p → q) ∧ (q → r )) → (p → r )
Formulas F1 and F2 are equivalent (written F1 ⇔ F2 ) iff for
all interpretations I , I |= F1 ↔ F2
F1 ⇔ F2 iff F1 ↔ F2 is valid
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Lecture 1: Introduction and Review of Basics
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Implication
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Thus, if we have a procedure for checking satisfiability, we can
also check equivalence.
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Lecture 1: Introduction and Review of Basics
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Example
Formula F1 implies F2 (written F1 ⇒ F2 ) iff for all
interpretations I , I |= F1 → F2
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Prove that F1 ∧ (¬F1 ∨ F2 ) implies F2 using semantic
argument method.
F1 ⇒ F2 iff F1 → F2 is valid
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Thus, if we have a procedure for checking satisfiability, we can
also check implication
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Caveat: F1 ⇔ F2 and F1 ⇒ F2 are not formulas (they are not
part of PL syntax); they are semantic judgments!
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Lecture 1: Introduction and Review of Basics
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Lecture 1: Introduction and Review of Basics
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Summary
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Next lecture:
Normal forms and algorithms for deciding satisfiability
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Optional reading:
Bradley & Manna texbook until Section 1.6
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