Computability and Complexity

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Computability and Complexity Andrei Bulatov
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Propositional Formulas
A propositional formula is an expression built
from

• variables
• parenthesis (, )
• logical connectives

• ? conjunction and
• ? disjunction or
• ? negation not
• ? implication if then

Examples
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Propositional Formulas Semantics
A truth assignment is an assignment of
variables in a formula ? with truth values 0
and 1 (or F and T, or FALSE and TRUTH)
Truth assignment T satisfies ?, written T ??
?

• if ? is a variable X, then T ?? ? if and
  only if T(X)1
• if ?? then T ?? ? if and only if T ??
  ?
• if then T ?? ? if
  and only if and
• if then T ?? ? if
  and only if or
• if then T ?? ?
  if and only if or

Examples T(X)1, T(Y)0, T(Z)0
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Types of Propositional Formulas
A formula ? is said to be valid if T ?? ?
for any truth assignment T
(tautology)
A formula ? is said to be satisfiable if T
?? ? for some truth assignment T
A formula ? is said to be unsatisfiable if T
?? ? for no truth assignment T
unsatisfiable
valid
s a t i s f i a b l e
Formulas ? and ? are said to be equivalent,
???, if they have the same satisfying
assignments
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Main Tautologies
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Main Equivalences
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Conjunctive Normal Form
A literal is a variable or its negation, X
or X
A clause is a disjunction of literals
A Conjunctive Normal Form (CNF) is a
conjunction of clauses
Examples
Theorem Every propositional formula is
equivalent to a CNF.
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Predicates and Quantifiers
A predicate on a set A is a function A ? A
?? A ? 0,1
Informally, a predicate expresses some property
of its argument
Examples P(X,Y) X ? Y
Q(X,Y,Z) Z is in between X and Y, that is
X lt Z lt Y or Y lt Z lt X
A function is a function A ? A ?? A ? A
Examples f(X,Y) X Y g(X,Y,Z) X
log(Y Z²) h(X) 3X X²
Quantifiers if a set A is fixed ?X
means for every X ? A ?X means
there exists X ? A
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First Order Syntax
A vocabulary is a collection of predicate and
function symbols, each of which is assigned a
non-negative number, the arity
Example (Number theory) Predicate symbols
(X,Y), i.e X Y
lt(X,Y), i.e. X lt Y Function
symbols (X,Y), i.e. X Y
?(X,Y), i.e. X ? Y
(X,Y), i.e.
?(X), i.e. X 1
0
Example (Graph theory) Predicate
symbols (X,Y), i.e X Y
E(X,Y),
i.e. X is connected to Y Function
symbols no
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A term is an expression built from variables
and function symbols
Examples ?((X,Y),(X,Z))
(X Y) ? (X Z)
We denote 1 ?(0), 2 ?(?(0)),
An atomic formula is a predicate symbol
followed by a list of terms in parenthesis the
number of terms in the list must match the arity
of the predicate symbol
Examples
(((X,T),(Y,T)),(Z,T))
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A first order formula is defined as follows

• an atomic formula is a formula
• if ? and ? are formulas, then (?????) is
  a formula
• if ? and ? are formulas, then (?????) is
  a formula
• if ? and ? are formulas, then (?????) is
  a formula
• if ? is a formula, then (??) is a formula
• if ? is a formula, then (?X ?) is a
  formula
• if ? is a formula, then (?X ?) is a
  formula

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Examples
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Free and Bound Variables
Any occurrence of X in an expression ?X ? or
?X ? is bound
Any occurrence which is not bound is free
A variable that has a free occurrence is called
free
A formula without free variables is called a
sentence