Logics for Data and Knowledge
Representation
Propositional Logic
Originally by Alessandro Agostini and Fausto Giunchiglia
Modified by Fausto Giunchiglia, Rui Zhang and Vincenzo Maltese
Outline



Introduction
Syntax
Semantics





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2
Truth valuation
Satisfiability
Validity
Entailment
Logical implication
Reasoning Services
Normal Forms
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Propositions

All knowledge representation languages deal with sentences
“All men are mortals”
“Obama is the president of the USA”

Sentences denote propositions (by definition), namely they express
something true or false (the mental image of what you mean when
you write the sentence)

Logic languages deal with propositions

Propositional logic is the simplest logic that does this (no individuals,
no quantifiers)
Mortal  Man
3
ObamaPresident
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Introduction: example
Mental
Model
World
Language
L
Monkey
SEMANTIC
GAP
4
PROPOSITION
SENTENCE
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Introduction: the famous triangle
CONCEPT
(what we have in mind)
refers to
symbolizes
Monkey
REFERENT
(the real object)
5
stands for
SYMBOL
(what we write)
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Language (Syntax)



Alphabet of symbols: The first step in setting up a formal language is
to list the symbols of the alphabet
Propositional Logic (PL) is a symbolic language
Σ0: the alphabet of a PL language
Σ0
Logical
, , , …
Descriptive
Constants
Variables
one proposition only
they can be substituted by any
proposition or formula
A, B, C …
P, Q, ψ …
NOTE: not only characters but also words (composed by several
characters) like “monkey” are descriptive symbols
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Additional Symbols

Auxiliary symbols
 Parentheses: ( )

Defined symbols
 Logical defined constants are, for all propositions P:
⊥ (falsehood symbol, false, bottom)
⊥ =df P∧¬P
T (truth symbol, true, top)
T =df ¬⊥


7
Defined symbols are not strictly necessary
However, they increase the usability by reducing the syntactic
complexity of propositions
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Formation Rules (FR): well formed formulas

Well formed formulas (wff) in PL can be described by the following
BNF grammar (codifying the FR):
<Atomic Formula> ::= A | B | ... | P | Q | ... | ⊥ | ⊤
<wff> ::= <Atomic Formula> | ¬<wff> | <wff>∧ <wff> | <wff> ∨ <wff>

Σ0 + FR define a propositional language

Atomic formulas are also called atomic propositions while wff are
propositional formulas (or just propositions)
A formula is correct if and only if it is a wff

ψ, PL
8
Yes, ψ is correct!
PARSER
No
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Example: the Monkey-Banana problem
“There is a monkey in a laboratory with some bananas hanging out of reach
from the ceiling. A box is available that will enable the monkey to reach the
bananas if he climbs on it. The monkey and box have height Low, but if the
monkey climbs onto the box he will have height High, the same as the
bananas. At this point the monkey can get a banana [...]”
Significant data: monkey, bananas, box
Significant knowledge: Low, High, Climb, GetBanana
All other data and knowledge are irrelevant!
Both data and knowledge are codified as (atomic or more complex)
formulas in the language





Language (any language you want to define):
L = {MonkeyLow, BananaHigh, MonkeyClimbBox, MonkeyGetBanana}
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Examples of formulas
Formula
inLab-Monkey
inlab-monkey
in-lab-monkey
inLab(Monkey)
(Monkey ∧ GetBanana) ∨ ¬ High
inLab(Monkey)∧
P ¬∧Q
Correct
Yes
(wff)
No
NOTE: In PL we do not have individuals (it is not so expressive).
Parentheses in inLab(Monkey) are not the auxiliary symbols like in e.g.
(Monkey ∧ GetBanana) ∨ ¬ High (problem of ambiguity)
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Propositional Theory
 Propositional
(or sentential) theory
A set of propositions
 It is a (propositional) knowledge base (containing true facts)
 It corresponds to a TBox (terminology) only, where no
meaning is specified yet, i.e. it is a syntactic notion

 Recall
that:
knowledge is a set of facts, i.e. statements
 we assumed statements are propositions

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Propositional Theories and Databases
A

propositional theory is not a database!
Propositions don’t show data explicitly (no individuals)
In PL the form P(a) must be interpreted as P-a (so the use of
form P(a) is ambiguous).
 Example: In the sentence ‘inLab(Monkey)’, the string ‘Monkey’
does not refer to an individual from some data domain

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The Monkey-Banana problem
“There is a monkey in a laboratory with some bananas hanging out of reach
from the ceiling. A box is available that will enable the monkey to reach the
bananas if he climbs on it. The monkey and box have height Low, but if the
monkey climbs onto the box he will have height High, the same as the
bananas. At this point the monkey can get a banana [...]”
L = {MonkeyLow, BananaHigh, MonkeyClimbBox, MonkeyGetBanana,
, , }
T = { (MonkeyLow  BananaHigh  MonkeyGetBanana)
 (MonkeyLow  MonkeyClimbBox)
 ( MonkeyLow   BananaHigh  MonkeyGetBanana)}
NOTE: We still need to assign semantics to the symbols in the language.
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Semantics means providing an interpretation
 So
far the elements of our propositional language are
simply strings of symbols
without formal meaning
 The
meanings which are intended to be attached to the
symbols and propositions form the intended
interpretation ν (nu) of the language
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The Monkey-Banana problem (II)
“There is a monkey in a laboratory with some bananas hanging out of reach
from the ceiling. A box is available that will enable the monkey to reach the
bananas if he climbs on it. The monkey and box have height Low, but if the
monkey climbs onto the box he will have height High, the same as the
bananas. At this point the monkey can get a banana [...]”
L = {A, B, C, D, , , }
T = { (A  B  D)
 (A  C)
 ( A   B  D)}
NOTE: Symbols are without meaning. What each symbol mean?
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Semantics: formal model

Intensional interpretation
We must make sure to assign the formal meanings out of our intended
interpretation to the (symbols of the) language, so that formulas
(propositions) really express what we intended (in terms of true/false)

The mental model: What we have in mind?
In our mind (mental model) we have a set of properties that we associate to
propositions. We need to make explicit (as much as possible) what we mean.

The formal model
This is done by defining a formal model M. Technically: we have to define a
pair (M,⊨) for our propositional language

Truth-values
In PL a sentence A is true (false) iff A denotes a formal object which satisfies
(does not satisfy) the properties of the object in the real world.
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The Monkey-Banana problem (III)
T = {  (A  C) }
L = {A, C}
Domain
D
17
D = {T, F}
Model
M
Formal model
M(A) = T
M(C) = F
Truth values
SEMANTIC
GAP
Mental
Model
Entailment
World
Intensional
Interpretation
I(A) = T
I(C) = F
Theory
T
M⊨T
Language
L
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Example: Banana (I)
18

Bananas may differ in many ways, but when using the
proposition Banana we have in mind some specific
properties: e.g. a given shape, a given color ...

These specific properties determine the intension of
the proposition Banana

The intentional interpretation of the proposition
Banana determines its truth-value (true or false).
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Example: Banana (II)
Mental
Model
World
Language
L
Language
L’
Banana
Banana 
(Yellow  Curve)
SENTENCE
SENTENCE
PROPOSITION
What we have in mind?
1) There is at least a banana
2) The banana is yellow
3) The banana has curve shape
4) No matter if opened
SEMANTIC
GAP
19
OUR MENTAL MODEL MATCHES
WITH THE REAL WORLD, SO THE
SENTENCES ARE TRUE
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Example: Banana (III)
Mental
Model
World
Language
L
Language
L’
Banana
Banana 
(Yellow  Curve)
SENTENCE
SENTENCE
PROPOSITION
What we have in mind?
1) There is at least a banana
2) The banana is yellow
3) The banana has curve shape
4) No matter if opened
SEMANTIC
GAP
20
OUR MENTAL MODEL DOES NOT
MATCH WITH THE REAL WORLD, SO
THE SENTENCES ARE FALSE
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Example: Banana (IV)
THE WORLD
MENTAL MODEL
I mean banana as
something yellow and
curve, no matter if
partially opened
MENTAL MODEL
T
F
FORMAL MODEL M1
21
T
F
FORMAL MODEL M2
I mean banana as
something yellow or red,
which can be one or many
together, no matter how
big they are, and whether
they are opened or not
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Truth-values

Definition: a truth valuation on a propositional language L is a
mapping ν assigning to each formula A of L a truth value ν(A),
namely in the domain D = {T, F}


ν(A)
ν(¬A)
ν(A∧B)
ν(A∨B)
= T or F according to the modeler, with A atomic
= T iff ν(A) = F
= T iff ν(A) = T and ν(B) = T
= T iff ν(A) = T or ν(B) = T

ν(⊥)
= F (since ⊥=df P∧¬P)

ν(⊤)
= T (since ⊤=df ¬⊥)


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Example of truth valuation ν
L = {MonkeyLow, BananaHigh, MonkeyClimbBox, MonkeyGetBanana,
, , }
T = { (MonkeyLow  BananaHigh  MonkeyGetBanana)
 (MonkeyLow  MonkeyClimbBox)
 ( MonkeyLow   BananaHigh  MonkeyGetBanana)}
MODEL
23
Informal
Semantics:
Formal Semantics:
“If the monkey is low
and the banana is high
in position, then the
monkey cannot get the
banana. “
ν(BananaHigh)
=T
ν(MonkeyClimbBox) = F
ν(MonkeyGetBanana) = F
ν(MonkeyLow)
=T
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Truth Tables

To compute (in polynomial time) truth valuations, the method of
truth tables was introduced (Wittgenstein, 1921).

Truth tables are well-known:
Truth-table
(TT) for the
logical
constants
¬ , ∧, ∨
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Truth Relation (Satisfaction Relation)

Let ν be a truth valuation on language L, we define the truthrelation (or satisfaction-relation) ⊨ and write
ν ⊨A
(read: ν satisfies A) iff ν(A) = True

Given a set of propositions Γ, we define
ν⊨Γ
iff if ν ⊨ θ for all formulas θ ∈ Γ
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Model and Satisfiability
 Let
ν be a truth valuation on language L. ν is a model of a
proposition P (set of propositions Γ) iff ν satisfies P (Γ).
P
(Γ) is satisfiable if there is some (at least one) truth
valuation ν such that ν ⊨ P (ν ⊨ Γ).
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Truth and Validity
 Let
P
ν be a truth valuation on language L.
is true under ν if ν ⊨ P
is valid if ν ⊨ P for all ν (notation: ⊨ P)
 P is called a tautology
P
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Example
L = {MonkeyLow, BananaHigh, MonkeyClimbBox, MonkeyGetBanana, , , }
MODEL
Informal
Semantics:
Formal Semantics:
“If the monkey is low
and the banana is high
in position, then the
monkey cannot get the
banana. “
ν(BananaHigh)
=T
ν(MonkeyClimbBox) = F
ν(MonkeyGetBanana) = F
ν(MonkeyLow)
=T

ν ⊨ MonkeyLow and ν ⊨ BananaHigh

ν ⊨  MonkeyClimbBox and ν ⊨  MonkeyGetBanana [SATISFIABLE]

MonkeyLow   MonkeyLow is a tautology
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[SATISFIABLE]
[VALID]
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Entailment
 Propositional
entailment
⊨ψ
where  = {θ1, ..., θn} is a finite set of propositions
ν ⊨ θi for all θi in  implies ν ⊨ ψ
We call entailment in a model v if it holds for one model
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Example
L = {MonkeyLow, BananaHigh, MonkeyClimbBox, MonkeyGetBanana, , , }
MODEL

Informal
Semantics:
Formal Semantics:
“If the monkey is low
and the banana is high
in position, then the
monkey cannot get the
banana. “
ν(BananaHigh)
=T
ν(MonkeyClimbBox) = F
ν(MonkeyGetBanana) = F
ν(MonkeyLow)
=T
 = {MonkeyLow, BananaHigh}
ν ⊨ MonkeyLow and ν ⊨ BananaHigh implies ν ⊨ MonkeyLow  BananaHigh
therefore  ⊨ MonkeyLow  BananaHigh
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[ENTAILMENT]
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Entailment
 Difference
in notation
⊨ ψ iff  ⊨ ψ
Left side means we have already filtered out all the
models that do not satisfy 
Right side means that we start from all models.
In any case the final result is the same.
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Implication: Premise (I)
 Propositional
entailment in the finite, i.e.
⊨ψ
where  = {θ1, ..., θn} is a finite set of propositions

How to embed such a notion directly in the language?
As we will see, this would allow us to use truth tables and
directly reformulate entailment in terms of the other reasoning
services
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Implication: Premise (II)
 Entailment
can be viewed as the logical implication
(θ1 ∧ θ2 ∧ ... ∧ θn) → ψ
to be read θ1 ∧ θ2 ∧ ... ∧ θn logically implies ψ
→
33
is a new symbol that we add to the language
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Implication and equivalence

We extend our alphabet of symbols with the following defined logical
constants:
→ (implication)
↔ (double implication or equivalence)
<Atomic Formula> ::= A | B | ... | P | Q | ... | ⊥ | ⊤
<wff> ::= <Atomic Formula> | ¬<wff> | <wff>∧ <wff> | <wff>∨ <wff> |
<wff> → <wff> | <wff> ↔ <wff> (new rules)


The operator precedence is (from higher to lower): ¬, ∧, ∨, →, ↔
Let propositions ψ, θ, and finite set {θ1,...,θn} of propositions be given.
We define:
 ⊨ θ → ψ iff θ ⊨ ψ
 ⊨ (θ1∧...∧θn) → ψ iff {θ1,...,θn} ⊨ ψ
 ⊨ θ ↔ ψ iff θ → ψ and ψ → θ
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Truth Table of →
defining the semantics of → in terms of ⊨ we have the
following truth-table for the logical implication P→Q:
 By
REMEMBER: Given ν, P ⊨ Q iff ν ⊨ P then ν ⊨ Q.
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Three Properties of Logical Implication
 Property
1:
For all ψ and θ: ψ → θ iff ¬ψ ∨ θ
 Property
2:
For all ψ: ¬ψ iff ψ →⊥
 Property
3:
For all ψ: ⊥ ⊨ ψ
(inconsistent theories imply any proposition)
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Examples (tautologies)

Laws for ∧, ¬ and →:
(A ∧ ¬B) → ¬(A → B)
¬(A → B) → (A ∧ ¬B)

Pierce’s law:
((A → B) → A) → A

De Morgan’s laws:
¬(A ∨ B) → (¬A ∧ ¬B)
(¬A ∧ ¬B) → ¬(A ∨ B)
¬(A ∧ B) → (¬A ∨ ¬B)
(¬A ∨ ¬B) → ¬(A ∧ B)
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Exercise (tautologies)
 Prove
the following double implications (i.e., logical
equivalences) by using truth tables.
38
(α ∧ β) ↔ (β ∧ α)
Commutativity of ∧
(α ∨ β) ↔ (β ∨ α)
Commutativity of ∨
((α ∧ β) ∧ γ) ↔ (α ∧ (β ∧ γ))
Associativity of ∧
((α ∨ β) ∨ γ) ↔ (α ∨ (β ∨ γ)):
Associativity of ∨
¬(¬α) ↔ α
Double-negation elimination
(α →β) ↔ (¬β →¬α)
Contraposition
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Exercise (tautologies)
 Prove
the following double implications (i.e., logical
equivalences) by using truth tables.
39
(α → β) ↔ (¬α ∨ β)
→-elimination
(α ↔ β) ↔ ((α → β) ∧ (β → α))
↔-elimination
¬(α ∧ β) ↔ (¬α ∨ ¬β)
DeMorgan Law for ∧
¬(α ∨ β) ↔ (¬α ∧ ¬β)
DeMorgan Law for ∨
(α ∧ (β ∨ γ)) ↔ ((α ∧ β) ∨ (α ∧ γ))
Distributivity of ∧ over ∨
(α ∨ (β ∧ γ)) ↔ ((α ∨ β) ∧ (α ∨ γ))
Distributivity of ∨ over ∧
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Reasoning Services
 The
basic reasoning tasks (or “services”) we can
represent (and compute) using ⊨ are:
Model Checking
 Satisfiability
 Unsatisfiability
 Validity
 Entailment

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(EVAL)
(SAT)
(UnSAT)
(VAL)
(ENT)
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Reasoning Services: EVAL
Model Checking (EVAL)
Is a proposition P true under a truth-valuation ν?
Check ν ⊨ P
P,ν
Yes
EVAL
No
Does ν ⊨ MonkeyLow  MonkeyGetBanana ?
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Reasoning Services: SAT
Satisfiability (SAT)
Is there a truth-valuation ν where P is true?
find ν such that ν ⊨ P
P
ν
SAT
No
Find a ν that ν ⊨ MonkeyLow  GetBanana
For instance, the truth-valuation ν where both are true
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Reasoning Services (unSAT)
 Unsatisfiability
is the impossibility to find a truth-valuation ν
Monkey is satisfiable.
Monkey ∨¬ Monkey is satisfiable.
Monkey ∧¬ Monkey is unsatisfiable (inconsistent).
In fact, for every truth-valuation ν, either
ν(A) = T, so ν(¬A) = F, hence ν(P) = F; or
ν(A) = F, so ν(¬A) = T, hence ν(P) = T.
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Reasoning Services: VAL
Validity (VAL)
Is P true according to all possible truth-valuation ν?
Check whether for all ν, ν ⊨ P
P
Yes
VAL
No
Is P = MonkeyLow ∨  MonkeyLow valid? Yes!
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Reasoning Services: ENT
Entailment (ENT)
All θ ∈ Γ true in ν (in all ν) implies ψ true in ν (in all ν)
check Γ ⊨ ψ in ν (in all ν) by checking that:
given that ν ⊨ θ for all θ ∈ Γ implies ν ⊨ ψ
Γ , ψ, ν
Yes
ENT
No
Does  MonkeyLow ⊨ MonkeyGetBanana ? No!
Does MonkeyLow  BananaHigh ⊨  MonkeyGetBanana? Yes!
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Reasoning Services (Entailment)
Exercises:

Demonstrate that {today} ⊨ today ∨ tomorrow

Demonstrate that {today} ⊭ tomorrow

Suppose P = today ∨ tomorrow. Then:


46
Define Γ such that Γ ⊨ P
Define Γ such that Γ ⊭ P
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Reasoning Services: properties (I)

EVAL is the easiest task. We just test one assignment.

SAT is NP complete. We need to test in the worst case all the
assignments. We stop when we find one which is true.

UnSAT is CO-NP. We need to test in the worst case all the
assignments. We stop when we find one which is true.

VAL is CO-NP. We need to test all the assignments and verify that
they are all true. We stop when we find one which is false.

ENT is CO-NP. It can be computed using VAL (see next slide)
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Reasoning Services: properties (II)

VAL(ψ) iff UnSAT( ψ)
ψ = (A   A) is valid, while ( A  A) is unsatisfiable

SAT(ψ)   VAL(ψ) iff SAT( ψ)   VAL( ψ)
ψ = A is satisfiable in all models where ν(A) = T. Informally, the left
side says that there is at least an assignment which makes A true
and there is at least an assignment which makes A false; the right
side says that there is at least an assignment which makes  A true
(in other words it makes A false) and there is at least an assignment
which makes  A false (in other words it makes A true).
This means that not all assignments make A true (false).

 ⊨ ψ, where  = {θ1, …, θn} iff VAL(θ1  …  θn  ψ)
Take A ⊨ A  B. the formula (A  A  B) is valid.
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Reasoning Services: properties (III)

EVAL is easy.

ENT can be computed using VAL

VAL can be computed using UnSAT

UnSAT is the opposite of SAT

All reasoning tasks can be reduced to SAT!!!
SAT is the most important reasoning service.
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Normal Forms: Literals

A literal is either an atomic proposition or the negation of an
atomic proposition, i.e., for all propositional variables P
P and ¬P are literals.
P is a positive literal
¬P is a negative literal
Monkey
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HighBanana
¬GetBanana
¬ClimbBox
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Normal Forms: Clauses

A (disjunctive) clause is a disjunction of literals α1  …  αn
B ∨ ¬C ∨ ¬D

A unit clause is a clause that contains only a single literal
B and ¬C are unit clauses

Monkey ∨ ¬Monkey
B ∨ ¬C is not
An empty clause does not contain literals
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Conjunctive Normal Form (CNF)
A
proposition is in conjunctive normal form (CNF) if it is
a (finite) conjunction of (disjunctive) clauses
Examples of formulas in CNF:
¬A ∧ B
(A ∨ ¬B) ∧ (B ∨ ¬C ∨ ¬D)
A ∧ (B ∨ C ∨ ¬D)
A  B  (¬ A  C)  (B  ¬ D  E)
Examples of formulas NOT in CNF:
¬(A ∧ B)
¬ is the outmost operator
A ∧ (B ∨ C ∧ ¬D)
∧ is nested within ∨
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Conversion of a formula to CNF

Theorem (e.g. Mendelson, 1987 [Prop. 1.4, Ex. 1.41(a)])
Every formula P has an equivalent CNF, namely P can be converted (in
polynomial time) into a proposition Q such that:




Q is in CNF
P and Q are equivalent (i.e., have the same truth table)
A conversion to a CNF that operates as in the theorem above is said
to be complete
The algorithm implementing the conversion to a CNF is based on the
tautologies about logical equivalence (or double implication):




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Double-negation law (i.e. elimination of ¬¬)
De Morgan’s laws
Distributive laws
Elimination of → and 
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Conversion to CNF: Basic Tautologies
(α → β) ↔ (¬α ∨ β)c
→-elimination
(α ↔ β) ↔ ((α → β) ∧ (β → α))
↔-elimination
¬(α ∧ β) ↔ (¬α ∨ ¬β)
DeMorgan Law for ∧
¬(α ∨ β) ↔ (¬α ∧ ¬β)
DeMorgan Law for ∨
(α ∧ (β ∨ γ)) ↔ ((α ∧ β) ∨ (α ∧ γ))
Distributivity of ∧ over ∨
(α ∨ (β ∧ γ)) ↔ ((α ∨ β) ∧ (α ∨ γ))
Distributivity of ∨ over ∧
¬(¬α) ↔ α
Double-negation elimination
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Example: Conversion to CNF of A ↔ (B ∨ C)
1. Eliminate ↔:
(A→(B ∨C)) ∧ ((B ∨C) →A)
2. Eliminate →:
(¬A∨B ∨C) ∧ (¬(B ∨C) ∨A)
3. Move ¬ inwards using de Morgan’s laws:
(¬A∨B ∨C) ∧ ((¬B ∧¬C) ∨A)
4. Apply distributivity of ∨ over ∧ and flatten:
(¬A∨B ∨C) ∧ (¬B ∨A) ∧ (¬C ∨A) (*)
(*) in CNF and equivalent to ‘A↔(B∨C)’.
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Normal Forms: Conjunctive clauses

A conjunctive clause is a conjunction of literals α1  …  αn
B  ¬C  ¬D

Monkey  ¬Monkey
NOTE: a conjunctive clause IS NOT a clause!
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Disjunctive Normal Form (DNF)
A
proposition is in disjunctive normal form (DNF) if it is
a (finite) disjunction of conjunctive clauses
Examples of formulas in DNF:
¬A ∨ B
(A ∧ ¬B) ∨ (B ∧ ¬C ∧ ¬D)
A ∨ (B ∧ C ∧ ¬D)
A ∨ B ∨ (¬ A ∧ C) ∨ (B ∧ ¬ D ∧ E)
Examples of formulas NOT in DNF:
¬(A ∨ B)
¬ is the outmost operator
A ∨ (B ∨ C ∧ ¬D)
∧ is nested within ∨
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Horn clauses

A Horn clause is a (disjunctive) clause with at most one positive
literal.
¬A
A¬B¬D
¬AB¬D
NOTE: ¬ A  ¬ B  D is equivalent to A  B  D
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