1. Logics for Data and Knowledge Representation
Propositional Logic
Originally by Alessandro Agostini and Fausto Giunchiglia
Modified by Fausto Giunchiglia, Rui Zhang and Vincenzo Maltese

2. Outline Introduction Syntax Semantics Logical implication
Truth valuation
Satisfiability
Validity
Entailment
Logical implication
Reasoning Services
Normal Forms
Changed 2

3. Propositions
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All knowledge representation languages deal with sentences
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)
“All men are mortals” “Obama is the president of the USA”
Mortal  Man ObamaPresident

4. Introduction: example
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World
Mental
Model
Language L
Monkey
SEMANTIC
GAP
PROPOSITION
SENTENCE 4

5. Introduction: the famous triangle
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CONCEPT
(what we have in mind)
refers to
symbolizes
Monkey
REFERENT
(the real object)
SYMBOL
(what we write)
stands for 5

6. they can be substituted by any proposition or formula
Language (Syntax)
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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
Descriptive
Logical
, , , …
Constants
one proposition only
A, B, C …
Variables
they can be substituted by any proposition or formula
P, Q, ψ …
NOTE: not only characters but also words (composed by several characters) like “monkey” are descriptive symbols

7. Additional Symbols Auxiliary symbols Parentheses: ( ) Defined symbols
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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 ¬⊥
Defined symbols are not strictly necessary
However, they increase the usability by reducing the syntactic complexity of propositions

8. Formation Rules (FR): well formed formulas
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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
PARSER
ψ, PL
Yes, ψ is correct! No

9. Example: the Monkey-Banana problem
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“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}

10. Examples of formulas
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Formula
Correct
inLab-Monkey
inlab-monkey
in-lab-monkey
inLab(Monkey)
(Monkey ∧ GetBanana) ∨ ¬ High
Yes
(wff)
inLab(Monkey)∧
P ¬∧Q 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)

11. Propositional Theory Propositional (or sentential) theory Recall that:
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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

12. Propositional Theories and Databases
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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

13. The Monkey-Banana problem
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“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.

14. Semantics means providing an interpretation
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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

15. The Monkey-Banana problem (II)
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“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?

16. Semantics: formal model
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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.
CHANGED

17. The Monkey-Banana problem (III)
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L = {A, C}
T = {  (A  C) }
Language L
Theory T
I(A) = T
I(C) = F
Intensional Interpretation
Entailment
World
Mental
Model
M ⊨ T
Domain D
Model M
SEMANTIC
GAP
Formal model M(A) = T
M(C) = F
D = {T, F}
Truth values 17

18. Example: Banana (I)
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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).
CHANGED

19. Example: Banana (II) World Banana Banana  (Yellow  Curve) SEMANTIC
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World
Mental
Model
Language L
Language L’
Banana
Banana 
(Yellow  Curve)
CHANGED
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
SENTENCE
SENTENCE
SEMANTIC
GAP
OUR MENTAL MODEL MATCHES WITH THE REAL WORLD, SO THE SENTENCES ARE TRUE 19

20. Example: Banana (III) World Banana Banana  (Yellow  Curve) SEMANTIC
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World
Mental
Model
Language L
Language L’
Banana
Banana 
(Yellow  Curve)
CHANGED
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
SENTENCE
SENTENCE
SEMANTIC
GAP
OUR MENTAL MODEL DOES NOT MATCH WITH THE REAL WORLD, SO THE SENTENCES ARE FALSE 20

21. Example: Banana (IV) T F T F
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THE WORLD
CHANGED
MENTAL MODEL
MENTAL MODEL
I mean banana as something yellow and curve, no matter if partially opened
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 T F T F
FORMAL MODEL M1
FORMAL MODEL M2 21

22. Truth-values
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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) = T or F according to the modeler, with A atomic
ν(¬A) = T iff ν(A) = F
ν(A∧B) = T iff ν(A) = T and ν(B) = T
ν(A∨B) = T iff ν(A) = T or ν(B) = T
ν(⊥) = F (since ⊥=df P∧¬P)
ν(⊤) = T (since ⊤=df ¬⊥)
ν(A  B) = T iff ν(A) = F or ν(B) = T

23. Example of truth valuation ν
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L = {MonkeyLow, BananaHigh, MonkeyClimbBox, MonkeyGetBanana,
, , }
T = { (MonkeyLow  BananaHigh  MonkeyGetBanana)
 (MonkeyLow  MonkeyClimbBox)
 ( MonkeyLow   BananaHigh  MonkeyGetBanana)}
MODEL
Informal Semantics:
“If the monkey is low and the banana is high in position, then the monkey cannot get the banana. “
Formal Semantics:
ν(MonkeyLow) = T
ν(BananaHigh) = T
ν(MonkeyClimbBox) = F
ν(MonkeyGetBanana) = F 23

24. Truth-table (TT) for the logical constants ¬ , ∧, ∨
Truth Tables
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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 ¬ , ∧, ∨

25. Truth Relation (Satisfaction Relation)
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Let ν be a truth valuation on language L, we define the truth- relation (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

26. Model and Satisfiability
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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).

27. Truth and Validity Let ν be a truth valuation on language L.
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Let ν be a truth valuation on language L.
P is true under ν if ν ⊨ P
P is valid if ν ⊨ P for all ν (notation: ⊨ P)
P is called a tautology

28. Example
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L = {MonkeyLow, BananaHigh, MonkeyClimbBox, MonkeyGetBanana, , , }
MODEL
Informal Semantics:
“If the monkey is low and the banana is high in position, then the monkey cannot get the banana. “
Formal Semantics:
ν(MonkeyLow) = T
ν(BananaHigh) = T
ν(MonkeyClimbBox) = F
ν(MonkeyGetBanana) = F
CHANGED
ν ⊨ MonkeyLow and ν ⊨ BananaHigh [SATISFIABLE]
ν ⊨  MonkeyClimbBox and ν ⊨  MonkeyGetBanana [SATISFIABLE]
MonkeyLow   MonkeyLow is a tautology [VALID] 28

29. Entailment ν ⊨ θi for all θi in  implies ν ⊨ ψ
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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
NEW!!! 29

30. Example
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L = {MonkeyLow, BananaHigh, MonkeyClimbBox, MonkeyGetBanana, , , }
MODEL
Informal Semantics:
“If the monkey is low and the banana is high in position, then the monkey cannot get the banana. “
Formal Semantics:
ν(MonkeyLow) = T
ν(BananaHigh) = T
ν(MonkeyClimbBox) = F
ν(MonkeyGetBanana) = F
NEW!!!
 = {MonkeyLow, BananaHigh}
ν ⊨ MonkeyLow and ν ⊨ BananaHigh implies ν ⊨ MonkeyLow  BananaHigh
therefore  ⊨ MonkeyLow  BananaHigh [ENTAILMENT] 30

31. Entailment Difference in notation ⊨ ψ iff  ⊨ ψ
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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.
NEW!!! 31

32. Implication: Premise (I)
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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
UPDATED 32

33. Implication: Premise (II)
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Entailment can be viewed as the logical implication
(θ1 ∧ θ2 ∧ ... ∧ θn) → ψ
to be read θ1 ∧ θ2 ∧ ... ∧ θn logically implies ψ
→ is a new symbol that we add to the language 33

34. Implication and equivalence
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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
Remark: ⊥ and ⊤ are syntactically defined symbols: ⊥ =df P∧¬P, ⊤=df ¬⊥, →,↔ are semantically defined symbols! 34 34

35. Truth Table of →
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By defining the semantics of → in terms of ⊨ we have the following truth-table for the logical implication P→Q:
REMEMBER: Given ν, P ⊨ Q iff ν ⊨ P then ν ⊨ Q. 35

36. Three Properties of Logical Implication
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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) 36

37. Examples (tautologies)
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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) 37

38. Exercise (tautologies)
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Prove the following double implications (i.e., logical equivalences) by using truth tables.
(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 38

39. Exercise (tautologies)
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Prove the following double implications (i.e., logical equivalences) by using truth tables.
(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 ∧ 39

40. Reasoning Services
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The basic reasoning tasks (or “services”) we can represent (and compute) using ⊨ are:
Model Checking (EVAL)
Satisfiability (SAT)
Unsatisfiability (UnSAT)
Validity (VAL)
Entailment (ENT)
CHANGED

41. Reasoning Services: EVAL
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Model Checking (EVAL)
Is a proposition P true under a truth-valuation ν?
Check ν ⊨ P
EVAL
P , ν
Yes No
Does ν ⊨ MonkeyLow  MonkeyGetBanana ? 41 41

42. Reasoning Services: SAT
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Satisfiability (SAT)
Is there a truth-valuation ν where P is true?
find ν such that ν ⊨ P
SAT P ν No
Find a ν that ν ⊨ MonkeyLow  GetBanana
For instance, the truth-valuation ν where both are true 42 42

43. Reasoning Services (unSAT)
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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. 43

44. Reasoning Services: VAL
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Validity (VAL)
Is P true according to all possible truth-valuation ν?
Check whether for all ν, ν ⊨ P
VAL P
Yes No
Is P = MonkeyLow ∨  MonkeyLow valid? Yes! 44 44

45. Reasoning Services: ENT
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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
ENT
Γ , ψ, ν
Yes No
Does  MonkeyLow ⊨ MonkeyGetBanana ? No!
Does MonkeyLow  BananaHigh ⊨  MonkeyGetBanana? Yes! 45 45

46. Reasoning Services (Entailment)
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Exercises:
Demonstrate that {today} ⊨ today ∨ tomorrow
Demonstrate that {today} ⊭ tomorrow
Suppose P = today ∨ tomorrow. Then:
Define Γ such that Γ ⊨ P
Define Γ such that Γ ⊭ P
Remark: The entailment is a relationship between wff’s that is based on semantics

47. Reasoning Services: properties (I)
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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)
NEW 47 47

48. Reasoning Services: properties (II)
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VAL(ψ) iff UnSAT( ψ)
SAT(ψ)   VAL(ψ) iff SAT( ψ)   VAL( ψ)
 ⊨ ψ, where  = {θ1, …, θn} iff VAL(θ1  …  θn  ψ)
ψ = (A   A) is valid, while ( A  A) is unsatisfiable
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).
NEW
Take A ⊨ A  B. the formula (A  A  B) is valid. 48 48

49. Reasoning Services: properties (III)
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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.
NEW 49 49

50. Normal Forms: Literals
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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 HighBanana ¬GetBanana ¬ClimbBox 50

51. Normal Forms: Clauses
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A (disjunctive) clause is a disjunction of literals α1  …  αn
A unit clause is a clause that contains only a single literal
An empty clause does not contain literals
B ∨ ¬C ∨ ¬D Monkey ∨ ¬Monkey
B and ¬C are unit clauses B ∨ ¬C is not 51

52. Conjunctive Normal Form (CNF)
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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 ∨ 52

53. Conversion of a formula to CNF
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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):
Double-negation law (i.e. elimination of ¬¬)
De Morgan’s laws
Distributive laws
Elimination of → and  53

54. Conversion to CNF: Basic Tautologies
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(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 54

55. Example: Conversion to CNF of A ↔ (B ∨ C)
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Eliminate ↔:
(A→(B ∨C)) ∧ ((B ∨C) →A)
Eliminate →:
(¬A∨B ∨C) ∧ (¬(B ∨C) ∨A)
Move ¬ inwards using de Morgan’s laws: (¬A∨B ∨C) ∧ ((¬B ∧¬C) ∨A)
Apply distributivity of ∨ over ∧ and flatten:
(¬A∨B ∨C) ∧ (¬B ∨A) ∧ (¬C ∨A) (*)
(*) in CNF and equivalent to ‘A↔(B∨C)’. 55

56. Normal Forms: Conjunctive clauses
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A conjunctive clause is a conjunction of literals α1  …  αn
NOTE: a conjunctive clause IS NOT a clause!
B  ¬C  ¬D Monkey  ¬Monkey 56

57. Disjunctive Normal Form (DNF)
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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 ∨ 57

58. Horn clauses
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A Horn clause is a (disjunctive) clause with at most one positive literal.
NOTE: ¬ A  ¬ B  D is equivalent to A  B  D
¬ A
A  ¬ B  ¬ D
¬ A  B  ¬ D 58