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Provides the ability to access individual assertions.

e.g. in Predicate calculus we may say:P denotes “It rained on Tuesday” but in

predicate calculus we may have a relation called weather(tuesday,rain).

Also we can have variables such as weather(X,rain)

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1. The set of letters, both upper and lower case, of English alphabet

2. The set of digits: 0, 1, … 93. The underscore ‘_’

Symbols begin with a letter and followed by any sequence of legal letters

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Constant symbols begin with “small” letters Symbols true and false are reserved

constants Variables are used to designate general

classes of objects or properties. Variables begin with “capital” letters

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They start with “small” letters

They have arity, e.g. father(?) has one arity while Plus(?,?) has 2 arity, …etc.

Examples: f(X,Y), father(david), price(house)

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Is a predicate of arity n followed by n terms enclosed in parentheses and separated by commas.

Examlpes:likes(ahmed,Chocolate) likes(X,ahmed)likes(X,Y)

friends(aly,ahmed)friends(father_of(ahmed),father_of(aly))

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¬ (universal quantifier)(existential quantifier)

Examples: Y friends(Y,peter)

X likes(X,ice_cream)

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Assume time and plus are function symbols of arity 2 and assume equal and good be predicate symbols of arity 2 and 3, respectively.

plus(two,three) is a function and thus not an atomic sentence

equal(plus(two,three),five) is an atomic sentenceequal(plus(2,3),seven) is a sentence (although

seems wrong computationally) X good(X,two,plus(two,three))

equal(plus(two,three),five) is a sentence(good(two,two,plus(two,three)))

(equal(plus(three,two),five) true ) is a sentence

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mother(amina,aly) mother(amina,mahmoud) father(hasan,aly) father(hasan,mahmoud)

X Y (father(X,Y) mother(X,Y) parent(X,Y))

X Y Z (parent(X,Y) parent(X,Z) brother(Y,Z))

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on(c,a)on(b,d)ontable(a)ontable(d)clear(b)clear(c )hand_emptyRule describing when a block is clear:X(¬ Y on(Y,X)) clear(X))(i.e. for all X, X is clear if there does not

exist a Y such that Y is on X)

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c

a

b

d

To stack one block on top of another:Assume we want to stack X on Y:1. Empty the hand2. Clear X3. Clear Y4. pick_up X and put_down X on YX Y ((hand_empty clear(X) clear(Y)

pick_up(X) put_down(X,Y)) stack(X,Y))

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An interpretation that makes a sentence true is said to satisfy that sentence.

An interpretation that satisfies every member of a set of expressions is said to satisfy the set.

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A proof procedure is a combination of an inference rule and an algorithm for applying that rule to a set of logical expressions to generate new sentences

Defn: a predicate calculus expression X logically follows from a set S of predicate calculus expressions if every interpretation and assignment that satisfies S also satisfies X.

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Modus ponens:If P is true and PQ is known to be true then

we can infer Q. Modus tollens:If P Q is known to be true and Q is known to

be false we can infer ¬P And eliminationIf PQ is true the P is true and Q is true And introduction:If P and Q are true then PQ is true Universal Instantiation:If a is from the domain of X, X p(X) lets us

infer p(a)

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“If it is raining then the ground will be wet” and we know “It is raining” then:

P denotes “It is raining” Q denotes “The ground is wet” i.e. PQ is true and P is true then we can

infer Q (i.e. The ground is wet)

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“All men are mortal and Socrates is a man” “Is Socrates mortal?”

This sentence can be represented as:X (man(X) mortal(X))man(socrates) If we substitute socrates for X we get:man(scorates) moral(socrates)Then we can now infer: mortal(socrates)

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Unification is an algorithm for determining the substitutions needed to make two predicate calculus expressions match.

Unification+inference rules (e.g. modus ponens) allow us to make inferences on a set of logical assertions.

To do this, the logical database must be expressed in an appropriate form.

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This is done by replacing all existentially quantified variables by their corresponding constants:

e.g. X parent(X,tom) is being replaced by parent(bob,tom) or parent(mary,tom).

Complication : X Y mother(X,Y) where the value of Y is dependent on X

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