AI Representation(models) and

Propositional Logic

Why do we need to go over these chapters?

Computer SolutionMain.Play(board) newboard=copy(board) do until winner newboard = find.best.move(board) newboard = you.move end main.play

find.best.move(board) generate.all.moves(board) find.legal.moves(board) find.best.move (board) ;by calling min.maxend find.best.move

How would you do it?

Go for the cornersGo for the middle

Another…mine sweeper

You have to find a way from a start field toan exit field on a board where mines arehidden. An electronic device tells you howmany mines are hidden on the neighboringfields (vertically, horizontally, anddiagonally). However, the device is unableto give the exact location of the mines that ithas detected. How do you proceed?

board

Final-board

Knowledge

• Once we have walked on the board for a while, how can we express the knowledge that we have acquired?

• The initial cell, for instance, has no bomb. We could express this type of knowledge with 0/1 variables that tell us, for each cell, whether there is a bomb or not: B13 = 0.

• When we see that our device can detect no mines in the proximity of the start field, we can express that knowledge as B12+B14+B22+B23+B24 = 0.

Knowledge base

In the given situation, among other things, we know that:• B11=B12=B13=B14=B21=B22=B23=B24=0.• B31+B32+B22+B12+B11 =0.• B31+B32+B33+B21+B23+B11+B12+B13=1.• Humans can easily deduce thatB31=B32=0 and B33=1.• Do you have an idea how a computer can perform

this inference?

Knowledge & reasoning

• To conduct rational inference, we first need to be able to state the rules and the known facts about a problem.

• The internal representation of facts and rules is called a knowledge base (KB).

• The KB is the starting point for all inference and reasoning that we may want to perform.

• Therefore, we need a kind of language that allows us to express what we know.

Dancing queen

• Here is another example:– If you have money, then you can buy an iPod.– If you have time and an iPod, then you can

download good dancing songs.– If you have time and money, you can learn how to

dance.– If you have an iPod and dancing songs and you

know how to dance, then you can throw a party and be a dancing queen.

Dancing Queen

Representations are Models of the world• If we are going to create programs that are

intelligent, then we need to figure out how to represent models of the world

• An important aspect of an AI agent is its model of the world

• As humans, we carry sophisticated models of the world in our heads

• They allow us to predict certain things about the future, play scenarios though in our heads to see what might happen, and not be constantly surprised by everything we see

The Role of a Model• Represent the environment• Provide a structure for the assimilation of new

knowledge• Be able to change in the light of new evidence,

but not too readily• Dry run without the need for sensor input or

actual actuator output• Be able to generate new facts that have not been

sensed from those that have

Logic

“Contrariwise,” continued Tweedledee, “if it was so, it might be, and if it were so, it would be; but as it isn’t, it ain’t. That’s logic!”

– Lewis Carroll

Propositional Logic• One way to represent or model the world is

propositional logic.• It can reason about such logical implications, we

introduce Propositional Logic (PL).• It allows us – to express simple truth facts like “I have money” or “I

have an iPod”, – to express logical statements like “If I have money then I

have an iPod”, and– to draw conclusions like “If I have money and (if I have

money then I have an iPod) then I have an iPod”.

What is a logic?

• A formal language– Syntax – what expressions are legal (well-formed)– Semantics – what legal expressions mean

• in logic the truth of each sentence with respect to each possible world.

• E.g the language of arithmetic– X+2 >= y is a sentence, x2+y is not a sentence– X+2 >= y is true in a world where x=7 and y =1– X+2 >= y is false in a world where x=0 and y =6

Logic consists of a representation…

• Logical constants: true, false• Proposition symbols: P, Q, R, ...• Logical connectives: & (or ^), , ¬, →, ↔ • Parentheses: ( )• Propositional logic is an extremely simple

representation

Propositional Logic - Syntax

• Sentences in Propositional Logic are defined in Backus-Naur Form (BNF):– A variable symbol (P,Q,R,…), and the constants

True, False are correct sentences.– Given correct sentences a, b: , ¬, →, ↔ • (¬ a) is a correct sentence. (negation)• (a ^ b) is a correct sentence. (conjunction)• (a b) is a correct sentence. (disjunction)• (a → b) is a correct sentence. (implication)• (a ↔ b) is a correct sentence. (equivalence)

Propositional Logic

• Represents facts as being either true or false– Formally represented by a letter, e.g. P or Q These

symbols can represent whole propositions such as Elvis is alive.

– Actually refer to facts about the environment, e.g. the speed limit in town is 30mph

• Single facts can be combined into sentences using Boolean operators such as (^ V )

• These sentences, if true, can become facts in the KB.• A KB is said to entail a sentence if it is true in the KB

Propositional Logic - Syntax

• To avoid the excessive use of parentheses, we agree to use the following convention on the order of precedence of operators:

¬ (highest), ^ , V , => , (lowest)• Now, we can write ((((¬P) V (Q ^ R)) => True) S)as ¬P V Q ^ R => True S• Even though these sentences are ambiguous in syntax because

of their unique semantics (to be defined next) we allow sentences like P V Q V R, P ^ Q ^ R, and P Q R. Sentences like P => Q => R are not allowed.

Propositional Logic - Syntax• A model or instantiation to a sentence in

propositional logic is an assignment of truth values to each variable: – For the sentence ¬P Q ^ R → True ↔ S

potential models are:– m1 = { P=true, Q=true, R=false, S=false }– m2 = { P=false, Q=true, R=false, S=true}

• We write a|m to denote the evaluation of a under model m.

Models

• Logicians typically think in terms of models, which are formally structured worlds with respect to which truth can be evaluated.

• m is a model of a sentence if is true in m• M() is the set of all models of

Syntax rules for propositional logic

• The constants true and false are propositions by themselves.

• A proposition symbol such as P or Q is a proposition by itself.

• Wrapping parentheses around a proposition produces proposition.

• A proposition can be formed by using a logical connective to combine two propositions.

Constants or proposition symbols by themselves make atomic propositions. Propositions containing a connective are called complex propositions.

Semantics

• A proposition symbol can mean anything at all. The logical system has no ”understanding” of its meaning.

• The semantics of propositional logic specifies the meaning of the logical connectives and how they combine propositions.

• A truth table can be used to specify the semantics of the logical

For example

Propositional Logic - Semantics• Given a model m for a sentence in BNF, the truth value of the

sentence is evaluated as follows:– Variable symbols P,Q,R,… are evaluated according to m.

True is evaluated as true and False as false.– For sub-sentences a and b we define:• (¬ a) is true iff a is false.• (a ^ b) is true iff a and b are both true.• (a b) is true iff a or b (or both) are true.• (a ¬, → b) is true iff a is false or b is true.• (a ↔ b) is true iff a and b are both true or both false.

• The semantics imply that a → b is logically equivalent to ¬a V b. Moreover, a b is logically equivalent to (¬a , b) ^ (a ¬b).

Propositional Logic

• Now what does “logically equivalent” or “iff (if and only if)” mean?

• We run into the problem here that anything that we may want to state about propositional logic involves the same language as propositional logic itself!

• Therefore, when making statements about PL, we use terms like “implies” or “iff” so as not to confuse statements that we make about PL with sentences in PL.

Propositional Logic - Semantics• Definition– When saying that two sentences a and b are

logically equivalent we mean to say that both sentences evaluate to the same truth value no matter what model we use for evaluation!

– We say that a sentence is valid or a tautology iff it evaluates to true under all models.

• A sentence a is satisfiable iff there exists a model under which a evaluates to true. Then, the model m such that a|m is true is said to satisfy a.

Propositional Logic - Semantics• Theorem: , ¬, →, ↔ – commutativity: a ^ b iff b ^ a, and a b iff b a.– associativity: (a ^ b) ^ c iff a ^ (b ^ c), – (a b) c iff a (b c), and – (a ↔ b) ↔ c iff a ↔ (b ↔ c).– double negation elimination: ¬(¬a) iff a.– contraposition: a → b iff ¬b → ¬a.– implication elimination: a → b iff (¬a b).– equivalence elimination: a ↔ b iff (¬a ^ ¬ b) (a ^ b).– De Morgan: ¬(a ^ b) iff ¬a ¬b, and ¬(a b) iff ¬a ^ ¬b.– distributivity: a ^ (b c) iff a ^c V a ^ b, and– a V (b ^ c) iff (a V c) ^ (a V b)

Propositional Logic - Semantics

• A model or instantiation to a sentence in propositional logic is an assignment of truth values to each variable: – For the sentence ¬P Q ^ R → True ↔ S potential

models are:• m1 = { P=true, Q=true, R=false, S=false }• m2 = { P=false, Q=true, R=false, S=true}

– Now, under which models does this sentence evaluate to be true?

• Why did we not allow sentences like P=> Q => R?