tp2623 : knowledge representation & reasoning

TOPIC 2 : PROPOSITIONAL LOGIC

TP2623 : Knowledge Representation &

Reasoning

By : Shereena ArifRoom : T2/8, Blok HEmail : shereen@ftsm.ukm.my / shereen.ma@gmail.com

Learning Outcomes

TP2623, shereen@ftsm.ukm.my

At the end of this topic, students will:Understand the concept of knowledge-bases

and propositional logicAble to construct knowledge base according

to propositional logic representation.

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Introduction

06/05/2011TP2623, shereen@ftsm.ukm.my

What is ‘Knowledge Representation & Reasoning’ ? Also known as KRR. Is a sub-area in AI- Artifical Intelligence.

Fundamental goal to represent knowledge in a manner that facilitates inferencing (i.e. drawing conclusions) from knowledge.

It analyzes how to formally think – how to use a symbol system to represent a domain of discourse

(that which can be talked about), along with functions that allow inference (formalized reasoning) about the objects

These concepts are central to the entire field of AI: The representation of knowledge The reasoning processes that bring knowledge to life

The importance of KR&R : To artificial agents successful behavior that would be very

hard to achieve otherwise. E.g : permainan catur.

In partially observable situation E.g : doktor dan pesakit

Understanding natural language E.g : Hamzah tertarik dengan sebentuk cincin di dalam kotak

pameran dan berhasrat membelinya. Flexibility to changes in the environment by updating the

relevant knowledge.

Knowledge-Based Agents

The central component of a knowledge-based agent is its knowledge base, KB.

Informally, a KB is a set of sentences related, but not identical to natural languages.

Each sentence is expressed in a language called knowledge representation language.

How to : Add new sentences into KB TELL Query about something already known ASK

Both tasks may involved inference deriving new sentences from old ones.

In logical agents, inference must obey the requirement : When one ASKs a question of the KB, The answer should follow from what has been told (or

TELLed) to the KB previously.

function KB-AGENT (percept) returns an action static : KB, a knowledge base

t, a counter, initially 0, indicating time

TELL (KB, MAKE-PERCEPT-SENTENCE(percept, t)) action ASK(KB, MAKE-ACTION-QUERY(t)) TELL(KB, MAKE-ACTION-SENTENCE(action, t)) t t + 1 return action

Agent takes percept as input and returns an action.

Agent maintains a knowledge base, KB, which may initially contain some background knowledge.

Each time the agent program is called it does two things. It TELLs the knowledge base what it perceives. It ASKs the knowledge base what action it should

perform.

In the process of answering, extensive reasoning may be done about the current state, about outcomes of possible actions, and so on.

Once the action is chosen, the agent records its choice with TELL and executes the action.

The wumpus world

The wumpus world is a cave consisting of rooms connected by passageways.

Lurking somewhere in the cave is the wumpus, a beast that eats anyone who enters its room

The wumpus can be shot by an agent, but the agent has only one arrow.

Some rooms contain bottomless pits that will trap anyone who wanders into these room (except the wumpus, who are too big).

Why ??.. The possibility of finding a heap of gold.Wumpus world an excellent testbed

environment for intelligent agents.

Wumpus World PEAS description

Performance measure gold +1000, death -1000 -1 per step, -10 for using the arrow

Environment Squares adjacent to wumpus are smelly Squares adjacent to pit are breezy Glitter if gold is in the same square Shoot kills wumpus if you are facing it Shooting uses up the only arrow Grab picks up gold if in same square Release drops the gold in same square

Sensors: The percept [Stench, Breeze, Glitter, Bump, Scream]

Actuators: Left turn, Right turn, Forward, Grab, Release, Shoot

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Exploring a wumpus world

[None, None, None, None, None] [None, Breeze, None, None, None]

[Stench, None, None, None, None] [Stench, Breeze, Glitter, None, None]

From the fact that there was no stench or breeze in [1,1], the agent can infer that [1,2] and [2,1] are free of dangers.

From the fact that it is still alive, it can infer than [1,1] is also OK.

A cautious agent will only move into a square that it knows is OK.

Let us suppose the agent moves to [2,1], giving us (b) in page 11.

The agent detects a breeze in [2,1], so there must be a pit in a neighbouring square, either [2,2] or [3,1]. (The notation P? indicates a possible pit.)

The pit cannot be in [1,1] because the agent has already been there and did not fall in.

At this point, there is only one known square that is OK and has not yet been visited.

So the prudent agent turns back and goes to [1,1] and then [1,2], giving us (a) in page 12 below.

The stench in [1,2] means that there must be a wumpus nearby.

But the wumpus cannot be in [1,1] and it cannot be in [2,2] (or the agent would have detected a stench when it was in [2,1].

Therefore, the agent can infer that the wumpus is in [1,3].

The notation W! indicates this.Moreover, the lack of a Breeze in [1,2] implies that there

is no pit in the [2,2]. Yet, we already inferred that there must be a pit in either

[2,2] or [3,1], so this means it must be in [3,1].This is fairly a difficult inference because it combines

knowledge gained at different times in different places.When the agent moved to [2,3], it detects a glitter, which

means that the gold must be there.

In each case where the agent draws a conclusion from the available information, that conclusion is guaranteed to be correct if the available information is correct fundamental property of reasoning.

In the rest of discussion, we will learn how to build logical agents that can represent the necessary information and draw the conclusions that were described.

Logic in general

Logics are formal languages for representing information so that conclusions can be drawn

Syntax defines the sentences in the languageSemantics define the "meaning" of sentences;

i.e., define truth of a sentence to each possible world

E.g x+y =4 is true if x=y=2, but false in x=y=1E.g., the language of arithmetic

x+2 ≥ y is a sentence; x2+ y > {} is not a sentence x+2 ≥ y is true if the number x+2 is no less than

the number y x+2 ≥ y is true in a world where x = 7, y = 1 x+2 ≥ y is false in a world where x = 0, y = 6

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Entailment

Entailment means that one thing follows from another:

KB ╞ αKnowledge base KB entails sentence α if and

only if in all worlds where KB is true, α is also true

Another way to say this is if KB is true, then α must also be true

E.g., the KB containing “the Rumah Kuning won” and “the Rumah Merah won” entails “Either the Rumah Kuning won or the Rumah Merah won”

E.g., x+y = 4 entails 4 = x+y Entailment is a relationship between sentences (i.e.,

syntax) that is based on semantics

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Entailment in the wumpus world

Situation after detecting nothing in [1,1], moving right, breeze in [2,1]

Consider possible models for KB assuming only pits

3 Boolean choices 8 possible models

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1 2 3 4

Wumpus models

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Wumpus models

KB = wumpus-world rules + observations

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Wumpus models

KB = wumpus-world rules + observationsα1 = "[1,2] is safe", KB ╞ α1, proved by model checkingIn every model in which KB is true, α1 is also true

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Wumpus models

KB = wumpus-world rules + observations

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Wumpus models

KB = wumpus-world rules + observations In some models in which KB is true, α2 is false α2 = "[2,2] is safe", KB ⊭ α2 The agent cannot conclude that [2,2] is safe.The example shows how definition of entailment can be

used to derive conclusions logical inference

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Inference

Entailment & inference : KB is a haystack & α is the needle Entailment needle in the haystack Inference how to find the needle

If an inference i can derive α from KB : KB ├i α “α can be derived from KB by procedure i” “i derives α from KB”

Soundness/Truth preserving : An inference algo is sound if it derives only entailed sentences i is sound if whenever KB ├i α, it is also true that KB╞ α an unsound inference procedure makes things up as it goes along

announces the discovery nonexistent needles. Model checking (if exist) is a sound procedure

Completeness An inference algo is complete if it can derive any sentence that is

entailed i is complete if whenever KB╞ α, it is also true that KB ├i α

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Recap : Described a reasoning process whose conclusions are

guaranteed to be true in any world in which the premises are true.

“if KB is true in the real world, then any sentence α is also true in the real world”

Sentences physical configurations of the agent Reasoning a process of constructing new physical

configurations from the old ones. Logical reasoning ensures that the new ones actually follow

the old ones.Preview: we will define a logic (first-order logic) which

is expressive enough to say almost anything of interest, and for which there exists a sound and complete inference procedure.

That is, the procedure will answer any question whose answer follows from what is known by the KB.

Propositional Logic or Calculus

Propositional logic A very simple logic Also called as Boolean logic. Syntax & Semantics The way in which the truth of

sentences is determined.

Syntax Defines allowable sentences.

Two types of sentences : Atomic sentences Complex sentences

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Atomic sentences (indivisible) consist of proposition symbol. Each symbol stands for a proposition that can be true

or false P, Q, R etc. Ex : W1,3 Wumpus is in [1,3] Two proposition symbols with fixed meaning : True &

False.

Complex sentences Constructed from simpler sentences. The sentences connected by logical connectives.

, , , ,

Logical Connectives

Not (negation) Ex : W1,3 negation of W1,3

And (conjunction) A B A and B

Or (disjunction) A B A or B

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Logical Connectives

Implies also known as rules or if-then statements. Conditional of A and B Implication of B given A Sometimes written as → or ⊃ Ex : (W1,3 P3,1 ) W2,2

Premise/Antecedent : (W1,3 P3,1 ) Conclusion/Consequent : W2,2

Ex : A B If A, then B A implies B B when A A only if B A is antecedent, B is consequent

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Logical Connectives

If and only if (biconditional or equivalence)A B is the same as (A B) (A B) if A, then B, and if B, then A

| (Sentence Sentence) | (Sentence V Sentence) | (Sentence Sentence) | (Sentence Sentence)

Propositional Logic Syntax ORValid Sentences

Logical constants by themselves T or F

Propositional symbol by itself P or Q

Parents around a sentence (P)

Sentences can be combined with logical connectives (P Q)

Order of precedence (highest to lowest) So P Q R S is the same as ((P) (Q R)) S

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Propositional logic: Syntax

Propositional logic is the simplest logic – illustrates basic ideas

The proposition symbols P1, P2 etc are sentences

If S is a sentence, S is a sentence (negation)

If S1 and S2 are sentences, S1 S2 is a sentence (conjunction)

If S1 and S2 are sentences, S1 S2 is a sentence (disjunction)

If S1 and S2 are sentences, S1 S2 is a sentence (implication)

If S1 and S2 are sentences, S1 S2 is a sentence (biconditional)

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Example Expressions

AFAA AA FAB(A B) (B)

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Propositional logic: Semantics

Semantics rules for determining the truth of a sentence with respect to a particular model

Each model specifies true/false for each proposition symbol Ex : P1,2 P2,2 P3,1

false true false m1 = {P1,2 = false, P2,2 = true, P3,1 = false}

The semantics for propositional logic must specify how to compute the truth value of any sentence, given a model.

Sentences are constructed from atomic sentences and the 5 connectives : Atomic sentences true in every model or false? Complex sentences rules, summarized in a truth

table.

Rules for evaluating truth with respect to a model m:S is true iff S is false S1 S2 is true iff S1 is true and S2 is trueS1 S2 is true iff S1is true or S2 is trueS1 S2 is true iff S1 is false or S2 is true i.e., is false iff S1 is true and S2 is falseS1 S2 is true iff S1S2 is true and S2S1 is true

Simple recursive process evaluates an arbitrary sentence, e.g.,P1,2 (P2,2 P3,1) = true (true false) = true true = true

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Truth Table

P QP Q P P Q P Q P Q

TF F T F F T

FF T T F T T

FT F F F T F

TT T F T T T

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Solving Problems with Truth Tables(Enumeration Method)

Problem If Ali starts playing with his Playstation, then he will stay

up late. If he stays up late, he will miss his morning class. He was in class.

Did he start playing his playstation?Propositions

P1: Ali plays playstation P2: Ali stays up late P3: Ali attends class

Premises P1 -> P2 P2 -> -P3 P3

Answer

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Truth Tables

Truth tables are effective, but impractical n propositions requires a table with 2n rows

More efficient methods of proving conclusions from premises exist E.g. Wang’s Algorithm

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Some Exercises Using Truth Table

(T F) T Answer:

(F F) T Answer:

((F T) F) ((T F)) Answer:

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Things we can say with Propositional Calculus

The sky is grey = S1It will rain = R1I have three arms = A1If S1 and R1 are T, then The sky is grey It will rain is also TIf S1 is T and A1 is F then The sky is grey I have three arms is T

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Combining expressions in Propositional Calculus

(A1 B1) C1(A2 ( C1 B4) C3(A4 B2) C2

Is equivalent to

((A1 B1) C1) ((A2 ( C1 B4) C3) ((A4 B2) C2)

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Logical equivalence

Two sentences are logically equivalent} iff true in same models: α ≡ ß iff α╞ β and β╞ α

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Logical equivalence

Two sentences are logically equivalent} iff true in same models: α ≡ ß iff α╞ β and β╞ α

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Validity and satisfiability

A sentence is valid if it is true in all models, also known as tautologye.g., True, A A, A A, (A (A B)) B

Validity is connected to inference via the Deduction Theorem:KB ╞ α if and only if (KB α) is valid

A sentence is satisfiable if it is true in some modele.g., A B, Ce.g, if sentence α is true in a model m m satisfies α, or m is a

model of α

A sentence is unsatisfiable if it is true in no modelse.g., AAα is valid iff α is unsatisfiable

Satisfiability is connected to inference via the following:KB ╞ α if and only if (KB α) is unsatisfiable

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Entailment: Propositional Inference(Enumeration Method)

Let = AB and KB = (AB)(B C)Is it the case that KB ╞ ?Check all possible models - must be true wherever KB is true

A B C AB B C KBF F FF F TF T FF T TT F FT F TT T FT T T

Answer

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The Method of Deduction

Previously we used the truth table to test arguments for validity

Here we develop an alternative approach the method of deduction

We can use rules of inference and rules of replacement

Covers standard patterns of inference that can be applied to derive chains of conclusions that lead to the desired goal inference rules

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Inference & Replacement with Propositional Calculus

Modus Ponens (Implication elimination)

Assume P Q, Given P ----------------- Then Q

For example:If the statement “If it is raining, then you get

wet” is true, and the statement “It is raining” is true, Then you can infer that “you get wet.”

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Disjunctive Syllogism (or introduction) Assume P -------------- Then P Q

For example:If the statement “It is raining” is true, Then you can infer that the compound statement “It

is raining” OR <anything> will be true

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Resolution

Assume P Q, Given P Z ----------------- Then Q Z

For example:If we assume “It is raining, or you go outside”, and the statement “It is not raining, or you stay

inside”, Then you can infer that “you go outside or you

stay inside.”

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And Elimination From a conjunction, you can infer any of the conjuncts E.g from (Wumpus Ahead ∧ Wumpus Alive), Wumpus

Alive can be inferred.

And introduction From a list of sentences, you can infer their

conjunction

Double negation elimination A double-negative makes a positive

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THANK YOU

… to be continued

06/05/2011

TP2623, shereen@ftsm.ukm.my 55

Let = AB and KB = (AB)(B C)

Is it the case that KB ╞ ?

Check all possible models - must be true wherever KB is true

06/05/2011

tp2623 : knowledge representation & reasoning

representation of knowledge

concept of knowledge

relevant knowledge

knowledge baset

background knowledge

action static

t action askkb

agent percept

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