wumpus world propositional logic

8/10/2019 Wumpus World Propositional Logic

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Agents thatReason Logically


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Knowledge and Reasoning

Reason for successful behaviors Perform well in complex environments Can adapt to changes in environment (Flexible)

Can combine general knowledge with current percepts to infer hidden aspects

Infer hidden states : ex A physician diagnose a disease not directly observable Natural Language understanding

John saw the diamond thru the window and coveted it John threw the stone thru the window and broke it

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A knowledge-based agent A knowledge-based agent includes a knowledge base and an

inference system . A knowledge base is a set of representations of facts of the world. Each individual representation is called a sentence . The sentences are expressed in a knowledge representation

language .

The agent operates as follows:1. It TELLs the knowledge base what it perceives.

2. It ASKs the knowledge base what action it should perform.3. It performs the chosen action.

Examples of sentencesThe moon is made of green cheeseIf A is true then B is trueA is false

All humans are mortalConfucius is a human


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Architecture of aknowledge-based agent

Knowledge Level. The most abstract level: describe agent by saying what it knows. Example: A taxi agent might know that the Sea Link Bridge

connects Bandra with Worli. Logical Level.

The level at which the knowledge is encoded into sentences. Example: Links(Sea Link, Bandra Reclamation, Worli).

Implementation Level. The physical representation of the sentences in the logical level. Example: (Map and routes of the area, Turn

right from SV Roadetc)


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The Inference engine derives new sentences from theinput and KB

The inference mechanism depends on representation in KB The agent operates as follows:

1. It receives percepts from environment2. It computes what action it should perform (by IE and KB)3. It performs the chosen action (some actions are simply

inserting inferred new facts into KB).

KnowledgeBase

InferenceEngine

Input fromenvironment

Output(actions)

Learning(KB update)


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Knowledge Level. The most abstract level -- describe agent by saying what it knows. Example: A taxi agent might know that the Golden Gate Bridge

connects San Francisco with the Marin County.

Logical Level. The level at which the knowledge is encoded into sentences. Example: Links(GoldenGateBridge, SanFrancisco,

MarinCounty).

Implementation Level. The physical representation of the sentences in the logical level. Example: (Links GoldenGateBridge, SanFrancisco,

MarinCounty)

KB can be viewed at different levelsreview


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The Wumpus World environment

The Wumpus computer game The agent explores a cave consisting of rooms connected by

passageways.

Lurking somewhere in the cave is the Wumpus , a beast thateats any agent that enters its room.

Some rooms contain bottomless pits that trap any agent thatwanders into the room.

Occasionally, there is a heap of gold in a room.

The goal is: to collect the gold and exit the world without being eaten


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Jargon file on Hunt the Wumpus WUMPUS /wuhm'p*s/ n. The central monster (and, in manyversions, the name) of a famous family of very early computer gamescalled Hunt The Wumpus, dating back at least to 1972 The wumpus lived somewhere in a cave with the topology of adodecahedron's edge/vertex graph

(later versions supported other topologies, including an icosahedron and

Mobius strip). The player started somewhere at random in the cave with fivecrooked arrows;

these could be shot through up to three connected rooms, and would kill the

wumpus on a hit (later versions introduced the wounded wumpus, which got very angry).


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Jargon file on Hunt the Wumpus (cont)

Unfortunately for players, the movement necessary to map the maze was made hazardous not merely by the wumpus

(which would eat you if you stepped on him)

There are also bottomless pits and colonies of super bats that would pick you upand drop you at a random location

(later versions added anaerobic termites that ate arrows, bat migrations, andearthquakes that randomly change pit locations).

This game appears to have been the first to use a non-random graph-structuredmap (as opposed to a rectangular grid like the even older Star Trek games).

In this respect, as in the dungeon-like setting and its terse, amusing messages, it prefigured ADVENT and Zork.It was directly ancestral to both.

(Zork acknowledged this heritage by including a super-bat colony.) Today, a port is distributed with SunOS and as freeware for the Mac.

A C emulation of the original Basic game is in circulation as freeware on the net.


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A typical Wumpus world

The agent always startsin the field [1,1].

The task of the agentis to find the gold,return to the field [1,1]and climb out of thecave.


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Agent in a Wumpus world: Percepts

The agent perceives a stench in the square containing the wumpus and in theadjacent squares (not diagonally)

a breeze in the squares adjacent to a pit

a glitter in the square where the gold is a bump , if it walks into a wall a woeful scream everywhere in the cave, if the wumpus is

killed

The percepts will be given as a five-symbol list : If there is a stench, and a breeze, but no glitter, no bump, and no scream,

the percept is[Stench, Breeze, None, None, None]

The agent can not perceive its own location .


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The actions of the agent in Wumpus game are:

go forward

turn right 90 degrees turn left 90 degrees grab means pick up an object that is in the same square as the agent shoot means fire an arrow in a straight line in the direction the agent

is looking. The arrow continues until it either hits and kills the wumpus or hits the wall. The agent has only one arrow. Only the first shot has any effect.

climb is used to leave the cave. Only effective in start field.

die , if the agent enters a square with a pit or a live wumpus. (No take-backs!)


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The agents goal

The agents goal is to find the gold and bring it back to the start as quickly as possible,without getting killed .

1000 points reward for climbing out of thecave with the gold

1 point deducted for every action taken

10000 points penalty for getting killed


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The Wumpus agents first step


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Later


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World-wide web wumpuses

http://scv.bu.edu/wcl http://216.246.19.186

http://www.cs.berkeley.edu/~russell/code/doc/overview-AGENTS.html
http://scv.bu.edu/wclhttp://216.246.19.186/http://www.cs.berkeley.edu/~russell/code/doc/overview-AGENTS.htmlhttp://www.cs.berkeley.edu/~russell/code/doc/overview-AGENTS.htmlhttp://www.cs.berkeley.edu/~russell/code/doc/overview-AGENTS.htmlhttp://www.cs.berkeley.edu/~russell/code/doc/overview-AGENTS.htmlhttp://www.cs.berkeley.edu/~russell/code/doc/overview-AGENTS.htmlhttp://216.246.19.186/http://scv.bu.edu/wcl


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Representation, reasoning, and logic The object of knowledge representation is to express knowledge in

a computer-tractable form , so that agents can perform well. A knowledge representation language is defined by:

its syntax , which defines all possible sequences of symbols thatconstitute sentences of the language. Examples: Sentences in a book, bit patterns in computer memory. X+Y = 4

its semantics , which determines the facts in the world to whichthe sentences refer. Each sentence makes a claim about the world. An agent is said to believe a sentence about the world.


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The connection betweensentences and facts

Semantics maps sentences in logic to facts in the world.The property of one fact following from another is mirrored

by the property of one sentence being entailed by another .


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Logic as a Knowledge-Representation (KR)language

Propositional Logic

First Order

Higher Order

Modal

FuzzyLogic

Multi-valuedLogic

ProbabilisticLogic

Temporal Non-monotonicLogic


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Ontology and epistemology Ontology is the study of what there is,

an inventory of what exists. An ontological commitment is a commitment to an existence claim.

Epistemology is major branch of philosophy that concerns the forms, nature, and preconditions of knowledge.


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Propositional logic

Logical constants : true, false Propositional symbols : P, Q, S, ...

Wrapping parentheses : ( ) Sentences are combined by connectives :

...and

...or

...implies

..is equivalent

...not


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Propositional logic (PL)

A simple language useful for showing key ideas and definitions User defines a set of propositional symbols , like P and Q. User defines the semantics of each of these symbols, e.g.:

P means "It is hot" Q means "It is humid"

R means "It is raining" A sentence (aka formula, well-formed formula, wff ) defined as:

A symbol

If S is a sentence, then ~S is a sentence (e.g., "not) If S is a sentence, then so is (S) If S and T are sentences, then (S v T) , (S ^ T) , (S => T) , and (S T)

are sentences (e.g., "or," "and," "implies," and "if and only if)

A finite number of applications of the above


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Examples of PL sentences

(P ^ Q) => R If it is hot and humid, then it is raining

Q => P If it is humid, then it is hot

QIt is humid.

A better way:Ho = It is hot Hu = It is humid R = It is raining


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A BNF grammar of sentences in

propositional logicS := ;

:= | ;

:= "TRUE" | "FALSE" |

"P" | "Q" | "S" ; := "(" ")" |

|

"NOT" ;

:= "NOT" | "AND" | "OR" | "IMPLIES" |"EQUIVALENT" ;


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Terms: semantics, interpretation, model, tautology,contradiction, entailment

The meaning or semantics of a sentence determines its interpretation .

Given the truth values of all of symbols in a sentence, it can beevaluated to determine its truth value (True or False).

A model for a KB is a possible world in which each sentence in theKB is True.

A valid sentence or tautology is a sentence that is True under allinterpretations, no matter what the world is actually like or what thesemantics is. Example: Its raining or its not raining.

An inconsistent sentence or contradictio n is a sentence that is Falseunder all interpretations. The world is never like what it describes, as in Its raining and it's not raining.

P entails Q , written P |= Q, means that whenever P is True, so is Q.

In other words, all models of P are also models of Q.


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Truth tables


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Truth tables II

The five logical connectives:

A complex sentence:


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Models of complex sentences


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Inference rules

Logical inference is used to create new sentences thatlogically follow from a given set of predicate calculussentences (KB).

An inference rule is sound if every sentence X produced

by an inference rule operating on a KB logically followsfrom the KB. (That is, the inference rule does not create any contradictions )

An inference rule is complete if it is able to produceevery expression that logically follows from the KB. We also say - expression is entailed by KB. Pleas note the analogy to complete search algorithms .


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Sound rules of inference

Here are some examples of sound rules of inference . Each can be shown to be sound using a truth table :

A rule is sound if its conclusion is true whenever the premise is true .

RULE PREMISE CONCLUSION Modus Ponens A, A => B BAnd Introduction A, B A ^ BAnd Elimination A ^ B A

Double Negation ~~A AUnit Resolution A v B, ~B AResolution A v B, ~B v C A v C


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Sound Inference Rules (deductive rules)

Here are some examples of sound rules of inference . Each can be shown to be sound using a truth table -- a rule is

sound if its conclusion is true whenever the premise is true.

RULE PREMISE CONCLUSION

Modus Tollens ~B, A => B ~AOr Introduction A A v BChaining A => B, B => C A => C

d f d


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Soundness of modus ponens

A B A B OK?

True True True

True False False

False True True

False False True

premise conclusion premise


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Soundness of the

resolution inference rule

Whenever premise is true, theconclusion is also true

But it may be also in other cases


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Resolution ruleUnit Resolution A v B, ~A B

Resolution A v B, ~B v C A v C

Operates on two disjunctions of literals The pair of two opposite literals cancel each

other , all other literals from the two disjuncts are combined toform a new disjunct as the inferred sentence

Resolution rule can replace all other inference rulesModus Ponens A, ~A v B BModus Tollens ~B, ~A v B ~AChaining ~A v B, ~B v C ~A v C

k ik i

k i

...... ...,~ ,... Thenliterals. be...,,...Let

1111

11

)~ and (

These oldrules arespecialcases of

resolution


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Proving things : what are proofs andtheorems?

A proof is a sequence of sentences, where each sentence is eithera premise or a sentence derived from earlier sentences in the proof

by one of the rules of inference. The last sentence is the theorem (also called goal or query) that

we want to prove. Example for the weather problem given above.

1 Hu Premise It is humid 2 Hu=>Ho Premise If it is humid, it is hot 3 Ho Modus Ponens(1,2) It is hot 4 (Ho^Hu)=>R Premise If its hot & humid, its raining 5 Ho^Hu And Introduction(1,2) It is hot and humid

6 R Modus Ponens(4,5) It is raining


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Proof by resolution

Theorem proving as search Start node: the set of given premises/axioms (KB + Input) Operator: inference rule (add a new sentence into parent node) Goal: a state that contains the theorem asked to prove Solution: a path from start node to a goal

Q ~Q v P ~P v ~Q v R premises

P

~Q v R

R theorem


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Normal forms of PL sentences

Disjunctive normal form ( DNF ) Any sentence can be written as a disjunction of conjunctions of literals. Examples: P ^ Q ^ ~R; A^B v C^D v P^Q^R; P Widely used in logical circuit design (simplification)

Conjunctive normal form ( CNF ) Any sentence can be written as a conjunction of disjunctions of literals.

Examples: P v Q v ~R; (A v B) ^ (C v D) ^ (P v Q v R); P Normal forms can be obtained by applying equivalence laws

[(A v B) => (C v D)] => P ~[~(A v B) v (C v D)] v P // law for implication

[~~(A v B) ^ ~(C v D)] v P // de Morgans law [(A v B)^(~C ^ ~D)] v P // double negation and de Morgans law (A v B v P)^(~C^~D v P) // distribution law (A v B v P)^(~C v P)^(~D v P) // a CNF


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Horn sentences

A Horn sentence or has the form:P1 ^ P2 ^ P3 ... ^ Pn => Q Horn clause

or alternatively~P1 v ~P2 v ~P3 ... V ~Pn v Q

where Ps and Q are non-negated atoms Horn Clause is a disjunction of literals of which

atmost one is positive. To get a proof for Horn sentences, apply Modus

Ponens repeatedly until nothing can be done

(P => Q) = (~P v Q)


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Horn clauses Horn Clauses with exactly one positive literal are called

Definite Clauses ( form the basis of Logic Programming) The Positive Literal is called the Head

The Negative Literals form the Body A definite clause with no negative literals simply assrets a

given proposition called a Fact Sentences with no positive literals are called integrity

constraints

40


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Entailment and derivation

Entailment: KB |= Q Q is entailed by KB (a set of premises or assumptions) if and

only if there is no logically possible world in which Q is false

while all the premises in KB are true. Or, stated positively, Q is entailed by KB if and only if theconclusion is true in every logically possible world in which allthe premises in KB are true.

Derivation: KB |- Q We can derive Q from KB if there is a proof consisting of a

sequence of valid inference steps starting from the premises inKB and resulting in Q


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Two important properties for inference

Soundness: If KB |- Q then KB |= Q If Q is derived from a set of sentences KB using a given set of rules of

inference, then Q is entailed by KB. Hence, inference produces only real entailments,

or any sentence that follows deductively from the premises is valid .

Completeness: If KB |= Q then KB |- Q If Q is entailed by a set of sentences KB, then Q can be derived from KB

using the rules of inference.

Hence, inference produces all entailments, or all valid sentences can be proved from the premises.

P iti l l i i k l


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Propositional logic is a weak language Hard to identify "individuals ." E.g., Mary, 3

Individuals cannot be PL sentences themselves.

Cant directly talk about properties of individuals or relations betweenindividuals. (hard to connect individuals to class properties). E.g., property of being a human implies property of being mortal E.g. Bill is tall

Generalizations, patterns, regularities cant easily be represented . E.g., all triangles have 3 sides All members of a class have this property Some members of a class have this property

A better representation is needed to capture the relationship (anddistinction) between objects and classes , including properties

belonging to classes and individuals.


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Confusius Example: weakness of PL

Consider the problem of representing the followinginformation: Every person i s mortal. Confucius is a person. Conf ucius is mor tal.

How can these sentences be represented so that we can inferthe third sentence from the first two?


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Example continued

In PL we have to create propositional symbols to stand for all or part of each sentence. For example, we might do:

P = person; Q = mortal; R = Confucius

so the above 3 sentences are represented as:P => Q; R => P; R => Q

Although the third sentence is entailed by the first two, we neededan explicit symbol, R , to represent an individual, Confucius, whois a member of the classes person and mortal.

To represent other individuals we must introduce separatesymbols for each one, with means for representing the fact that allindividuals who are people are also "mortal.

PL is Too Weak a Representational


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PL is Too Weak a RepresentationalLanguage

Consider the problem of representing the following information: Every person is mortal. (S1) Confucius is a person. (S2) Confucius is mortal. (S3)

S3 is clearly a logical consequence of S1 and S2. But how can these sentences be represented using PL so that we can infer the third

sentence from the first two?

We can use symbols P, Q, and R to denote the three propositions, but this leads us to nowhere because knowledge important to infer R from P

and Q (i.e., relationship between being a human and mortality, and the membership

relation between Confucius and human class) is not expressed in a way thatcan be used by inference rules


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Alternatively, we can use symbols for parts of each sentence P = "person; M = "mortal; C = "Confucius" The above 3 sentences can be roughly represented as:

S2: C => P; S1: P => M; S3: C => M. Then S3 is entailed by S1 and S2 by the chaining rule.

Bad semantics

Confucius (and person and mortal) are not PL sentences ( not adeclarative statement ) and cannot have a truth value. What does P => M mean?

We need infinite distinct symbols X for individual persons , andinfinite implications to connect these X with P (person) and M

(mortal) because we need a unique symbol for each individual.Person_1 => P; person_1 => M;Person_2 => P; person_2 => M;

... ...Person_n => P; person_n => M


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First-Order Logic (abbreviated FOL or FOPC) is expressiveenough to concisely represent this kind of situation by separatingclasses and individuals Explicit representation of individuals and classes, x, Mary, 3,

persons. Adds relations, variables, and quantifiers, e.g.,

Every person is mortal Forall X: person(X) => mortal(X) There is a white alligator There exists some X: Alligator(X) ^

white(X)


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The Hunt the Wumpus agent

Some Atomic PropositionsS12 = There is a stench in cell (1,2)B34 = There is a breeze in cell (3,4)W22 = The Wumpus is in cell (2,2)V11 = We have visited cell (1,1)OK11 = Cell (1,1) is safe.etc

Some rules(R1) ~S11 => ~W11 ^ ~W12 ^ ~W21(R2) ~S21 => ~W11 ^ ~W21 ^ ~W22 ^ ~W31(R3) ~S12 => ~W11 ^ ~W12 ^ ~W22 ^ ~W13(R4) S12 => W13 v W12 v W22 v W11etc

Note that the lack of variables requires us to give similarrules for each cell.


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After the third move

We can prove that theWumpus is in (1,3) usingthe four rules given.

See R&N section 6.5

Proving W13~S11(R1) ~S11 => ~W11 ^ ~W12 ^ ~W21


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Proving W13 Apply MP with ~S11 and R1:

~W11 ^ ~W12 ^ ~W21

Apply And-Elimination to this we get 3 sentences:~W11, ~W12, ~W21

Apply MP to ~S21 and R2, then applying And-elimination:~W22, ~W21, ~W31

Apply MP to S12 and R4 we obtain:W13 v W12 v W22 v W11

Apply Unit resolution on (W13 v W12 v W22 v W11) and ~W11W13 v W12 v W22

Apply Unit Resolution with (W13 v W12 v W22) and ~W22W13 v W12

Apply UR with (W13 v W12) and ~W12

W13 QED

~W11 ^ ~W12 ^ ~W21

(W13 v W12 v W22) ~W22

W13 v W12

~W12

W13

(W13 v W12 v W22 v W11) ~W11

S12 R4

~S21

R2~W22, ~W21, ~W31


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Problems with thepropositional Wumpus hunter

Lack of variables prevents stating more general rules.

E.g., we need a set of similar rules for each cell Change of the KB over time is difficult to represent

Standard technique is to index facts with the time

when theyre true This means we have a separate KB for every time

point.

S


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Summary Intelligent agents need knowledge about the world for making

good decisions. The knowledge of an agent is stored in a knowledge base in the

form of sentences in a knowledge representation language . A knowledge-based agent needs a knowledge base and an

inference mechanism . It operates by storing sentences in its knowledge base, inferring new sentences with the inference mechanism, and using them to deduce which actions to take.

A representation language is defined by its syntax and semantics,which specify the structure of sentences and how they relate to thefacts of the world.

The interpretation of a sentence is the fact to which it refers. If this fact is part of the actual world, then the sentence is true.


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wumpus world propositional logic

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