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Logical Agents

CS 171/271(Chapter 7)

Some text and images in these slides were drawn fromRussel & Norvig’s published material

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Logic and Knowledge Bases Logic: means of representation and

reasoning Knowledge Base (KB): set of

sentences (expressed in some language)

Inference: deriving new sentences from sentences in the KB

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Knowledge-BasedAgent Function

TELL: adds a sentence to the KBASK: queries the KB

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Example: Wumpus World 4 by 4 grid of rooms A room may contain:

Agent, Wumpus, Pit, Gold Agent can perceive pit or

wumpus from neighboring squares

Agent starts in lower left corner, can move to neighboring squares, or shoot an arrow N,E,W, or S

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Wumpus WorldPEAS Description

Performance measure: gold +1000 death –1000 -1 per step -10 for using up arrow

Environment: 4 by 4 grid of rooms one room contains the agent (initially at [1,1] facing right) one room (not [1,1]) contains the wumpus (and it stays

there) one room contains the gold the other rooms may contain a pit

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PEAS Description, continued Actuators: Left turn, Right turn, Forward, Grab, Shoot

Shooting kills wumpus if you are facing it Shooting uses up the only arrow Grabbing picks up gold if in same square Agent dies when it enters a room containing pit/live

wumpus Sensors: Stench, Breeze, Glitter, Bump, Scream

Squares adjacent to wumpus are smelly Squares adjacent to a pit are breezy Glitter perceived in square containing gold Bump perceived when agent hits a wall Scream perceived everywhere when wumpus is hit

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Wumpus World and Knowledge State of knowledge

What is known about the rooms at time t Associate one or more values to each room,

when known: A, B, G, OK, P, S, V, W(use ? to indicate possibility)

Contrast against what are actually in the rooms A move and resulting percept allow agent

to update the state of knowledge Next move would depend on what is known

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Example: Initial Stateand First Move

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

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Sample Action Sequence: forward, turn around, forward,turn right, forward, turn right, forward, turn left, forward

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Later Moves

Actions: forward, turn around, forward, turn right,forward, turn right, forward, turn left, forward

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Inference Agent can infer that there is a

wumpus in [1,3] Stench in [1,2] means wumpus is in [1,1],

[1,3], or [2,2] Wumpus not in [1,1] by the rules of the game Wumpus not in [2,2] because [2,1] had no

stench

Agent can also infer that there is a pit in [3,1] (how?)

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Logic Representation

Syntax: how well-formed sentences are specified

Semantics: “meaning” of the sentences; truth with respect to each possible world (model)

Reasoning Entailment: sentence following from

another sentence ( a ╞ b )

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Models and Entailment Logicians typically think in terms of models, with

respect to which truth can be evaluated model: a possible world

We say m is a model of a sentence α if α is true in m M(α) is the set of all models of α Then KB ╞ α iff M(KB) M(α)

E.g.KB = I am smart and you are pretty α = I am smart

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Models and Entailmentin the Wumpus WorldSituation 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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Wumpus Models

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Wumpus Models

KB = wumpus-world rules + observations

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Wumpus Models

α1 = "[1,2] is safe", KB ╞ α1

proved by model checking

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Wumpus Models

α2 = "[2,2] is safe", KB ╞ α2

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Inference Algorithm An inference algorithm i is a

procedure that derives sentences from a knowledge base: KB ├i s

i is sound if it derives only entailed sentences

i is complete if it can derive any sentence that is entailed

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Propositional Logic (PL) PL: logic that consists of proposition

symbols and connectives Each symbol is either true or false Syntax: describes how the symbols

and connectives form sentences Semantics: describes rules for

determining the truth of a sentence wrt to a model

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Syntax A sentence in Propositional Logic is either

Atomic or Complex Atomic Sentence

Symbol: e.g., P, Q, R, … True False

Complex Sentence Let S and T be sentences (atomic or complex) The following are also sentences:

S, S T, S T, S T, S T

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Connectives S: negation

if P is a symbol, P and P are called literals S T: conjunction

S and T are called conjuncts S T: disjunction

S and T are called disjuncts S T: implication

S is called the premise, T is called the conclusion

S T: biconditional

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Back to the Wumpus World Start with a vocabulary of proposition

symbols, for example: Pi,j: there is a pit in room [i,j] Bi,j: there is a breeze in room [i,j]

Sample sentences (could be true or false) P1,2

B2,2 P2,3

P4,3 B3,3 B4,2 B4,4

P3,4 B1,3

Note issue of precedence with connectives

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Semantics Truth of symbols are specified in

the model Truth of complex sentences can be

determined using truth tables

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Knowledge Base forthe Wumpus World Rules constitute the initial KB and can be

expressed in PL; for example: P1,1

P4,4 B3,4 B4,3

As the agent progresses, it can perceive other facts and incorporate it in its KB; for example:

B1,1 if it doesn’t perceive a breeze in room [1,1] B2,1 if it perceives a breeze in room [2,1]

Can view the KB as a conjunction of all sentences asserted as true so far

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Inference in theWumpus World We want to decide on the existence

of pits in the rooms; i.e. does KB╞ Pi,j ?

Suppose we have already perceived B1,1 and B2,1

KB contains the rules and these facts What can we say about:

P1,1, P1,2, P2,1, P2,2, P3,1 ?

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Truth Table Depicting128 Possible Models

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Inference Examples KB is true when the rules hold—only for

three rows in the table The three rows are models of KB

Consider the value of P1,2 for these 3 rows P1,2 is false in all rows

(the rows are models of α1 = P1,2) Thus, there is no pit in room [1,2]

Consider the value of P2,2 for these 3 rows P1,2 false in one row, true for 2 rows Thus, there may be a pit in room [2,2]

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Inference by Enumeration We want an algorithm that determines

whether KB entails some sentence α Strategy:

Enumerate all possible models (true-false combinations of symbols in KB)

Consider only those models of KB (models where KB is true)

Return true if α is true for all such models

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Inference by Enumeration

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Analysis Inference by Enumeration is sound

and complete By definition of sound and complete

Runs in exponential time - O(2n) Requires linear space - O(n)

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To be continued…What’s next? Other Logical Inference Algorithms:

can’t really do better than exponential, but there are algorithms that do reasonably better in practice

First-order Logic (FOL):deals with a world of objects, functions, and relations, rather than just facts (PL)