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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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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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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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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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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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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)