Transcript CMSC 723: Introduction to Computational Linguistics

Notes adapted from lecture notes for CMSC 421 by B.J. Dorr
logic agents
“Thinking Rationally”
• Computational models of human “thought”
processes
• Computational models of human behavior
• Computational systems that “think” rationally
• Computational systems that behave rationally
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Logical Agents
• Reflex agents find their way from Arad to
Bucharest by dumb luck
• Chess program calculates legal moves of its
king, but doesn’t know that no piece can be
on 2 different squares at the same time
• Logic (Knowledge-Based) agents combine
general knowledge with current percepts to
infer hidden aspects of current state prior to
selecting actions
– Crucial in partially observable environments
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Outline
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Knowledge-based agents
Wumpus world
Logic in general
Propositional and first-order logic
– Inference, validity, equivalence and satifiability
– Reasoning patterns
• Resolution
• Forward/backward chaining
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Knowledge Base
Knowledge Base : set of sentences represented in a
knowledge representation language and represents
assertions about the world.
tell
ask
Inference rule: when one ASKs questions of the KB, the
answer should follow from what has been TELLed to the
KB previously.
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Generic KB-Based Agent
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Generic KB-Based Agent
• Takes percent as input and returns an action
• agent maintains a knowledge base KB
• Each time agent program is called it does
– TELLs knowledge base what It perceives
– ASKs knowledge base what action it should
perform
– Agent records its choice with TELL an executes
action
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Abilities KB agent
• Agent must be able to:
– Represent states and actions,
– Incorporate new percepts
– Update internal representation of the world
– Deduce hidden properties of the world
– Deduce appropriate actions
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Desription level
• The KB agent is similar to agents with internal
state
• Agents can be described at different levels
– Knowledge level
• What they know, regardless of the actual
implementation. (Declarative description)
– Implementation level
• Data structures in KB and algorithms that manipulate
them e.g propositional logic and resolution.
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A Typical Wumpus World
Wumpus
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A Typical Wumpus World
• Cave consisting of rooms connected by
passageways
• Agent has only one arrow
• Some of rooms contain bottomless pit that will
trap anyone who wanders into these rooms –
except for wumpus
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Wumpus World PEAS Description
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Wumpus World Characterization
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Observable?
Deterministic?
Episodic?
Static?
Discrete?
Single-agent?
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Wumpus World Characterization
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Observable? No, only local perception
Deterministic?
Episodic?
Static?
Discrete?
Single-agent?
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Wumpus World Characterization
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Observable? No, only local perception
Deterministic? Yes, outcome exactly specified
Episodic?
Static?
Discrete?
Single-agent?
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Wumpus World Characterization
• Observable? No, only local perception
• Deterministic? Yes, outcome exactly specified
• Episodic? No, sequential at the level of
actions
• Static?
• Discrete?
• Single-agent?
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Wumpus World Characterization
• Observable? No, only local perception
• Deterministic? Yes, outcome exactly specified
• Episodic? No, sequential at the level of
actions
• Static? Yes, Wumpus and pits do not move
• Discrete?
• Single-agent?
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Wumpus World Characterization
• Observable? No, only local perception
• Deterministic? Yes, outcome exactly specified
• Episodic? No, sequential at the level of
actions
• Static? Yes, Wumpus and pits do not move
• Discrete? Yes
• Single-agent?
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Wumpus World Characterization
• Observable? No, only local perception
• Deterministic? Yes, outcome exactly specified
• Episodic? No, sequential at the level of
actions
• Static? Yes, Wumpus and pits do not move
• Discrete? Yes
• Single-agent? Yes, Wumpus is essentially a
natural feature.
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Exploring the Wumpus World
[1,1] The KB initially contains the rules of the environment. The first percept is [none,
none,none,none,none], move to safe cell e.g. 2,1
[none, none,none,none,none] – no stench, no breeze, no glitter, no wall in front,
no scream
[2,1] breeze which indicates that there is a pit in [2,2] or [3,1], return to [1,1] to try next
safe cell
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Exploring the Wumpus World
[1,2] Stench in cell which means that wumpus is in [1,3] or [2,2]
YET … not in [1,1]
YET … not in [2,2] or stench would have been detected in [2,1]
THUS … wumpus is in [1,3]
THUS [2,2] is safe because of lack of breeze in [1,2]
THUS pit in [1,3]
move to next safe cell [2,2]
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Exploring the Wumpus World
[2,2] move to [2,3]
[2,3] detect glitter , smell, breeze
THUS pick up gold
THUS pit in [3,3] or [2,4]
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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
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Entailment
• One thing follows from another
KB | = 
• KB entails sentence  if and only if  is
true in worlds where KB is true.
 E.g. x+y=4 entails 4=x+y
• Entailment is a relationship between
sentences that is based on semantics.
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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 
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Wumpus world model
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Wumpus world model
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Wumpus world model
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Wumpus world model
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Wumpus world model
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Wumpus world model
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Logical inference
• The notion of entailment can be used for logic
inference.
– Model checking (see wumpus example): enumerate all
possible models and check whether  is true.
• If an inference algorithm i can derive  from KB
– KB |-i 
– Sentence  is derived from KB by i or i derives  from KB
• Soundness: i is sound if whenever KB |-i , it is also
true that KB |= 
• Completeness : i is complete if whenever KB |=  it
is also true that KB|-i 
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Schematic perspective
If KB is true in the real world, then any sentence  derived
From KB by a sound inference procedure is also true in the
real world.
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Logical inference
• 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.
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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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Truth tables for connectives
P
Q
false false
false true
true false
true true
P
PQ
PQ
PQ
PQ
true
true
false
false
false
false
false
true
false
true
true
true
true
true
false
true
true
false
false
true
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Propositional logic: Semantics
• Each model species true/false for each proposition symbol
• E.g. P1,2 P2,2 P3,1
true true false
• (With these symbols, 8 possible models, can be enumerated
automatically.)
• 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 true
• S1  S2 is true iff S1 is true or S2 is true
• S1  S2 is true iff S1 is false or S2 is true
• i.e., is false iff S1 is true and S2 is false
• S1  S2 is true iff  S2 is true and S2  S1 is true
• Simple recursive process evaluates an arbitrary sentence,
e.g.,
•  P1,2  (P2,2  P3,1) = true  (false  true)=true  true=true
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Wumpus world sentences
• Let Pi,j be true if there is a pit in [i, j].
• Let Bi,j be true if there is a breeze in [i, j].
 P1,1
 B1,1
B2,1
• “Pits cause breezes in adjacent squares"
B1,1  (P1,2  P2,1)
B2,1  (P1,1  P2,2  P3,1)
“A square is breezy if and only if there is an adjacent pit"
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Truth tables for inference
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Logical equivalence
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Validity and satisability
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Inference Rules
Modus ponens
, 

And elimination


Logical equivalences on pg 40 apply
Eg biconditional elimination yields 2 inference rules

and
()  ()
()  ()

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Inference Rules
• All inferences may not apply in the opposite
direction
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Inference Rules
• Wumpus world KB
There is no pit in [1,1] etc
R1:  P1,1
R2: B1,1  (P1,2  P2,1)
R3: B2,1  (P1,1  P2,2  P3,1)
R4:  B1,1
R5: B2,1
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Inference Rules
• Show there is no pit in [1,2]
• biconditional elimination to R2
R6: (B1,1(P1,2  P2,1))  ((P1,2  P2,1 )  B1,1)
And elimination to R6
R7: ((P1,2  P2,1 )  B1,1)
logical equivalence for contrapositives
R8: ( B1,1 (P1,2  P2,1))
Apply modus ponens with R8 with precept R4
R9: (P1,2  P2,1))
De Morgan
R10: P1,2   P2,1)
Neither [1,2] nor [2,1] contains a pit
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Searching for Proofs
• Finding proofs is exactly like finding solutions to
search problems.
• Can search forward (forward chaining) to
derive goal or search backward (backward
chaining) from the goal.
• Searching for proofs is not more efficient than
enumerating models, but in many practical
cases, it’s more efficient because we can
ignore irrelevant propositions
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Resolution
• So far – soundness of inference algorithm
– soundness – truth preserving – inference
algorithm derives only entailed sentences
– Unsound – makes things up as it goes along
• Completeness of inference algorithms?
– Completeness – algorithm can derive any
sentence that is entailed
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Resolution
• Wumpus world
– agent returns from [2,1] to [1,1], then goes
to [1,2] – stench, but no breeze
– Add facts to knowledge base
• R11: B1,2
• R12: B1,2  (P1,1  P2,2  P1,3)
– Same process that led to R10 – we can
derive
• R13: P2,2
• R14: P1,3
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Resolution
– Biconditional elimination to R3, …
• R15: P1,1  P2,2  P3,1
– Resolution
• P2,2 in R13 resolves with P2,2 in R15 to give
R16: P1,1  P3,1
In other words, if there Is a pit in one of
[1,1],[2,2],[3,1], and it is not in [2,2] then it is in
[1,1] or [3,1]
• Similarly, P1,1 in R1 resolves with P1,1 in R16 to
give
R17: P3,1
In other words, if there Is a pit of [1,1] or [3,1],
and it is not in [1,1] then it is in [3,1]
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Resolution
• We used the unit resolution inference rule
above
li and m are complimentary literals (one is the
negative of the other)
• A Full Resolution Rule is a generalization of
this rule
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Resolution
• For clauses of length two:
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Conjunctive normal form
• Conjunctive Normal Form is a disjunction of
literals
• Example:
literals
(A 
B   C)  (B  D)  ( A)  (B  C)
clause - Disjunction of literals
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Conjunctive normal form
• Convert R2 (B1,1  (P1,2  P2,1) into CNF
• Steps
– Eliminate , replacing    with ()()
(B1,1  (P1,2  P2,1 ))  ((P1,2  P2,1 )) B1,1)
– Eliminate , replacing    with 
(B1,1  P1,2  P2,1)  ((P1,2  P2,1 ))  B1,1)
–  must appear only in literals
(B1,1  P1,2  P2,1)  ((P1,2   P2,1 )  B1,1)
– Distribute
(B1,1P1,2 P2,1)  (P1,2  B1,1)  (  P2,1B1,1)
• Conjunction of three clauses – CNF form
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Resolution Algorithm
• To show KB |= , we show (KB   ) is
unsatisfiable.
• This is a proof by contradiction.
• First convert (KB   ) into CNF.
• Then apply resolution rule to resulting clauses.
• The process continues until:
– there are no new clauses that can be added
(KB does not entail )
– two clauses resolve to yield empty clause
(KB entails )
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Simple Inference in Wumpus World
• Agent is in [1,1]
– KB = R2  R4 = (B1,1  (P1,2  P2,1))   B1,1
– Wish to prove  which is, say,  P1,2
– Convert KB   P1,2 to CNF
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Horn Clauses
• A disjunction of literals where at most one is
positive
( L1,1  Breeze  B1,1) is a horn clause
( B1,1  P1,2  P2,1)) is not a horn clause
• every Horn clause can be written as an
implication whose premise is a conjunction of
positive literals and whose conclusion is a
single positive literal
( L1,1  Breeze  B1,1) can be written as
(L1,1  Breeze)  B1,1
( W1,1   W1,2 ) can be written as
( W1,1  W1,2)  False
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Horn Clauses
• Inference with Horn clauses can be done
through the forward chaining and backward
chaining algorithms
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Forward Chaining
• Fire any rule whose premises are satisfied in
the KB.
• Add its conclusion to the KB until query is
found.
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Forward Chaining Example
PQ
LMP
BLM
APL
ABL
A
B
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Forward Chaining Example
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Forward Chaining Example
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Forward Chaining Example
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Forward Chaining Example
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Forward Chaining Example
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Forward Chaining Example
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Forward Chaining Example
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Backward Chaining
• Motivation: Need goal-directed reasoning in
order to keep from getting overwhelmed with
irrelevant consequences
• Main idea:
– Work backwards from query q
– To prove q:
• Check if q is known already
• Prove by backward chaining all premises of some rule
concluding q
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Backward Chaining Example
PQ
LMP
BLM
APL
ABL
A
B
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Backward Chaining Example
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Backward Chaining Example
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Backward Chaining Example
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Backward Chaining Example
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Backward Chaining Example
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Backward Chaining Example
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Backward Chaining Example
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Backward Chaining Example
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Backward Chaining Example
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Backward Chaining Example
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Forward Chaining vs. Backward
Chaining
• FC is data-driven—it may do lots of work
irrelevant to the goal
• BC is goal-driven—appropriate for problemsolving
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