Transcript Logic presentation.

CS 8520: Artificial Intelligence
Logical Agents and First
Order Logic
Paula Matuszek
Fall, 2008
CSC 8520 Fall, 2008. Paula Matuszek. Slides in part from aima.eecs.berkeley.edu/slides-ppt, chs 7-9
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Outline
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Knowledge-based agents
Wumpus world
Logic in general - models and entailment
Propositional (Boolean) logic
Equivalence, validity, satisfiability
Inference rules and theorem proving
– forward chaining
– backward chaining
– resolution
CSC 8520 Fall, 2008. Paula Matuszek. Slides in part from aima.eecs.berkeley.edu/slides-ppt, chs 7-9
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Knowledge bases
• Knowledge base = set of sentences in a formal languageDeclarative
approach to building an agent (or other system):
– Tell it what it needs to know
• Then it can Ask itself what to do - answers should follow from the KB
• Agents can be viewed at the knowledge level
i.e., what they know, regardless of how implemented
• Or at the implementation level
– i.e., data structures in KB and algorithms that manipulate them
CSC 8520 Fall, 2008. Paula Matuszek. Slides in part from aima.eecs.berkeley.edu/slides-ppt, chs 7-9
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A simple knowledge-based agent
• This agent tells the KB what it sees, asks the KB what to do, tells the
KB what it has done (or is about to do).
• The agent must be able to:
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Represent states, actions, etc.
Incorporate new percepts
Update internal representations of the world
Deduce hidden properties of the world
Deduce appropriate actions
CSC 8520 Fall, 2008. Paula Matuszek. Slides in part from aima.eecs.berkeley.edu/slides-ppt, chs 7-9
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Wumpus World PEAS Description
• Performance measure
– gold +1000, death -1000
– -1 per step, -10 for using the arrow
• Environment
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Squares adjacent to wumpus are smelly
Squares adjacent to pit are breezy
Glitter iff gold is in the same square
Shooting kills wumpus if you are facing it
Shooting uses up the only arrow
Grabbing picks up gold if in same square
Releasing drops the gold in same square
• Sensors: Stench, Breeze, Glitter, Bump, Scream
• Actuators: Left turn, Right turn, Forward, Grab, Release, Shoot
CSC 8520 Fall, 2008. Paula Matuszek. Slides in part from aima.eecs.berkeley.edu/slides-ppt, chs 7-9
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Wumpus world characterization
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Fully Observable No – only local perception
Deterministic Yes – outcomes exactly specified
Episodic No – sequential at the level of actions
Static Yes – Wumpus and Pits do not move
Discrete Yes
Single-agent? Yes – The wumpus itself is
essentially a natural feature, not another agent
CSC 8520 Fall, 2008. Paula Matuszek. Slides in part from aima.eecs.berkeley.edu/slides-ppt, chs 7-9
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Exploring a wumpus world
Directly observed:
S: stench
B: breeze
G: glitter
A: agent
V: visited
Inferred (mostly):
OK: safe square
1,1
2,1
3,1
4,1
P: pit
W: wumpus
CSC 8520 Fall, 2008. Paula Matuszek. Slides in part from aima.eecs.berkeley.edu/slides-ppt, chs 7-9
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Exploring a wumpus world
In 1,1 we don’t get B
or S, so we know 1,2
and 2,1 are safe.
Move to 1,2.
In 1,2 we feel a
breeze.
CSC 8520 Fall, 2008. Paula Matuszek. Slides in part from aima.eecs.berkeley.edu/slides-ppt, chs 7-9
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Exploring a wumpus world
In 1,2 we feel a
breeze. So we know
there is a pit in 1,3 or
2,2.
CSC 8520 Fall, 2008. Paula Matuszek. Slides in part from aima.eecs.berkeley.edu/slides-ppt, chs 7-9
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Exploring a wumpus world
So go back to
1,1, then to 1,2,
where we smell a
stench.
CSC 8520 Fall, 2008. Paula Matuszek. Slides in part from aima.eecs.berkeley.edu/slides-ppt, chs 7-9
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Exploring a wumpus world
We don't feel a breeze
in 2,1, so 2,2 can't be
a pit, so 1,3 must be a
pit.
We don't smell a
stench in 1,2, so 2,2
can't be the wumpus,
so 1,3 must be the
wumpus.
2,2 has neither pit nor
wumpus and is
therefore okay.
CSC 8520 Fall, 2008. Paula Matuszek. Slides in part from aima.eecs.berkeley.edu/slides-ppt, chs 7-9
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Exploring a wumpus world
We move to 2,2. We
don’t get any sensory
input.
CSC 8520 Fall, 2008. Paula Matuszek. Slides in part from aima.eecs.berkeley.edu/slides-ppt, chs 7-9
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Exploring a wumpus world
So we know
that 2,3 and 3,2
are ok.
CSC 8520 Fall, 2008. Paula Matuszek. Slides in part from aima.eecs.berkeley.edu/slides-ppt, chs 7-9
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Exploring a wumpus world
Move to 3,2,
where we
observe stench,
breeze and
glitter!
At this point we
could infer the
existence of
another pit
(where?), but
since we have
found the gold
we don't bother.
We have won.
CSC 8520 Fall, 2008. Paula Matuszek. Slides in part from aima.eecs.berkeley.edu/slides-ppt, chs 7-9
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Logic in general
• Logics are formal languages for representing information
such that conclusions can be drawn
• Syntax defines the sentences in the language
• Semantics define the "meaning" of sentences;
– i.e., define truth of a sentence in a world
• E.g., the language of arithmetic
• x+2 ≥ y is a sentence; x2+y > {} is not a sentence
– x+2 ≥ y is true iff the number x+2 is no less than the number yx+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
CSC 8520 Fall, 2008. Paula Matuszek. Slides in part from aima.eecs.berkeley.edu/slides-ppt, chs 7-9
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Entailment
• Entailment means that one thing follows from
another:
• Knowledge base KB entails sentence S if and only
if S is true in all worlds where KB is true
– E.g., the KB containing “the Giants won” and “the
Reds won” entails “Either the Giants won or the Reds
won”
– E.g., x+y = 4 entails 4 = x+y
– Entailment is a relationship between sentences (i.e.,
syntax) that is based on semantics
CSC 8520 Fall, 2008. Paula Matuszek. Slides in part from aima.eecs.berkeley.edu/slides-ppt, chs 7-9
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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:
there are 3 Boolean choices
 8 possible models
(ignoring sensory data)
CSC 8520 Fall, 2008. Paula Matuszek. Slides in part from aima.eecs.berkeley.edu/slides-ppt, chs 7-9
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Wumpus models for pits
CSC 8520 Fall, 2008. Paula Matuszek. Slides in part from aima.eecs.berkeley.edu/slides-ppt, chs 7-9
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Wumpus models
• KB = wumpus-world rules + observations. Only
the three models in red are consistent with KB.
CSC 8520 Fall, 2008. Paula Matuszek. Slides in part from aima.eecs.berkeley.edu/slides-ppt, chs 7-9
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Wumpus Models
KB plus hypothesis S1 that pit is not in 1.2.
KB is solid red boundary. S1 is dotted yellow
boundary. KB is contained within S1, so KB
entails S; in every model in which KB is true, so
is S. We can conclude that the pit is not in1.2.
KB plus hypothesis S2 that pit is not in 2.2 .
KB is solid red boundary. S2 is dotted brown
boundary. KB is not within S1, so KB does not
entail S2; nor does S2 entail KB. So we can't
conclude anything about the truth of S2 given
KB.
CSC 8520 Fall, 2008. Paula Matuszek. Slides in part from aima.eecs.berkeley.edu/slides-ppt, chs 7-9
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Inference
• KB entailsi S = sentence S can be derived from KB by procedure i
• Soundness: i is sound if it derives only sentences S that are entailed by
KB
• Completeness: i is complete if it derives all sentences S that are
entailed by KB.
• First-order logic:
– Has a sound and complete inference procedure
– Which will answer any question whose answer follows from what is
known by the KB.
– And is richly expressive
• We must also be aware of the issue of grounding: the connection
between our KB and the real world.
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Straightforward for Wumpus or our adventure games
Much more difficult if we are reasoning about real situations
Real problems seldom perfectly grounded, because we ignore details.
Is the connection good enough to get useful answers?
CSC 8520 Fall, 2008. Paula Matuszek. Slides in part from aima.eecs.berkeley.edu/slides-ppt, chs 7-9
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Propositional logic: Syntax
• Propositional logic is the simplest logic – illustrates basic
ideas
• The proposition symbols P1, P2 etc are sentences
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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)
CSC 8520 Fall, 2008. Paula Matuszek. Slides in part from aima.eecs.berkeley.edu/slides-ppt, chs 7-9
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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 
B2,1 
(P1,2  P2,1)
(P1,1  P2,2  P3,1)
CSC 8520 Fall, 2008. Paula Matuszek. Slides in part from aima.eecs.berkeley.edu/slides-ppt, chs 7-9
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Inference by enumeration
• Depth-first enumeration of all models is sound and complete
• For n symbols, time complexity is O(2n), space complexity is O(n)
CSC 8520 Fall, 2008. Paula Matuszek. Slides in part from aima.eecs.berkeley.edu/slides-ppt, chs 7-9
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Inference-based agents in the wumpus world
A wumpus-world agent using propositional logic:
P1,1
W1,1
Bx,y  (Px,y+1  Px,y-1  Px+1,y  Px-1,y)
Sx,y  (Wx,y+1  Wx,y-1  Wx+1,y  Wx-1,y)
W1,1  W1,2  …  W4,4
W1,1  W1,2
W1,1  W1,3
…
 64 distinct proposition symbols, 155 sentences
CSC 8520 Fall, 2008. Paula Matuszek. Slides in part from aima.eecs.berkeley.edu/slides-ppt, chs 7-9
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CSC 8520 Fall, 2008. Paula Matuszek. Slides in part from aima.eecs.berkeley.edu/slides-ppt, chs 7-9
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Expressiveness limitation of
propositional logic
• Rapid proliferation of clauses.
– For instance, Wumpus KB contains "physics" sentences for every
single square
• Very bushy inference, especially if forward chaining.
• Not trivial to express complex relationships; people don't
naturally think in logical terms.
• Monotonic: if something is true it stays true
• Binary: something is either true or false, never maybe or
unknown
CSC 8520 Fall, 2008. Paula Matuszek. Slides in part from aima.eecs.berkeley.edu/slides-ppt, chs 7-9
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Pros and cons of propositional logic
 Propositional logic is declarative
 Propositional logic allows
partial/disjunctive/negated information (unlike
most data structures and databases)
 Propositional logic is compositional:
– meaning of B1,1  P1,2 is derived from meaning of B1,1 and of P1,2
 Meaning in propositional logic is contextindependent (unlike natural language, where meaning
depends on context)
 Propositional logic has very limited expressive
power (unlike natural language)
– E.g., cannot say "pits cause breezes in adjacent squares“
• except by writing one sentence for each square
CSC 8520 Fall, 2008. Paula Matuszek. Slides in part from aima.eecs.berkeley.edu/slides-ppt, chs 7-9
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Summary
• Logical agents apply inference to a knowledge base to derive new
information and make decisions
• Basic concepts of logic:
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syntax: formal structure of sentences
semantics: truth of sentences wrt models
entailment: necessary truth of one sentence given another
inference: deriving sentences from other sentences
soundness: derivations produce only entailed sentences
completeness: derivations can produce all entailed sentences
• Wumpus world requires the ability to represent partial and negated
information, reason by cases, etc.
• Resolution is complete for propositional logic
Forward, backward chaining are linear-time, complete for Horn
clauses
• Propositional logic lacks expressive power
CSC 8520 Fall, 2008. Paula Matuszek. Slides in part from aima.eecs.berkeley.edu/slides-ppt, chs 7-9
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First-order logic
• Whereas propositional logic assumes the
world contains facts,
• first-order logic (like natural language)
assumes the world contains
– Objects: people, houses, numbers, colors,
baseball games, wars, …
– Relations: red, round, prime, brother of, bigger
than, part of, comes between, …
– Functions: father of, best friend, one more than,
plus, …
CSC 8520 Fall, 2008. Paula Matuszek. Slides in part from aima.eecs.berkeley.edu/slides-ppt, chs 7-9
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Syntax of FOL: Basic elements
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Constants
Predicates
Functions
Variables
Connectives
Equality
Quantifiers
KingJohn, 2, Villanova,...
Brother, >,...
Sqrt, LeftLegOf,...
x, y, a, b,...
, , , , 
=
, 
CSC 8520 Fall, 2008. Paula Matuszek. Slides in part from aima.eecs.berkeley.edu/slides-ppt, chs 7-9
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Terms and Atomic sentences
• A term is a logical expression that refers to an
object.
– Book(Naomi). Naomi's book.
– Textbook(8520). Textbook for 8520.
• An atomic sentence states a fact.
– Student(Naomi).
– Student(Naomi, Paula).
– Student(Naomi, AI).
Note that the interpretation of these is different; it depends
on how we consider them to be grounded.
CSC 8520 Fall, 2008. Paula Matuszek. Slides in part from aima.eecs.berkeley.edu/slides-ppt, chs 7-9
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Complex sentences
• Complex sentences are made from atomic
sentences using connectives
S, S1  S2, S1  S2, S1  S2, S1  S2,
E.g. Sibling(KingJohn,Richard) 
Sibling(Richard,KingJohn)
CSC 8520 Fall, 2008. Paula Matuszek. Slides in part from aima.eecs.berkeley.edu/slides-ppt, chs 7-9
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Truth in first-order logic
• Sentences are true with respect to a model and an
interpretation
• Model contains objects (domain elements) and
relations among them
• Interpretation specifies referents for
constant symbols →
predicate symbols →
function symbols →
objects
relations
functional relations
• An atomic sentence predicate(term1,...,termn) is
true iff the objects referred to by term1,...,termn
are in the relation referred to by predicate
CSC 8520 Fall, 2008. Paula Matuszek. Slides in part from aima.eecs.berkeley.edu/slides-ppt, chs 7-9
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Knowledge base for the wumpus world
• Perception
– t,s,b Percept([s,b,Glitter],t)  Glitter(t)
• Reflex
– t Glitter(t)  BestAction(Grab,t)
CSC 8520 Fall, 2008. Paula Matuszek. Slides in part from aima.eecs.berkeley.edu/slides-ppt, chs 7-9
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Deducing hidden properties
• x,y,a,b Adjacent([x,y],[a,b]) 
[a,b]  {[x+1,y], [x-1,y],[x,y+1],[x,y-1]}
Properties of squares:
• s,t At(Agent,s,t)  Breeze(t)  Breezy(s)
Squares are breezy near a pit:
Diagnostic rule---infer cause from effect
s Breezy(s)   r Adjacent(r,s)  Pit(r)
Causal rule---infer effect from cause
r Pit(r)  [s Adjacent(r,s)  Breezy(s)]
CSC 8520 Fall, 2008. Paula Matuszek. Slides in part from aima.eecs.berkeley.edu/slides-ppt, chs 7-9
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Knowledge engineering in FOL
1. Identify the task
2. Assemble the relevant knowledge
3. Decide on a vocabulary of predicates, functions,
and constants
4. Encode general knowledge about the domain
5. Encode a description of the specific problem
instance
6. Pose queries to the inference procedure and get
answers
7. Debug the knowledge base
CSC 8520 Fall, 2008. Paula Matuszek. Slides in part from aima.eecs.berkeley.edu/slides-ppt, chs 7-9
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Summary
• First-order logic:
– objects and relations are semantic primitives
– syntax: constants, functions, predicates,
equality, quantifiers
• Increased expressive power: sufficient to
define wumpus world
CSC 8520 Fall, 2008. Paula Matuszek. Slides in part from aima.eecs.berkeley.edu/slides-ppt, chs 7-9
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Inference
• When we have all this knowledge we want
to DO SOMETHING with it
• Typically, we want to infer new knowledge
– An appropriate action to take
– Additional information for the Knowledge Base
• Some typical forms of inference include
– Forward chaining
– Backward chaining
– Resolution
CSC 8520 Fall, 2008. Paula Matuszek. Slides in part from aima.eecs.berkeley.edu/slides-ppt, chs 7-9
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Example knowledge base
• The law says that it is a crime for an American to sell
weapons to hostile nations. The country Nono, an enemy
of America, has some missiles, and all of its missiles were
sold to it by Colonel West, who is American.
• Prove that Col. West is a criminal
CSC 8520 Fall, 2008. Paula Matuszek. Slides in part from aima.eecs.berkeley.edu/slides-ppt, chs 7-9
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Inference
• We need to DO SOMETHING with our
knowledge.
– Forward chaining
– Backward chaining
– Resolution
CSC 8520 Fall, 2008. Paula Matuszek. Slides in part from aima.eecs.berkeley.edu/slides-ppt, chs 7-9
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Example knowledge base contd.
... it is a crime for an American to sell weapons to hostile nations:
American(x)  Weapon(y)  Sells(x,y,z)  Hostile(z)  Criminal(x)
Nono … has some missiles, i.e., x Owns(Nono,x)  Missile(x):
Owns(Nono,M1) and Missile(M1)
… all of its missiles were sold to it by Colonel West
Missile(x)  Owns(Nono,x)  Sells(West,x,Nono)
Missiles are weapons:
Missile(x)  Weapon(x)
An enemy of America counts as "hostile“:
Enemy(x,America)  Hostile(x)
West, who is American …
American(West)
The country Nono, an enemy of America …
Enemy(Nono,America)
CSC 8520 Fall, 2008. Paula Matuszek. Slides in part from aima.eecs.berkeley.edu/slides-ppt, chs 7-9
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Forward chaining algorithm
CSC 8520 Fall, 2008. Paula Matuszek. Slides in part from aima.eecs.berkeley.edu/slides-ppt, chs 7-9
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Forward chaining proof
CSC 8520 Fall, 2008. Paula Matuszek. Slides in part from aima.eecs.berkeley.edu/slides-ppt, chs 7-9
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Forward chaining proof
CSC 8520 Fall, 2008. Paula Matuszek. Slides in part from aima.eecs.berkeley.edu/slides-ppt, chs 7-9
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Forward chaining proof
CSC 8520 Fall, 2008. Paula Matuszek. Slides in part from aima.eecs.berkeley.edu/slides-ppt, chs 7-9
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Properties of forward chaining
• Sound and complete for first-order definite clauses
• Datalog = first-order definite clauses + no functions
• FC terminates for Datalog in finite number of iterations
• May not terminate in general if α is not entailed
• This is unavoidable: entailment with definite clauses is
semidecidable
CSC 8520 Fall, 2008. Paula Matuszek. Slides in part from aima.eecs.berkeley.edu/slides-ppt, chs 7-9
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Efficiency of forward chaining
Incremental forward chaining: no need to match a rule on
iteration k if a premise wasn't added on iteration k-1
 match each rule whose premise contains a newly added positive
literal
Matching itself can be expensive:
Database indexing allows O(1) retrieval of known facts
– e.g., query Missile(x) retrieves Missile(M1)
Forward chaining is widely used in deductive databases
CSC 8520 Fall, 2008. Paula Matuszek. Slides in part from aima.eecs.berkeley.edu/slides-ppt, chs 7-9
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Backward chaining algorithm
CSC 8520 Fall, 2008. Paula Matuszek. Slides in part from aima.eecs.berkeley.edu/slides-ppt, chs 7-9
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Backward chaining example
CSC 8520 Fall, 2008. Paula Matuszek. Slides in part from aima.eecs.berkeley.edu/slides-ppt, chs 7-9
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Backward chaining example
CSC 8520 Fall, 2008. Paula Matuszek. Slides in part from aima.eecs.berkeley.edu/slides-ppt, chs 7-9
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Backward chaining example
CSC 8520 Fall, 2008. Paula Matuszek. Slides in part from aima.eecs.berkeley.edu/slides-ppt, chs 7-9
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Backward chaining example
CSC 8520 Fall, 2008. Paula Matuszek. Slides in part from aima.eecs.berkeley.edu/slides-ppt, chs 7-9
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Backward chaining example
CSC 8520 Fall, 2008. Paula Matuszek. Slides in part from aima.eecs.berkeley.edu/slides-ppt, chs 7-9
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Backward chaining example
CSC 8520 Fall, 2008. Paula Matuszek. Slides in part from aima.eecs.berkeley.edu/slides-ppt, chs 7-9
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Backward chaining example
CSC 8520 Fall, 2008. Paula Matuszek. Slides in part from aima.eecs.berkeley.edu/slides-ppt, chs 7-9
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Backward chaining example
CSC 8520 Fall, 2008. Paula Matuszek. Slides in part from aima.eecs.berkeley.edu/slides-ppt, chs 7-9
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Properties of backward chaining
• Depth-first recursive proof search: space is linear
in size of proof
• Incomplete due to infinite loops
–  fix by checking current goal against every goal on
stack
• Inefficient due to repeated subgoals (both success
and failure)
–  fix using caching of previous results (extra space)
• Widely used for logic programming
CSC 8520 Fall, 2008. Paula Matuszek. Slides in part from aima.eecs.berkeley.edu/slides-ppt, chs 7-9
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Forward vs. backward chaining
• FC is data-driven, automatic, unconscious processing,
– e.g., object recognition, routine decisions
• May do lots of work that is irrelevant to the goal
• BC is goal-driven, appropriate for problem-solving,
– e.g., Where are my keys? How do I get into a PhD program?
• Complexity of BC can be much less than linear in size of
KB
• Choice may depend on whether you are likely to have
many goals or lots of data.
CSC 8520 Fall, 2008. Paula Matuszek. Slides in part from aima.eecs.berkeley.edu/slides-ppt, chs 7-9
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