Transcript Document

Logical Agents
Russell & Norvig Chapter 7
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Programming Language Moment
• We often say a programming language is
procedural or declarative.
• What is meant by that?
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Declarative Statements in Prolog
(variables begin with capital letter)
fun(X) :red(X),
car(X).
fun(X) :blue(X),
bike(X).
fun(ice_cream).
/* an item is fun if */
/* the item is red */
/* and it is a car */
/* or an item is fun if */
/* the item is blue */
/* and it is a bike */
/* ice cream is also fun. */
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Knowledge bases
• Knowledge base = set of sentences in a formal language
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• Declarative approach to building an agent (or other system):
– Tell it what it needs to know
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• Then it can Ask itself what to do - answers should follow from the
KB
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• 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
–
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Terms
• percept - information I sense or
“perceive”
• Tell – way of updating the contents of the
knowledge base
• Ask – ask KB to tell you want to do
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A simple knowledge-based agent
• The agent must be able to:
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– Represent states, actions, etc.
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– Incorporate new percepts
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– Update internal representations of the world
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Wumpus World PEAS (performance,
environment, sensors, actuators) description
• Performance measure
– gold +1000, death -1000
– -1 per step, -10 for shooting
• 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
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Categorization of Environments
• Accessible vs inaccessible – have complete information
about environment state?
• Deterministic vs Nondeterministic outcomes exactly
specified?
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• Episodic vs history sensitive (sequential) – actions in
episode have no relationship to actions in other episodes
• Static vs dynamic changes only by actions of agent?
• Discrete vs continuous- fixed, finite number of
actions/percepts?
• Single-agent vs multi-agent? How many agents?
• How would you categorize wumpus world?
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Wumpus world characterization
• Fully Observable/Accessible No – only local
perception
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• Deterministic Yes – outcomes exactly specified
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• Episodic Yes – environment doesn’t respond to
runs, only state.
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• Static Yes – Wumpus and Pits do not move
•
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How would you direct the agent
what to do?
What does this remind you of?
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Exploring a wumpus world
Upper Left indicates precepts
B: Breeze
S: Stench
G: Glitter
Upper Right Status
OK: Safe, no pit or wumpus
?: Unknown
P?: Could be Pit
In Center
[A]: Agent has been there
P: is a Pit
W: is a Wumpus
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Exploring a wumpus world
Sense a breeze, so what do I know?
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Exploring a wumpus world
So what should I do?
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Exploring a wumpus world
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Exploring a wumpus world
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Exploring a wumpus world
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Exploring a wumpus world
What is sensor in
2,2?
Do sensors note
Pit or Wumpus in
diagonal square?
Do I really know
2,3 and 3,2 are
okay?
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Exploring a wumpus world
Will I always know
exactly what to do?
Give an example?
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Logic in general
• Logics are formal languages for representing information
such that conclusions can be drawn
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• Syntax defines the sentences in the language
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• Semantics define the "meaning" of sentences;
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– i.e., define truth of a sentence in a world
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• E.g., the language of arithmetic
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– x+2 ≥ y is a sentence; x2+y > {} is not a sentence
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– x+2 ≥ y is true iff the number x+2 is no less than the number y
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Entailment
• Entailment means that one thing follows from another:
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KB ╞ α
• Knowledge base KB entails sentence α if and only if α is
true in all worlds where KB is true
– Playing mud football “entails” getting muddy
– E.g., the KB containing “the Giants won” and “the Reds won”
entails “Either the Giants won or the Reds won”
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– E.g., x+y = 4 entails 4 = x+y
– E.g. xy=1 entails what?
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– Entailment is a relationship between sentences (i.e., syntax) that
is based on semantics
Entailment doesn’t say you can prove it – only that 20
it is true!
Models
• Logicians typically think in terms of models, which are formally
structured worlds with respect to which truth can be evaluated
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• 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 – or
all possible worlds where α is true
• Then KB ╞ α iff M(KB)  M(α)
•
In other words, the KB doesn’t contain
any models in which α isn’t true.
E.g.
– KB = Giants won and Reds
won
– α = Giants won
–
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Example
• : fruits are low calorie
• A model of the sentence
is a world with tomatoes.
• M() – all models where
 is true.
• M(KB) - all models legal
in our knowledge base
All Fruity worlds
dried fruits
papaya
apple
cherries
grapefruit
mango
banana
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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
Can’t go diagonal, so only
consider choices from [A]
squares.
3 Boolean choices for ? squares
 8 possible models
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All Possible Wumpus models
(Pits only – show all “possible”)
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Wumpus models
• KB (red)= wumpus-world rules + observations
• Blue can’t happen as inconsistent with percepts
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Wumpus models
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Remember KB ╞ α iff M(KB)  M(α)
KB = wumpus-world rules + observations
α1 = "[1,2] is safe", KB ╞ α1, proved by model checking
Notice set of α1 includes some models not allowed in KB
In all possible models in KB, [1,2] is safe
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Wumpus models
• Can we prove [2,2] is safe?
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Wumpus models
• KB = wumpus-world rules + observations
• α2 = "[2,2] is safe", KB ╞ α2
•
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Inference
• KB ├i α = sentence α can be derived from KB by
procedure (set of rules) i
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• Soundness: i is sound if whenever KB ├i α, it is also true
that KB╞ α (only true things are derived)
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• Completeness: i is complete if whenever KB╞ α, it is also
true that KB ├i α (if α is true, we can derive it)
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• 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.
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• That is, the procedure will answer any question whose
Propositional logic: Syntax
• Propositional logic is the simplest logic – illustrates basic
ideas
• Each symbol can be true or false
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• The proposition symbols P1, P2 etc are sentences
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–
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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)
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Propositional (boolean) logic:
Semantics
Each model specifies true/false for each proposition symbol
E.g.
false
P1,2
true
P2,2
false
P3,1
With these symbols, 8 possible models, can be enumerated automatically.
Rules for evaluating truth with respect to a model m:
S
S1  S2
S1  S2
S1  S2
i.e.,
S1  S2
is true iff
is true iff
is true iff
is true iff
is false iff
is true iff
S is false
S1 is true and
S2 is true
S1is true or
S2 is true
S1 is false or
S2 is true
S1 is true and
S2 is false
S1S2 is true andS2S1 is true
Simple recursive process evaluates an arbitrary sentence, e.g.,
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Truth tables for connectives
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Exploring a wumpus world
What Predicates are known?
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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)
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Idea
• Consider all possible values of
propositions
• Throw out ones made invalid by KB
• Examine  in remaining possibilities
• Would work, but is exponential, right?
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Truth tables for inference
 P1,1
B1,1
B2,1 (actually different than world of previous slide)
 P2,1 (as you are there and feel breeze)
 is P1,2
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Inference by enumeration –
consider all models of KB
• Depth-first enumeration of all models is sound and complete
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• For n symbols, time complexity is O(2n), space complexity is O(n)
• Notice – try each possible value for symbol
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• PL-True?(sentence,model) returns true if sentence holds within
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Logical equivalence
• Two sentences are logically equivalent} iff true in same
models: α ≡ ß iff α╞ β and β╞ α
•
•
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Deduction theorem
• Known to the ancient Greeks:
• For any sentences  and ,
( ╞ ) if and only if the sentence
(  ) is valid.
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Validity and satisfiability
A sentence is valid if it is true in all models,
e.g., True,
A A, A  A, (A  (A  B))  B
Validity is connected to inference via the Deduction Theorem:
KB ╞ α if and only if (KB  α) is valid
example: : Ali will be accepted to USU
A sentence is satisfiable if it is true in some model
e.g., A B, C
I can make it true; some way of satisfying
example: : Ali will get an A in 6100
Many problems in computer science are really satisfiability problems. For
example, constraint satisfaction.
With appropriate transformations, search problems can be solved by
constraint satisfaction.
For example, “Find the shortest path to WalMart” becomes “Is there a path
to WalMart of less than two miles?”
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Validity and satisfiability
A sentence is unsatisfiable if it is true in no
models
e.g., AA
example: : Ali will get an A in 6411 (No such
class exists for him to take.)
We are familiar with this as proof by
contradiction.
Satisfiability is connected to inference via
the following:
KB ╞ α if and only if (KB α) is unsatisfiable
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Proof methods
• Proof methods divide into (roughly) two kinds:
Every known inference algorithm for propositional logic has a worst
case complexity that is exponential in the size of the input
– Model checking (as we saw)
• truth table enumeration (always exponential in n)
•
• improved backtracking, e.g., Davis--Putnam-Logemann-Loveland (DPLL)
– – Can I find an assignment of variables which makes this true?
• heuristic search in model space (sound but incomplete: doesn’t lie, but may
not find a solution when one exists)
e.g., min-conflicts (guess something that makes the fewest clauses
unhappy) hill-climbing algorithms
– Application of inference rules
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• Legitimate (sound) generation of new sentences from old
•
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• Proof = a sequence of inference rule applications
Can use inference rules as operators in a standard search algorithm
Reasoning patterns
• Modus Ponens
 , 
______________

The horizontal line separates the
“if” part from
the “then” part.
• And Elimination

______________

• Can be used wherever they apply without the need for enumerating
models.
• A series of inferences (a proof) is an alternative to enumerating
models.
• In the worst case, searching for proofs is no more efficient than
enumerating models. However, finding a proof can be highly
efficient
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Resolution
Conjunctive Normal Form (CNF)
conjunction of disjunctions of literals
clauses
E.g., (A  B)  (B  C  D)
•
•
Think, if B is true, what do I know?
If B is false, what do I know?
Since B must be true or false,
what do I now know?
Resolution inference rule (for CNF):
li …  lk,
m1  …  mn
li  …  li-1  li+1  …  lk  m1  …  mj-1  mj+1 ...  mn
where li and mj are complementary literals.
E.g., P1,3  P2,2,
P2,2
P1,3
This single rule, Resolution, coupled with
a search algorithm yields a
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Refutation Completeness
• Given A is true, we cannot use resolution
to generate that AB is true
• We can use resolution to argue whether
AB is true. This is termed refutation
completeness.
• but first, we need to get our rules in a form
that resolution works on
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Conversion to CNF
(conjunctive normal form, conjunction of
disjunctions)
B1,1  (P1,2  P2,1)
1. Eliminate , replacing α  β with (α  β)(β  α).
2.
(B1,1  (P1,2  P2,1))  ((P1,2  P2,1)  B1,1)
2. Eliminate , replacing α  β with α β.
(B1,1  P1,2  P2,1)  ((P1,2  P2,1)  B1,1)
3. Move  inwards using de Morgan's rules and doublenegation:
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Resolution Algorithms
• In order to show KB ╞ , we show (KB
) is unsatisfiable. We do this by
proving a contradiction
• What will a contradiction look like in
resolution
 ,   

• What if we have
So what does deriving NOTHING mean?
 , 
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Resolution algorithm
• Proof by contradiction, i.e., show KBα unsatisfiable
•
Notice the contradiction
so getting an empty means
we got a contradiction
and KB does imply 
So when we return true, we mean the clause IS unsatisfiable
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Resolution example
•
•
•
•
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We want to show that  B1,1 P1,2
KB = (B1,1  (P1,2 P2,1))  B1,1
α = P1,2
Suppose  is true and show contradiction
(B1,1  P1,2  P2,1)  (P1,2  B1,1)  (P2,1  B1,1)
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The Ground Resolution Theorem
• If a set of clauses is unsatisfiable, then the
resolution closure of those clauses contains the
empty clause.
• Let’s take a break from this chapter
and go back into our text!
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Forward and backward chaining
• Horn Form (restricted)
KB = conjunction of Horn clauses
– Horn clause =
• proposition symbol; or
• (conjunction of symbols)  symbol
– E.g., C  (B  A)  (C  D  B)
–
– Sometimes we say Horn Clauses have at most ONE positive. This is
because α1  …  αn  β is the same as  (α1  …  αn) V β
which is  α1 V… V  αn V β
• Modus Ponens (for Horn Form): complete for Horn KBs
•
α1, … ,αn,
α 1  …  αn  β
β
• Can be used with forward chaining or backward chaining.
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Forward chaining
• Idea: 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 algorithm
• Forward chaining is sound and complete for Horn KB
•
• Head[c] is the clause that is implied
• count[c] is number of unsatisfied premises
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Forward chaining example
Red indicates number of
unsatisfied premises.
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Forward chaining example
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Forward chaining example
L was proved in one of two
ways
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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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Proof of completeness
•
Forward Chaining (FC) derives every atomic
sentence that is entailed by KB
•
1. FC reaches a fixed point where no new atomic
sentences are derived
2.
2. Consider the final state as a model m, assigning
true/false to symbols
3.
3. Every clause in the original KB is true in m
4.
a1  …  ak  b
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Backward chaining (BC)
Idea: work backwards from the query q:
to prove q by BC,
check if q is known already, or
prove by BC all premises of some rule concluding q
Avoid loops: check if new subgoal is already on the goal
stack
Avoid repeated work: check if new subgoal
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Backward chaining example
Green is our goal.
Red is what we already know
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Backward chaining example
Solid green means we
don’t need to worry
about this proof as long
as we can prove all the
other “open” green predicates.
It’s trueness or falseness will be
the result of other goals.
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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 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
•
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Efficient propositional inference
Two families of efficient algorithms for propositional
inference:
Backtracking search algorithms
• DPLL algorithm (Davis, Putnam, Logemann, Loveland)
•
• Incomplete local search algorithms
– WalkSAT algorithm
–
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The DPLL algorithm
Determine if an input propositional logic sentence (in CNF) is
satisfiable.
Improvements over truth table enumeration:
1. Early termination (short circuit evaluation)
A clause is true if any literal is true.
A sentence is false if any clause is false.
2. Pure symbol heuristic
Pure symbol: always appears with the same "sign" in all clauses.
e.g., In the three clauses (A  B), (B  C), (C  A), A and B are pure, C is
impure.
Make a pure symbol literal true – as no clause is inconvenienced if it isn’t.
3. Unit clause heuristic
Unit clause: only one literal in the clause
The only literal in a unit clause must be true.
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The DPLL algorithm
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The WalkSAT algorithm
• Incomplete, local search algorithm
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• Guesses at a solution and then tries to make it better.
• Makes better by finding one clause that fails and
changing a value to make it succeed.
• Evaluation function to decide which value in the clause
to flip: The min-conflict heuristic of minimizing the
number of unsatisfied clauses
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• Balance between greediness and randomness
•
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The WalkSAT algorithm
Note, else is associated with the “with probability p” – so you either
randomly change a value or pick the one which causes less grief.
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Hard satisfiability problems
• WalkSat works best if there are LOTS of
solutions, as it is easier to land on one.
• For example – each of you randomly pick
values for
A,B,C,D,E
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Hard satisfiability problems
• For your values, is the following CNF
true?:
(A  B)  (C  D  E)
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Hard satisfiability problems
• In general, what makes a problem hard?
• Consider random 3-CNF sentences. e.g.,
•
(D  B  C)  (B  A  C)  (C 
B  E)  (E  D  B)  (B  E  C)
m = number of clauses
n = number of symbols
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– Hard problems seem to cluster near m/n = 4.3
Hard satisfiability problems
what is probability they can be satisfied?
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Hard satisfiability problems
• Median runtime for 100 satisfiable random 3CNF sentences, n = 50
•
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Inference-based agents in the
4x4 wumpus world
A wumpus-world agent using propositional logic:
P1,1 First cell can’t be pit
W1,1 First cell can’t be wumpus
Bx,y  (Px,y+1  Px,y-1  Px+1,y  Px-1,y) Breeze means wumpus close
Sx,y  (Wx,y+1  Wx,y-1  Wx+1,y  Wx-1,y) Stench means wumpus
close
W1,1  W1,2  …  W4,4 Wumpus is somewhere
W1,1  W1,2 Wumpus can’t be in neighboring cells as only 1
W1,1  W1,3
…
 64 distinct proposition symbols, 155 sentences
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Wumpus Algorithm
• At seats: In the following algorithm, what
does each piece do?
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Expressiveness limitation of
propositional logic
• KB contains "physics" sentences for every single square
– can’t easily say “This principle is true everywhere!”
•
• For every time t and every location [x,y],
•
Lx,y  FacingRightt  Forwardt  Lx+1,y
t
t
• Rapid proliferation of clauses
•
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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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–
–
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–
–
–
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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 89
information, reason by cases, etc.