m7-logic.ppt

Outline
• 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
–

Knowledge bases
• Knowledge base = set of sentences in a formal language
•
• Declarative 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
–

A simple knowledge-based agent
• The agent must be able to:
•
– Represent states, actions, etc.
–
– Incorporate new percepts
–
– Update internal representations of the world
–

Wumpus World PEAS
description
• Performance measure
– gold +1000, death -1000
– -1 per step, -10 for using the arrow
• Environment
•
– 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
–

Wumpus world characterization
• 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
•

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
–

Entailment
• Entailment means that one thing follows from
another:
•
KB ╞ α
• Knowledge base KB entails sentence α if and
only if α 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
–

Models
• Logicians typically think in terms of models, which are formally
structured worlds with respect to which truth can be evaluated
•
• 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 = Giants won and Reds
won α = Giants won
–

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
3 Boolean choices  8
possible models

Wumpus models
• KB = wumpus-world rules + observations
•

Wumpus models
• α1 = "[1,2] is safe", KB ╞ α1, proved by model checking
•
•

Wumpus models

Wumpus models
• α2 = "[2,2] is safe", KB ╞ α2
•

Inference
• KB ├i α = sentence α can be derived from KB by
procedure i
•
• Soundness: i is sound if whenever KB ├i α, it is also true
that KB╞ α
•
• Completeness: i is complete if whenever KB╞ α, it is also
true that KB ├i α
•
• 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.
•

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)
–

Propositional logic: Semantics
Each model specifies true/false for each proposition symbol
E.g. P1,2 P2,2 P3,1
false 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 S1is 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 S1S2 is true andS2S1 is true
Simple recursive process evaluates an arbitrary sentence, e.g.,

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)

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)
•

Logical equivalence
• Two sentences are logically equivalent} iff true in same
models: α ≡ ß iff α╞ β and β╞ α
•
•

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
A sentence is satisfiable if it is true in some model
e.g., A B, C
A sentence is unsatisfiable if it is true in no models
e.g., AA
Satisfiability is connected to inference via the following:
KB ╞ α if and only if (KB α) is unsatisfiable

Proof methods
• Proof methods divide into (roughly) two kinds:
– Application of inference rules
–
• Legitimate (sound) generation of new sentences from old
•
• Proof = a sequence of inference rule applications
Can use inference rules as operators in a standard search
algorithm
•
• Typically require transformation of sentences into a normal form
– Model checking
• truth table enumeration (always exponential in n)
•

Resolution
Conjunctive Normal Form (CNF)
conjunction of disjunctions of literals
clauses
E.g., (A  B)  (B  C  D)
• 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

Resolution
Soundness of resolution inference rule:
(li  …  li-1  li+1  …  lk)  li
mj  (m1  …  mj-1  mj+1 ...  mn)
(li  …  li-1  li+1  …  lk)  (m1  …  mj-1  mj+1 ...  mn)

Conversion to CNF
B1,1  (P1,2  P2,1)β
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 double-
negation:

Resolution algorithm
• Proof by contradiction, i.e., show KBα unsatisfiable
•

Resolution example
• KB = (B1,1  (P1,2 P2,1))  B1,1 α = P1,2
•

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)
–
• Modus Ponens (for Horn Form): complete for Horn KBs
•
α1, … ,αn, α1  …  αn  β
β
• Can be used with forward chaining or backward chaining.
• These algorithms are very natural and run in linear time
•

Forward chaining
• Idea: fire any rule whose premises are satisfied in the
KB,
– add its conclusion to the KB, until query is found

Forward chaining algorithm
• Forward chaining is sound and complete for
Horn KB
•

Proof of completeness
• 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

Backward chaining
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

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
•

Efficient propositional inference
Two families of efficient algorithms for propositional
inference:
Complete backtracking search algorithms
• DPLL algorithm (Davis, Putnam, Logemann, Loveland)
•
• Incomplete local search algorithms
– WalkSAT algorithm
–

The DPLL algorithm
Determine if an input propositional logic sentence (in CNF) is
satisfiable.
Improvements over truth table enumeration:
1. Early termination
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.
3. Unit clause heuristic
Unit clause: only one literal in the clause
The only literal in a unit clause must be true.

The WalkSAT algorithm
• Incomplete, local search algorithm
•
• Evaluation function: The min-conflict heuristic of
minimizing the number of unsatisfied clauses
•
• Balance between greediness and randomness
•

Hard satisfiability problems
• 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

Hard satisfiability problems
• Median runtime for 100 satisfiable random 3-
CNF sentences, n = 50
•

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

• KB contains "physics" sentences for every single square
•
• For every time t and every location [x,y],
•
Lx,y  FacingRightt  Forwardt  Lx+1,y
• Rapid proliferation of clauses
•
Expressiveness limitation of
propositional logic
t
t

Summary
• Logical agents apply inference to a knowledge base to derive new
information and make decisions
•
• Basic concepts of logic:
•
– 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.

m7-logic.ppt

More Related Content

Similar to m7-logic.ppt (20)

Recently uploaded (20)

m7-logic.ppt