Logical Agents (pt 2)

1
Logical Agents (pt 2)

• Chapter 7
• March 2, 2007

2
Logical Agents
REVIEW

• 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 with respect to
  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
  

3
Logic
REVIEW

• 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
• Entailment means that one thing follows from
  another
  
• KB a
• Knowledge base KB entails sentence a if and only
  if a is true in all worlds where KB is true

4
Models
REVIEW

• m is a model of a sentence a if a is true in m
• M(a) is the set of all models of a
• KB a iff M(KB) ? M(a)

5
Inference Procedures
REVIEW

• KB i a sentence a can be derived
  from KB by procedure i
  
• Soundness
• i is sound if whenever KB i a,it is also true
  that KB a
  
• unsound procedures make things up
• Completeness
• i is complete if whenever KB a,it is also true
  that KB i a
  
• incomplete procedures miss things

6
Propositional Logic Syntax
REVIEW

• Propositional logic is the simplest logic
• aka Boolean logic
• illustrates basic ideas
• less expressive than first-order logic (next
  chapter)
  
• 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)
  

7
Propositional Logic Semantics
REVIEW

• Each model specifies true/false value for each
  proposition symbol
  
• E.g. P1,2 is false, P2,2 is true, P3,1 is false
• With these symbols, 8 possible models, could 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
• S1 ? S2 is true iff S1?S2 is true and S2?S1 is
  true
  
• Simple recursive process evaluates an arbitrary
  sentence, e.g.,
• ?P1,2 ? (P2,2 ? P3,1) true ? (true ? false)
  true ? true true
  

8
Truth Tables for Connectives
REVIEW
9
Wumpus World Example

• Situation after detecting nothing in 1,1,
  moving right, breeze in 2,1

10
Wumpus World KB

• Propositions
• 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.
• Knowledge Base
• 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

R1, R2, R3 are true in all wumpus worlds
R4, R5 result from percepts in current wumpus
world
11
Inference by Truth Table
a1 P1,2
possible models are enumerated by listing all
possible values for the 7 propositions
KB a1?
12
Inference by Truth Table
KB R1 ? R2 ? R3 ? R4 ? R5 (all sentences
must be true)
KB is true onlyin three models
13
Inference by Truth Table
in all models where KB is true, a1 P1,2 is
also true,therefore, KB a1 (there is no pit in
1,2)
14
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)
  

15
Logical Equivalence

• Two sentences are logically equivalent if they
  are true in the same set of models
• a ß iff a ß and ß a
• Equivalence can be determined using truth tables
  

P Q P ? Q Q ? P
false false false false
false true false false
true false false false
true true true true
16
Standard Equivalences
17
Validity

• A sentence is valid if it is true in all models
• The following sentences are valid
• True
• A ??A
• A ? A
• (A ? (A ? B)) ? B
• Valid sentences are also called tautologies
• Deduction Theorem
• (KB a) if and only if (KB ? a) is valid

18
Satisfiability

• A sentence is satisfiable if it is true in some
  model
• The following are satisfiable
• A? B
• R1 ? R2 ? R3 ? R4 ? R5 (in the wumpus example)
• A sentence is unsatisfiable if it is true in no
  models
• The following is unsatisfiable
• A??A
• Satisfiability can be checked by enumerating
  possible models until one is found in which
  sentence is true (NP-complete)

satisfiability is the first problem that was
proven to be NP-complete.
19
Satisfiability

• Results
• a is valid iff ?a is unsatisfiable
• ?a is satisfiable iff a is not valid
• KB a if and only if (KB ??a) is unsatisfiable
• Proving that a ß by checking the
  unsatisfiability of (a ?? ß) is the standard
  proof technique of reductio ad absurdum (aka
  refutation or proof by contradiction)
• assume ß is false and show that this leads to a
  contradiction with some axiom a

20
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
  applicationsCan use inference rules as operators
  in a standard search algorithm
  
• Typically requires transformation of sentences
  into a normal form
• Model checking
• truth table enumeration (always exponential in n)
  
• improved backtracking, e.g., DPLL
• heuristic search in model space (sound but
  incomplete)
• e.g., min-conflicts-like hill-climbing
  algorithms
  

21
Wumpus World KB

• Propositions
• 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.
• Knowledge Base
• 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

R1, R2, R3 are true in all wumpus worlds
R4, R5 result from percepts in current wumpus
world
22
Wumpus World Inference

• 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
• R6 B1,1 ? (P1,2 ? P2,1) ? (P1,2 ? P2,1) ?
  B1,1 (from R2)
• R7 (P1,2 ? P2,1) ? B1,1 (from R6)
• R8 B1,1 ? (P1,2 ? P2,1) (from R7)
• R9 (P1,2 ? P2,1) (from R8 and R4)
• R10 P1,2 ? P2,1 (from R9)
• neither 1,2 nor 2,1 contains a pit