1. Logical Agents Logic Propositional Logic Summary
Knowledge-Based Agents
Logic
Propositional Logic
Summary

2. Knowledge-Based Agents
Central component is a knowledge-base
Knowledge is made of sentences
Sentences are expressed in a language
We wish to derive new sentences through
inference.

3. Knowledge-Based Algorthim
Function KB (percept) returns action
TELL (KB; PERCEPT-SENTENCE(percept,t))
action = ASK(KB,MAKE-ACTION-QUERY(t))
TELL (KB;ACTION-SENTENCE(action,t))
T = t + 1
Return action

4. Declarative vs Procedural
Two approaches to store knowledge:
Declarative.
Sentences representing knowledge
Procedural.
Encode behavior directly as code.

5. An Example Moving on a square to reach a goal
state while avoiding risks.

9. Logical Agents Logic Propositional Logic Summary
Knowledge-Based Agents
Logic
Propositional Logic
Summary

10. Logic Important terms: Syntax .- rules to represent well-formed
sentences
Semantics.- define the truth of each sentence.
Model.- Possible world (m is a model of a)
Entailment:
a |= b a entails b means:
In every model where a is true, b is also true.

11. Logic Sentence a is derived from KB by algorithm i: KB |=i a
Algorithm i is called sound (truth preserving) if
it derives only entailed sentences.
Algorithm i is called complete if it derives all
entailed sentences.

12. Physical configuration
Sentence Sentence
entails
Representation
semantics
semantics
Real World Real World
follows

13. Allen Newell Born in 1927 – died in 1992 1982 “The Knowledge Level”
Rational agents can be described
by the knowledge they contain
With Herbert Simon he developed the Logic Theorist (first real AI program).
Winner of the Turing Award

14. Logical Agents Logic Propositional Logic Summary
Knowledge-Based Agents
Logic
Propositional Logic
Summary

15. Syntax Atomic sentences: propositional symbols
Complex sentences use logical connectives
Negation,conjunction,disjunction
Implication and biconditional.

16. Syntax Sentence  Atom | Complex Structure
Atom  True | False | Symbol
Symbol  P | Q | R
Complex S  sentence
| ( sentence ^ sentence )
| ( sentence V sentence )
| ( sentence  sentence)
| ( sentence  sentence)

17. Semantics How to define the truth value of statements?
Connectives associate to truth tables.
The knowledge base of an agent grows by
telling it of new statements:
TELL(KB,S1), … TELL(KB,Sn)
KB = S1 ^ … ^ Sn

18. Inference We want to know if KB entails a: KB |= a
Approach 1: Try all models
KB: {b , b  a}
b a Models consistent with KB
t t YES
t f NO
f t NO
f f NO

19. Check All Models Method is characterized as:
sound (directly implements entailment)
complete (it works for any KB and a and
it always terminates.
Problem: With n symbols there are 2n models.
Time complexity is O(2n)
Space complexity is O(n)

20. Some concepts Equivalence: a = b  a |= b and b |= a Validity
A sentence is valid if it is true in all models
Example p V -p
Satisfiability
A sentence is satisfiable if it is true in some
model

21. Reasoning Patterns Modus Ponens a  b, a b And-Elimination a ^ b a
Commutativity of ^ and V
de Morgan’s Laws, etc.

22. Proofs Applying a sequence of rules is called a proof.
Equivalent to searching for a solution
Monotonicity: if KB |= a then KB ^ b |= a
{b, b a} a
Modus ponens

23. Resolution Complete and sound inference algorithm a v b ~a v c b v c
Cannot derive all conclusions but can tell
if they are true or false (refutation completeness).

24. Applying Resolution Resolution only works on disjunction of literals.
How can we apply resolution to all clauses?
Solution: convert to conjunctive normal form (CNF)
CNF : (L11 V .. V L1k) ^ … ^ (Ln1 V … V Lnk)

25. Resolution How does it work? To show that KB |= a
We show that (KB ^ ~a) is unsatisfiable.
Every pair that contains complementary literals is
resolved.
Continue until there are no new clauses (KB |= a)
Or we derive the empty clause (a V ~a) (KB |= a)

26. Example
a v b v ~c d v c KB
a v b v d ~b
a v d ~d
a ~a
KB |= b

27. Horn Clauses In practice we restrict our clauses to Horn Clauses.
(base of logic programming)
Horn Clause: disjunction of literals with at most
one of them being positive.
Example: (~a v ~b v c) (definite clause)
Equivalent to ( a ^ b  c )

28. Forward Chaining Begins from known facts in the knowledge base.
If premises of implication are known derive
conclusion.
Continue until query is added or no more
inferences can be made.
a ^ b ^ c  d a,b,c conclude d

29. Example Q P  Q L ^ M  P P B ^ L  M A ^ P  L M A ^ B  L A B L A B
AND-OR Graph

30. Backward Chaining Works backwards from the query q
Find implications that conclude q
If all premises can be proved true then q is true
a ^ b ^ c  q
How do I prove a? and b? and c?

31. Logical Agents Logic Propositional Logic Summary
Knowledge-Based Agents
Logic
Propositional Logic
Summary

32. Summary Rational agents need to work with knowledge bases.
A language is defined by syntax and semantics
KB |= a if a is true in all models where KB is true
Inference techniques can be characterized based
on soundness and completeness.
Simplest logic is propositional logic
A strong inference rule is called “resolution”
When using Horn clauses, reasoning methods
are forward and backward chaining.