1. CS 416 Artificial Intelligence
Lecture 11
Logical Agents
Chapter 7

2. Midterm Exam Midterm will be on Thursday, March 13th
It will cover material up until Feb 27th

3. Reasoning w/ propositional logic
Remember what we’ve developed so far
Logical sentences
And, or, not, implies (entailment), iff (equivalence)
Syntax vs. semantics
Truth tables
Satisfiability, proof by contradiction

4. Logical Equivalences
Know these equivalences

5. Reasoning w/ propositional logic
Inference Rules
Modus Ponens:
Whenever sentences of form a => b and a are given the sentence b can be inferred
R1: Green => Martian
R2: Green
Inferred: Martian

6. Reasoning w/ propositional logic
Inference Rules
And-Elimination
Any of conjuncts can be inferred
R1: Martian ^ Green
Inferred: Martian
Inferrred: Green
Use truth tables if you want to confirm inference rules

7. Example of a proof P? P?
~P ~B B P? P?

8. Example of a proof P? ~P
~P ~B B ~P P?

9. Constructing a proof Proving is like searching
Find sequence of logical inference rules that lead to desired result
Note the explosion of propositions
Good proof methods ignore the countless irrelevant propositions

10. Monotonicity of knowledge base
Knowledge base can only get larger
Adding new sentences to knowledge base can only make it get larger
If (KB entails a)
((KB ^ b) entails a)
This is important when constructing proofs
A logical conclusion drawn at one point cannot be invalidated by a subsequent entailment

11. How many inferences?
Previous example relied on application of inference rules to generate new sentences
When have you drawn enough inferences to prove something?
Too many make search process take longer
Too few halt logical progression and make proof process incomplete

12. Resolution Unit Resolution Inference Rule
If m and li are complementary literals

13. Resolution Inference Rule
Also works with clauses
But make sure each literal appears only once

14. Resolution and completeness
Any complete search algorithm, applying only the resolution rule, can derive any conclusion entailed by any knowledge base in propositional logic
More specifically, refutation completeness
Able to confirm or refute any sentence
Unable to enumerate all true sentences

15. What about “and” clauses?
Resolution only applies to “or” clauses
Every sentence of propositional logic can be transformed to a logically equivalent conjunction of disjunctions of literals
Conjunctive Normal Form (CNF)
A sentence expressed as conjunction of disjunction of literals
k-CNF: exactly k literals per clause

16. CNF

17. An algorithm for resolution
We wish to prove KB entails a
Must show (KB ^ ~a) is unsatisfiable
No possible way for KB to entail (not a)
Proof by contradiction

18. An algorithm for resolution
KB ^ ~a is put in CNF
Each pair with complementary literals is resolved to produce new clause which is added to KB (if novel)
Cease if no new clauses to add (~a is not entailed)
Cease if resolution rule derives empty clause (~a is entailed)

19. Example of resolution Proof that there is not a pit in P1,2: ~P1,2
KB ^ P1,2 leads to empty clause
Therefore ~P1,2 is true

20. Formal Algorithm

21. Horn Clauses Horn Clause
Disjunction of literals with at most one is positive
(~a V ~b V ~c V d)
(~a V b V c V ~d) Not a Horn Clause

22. Horn Clauses Can be written as a special implication
(~a V ~b V c)  (a ^ b) => c
(~a V ~b V c) == (~(a ^ b) V c) … de Morgan
(~(a ^ b) V c) == ((a ^ b) => c … implication elimination

23. Horn Clauses Permit straightforward inference determination
Forward chaining
Backward chaining

24. Horn Clauses
Permit determination of entailment in linear time (in order of knowledge base size)

25. Forward Chaining

26. Forward Chaining Properties Data Driven Sound Complete
All entailed atomic sentence will be derived
Data Driven
Start with what we know
Derive new info until we discover what we want

27. Backward Chaining Start with what you want to know, a query (q)
Look for implications that conclude q
Goal-Directed Reasoning