CS 416 Artificial Intelligence

1. CS 416Artificial Intelligence Lecture 10 Logic Chapters 7 and 8

2. Homework Assignment • First homework assignment is out • Due October 20th

3. Midterm • October 25th • Up through chapter 9 (excluding chapter 5)

4. Review • Store information in a knowledge base • Backus-Naur Form (BNF) • Use equivalences to isolate what you want • Use inference to relate sentences • Modus Ponens • And-Elimination

5. 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 • The two inference rules are sound but not complete!

6. Completeness • Recall that a complete inference algorithm is one that can derive any sentence that is entailed • Resolution is a single inference rule • Guarantees the ability to derive any sentence that is entailed • i.e. it is complete • It must be partnered with a complete search algorithm

7. Resolution • Resolution is a single inference rule • Guarantees the ability to derive any sentence that is entailed • i.e. it is complete • It must be partnered with a complete search algorithm

8. Resolution • Unit Resolution Inference Rule • If m and li arecomplementaryliterals

9. Resolution Inference Rule • Also works with clauses • But make sure each literal appears only once • We really just want the resolution to return A

10. 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

11. What about “and” clauses? • Resolution only applies to “or” clauses (disjunctions) • 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

12. CNF

13. An algorithm for resolution • We wish to prove KB entails a • That is, will some sequence of inferences create new sentences in knowledge base equal to a? • If some sequence of inferences created ~ a, we would know the knowledge base could not also entail a • a ^ ~ a is impossible

14. An algorithm for resolution • To show KB cannot resolve a • Must show (KB ^ ~a) is unsatisfiable • No possible way for KB to entail (not a) • Proof by contradiction

15. An algorithm for resolution • Algorithm • (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 • If resolution generates a… a ^ ~a = empty clause • (~a is entailed)

16. 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

17. Formal Algorithm

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

19. Horn Clauses • Why? • Every Horn clause can be written as an implication: • (a conjunction of positive literals) => single positive literal • Inference algorithms can use forward chaining and backward chaining • Entailment computation is linear in size of KB

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

21. Forward Chaining • Does KB (Horn clauses) entail q (single symbol)? • Keep track of known facts (positive literals) • If the premises of an implication are known • Add the conclusion to the known set of facts • Repeat process until no further inferences or q is added

22. Forward Chaining

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

24. Backward Chaining • Start with what you want to know, a query (q) • Look for implications that conclude q • Look at the premises of those implications • Look for implications that conclude those premises… • Goal-Directed Reasoning • Can be less complex search than forward chaining

25. Making it fast • Example • Problem to solve is in CNF • Is Marvin a Martian given --- M == 1 (true)? • Marvin is green --- G=1 • Marvin is little --- L=1 • (little and green) implies Martian --- (L ^ G) => M ~(L^G) V M ~L V ~G V M • Proof by contradiction… are there true/false values for G, L, and M that are consistent with knowledge base and Marvin not being a Martian? • G ^ L ^ (~L V ~G V M) ^ ~M == empty?

26. Searching for variable values • Want to find values such that: • Randomly consider all true/false assignments to variables until we exhaust them all or find match (model checking) • (G, L, M) = (1, 0, 0)… no = (0, 1, 0)… no = (0, 0, 0)… no = (1, 1, 0)… no • Alternatively… G ^ L ^ (~L V ~G V M) ^ ~M == 0

27. G L M Backtracking Algorithm • Davis-Putnam Algorithm (DPLL) • Search through possible assignments to (G, L, M) via depth-first search (0, 0, 0) to (0, 0, 1) to (0, 1, 0)… • Each clause of CNF must be true • Terminate consideration when clause evaluates to false • Use heuristics to reduce repeated computation of propositions • Early termination • Pure symbol heuristic • Unit clause heuristic Cull branches from tree

28. Searching for variable values • Other ways to find (G, L, M) assignments for: • G ^ L ^ (~L V ~G V M) ^ ~M == 0 • Simulated Annealing (WalkSAT) • Start with initial guess (0, 1, 1) • With each iteration, pick an unsatisfied clause and flip one symbol in the clause • Evaluation metric is the number of clauses that evaluate to true • Move “in direction” of guesses that cause more clauses to be true • Many local mins, use lots of randomness

29. WalkSAT termination • How do you know when simulated annealing is done? • No way to know with certainty that an answer is not possible • Could have been bad luck • Could be there really is no answer • Establish a max number of iterations and go with best answer to that point

30. So how well do these work? • Think about it this way • 16 of 32 possible assignments are models (are satisfiable) for this sentence • Therefore, 2 random guesses should find a solution • WalkSAT and DPLL should work quickly

31. What about more clauses? • If # symbols (variables) stays the same, but number of clauses increases • More ways for an assignment to fail (on any one clause) • More searching through possible assignments is needed • Let’s create a ratio, m/n, to measure #clauses / # symbols • We expect large m/n causes slower solution

32. What about more clauses • Higher m/n means fewer assignments will work • If fewer assignments work it is harder for DPLL and WalkSAT

33. Combining it all • 4x4 Wumpus World • The “physics” of the game • At least one wumpus on board • A most one wumpus on board (for any two squares, one is free) • n(n-1)/2 rules like: ~W1,1 V ~W1,2 • Total of 155 sentences containing 64 distinct symbols

34. Wumpus World • Inefficiencies as world becomes large • Knowledge base must expand if size of world expands • Preferred to have sentences that apply to all squares • We brought this subject up last week • Next chapter addresses this issue

35. Chapter 8: First-Order Logic

36. What do we like about propositional logic? • It is: • Declarative • Relationships between variables are described • A method for propagating relationships • Expressive • Can represent partial information using disjunction • Compositional • If A means foo and B means bar, A ^ B means foo and bar

37. What don’t we like about propositional logic? • Lacks expressive power to describe the environment concisely • Separate rules for every square/square relationship in Wumpus world

38. Natural Language • English appears to be expressive • Squares adjacent to pits are breezy • But natural language is a medium of communication, not a knowledge representation • Much of the information and logic conveyed by language is dependent on context • Information exchange is not well defined • Not compositional (combining sentences may mean something different) • It is ambiguous

39. But we borrow representational ideas from natural language • Natural language syntax • Nouns and noun phrases refer to objects • People, houses, cars • Properties and verbs refer to object relations • Red, round, nearby, eaten • Some relationships are clearly defined functions where there is only one output for a given input • Best friend, first thing, plus • We build first-order logic around objects and relations

40. Ontology • a “specification of a conceptualization” • A description of the objects and relationships that can exist • Propositional logic had only true/false relationships • First-order logic has many more relationships • The ontological commitment of languages is different • How much can you infer from what you know? • Temporal logic defines additional ontological commitments because of timing constraints