![Hard Satisfiability Problems
• Critical point: ratio of clauses/variables =
4.24 (empirical).
• If above, problems usually unsatsifiable.
• If below, problems usually satisfiable.
• Theorem: Critical range is bounded by
[3.0003, 4.598].](https://image.slidesharecdn.com/copyofpropositionallogic-230524163310-3d9c4f58/85/PropositionalLogic-ppt-27-320.jpg)
PropositionalLogic.ppt
- 1. Propositional Logic or how to reason correctly Chapter 8 (new edition) Chapter 7 (old edition)
- 2. Goals • Feigenbaum: In the knowledge lies the power. Success with expert systems. 70’s. • What can we represent? – Logic(s): Prolog – Mathematical knowledge: mathematica – Common Sense Knowledge: Lenat’s Cyc has a million statement in various knowledge – Probabilistic Knowledge: Bayesian networks • Reasoning: via search
- 3. History • 300 BC Aristotle: Syllogisms • Late 1600’s Leibnitz’s goal: mechanization of inference • 1847 Boole: Mathematical Analysis of Logic • 1879: Complete Propositional Logic: Frege • 1965: Resolution Complete (Robinson) • 1971: Cook: satisfiability NP-complete • 1992: GSAT Selman min-conflicts
- 4. Syllogisms • Proposition = Statement that may be either true or false. • John is in the classroom. • Mary is enrolled in 270A. • If A is true, and A implies B, then B is true. • If some A are B, and some B are C, then some A are C. • If some women are students, and some students are men, then ….
- 5. Concerns • What does it mean to say a statement is true? • What are sound rules for reasoning • What can we represent in propositional logic? • What is the efficiency? • Can we guarantee to infer all true statements?
- 6. Semantics • Model = possible world • x+y = 4 is true in the world x=3, y=1. • x+y = 4 is false in the world x=3, y = 1. • Entailment S1,S2,..Sn |= S means in every world where S1…Sn are true, S is true. • Careful: No mention of proof – just checking all the worlds. • Some cognitive scientists argue that this is the way people reason.
- 7. Reasoning or Inference Systems • Proof is a syntactic property. • Rules for deriving new sentences from old ones. • Sound: any derived sentence is true. • Complete: any true sentence is derivable. • NOTE: Logical Inference is monotonic. Can’t change your mind.
- 8. Proposition Logic: Syntax • See text for complete rules • Atomic Sentence: true, false, variable • Complex Sentence: connective applied to atomic or complex sentence. • Connectives: not, and, or, implies, equivalence, etc. • Defined by tables.
- 9. Propositional Logic: Semantics • Truth tables: p =>q |= ~p or q p q p =>q ~p or q t t t t t f f f t t t t t t t t
- 10. Implies => • If 2+2 = 5 then monkeys are cows. TRUE • If 2+2 = 5 then cows are animals. TRUE • Indicates a difference with natural reasoning. Single incorrect or false belief will destroy reasoning. No weight of evidence.
- 11. Inference • Does s1,..sk entail s? • Say variables (symbols) v1…vn. • Check all 2^n possible worlds. • In each world, check if s1..sk is true, that s is true. • Approximately O(2^n). • Complete: possible worlds finite for propositional logic, unlike for arithmetic.
- 12. Translation into Propositional Logic • If it rains, then the game will be cancelled. • If the game is cancelled, then we clean house. • Can we conclude? – If it rains, then we clean house. • p = it rains, q = game cancelled r = we clean house. • If p then q. not p or q • If q then r. not q or r • if p then r. not p or r (resolution)
- 13. Concepts • Equivalence: two sentences are equivalent if they are true in same models. • Validity: a sentence is valid if it true in all models. (tautology) e.g. P or not P. – Sign: Members or not Members only. – Berra: It’s not over till its over. • Satisfiability: a sentence is satisfied if it true in some model.
- 14. Validity != Provability • Goldbach’s conjecture: Every even number (>2) is the sum of 2 primes. • This is either valid or not. • It may not be provable. • Godel: No axiomization of arithmetic will be complete, i.e. always valid statements that are not provable.
- 15. Natural Inference Rules • Modus Ponens: p, p=>q |-- q. – Sound • Resolution example (sound) – p or q, not p or r |-- q or r • Abduction (unsound, but common) – q, p=>q |-- p – ground wet, rained => ground wet |-- rained – medical diagnosis
- 16. Natural Inference Systems • Typically have dozen of rules. • Difficult for people to use. • Expensive for computation. – e.g. a |-- a or b – a and b |-- a • All known systems take exponential time in worse case. (co-np complete)
- 17. Full Propositional Resolution • clause 1: x1 +x2+..xn+y (+ = or) • clause 2: -y + z1 + z2 +… zm • clauses contain complementary literals. • x1 +.. xn +z1 +… zm • y and not y are complementary literals. • Theorem: If s1,…sn |= s then s1,…sn |-- s by resolution. Refutation Completeness. Factoring: (simplifying: x or x goes to x)
- 18. Conjunctive Normal Form • To apply resolution we need to write what we know as a conjunct of disjuncts. • Pg 215 contains the rules for doing this transformation. • Basically you remove all and => and move “not’s” inwards. Then you may need to apply distributive laws.
- 19. Proposition -> CNF Goal: Proving R • P • (P&Q) =>R • (S or T) => Q • T • Distributive laws: • (-s&-t) or q (-s or q)&(-t or q). • P • -P or –Q or R • -S or Q • -T or Q • T • Remember:implicit adding.
- 20. Resolution Proof • P (1) • -P or –Q or R (2) • -S or Q (3) • -T or Q (4) • T (5) • ~R (6) • -P or –Q : 7 by 2 & 6 • -Q : 8 by 7 & 1. • -T : 9 by 8 & 4 • empty: by 9 and 5. • Done: order only effects efficiency.
- 21. Resolution Algorithm To prove s1, s2..sn |-- s 1. Put s1,s2,..sn & not s into cnf. 2. Resolve any 2 clauses that have complementary literals 3. If you get empty, done 4. Continue until set of clauses doesn’t grow. Search can be expensive (exponential).
- 22. Forward and Backward Reasoning • Horn clause has at most 1 positive literal. – Prolog only allows Horn clauses. – if a, b, c then d => not a or not b or not c or d – Prolog writes this: • d :- a, b, c. – Prolog thinks: to prove d, set up subgoals a, b,c and prove/verify each subgoal.
- 23. Forward Reasoning • From facts to conclusions • Given s1: p, s2: q, s3: p&q=>r • Rewrite in clausal form: s3 = (-p+-q+r) • s1 resolve with s3 = -q+r (s4) • s2 resolve with s4 = r • Generally used for processing sensory information.
- 24. Backwards Reasoning: what prolog does • From Negative of Goal to data • Given s1: p, s2: q, s3: p&q=>r • Goal: s4 = r • Rewrite in clausal form: s3 = (-p+-q+r) • Resolve s4 with s3 = -p +-q (s5) • Resolve s5 with s2 = -p (s6) • Resolve s6 with s1 = empty. Eureka r is true.
- 25. Davis-Putnam Algorithm • Effective, complete propositional algorithm • Basically: recursive backtracking with tricks. – early termination: short circuit evaluation – pure symbol: variable is always + or – (eliminate the containing clauses) – one literal clauses: one undefined variable, really special cases of MRV • Propositional satisfication is a special case of Constraint satisfication.
- 26. WalkSat • Heuristic algorithm, like min-conflicts • Randomly assign values (t/f) • For a while do – randomly select a clause – with probability p, flip a random variable in clause – else flip a variable which maximizes number of satisfied clauses. • Of course, variations exists.
- 27. Hard Satisfiability Problems • Critical point: ratio of clauses/variables = 4.24 (empirical). • If above, problems usually unsatsifiable. • If below, problems usually satisfiable. • Theorem: Critical range is bounded by [3.0003, 4.598].
- 28. What can’t we say? • Quantification: every student has a father. • Relations: If X is married to Y, then Y is married to X. • Probability: There is an 80% chance of rain. • Combine Evidence: This car is better than that one because… • Uncertainty: Maybe John is playing golf.