Propositional Logic or how to reason correctlyChapter 8 (new edition)Chapter 7 (old edition)

GoalsFeigenbaum: In the knowledge lies the power. Success with expert systems. 70s.What can we represent?Logic(s): PrologMathematical knowledge: mathematicaCommon Sense Knowledge: Lenats Cyc has a million statement in various knowledgeProbabilistic Knowledge: Bayesian networks Reasoning: via search

History300 BC Aristotle: SyllogismsLate 1600s Leibnitzs goal: mechanization of inference1847 Boole: Mathematical Analysis of Logic1879: Complete Propositional Logic: Frege1965: Resolution Complete (Robinson)1971: Cook: satisfiability NP-complete1992: GSAT Selman min-conflicts

SyllogismsProposition = 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 .

ConcernsWhat does it mean to say a statement is true?What are sound rules for reasoningWhat can we represent in propositional logic?What is the efficiency?Can we guarantee to infer all true statements?

SemanticsModel = possible worldx+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 S1Sn 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.

Reasoning or Inference SystemsProof 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. Cant change your mind.

Proposition Logic: SyntaxSee text for complete rulesAtomic Sentence: true, false, variableComplex Sentence: connective applied to atomic or complex sentence.Connectives: not, and, or, implies, equivalence, etc.Defined by tables.

Propositional Logic: SemanticsTruth tables: p =>q |= ~p or q

Implies =>If 2+2 = 5 then monkeys are cows. TRUEIf 2+2 = 5 then cows are animals. TRUEIndicates a difference with natural reasoning. Single incorrect or false belief will destroy reasoning. No weight of evidence.

InferenceDoes s1,..sk entail s?Say variables (symbols) v1vn.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.

Translation into Propositional LogicIf 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 qIf q then r. not q or rif p then r. not p or r (resolution)

ConceptsEquivalence: 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: Its not over till its over.Satisfiability: a sentence is satisfied if it true in some model.

Validity != ProvabilityGoldbachs 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.

Natural Inference RulesModus Ponens: p, p=>q |-- q.SoundResolution example (sound)p or q, not p or r |-- q or rAbduction (unsound, but common)q, p=>q |-- pground wet, rained => ground wet |-- rainedmedical diagnosis

Natural Inference SystemsTypically have dozen of rules.Difficult for people to use.Expensive for computation.e.g. a |-- a or ba and b |-- aAll known systems take exponential time in worse case. (co-np complete)

Full Propositional Resolutionclause 1: x1 +x2+..xn+y (+ = or)clause 2: -y + z1 + z2 + zmclauses contain complementary literals.x1 +.. xn +z1 + zmy 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)

Conjunctive Normal FormTo 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 nots inwards. Then you may need to apply distributive laws.

Proposition -> CNFGoal: Proving RP(P&Q) =>R(S or T) => QTDistributive laws:(-s&-t) or q (-s or q)&(-t or q).P-P or Q or R-S or Q-T or QTRemember:implicit adding.

Resolution ProofP (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 & 4empty: by 9 and 5.Done: order only effects efficiency.

Resolution Algorithm To prove s1, s2..sn |-- s Put s1,s2,..sn & not s into cnf.Resolve any 2 clauses that have complementary literalsIf you get empty, doneContinue until set of clauses doesnt grow.Search can be expensive (exponential).

Forward and Backward ReasoningHorn 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 dProlog writes this:d :- a, b, c. Prolog thinks: to prove d, set up subgoals a, b,c and prove/verify each subgoal.

Forward ReasoningFrom facts to conclusionsGiven 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 = rGenerally used for processing sensory information.

Backwards Reasoning: what prolog doesFrom Negative of Goal to dataGiven s1: p, s2: q, s3: p&q=>r Goal: s4 = rRewrite 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.

Davis-Putnam AlgorithmEffective, complete propositional algorithmBasically: recursive backtracking with tricks.early termination: short circuit evaluationpure symbol: variable is always + or (eliminate the containing clauses)one literal clauses: one undefined variable, really special cases of MRVPropositional satisfication is a special case of Constraint satisfication.

WalkSatHeuristic algorithm, like min-conflictsRandomly assign values (t/f)For a while dorandomly select a clausewith probability p, flip a random variable in clauseelse flip a variable which maximizes number of satisfied clauses.Of course, variations exists.

Hard Satisfiability ProblemsCritical 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].

What cant 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 becauseUncertainty: Maybe John is playing golf.