Limitations of First-Order Logic FOL is very expressive, but... consider how to translate these: "most students graduate in 4 years" x student(x) → duration(undergrad)years(4) ??? "only a few students switch majors" s,m1,m2,t1,t2 student(s)^major(s,m1,t1)major(s,m2,t2) m1m2  t1t2 ??? "all students must take Communications, except Joe" The problems involve: default rules & exceptions degrees of truth strength of rules

Solutions Closed-World Assumption in PROLOG Non-monotonic logics Semantic Networks Fuzzy Logic Bayesian Probability

Closed-World Assumption (CWA) in PROLOG every fact that is not asserted is assumed to be false - very handy example facts: query with negation: how is this implemented? how does this affect reasoning/inference? modify back-chaining to handle negative antecedents: "Negation-as-failure" when trying to prove ¬P(X) on goal stack, try proving P(X) and if fail then ¬P(X) succeeds

Non-monotonic Logics allow retractions later (popular for truth- maintenance systems) "birds fly", "penguins are birds that don't fly" x bird(x)→fly(x) x penguin(x)→bird(x), x penguin(x)→¬fly(x) bird(tweety), bird(opus) |= fly(opus) later, add that opus is a penguin, change inference penguin(opus) |= ¬fly(opus) Definition: A logic is monotonic if everything that is entailed by a set of sentences a is entailed by any superset of sentences ab opus example is non-monotonic

example syntax of default rule bird(x): fly(x) / fly(x) or bird(x) ≻ fly(x) semantics: "if PRECOND is satisfied and it is not inconsistent to believe CONSEQ, then CONSEQ" Circumscription add abnormal predicates to rules x bird(x)¬abnormal1(x)→fly(x) x penguin(x) ¬abnormal2(x) →bird(x) x penguin(x) ¬abnormal3(x) →¬fly(x) algorithm: minimize number of abnormals needed to make KB consistent {bird(tweety),fly(tweety),bird(opus),penguin(opus), ¬fly(opus)} is INCONSISTENT {bird(tweety),fly(tweety),bird(opus),penguin(opus), ¬fly(opus), abnormal1(opus)} is CONSISTENT

Semantic Networks graphical representation of knowledge nodes, slots, edges, "isa" links procedural mechanism for answering queries follow links different than formal definition of "entailment" inheritance can override defaults related to Description Logics popular for large medical knowledge bases of anatomy, diseases, injuries, procedures...

Fuzzy Logic some expressions involve "degrees" of truth, like "John is tall" membership function "most students with high SATs have high GPAs" inference by computing with membership funcs. "only days that are warm and not windy are good for playing frisbee" suppose today is 85 and the wind is 15 kts NE T(A^B) = min(T(A),T(B)) T(AvB) = max(T(A),T(B)) popular for control applications (like thermostats...)

warm not windy windy 1.0 wind speed 0.4 15 85 temp: 30 100 0kts 50 kts wind: speed temp

Probability conditional probabilities play role of rules people with a toothache are likely to have a cavity p(cavity|toothache) = 0.6 joint probabilities, priors Bayes Rule

Bayesian Networks graphical models where edges represent conditional probabilities popular for modern AI systems (expert systems) important for handling uncertainty =