Limitations of FirstOrder Logic FOL is very expressive

  • Slides: 10
Download presentation
Limitations of First-Order Logic • FOL is very expressive, but. . . • consider

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, m 1, m 2, t 1, t 2 student(s)^major(s, m 1, t 1) major(s, m 2, t 2) m 1 m 2 t 1 t 2 ? ? ? • "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

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

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 truthmaintenance systems) • "birds fly", "penguins

Non-monotonic Logics • allow retractions later (popular for truthmaintenance 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)

• 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) ¬abnormal 1(x)→fly(x) x penguin(x) ¬abnormal 2(x) →bird(x) x penguin(x) ¬abnormal 3(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), abnormal 1(opus)} is CONSISTENT • • •

Semantic Networks • graphical representation of knowledge • nodes, slots, edges, "isa" links •

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" •

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(Av. B) = max(T(A), T(B)) • popular for control applications (like thermostats. . . )

not windy 1. 0 0. 4 temp: 30 85 wind speed warm 15 100

not windy 1. 0 0. 4 temp: 30 85 wind speed warm 15 100 0 kts wind: speed 50 kts temp

Probability • conditional probabilities play role of rules • people with a toothache are

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

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