Artificial Intelligence

Artificial Intelligence CS 165A Tuesday, November 6, 2007 • Inference in FOL (Ch 9) Today 1 1 1

George Boole (1815-1864) British • More than 100 years later, he didn’t know about Leibniz, but proceeded to bring to life part of Leibniz’ dream • His insight: Logical relationships are expressible as a kind of algebra • Letters represent classes (rather than numbers) • So logic can be viewed as a form of mathematics • Published The Laws of Thought • He extended Aristotle's simple syllogisms to a broader range of reasoning • Syllogism: Premise_1, Premise_2  Conclusion • His logic: Propositional logic

Gottlob Frege (1848-1925) German • He provided the first fully developed system of logic that encompassed all of the deductive reasoning in ordinary mathematics. • He intended for logic to be the foundation of mathematics – all of mathematics could be based on, and derived from, logic • In 1879 he published Begriffsschrift, subtitled “A formula language, modeled upon that of arithmetic, for pure thought” • This can be considered the ancestor of all current computer programming languages • Made the distinction between syntax and semantics critical • He invented what we today call predicate calculus (or first-order logic)

Notes • HW#3 due Thursday • Midterm exam one week from today • Short-answer conceptual questions and HW-like questions • Topics: Reading, lectures, material through chapter 8 • What AI is, main issues and areas in AI • Problem solving and intelligent agents • Blind search, informed search, adversarial search • Logic and inference in propositional calculus • Basics of first-order logic • Closed book • You may bring one 8.5”x11” piece of paper with your notes • Some formulas, procedures will be given (posted in advance)

Reminder • Term • Constant, variable, function( ) • Atomic sentence • Predicate( ), term1 = term2 • Literal • An atomic sentence or a negated atomic sentence • Sentence • Atomic sentence, sentences with quantifiers and/or connectives

Bob is a buffalo Pat is a pig Buffaloes outrun pigs Does Bob outrun Pat? Buffalo(Bob) Pig(Pat) Buffalo(x)  Pig(y) Outrun(x,y) KB entails Outrun(Bob, Pat)? KB0 S KB0 KB1 KB2 KB3 Review Simple example of inference in FOL KB0 |– Buffalo(Bob)  Pig(Pat) (And-Introduction) KB1 |– Buffalo(Bob)  Pig(Pat) Outrun(Bob, Pat) (Universal Instantiation) [coming soon] KB2 |– Outrun(Bob, Pat) (Modus Ponens)

Using FOL to express knowledge • One can express the knowledge of a particular domain in first-order logic • Example: The “kinship domain” • Objects: people • Properties: gender, family relationships • Unary predicates: Male, Female, MotherOf, FatherOf • Binary predicates: Parent, Sibling, Brother, Sister, Son, Daughter, Father, Mother, Uncle, Aunt, Grandparent, Grandfather, Grandmother, Husband, Wife, Spouse, Brother-in-law, Stepmother, etc…. • Functions: MotherOf, FatherOf… • Note: There is usually (always?) more than one way to specify knowledge

Kinship domain • Write down what we know (what we want to be in the KB) • One’s mother is one’s female parent •  m, c Mother(m, c)  Female(m)  Parent(m, c) •  m, c TheMotherOf(c) = m  Female(m)  Parent(m, c) • One’s husband is one’s male spouse •  w, h Husband(h, w)  Male(h)  Spouse(h, w) • One is either male or female •  x Male(x)  Female(x) • Parent-child relationship •  p, c Parent(p, c)  Child(c, p) • Grandparent-grandchild relationship •  g, c Grandparent(g, c)  p Parent(g, p)  Parent(p, c) • Etc… • Now we can reason about family relationships. (How?)

Kinship domain (cont.) Assertions (“Add this sentence to the KB”) TELL( KB,  m, c Mother(c) = m  Female(m)  Parent(m, c)) TELL( KB,  w, h Husband(h, w)  Male(h)  Spouse(h, w)) TELL( KB,  x Male(x)  Female(x)) TELL( KB, Female(Mary)  Parent(Mary, Frank)  Parent(Frank, Ann)) • Note: TELL( KB, S1  S2) TELL( KB, S1) and TELL( KB, S2) (because of and-elimination and and-introduction) • Queries (“Does the KB entail this sentence?”) • ASK( KB, Grandparent(Mary, Ann) )  True • ASK( KB,  x Child(x, Frank))  True • But a better answer would be  { x / Ann } • This returns a substitution or binding

KB  KB i  Implementing ASK: Inference • We want a sound and complete inference algorithm so that we can produce (or confirm) entailed sentences from the KB • The resolution rule, along with a complete search algorithm, provides a complete inference algorithm to confirm or refute a sentence  in propositional logic(Sec. 7.5) • Based on proof by contradiction (refutation) • Refutation: To prove that the KB entails P, assume P and show a contradiction: • (KB  P  False)  (KB  P) Prove this!

Inference in First-Order Logic • Inference rules for propositional logic: • Modus ponens, and-elimination, and-introduction, or-introduction, resolution, etc. • These are valid for FOL also • But since these don’t deal with quantifiers and variables, we need new rules, especially those that allow for substitution (binding) of variables to objects • These are called lifted inference rules

Substitution and variable binding • Notation for substitution: • SUBST( Binding list, Sentence ) • Binding list: { var / ground term, var / ground term, … } • “ground term” = term with no variables • SUBST( {var/gterm}, Func(var) ) = Func(gterm) • SUBST(, p) • Examples: • SUBST( {x/Mary}, FatherOf(x) ) = FatherOf(Mary) • SUBST( {x/Joe, y/Lisa}, Siblings(x,y) ) = Siblings(Joe, Lisa)

Existential Instantiation k – constant that does not appear elsewhere in the knowledge base • Existential Introduction v – variable not in g – ground term in  Three new inference rules using SUBST(, p) • Universal Instantiation g – ground term

To Add to These Rules

Universal Instantiation – examples • x Sleepy(x) • SUBST({x/Joe}, ) • Sleepy(Joe) • x Mother(x)  Female(x) • SUBST({x/Mary}, ) • Mother(Mary)  Female(Mary) • SUBST({x/Dad}, ) • Mother(Dad)  Female(Dad) • x, y Buffalo(x)  Pig(y) Outrun(x,y) • SUBST({x/Bob}, ) • y Buffalo(Bob)  Pig(y) Outrun(Bob,y) g – ground term

Existential Instantiation – examples k – constant that does not appear elsewhere in the knowledge base • x BestAction(x) • SUBST({x/B_A}, ) • BestAction(B_A) • “B_A” is a constant; it is not in our universe of actions • y Likes(y, Broccoli) • SUBST({y/Truman}, ) • x Likes(Truman, Broccoli) • “Truman” is a constant; it is not in our universe of people

Existential Introduction – examples v – variable not in g – ground term in  • Likes(Jim, Broccoli) • SUBST({Jim/x}, ) • x Likes(x, Broccoli) • x Likes(x, Broccoli)  Healthy(x) • SUBST({Broccoli/y}, ) • y x Likes(x, y)  Healthy(x)

What’s our goal here? • Formulate a search process: • Initial state • KB • Operators • Inference rules • Goal test • KB contains S • What is a node? • KB + new sentences (generated by applying the inference rules) • In other words, the new state of the KB • What kind of search to use? • I.e., which node to expand next? • How to apply inference rules?   • Need to match the premise pattern 

Applying inference rules Path = proof Path shows reasoning (here’s why I think you have cancer) • FOL query • ASK(KB, Outruns(Bob, Pat)) • ASK(KB,  x BestAction(x, t)) • ASK(KB,  x Equals(x, AuntOf(HusbandOf(Mary)))) • Proof – From axioms (KB) to hypothesis (S) • Does the KB entail a sentence S – if so, what steps were taken? • We need to apply inference rules and appropriate substitutions to reach the desired sentence • Answer: Yes, SUBST({x/Grab}, Action(x, t)) • Answer: Yes, the KB entails S (and here’s how…) • Reasoning systems may require the solution path, not just the final answer

Concerns • Is the inference procedure sound and complete? • Is it efficient? • We need to be concerned about the branching factor • Generalized modus ponens (GMP) is one way of reducing branching factor • How do we know which operators (inference rules) are valid at any particular time? • Inference rule: Pig(x)  Cuddly(x) • KB: Pig(Bob) Pig(x) and Pig(Bob) are unified by the substitution  = {x/Bob}

Buffaloes outrun pigs example Problem of branching factor • 10 inference rules • Branching factor can be huge! • Common inference pattern: • And-Introduction • Universal Instantiation • Modus ponens • Let’s introduce a new inference rule which reduces the branching factor by combining these three steps into one • Generalized Modus Ponens

General rule Specific facts Instantiated conclusion Generalized Modus Ponens • For atomic sentences pi , pi' , and q , where there is a substitution  such that SUBST(, pi') = SUBST(, pi) for all i, then • GMP features: • Efficient – it combines several inferences into one • Sensible – substitutions are purposeful (unification) • Efficient – when sentences are in a certain canonical form (Horn clauses), this is the only inference rule necessary (!!!)

Generalized Modus Ponens Example • Go from • Buffalo(Bob), Pig(Pat), (Buffalo(x)  Pig(y) Outrun(x,y)) to • Outrun(Bob, Pat) • More generally, go from • p1', p2', (p1p2q) to • SUBST(, q) where SUBST(, pi') = SUBST(, pi) • Reminder: What is  ?

Unification • Unification takes two atomic sentences p and q and returns a substitution that would make p and q look the same (or else it fails) • UNIFY(p, q) =  where SUBST(, p) = SUBST(, q) •  is the unifier of the sentences p and q • Examples • p = Knows(John, x) q = Knows(John, Jane) •  = {x/Jane} • p = Knows(John, x) q = Knows(y, Fred) •  = {x/Fred, y/John} • p = Knows(John, x) q = Knows(y, MotherOf(y)) •  = {x/MotherOf(John), y/John} • p = Knows(John, x) q = Knows(x, Mary) •  = fail • p = Knows(John, x1) q = Knows(x2, Mary) •  = {x1/Mary, x2/John}

From GMP to GR • GMP requires sentences in Horn clause • Prolog is based on this: Horn clause form and GMP inference • Horn clause: an implication of positive literals • It is “reasonably powerful,” but • Cannot handle disjunctions or negations • Many legal sentences cannot be expressed in a Horn clause • I.e., this form is incomplete • It turns out we can do better by putting the sentences of the KB into a different format and using just one inference rule, which is both complete and sound! • KB Format: Conjunctive normal form (CNF) • Inference Rule: Generalized (first-order) resolution

Note: Completeness • An inference procedure i is complete iff • KB i whenever KB  • “The procedure finds the needle in the haystack when there is one” • An inference procedure using GMP is incomplete • There are sentences entailed by the KB that the procedure cannot infer • A complete inference procedure exists: GeneralizedResolution • Generalized resolution will produce  if KB entails  • However, what if  is not entailed by the KB? Will resolution tell us this? • No, there exists no certain procedure to show this in general • Related to the famous “Halting problem”

Simple resolution Let’s rewrite it: or: Review Resolution Remember: p  qp  q So resolution is really the “chaining” of implications….

Generalized (first-order) resolution • Generalized resolution is the basis for a complete inference procedure • Can derive new implications (P  Q) • GMP can only derive atomic conclusions (Q) • Any sentence can be put into its normal form • Just like “regular” resolution except • Uses UNIFY/SUBST rather than = • Deals with more than 2 disjuncts • GR is complete, but semidecidable • Not always able to show that a sentence is not entailed by the KB

Reminder: Atomic sentences vs. Literals • Atomic sentences • Predicate and its arguments (terms) • P(x), Q(y), Siblings(Bill, z), Equal(x, y) • Literals • Atomic sentences and negated atomic sentences • P(x), Q(y), Siblings(Bill, z), Equal(x, y)

For literals piand qi , where UNIFY(pj , qk) =  Disjunctions For atomic sentences pi, qi, ri, si , where UNIFY(pj , qk) =  Implications Two versions of Generalized Resolution

Artificial Intelligence