Announcements First-Order Logic Homework #2 is assigned, it is due Monday, July 7 (1 week from today) Project proposals are due today Burr H. Settles CS-540, UW-Madison www.cs.wisc.edu/~cs540-1 Read Chapter 9 in AI: A Modern Approach Summer 2003 for next time 1 2 General Logic PL Review: Truth Tables ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ∧ ∨ ¢ ∨ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ¢ ((A C) (A¢ C)) (C B) (A (B C)) ((A B) (A C)) ¡ £ £ £ £ £ £ £ £ £ £ £ £ £ £ £ ∧ ∨ £ ∨ £ £ £ £ £ £ £ £ £ £ £ £ £ £ £ Logics are characterized by what they consider to A B C ((A C) (A £ C)) (C B) A B C (A (B C)) ((A B) (A C)) be “primitives” T T T T T T T T T T F T T T F T T F T T T F T T T F F T T F F T Logic Primitives Available Knowledge F T T T F T T T Propositional facts true/false/unknown F T F T F T F T F F T T F F T T First-Order facts, objects, relations true/false/unknown F F F F F F F T Temporal facts, objects, relations, true/false/unknown times (A∨B)∧(A∨¬B)∧(¬A∨B)∧(¬A∨¬B) A B (A∨B)∧(A∨¬B)∧(¬A∨B)∧(¬A∨¬B) Probability Theory facts degree of belief 0…1 valid T T F satisfiable, Fuzzy degree of truth degree of belief 0…1 T F F but not valid F T F 3 unsatisfiable F F F 4 PL Review: Inference Rules First-Order Logic ¤ ¤ £ £ ¥ £ Modus Ponens £ , Given the following ¥ knowledge base: ¨ Propositional logic has advantages ¤ ¤ ¤ ∧ ∧ ∧ 1 ∧ 2 ∧ … ∧ n And-Elimination (AE): ¤ 1. P i – Simple § § § 2. P § R ¤ ¤ ¤ 1, 2, … , n – Inference is fast and easy ¤ ¤ ¤ § § ¬ § And-Introduction (AI): ∧ ∧ ∧ 3. R § ¬W 1 ∧ 2 ∧ … ∧ n 4. S ∨ R ¨ ¤ But PL is limited in key ways ∧ § § ∨ § i 5. (P ∧ R) § (S ∨ W) ¤ ¤ Or-Introduction (OI): ¤ ∨ ∨ ∨ 1 2 … n – Enumerate all facts as separate propositions Prove S using natural Double-Negation ¬ ¬ ¤ – No concept of individuals or objects ¤ deduction with these rules. Elimination (DNE): – Can’t express relationships easily ¤ ∨ ¬ 6. R (MP: 1,2) ¨ ¥ ∨ ¥ , ¬ Unit Resolution (UR): ¤ 7. ¬W (MP: 3,6) First-Order Logic is a logic language designed to ¤ ∧ ¥ ¦ ∨ ¥ , ¬ ∨ 8. P R (AI: 1,6) Resolution (R): ¤ remedy these problems ∨ ¦ 9. S ∨ W (MP: 5,8) 10. S (UR: 7,9) ¤ deMorgan’s Law (DML): ¬(¬ ∨ ¥ ) ¤ ∧ ¬¥ ∧ ¬ 5 6 1 FOL Syntax: Basic FOL Syntax: Basic ¨ A term is used to denote an object in the world ¨ An atom is smallest expression to which a truth – Constant: Bob, 2, Madison, Green, … value can be assigned – Variable: x, y, a, b, c, … – Predicate(term1, …, termn): – Function(term1, …, termn): • e.g. teacher(Burr,You), lte(sqrt(2),sqrt(7)) • e.g. sqrt(9), distance(Madison,Chicago) • Maps one or more objects to a truth value • Maps one or more objects to another object • Represents a user defined relation • Can refer to an unnamed object: e.g. leftLegOf(John) • Represents a user defined functional relation – Term1 = Term2: • e.g. height(Burr) = 73in, 1 = 2 ¨ • Represents the equality relation when A ground term is a term with no variables two terms refer to the same object 7 8 FOL Syntax: Basic FOL Syntax: Basic ¡ ¨ A sentence represents a fact in the world that is Sentences are assigned a truth value with respect assigned a truth value to a model and an interpretation – Atom ∧ ∨ ¬ ⇔ ¨ ∧ ∨ ¬ ⇔ – Complex sentence using connectives: The model contains the objects and the relations ¢ ¢ ¢ • e.g. spouse(Burr,Nat) ¢ spouse(Burr,Nat) among them • e.g. less(11,22) ∧ less(22,33) ¨ The interpretation specifies what symbols refer to: – Complex sentence using quantified variables: ∀ ∃ – Constants symbols refer to objects • More about these in a bit… – Predicate symbols refer to relations – Functional symbols refer to functional relations 9 10 FOL Semantics: Assigning Truth FOL Syntax: Quantifiers ¡ ∀ The atom predicate(term1, …, termn) is true iff the The universal quantifier: objects referred to by term1, …, termn are in the relation referred to by the predicate ¡ Sentence holds true for all values of x in the domain of variable x ¨ What is the truth value for s(B,N)? ¨ – Model: Main connective typically ¡ forming if-then rules • Objects: Burr, Nat, Thom, Mark in FOL becomes: • Relation: spouse {<Burr,Nat>,<Nat,Burr>} – “All humans are mammals” ¢ ∀ ¢ ¢ – Interpretation: x human(x) ¢ mammal(x) • B means Burr, N means Nat, T means Thom, etc. – Means if x is a human then x is a mammal • s(term1,term2) means term1 is the spouse of term2 11 12 2 FOL Syntax: Quantifiers FOL Syntax: Quantifiers ¡ ¡ ¡ ¨ ∀x human(x) ¡ mammal(x) Common mistake is to use ∧ as main connective – Results in a blanket statement about everything ¨ Equivalent to the conjunction of all the instantiations of variable x: ¨ For example: ∀x human(x) ∧ mammal(x) ∧ (human(Burr) ∧ mammal(Burr)) ∧ ¢ ¢ ¢ (human(Burr) ¢ mammal(Burr)) ∧ ∧ (human(Nat) ∧ mammal(Nat)) ∧ ¢ ¢ ¢ (human(Nat) ¢ mammal(Nat)) ∧ ∧ (human(Thom) ∧ mammal(Thom)) ∧ … ¢ ¢ ¢ (human(Thom) ¢ mammal(Thom)) ∧ … – But this means everything is human and a mammal! 13 14 FOL Syntax: Quantifiers FOL Syntax: Quantifiers The existential quantifier: ∃ ∃x human(x) ∧ male(x) ¡ Sentence holds true for some value of x in the ¨ Equivalent to the disjunction of all the domain of variable x instantiations of variable x: (human(Burr) ∧ male(Burr)) ∨ ¨ Main connective typically ∧ (human(Nat) ∧ male(Nat)) ∨ – “Some humans are male” in FOL becomes: (human(Thom) ∧ male(Thom)) ∨ … ∃x human(x) ∧ male(x) – Means x is some human and x is a male 15 16 FOL Syntax: Quantifiers FOL Syntax: Quantifiers ¨ ¨ Common mistake is to use ¡ as main connective. Properties of quantifiers: – Results in a weak statement – ∀x ∀y is the same as ∀y ∀x – ∃x ∃y is the same as ∃y ∃x ¨ ¡ ¡ ¡ For example: ∃x human(x) ¡ male(x) ¨ ¢ ¢ ∨ ¢ (human(Burr) ¢ male(Burr)) Why? ¢ ¢ ∨ ¢ (human(Nat) ¢ male(Nat)) ∨ – ∀x ∀y likes(x,y) ∨ ¢ ¢ ∨ ¢ (human(Thom) ¢ male(Thom)) … the active voice: “Everyone likes everyone.” – Can be true if there is something not human! – ∀y ∀x likes(x,y) the passive voice: “Everyone is liked by everyone.” 17 18 3 FOL Syntax: Quantifiers FOL Syntax: Quantifiers ¨ ¨ Properties of quantifiers: Properties of quantifiers: – ∀x ∃y is not the same as ∃y ∀x – ∀x P(x) is the same as ¬∃x ¬P(x) – ∃x ∀y is not the same as ∀y ∃x – ∃x P(x) is the same as ¬∀x ¬P(x) ¨ ¨ Why? Why? – ∀x ∃y likes(x,y) – ∀x sleep(x) “Everyone has someone they like.” “Everybody sleeps.” – ∃y ∀x likes(x,y) – ¬∃x ¬sleep(x) “There is someone who is liked by everyone.” double negative: “Nobody don’t sleep.” 19 20 FOL Syntax: Quantifiers FOL Syntax: Basics ¨ Properties of quantifiers: A free variable is a variable that isn’t bound – ∀x P(x) when negated is ∃x ¬P(x) by a quantifier – ∃x P(x) when negated is ∀x ¬P(x) – i.e. ∃y Likes(x,y): x is free, y is bound ¨ Why? – ∀x sleeps(x) A well-formed formula is a sentence where “Everybody sleeps.” all variables are quantified (none are free) – ∃x ¬sleeps(x) negated: “Somebody doesn’t sleep.” 21 22 Summary So Far Summary So Far ¨ Constants: Bob, 2, Madison, … Term: Constant, variable, or function… denotes an object in the world (a ground term has no variables) ¨ Variables: x, y, a, b, c, … ¨ Functions: Income, Address, Sqrt, … Atom: Is smallest expression assigned a truth value ¨ – e.g. Predicate(term , …, term ), term = term Predicates: Teacher, Sisters, Even, Prime… 1 n 1 2 ¨ ¡ Connectives: ∧ ∨ ¬ ⇔ Sentence: An atom, quantified sentence with variables, or ¨ Equality: = complex sentence using connectives; assigned a truth value ¨ ∀ ∃ Quantifiers: Well-Formed Formula (wff): A sentence where all variables are quantified 23 24 4 Thinking in Logical Sentences Thinking in Logical Sentences Convert the following sentences into FOL: We can also do this with relations: ¨ ¨ “Bob is a fish.” “America bought Alaska from Russia.” – What is the constant? – What are the constants? • Bob • America, Alaska, Russia – What is the predicate? – What are the relations? • is a fish • bought – Answer: fish(Bob) – Answer: bought(America, Alaska, Russia) ¨ ¨ “Burr and Mark are grad students.” “Warm is between cold and hot.” ¨ ¨ “Burr, Mark, or Nat is not a rat.” “Burr and Nat are married.” 25 26 Thinking in Logical Sentences Thinking in Logical Sentences Now let’s think about quantification: ¨ All ¨ “Burr likes everything.” – Things: anything, everything, whatever – What is the constant? – Persons: anybody, anyone, everybody, everyone, whoever • Burr ¨ Some (at least one) – How are they variables quantified? – Things: something • All/universal – Persons: somebody, someone ∀ – Answer: x likes(Burr, x) ¨ None – i.e. likes(Burr, IceCream) ∧ likes(Burr, Nat) ∧ likes(Burr, Armadillos) ∧ … – Things: nothing ¨ “Burr likes something.” – Persons: nobody, no one ¨ “Somebody likes Burr.” 27 28 Thinking in Logical Sentences Thinking in Logical Sentences We can also have multiple quantifiers: Let’s allow more complex quantified relations: ¨ ¨ “Somebody heard something.” “All stinky shoes are allowed.” – What are the variables? – How are ideas connected? • being a shoe and being stinky implies that it is allowed • somebody and something ∀ ∧ – Answer: ∀x shoe(x) ∧ stinky(x) allowed(x) – How are they quantified? ¨ • both are at least one/existential “No stinky shoes are allowed.” ¬∃ ∧ ∧ – Answer: ∃x,y heard(x,y) – Answer: x shoe(x) stinky(x) allowed(x) ¨ ¨ “Everybody heard everything.” The equivalent: “Stinky shoes are not allowed.” ¨ “Somebody did not hear everything.” ∀ ∧ ¬ – Answer: ∀x shoe(x) ∧ stinky(x) ¬allowed(x) 29 30 5 Thinking in Logical Sentences Thinking in Logical Sentences And some more complex relations: And some really complex relations: ¨ ¨ “No one sees everything.” “Any good amateur can beat some professional.” – What are the variables and quantifiers? – Lets break this down: ¢ ∀ ¢ ¢ • nothing and everything • x [ (x is a good amateur) ¢ (x can beat some professional) ] • not one (i.e.