Logic. The World. First Order Predicate Calculus (FOPC)

Size: px

Start display at page:

Download "Logic. The World. First Order Predicate Calculus (FOPC)"

Norah Bryant
5 years ago
Views:

1 Logic Logic was developed to account for why people can make certain types of inferences, called deductions. It is a language, in which it is possible to write down facts and derive new facts from them. The World A conceptualization of the world consists of objects, functions, and relations An object: a thing, a set of things, an abstract thing, an event, a set of events,.... The set of all objects in a conceptualization is the universe of discourse A function: maps one set of objects onto another object (or set of objects) (e.g., motherof, children-of) A relation: A set of n-tuples (e.g. EMPLOYER-EMPLOYEE = {<DePaul, Lytinen>, <DePaul, Minogue>, <US-PEOPLE, Clinton>,...}. Degenerate relations can be 1-tuples, and are properties. First Order Predicate Calculus (FOPC) Logic statements consist of a mixture of terms, predicates, connectives, and quantifiers. A term (object constant) is a symbol that refers to an object (or set of objects) in the world. Example: book3 might refer to a particular book. A predicate (relation constant) is a symbol that represents an attribute of an object, or a relationship between 2 or more objects. Examples: inst: a relationship, as in inst(book3, book). can-fly: a property, as in can-fly(tweety2). A predicate with its arguments is a formula or proposition or sentence. The predicate always comes first (with some exceptions). Example: inst(book3, book). There are 4 connectives: &,, ( ), and. There are 2 quantifiers: and. A function (function constant) is another kind of term. Terms A term is a symbol that refers to an object (or set of objects) in the world. A different symbol is used for each different object (though 2 symbols can refer to the same object) 1

2 Example: book1, book2 might refer to 2 different books Kinds of terms: 1. Concrete terms refer to particular objects or events Example: book2, holiday12, robbery7 2. Compound terms refer to more complicated objects Example: book-set4 (e.g., World Book Encyclopedia) 3. Abstract terms refer to general concepts: classes of objects or types of events Example: book, robbery, blue 2

3 Names Don t Matter Which term is used to represent an object isn t important (except for readability). 1. Concrete terms refer to particular objects or events Example: dfij2, sdfoij12, woi7 2. Compound terms refer to more complicated objects Example: spioj4 (e.g., World Book Encyclopedia) 3. Abstract terms refer to general concepts: classes of objects or types of events Example: pdoij, zjoif, wreoui Predicates and Formulas Predicates express relationships between terms, or properties of terms. They correspond to relations in the world. They usually take one or more arguments. A predicate with its arguments is called a formula or a proposition. This is the simplest kind of formula, so it s also called an atomic formula or literal. Examples: inst(book3, book) paperback(book3) subject(book3, computer-science) owns(pat2, book3) name(pat2, pat) bought(pat2, book3, store12) How do you know what predicates to use? You don t you just make them up Connectives More complicated formulas can be written using connectives. There are 4 of them: & and or implies (if...then) not Connectives are always written between formulas Examples: inst(book3, book) & subject(book3, computer-science) subject(book3, ai) subject(book3, psychology) inst(book3, book) & owns(pat3, book3) can-read(pat3) subject(book3, computer-science) 3

4 Quantifiers 1. : universal quantifier allows us to make statements about classes of objects Example: all books are made of paper really the same as: for all things, if that thing is a book, then it s made of paper x inst(x, book) made-of(x, paper) x is a term, but it s a variable term 2. : existential quantifier allows us to talk about an object without knowing the identify of the object Example: Someone is outside the door Don t know who, just know that someone is there. x inst(x, person) & outside(x, door3) Combining quantifiers Sometimes, it s necessary to use several quantifiers in same statement. Examples: All cats and dogs hate each other. x y (inst(x, dog) & inst(y, cat)) hate(x, y) & hate(y, x) Every student is enrolled in (at least) one course. x inst(x, student) y inst(y, course) & enrolled-in(x, y) Functions May appear wherever terms can Look just like formulas, but they aren t they re terms Example: Every person has a mother and a father as a parent. x inst(x,person) parent(x,mother-of(x)) & parent(x,father-of(x)) Functions do not return an answer. Think of mother-of(pat2) as a term whose name has parentheses in it. Some rules of thumb 1. Always follow or with one or more variables 4

5 2. If an English sentence says All... or Every... then use. 3. and often go together. Example: All birds can sing. x inst(x, bird) can-sing(x) 4. If an English sentence says Some... or There is... then use. 5. and & often go together. Example: Some birds can t fly. (x) inst(x, bird) & can-fly(x) 6. If you can say All x s have a y... or For all x s, there s a y... use both and Example: All birds have beaks. (i.e., All birds have a beak) x inst(x, bird) y inst(y, beak) & has(x,y) 1. Pat is a student and is in CSC All students in CSC480 are intelligent. Examples of FOPC inst(pat0, student) & in(pat0,csc480) x inst(x,student) & in(x,csc480) intelligent(x) 3. All students who are intelligent get A s in all their courses. x y ( inst(x,student) & intelligent(x) & in(x,y)) grade(x,y,a) 4. Your mother s mother is your grandmother. x inst(x,human) grandmother(mother-of(mother-of(x)), x) Other definitions A sentence is satisfiable iff there is some interpretation which satisfies it (i.e., if it could possibly be true). A sentence is unsatisfiable (or inconsistent) iff it cannot be true under any circumstances. For example: inst(pat0,human) & inst(pat0,human) A sentence is valid iff it is always true, no matter what the circumstances. For example: x inst(x,human) inst(x,human) 5

6 Logical Equivalence Two propositions in logic are equivalent iff they are true under exactly the same circumstances. Example: All humans are mortal. All immortal beings are not humans. One proposition entails another if its models are a subset of the other s 6

7 Connectives and Logical Equivalence Can think of connectives being defined in terms of truth tables 1. P Q = P Q 2. (P&Q) = P Q 3. P (Q&R) = (P Q)&(P R) & P Q P&Q P Q P Q T T T T T T T F F T F T F T F F T T F F F F F F P Q P Q P P T T T T F T F F F T F T T F F T Redundancy in connectives P Q P Q P P Q T T T F T T F F F F F T T T T F F T T T P Q P&Q (P&Q) P Q P Q T T T F F F F T F F T F T T F T F T T F T F F F T T T T P Q R Q&R P P Q P R (P Q)& Q&R (P R) T T T T T T T T T T F F F T F F T F T F F F T F T F F F F F F F F T T T T T T T F T F F T T T T F F T F T T T T F F F F T T T T 7

8 Logical Equivalence and Quantifiers The universal quantifier is shorthand for an infinite conjunction: x Px = Pt 0 &Pt 1 &Pt 2 &..., for all terms t n. The existential quantifier is shorthand for an infinite disjunction: x Px = Pt 0 Pt 1 Pt 2..., for all terms t n. 1. x Px = x Px 2. P x Qx = x P Qx x inst(x,person) y parent(x,y) = x y inst(x,person) parent(x,y) x Px = (Pt 0 & Pt 1 & Pt 2...) = Pt 0 Pt 1 Pt 2... = x Px P x Qx = P (Qt 0 Qt 1 Qt 2...) = ( P Qt 0 ) ( P Qt 1 ) ( P Qt 2 )... = (P Qt 0 ) (P Qt 1 )... = x P Qx Deduction We can deduce Q from P whenever we know that P s truth conditions are subsumed by Q s and P is true. Rules of deduction can also be derived from truth tables. 1. Modus ponens Sometimes written: 2. And elimination P Q P Q (P Q)&P ((P Q)&P) Q T T T T T F F F F T T F F F T F P Q P Q 8

9 P Q P&Q (P&Q) P T T T T F F F T F F F F Sometimes written: P&Q P 3. Modus tolens Sometimes written: Logic is redundant P Q P Q Q (P Q) & Q ((P Q) & Q) P T T T F F T F F T F F T T F F F F T T T P Q Q P Skolemization Want equivalent facts to be represented the same Skolemization: canonical form for quantifiers The Rules of Skolemization 1. Replace universally quantified variables with Skolem variables 2. Replace existentially quantified variables with Skolem functions 3. Within negation, reverse rules 1 and 2 Everyone likes Pat. human(?x) likes(?x, Pat0) Someone likes Pat. human(sk0) & likes(sk0, Pat0) Examples 9

10 Everyone likes someone. x human(x) y likes(x,y) x y human(x) likes(x,y) human(?x) likes(?x, sk1(?x)) There are no unicorns. x unicorn(x) x unicorn(x) unicorn(?x) Not everyone likes someone. x human(x) y likes(x,y) x human(x) y likes(x,y) ( human(sk2) likes(sk2,?y)) human(sk2) & likes(sk2,?y) Microsoft has released a version of Word for every operating system on every PC. x y pc-type(x) & os(y) z version(z,word) & released(microsoft,z) pc-type(?x) & os(?y) version(sk4(?x,?y),word) & released(microsoft,sk4(?x,?y)) Cannonical form for connectives Also called conjunctive normal form Only use,, and & (implicitly) Clause Form Only literals (atomic formulas) may be negated Only literals may be disjuncts (negations are ok) Converting to Clause Form 1. Skolemize; change variable names to make them unique 2. p q = p q 3. Distribute negations, using DeMorgan s laws (p&q) = p q (p q) = p& q 4. Distribute disjunctions, using (p&q) r = (p r)&(q r) 10

11 All cities have mayors. x inst(x,city) y mayor-of(x,y) inst(?x,city) mayor-of(?x,sk0(?x)) All big cities have suburbs. More Examples of Logic x inst(x,city) & big(x) y suburb-of(x,y) inst(?x,city) big(?x) suburb-of(?x,sk1(?x)) All cities that are not big do not have suburbs. x inst(x,city) & big(x) y suburb-of(x,y) x y inst(x,city) & big(x) suburb-of(x,y) inst(?x,city) big(?x) suburb-of(?x,?y) All suburbs are cities that are not big. x y suburb-of(x,y) inst(y,city) & big(y) suburb-of(?x,?y) inst(?y,city) suburb-of(?x,?y) big(?y) Logic and Deduction One proposition entails another if its models are a subset of the other s Finding other propositions that are entailed by what you already know is called deduction (a kind of inference). Precisely Defining Inference Let d be a database which consists of a finite sequence of sentences. An inference procedure is a function step that maps an initial database d from the set D and a positive integer n into the database for th n-th step of inference. step : D N D step(, 1) = A proposition P can be derived from iff P step(, i) for some positive integer i An inference procedure is complete iff any sentence which is logically entailed by the sentences in can be derived from An inference procedure is sound iff any sentence which can be derived from is entailed by the sentences in 11

12 Markov inference procedure: choice of database on each step is determined entirely by the database for the last step: next : D D { if n = 1 step(, n) = next(step(, n 1)) otherwise An incremental inference procedure is one in which each new database is obtained from the previous database by adding zero or more new conclusions (monotonic): new : D N D step(, n) = A rule of deduction: Or: P Q P R Q R Truth table: (P Q)&( P R) (Q R) { if n = 1 append(step(, n 1), new(, n 1)) otherwise Resolution P Q R P Q P R (P Q)&( P R) Q R (P Q)&( P R) Q R T T T T T T T T T F T F F T T F T T T T T T F F T F F F F T T T T T T F T F T T T T F F T F T F T F F F F T F F Incremental Markov inference procedure Sound and complete Resolution and Variables Implicitly, resolution includes universal elimination: inst(?x,human) mortal(?x) inst(socrates,human) mortal(socrates) 12

13 Resolving 2 clauses involves unification and instantiation unification: making sure variables are compatible, and computing their bindings instantiation: replacing variables with their bindings Unification the process of determining whether two expressions can be made identical by appropriate substitutions for their variables. substitution: any finite set of associations between variables and expressions in which: 1. each variable is associated with at most one expression 2. no variable with an associated expression occurs within any of the associated expressions Example: {?x/a,?y/f(b),?z/?w} is ok, {?x/g(?y),?y/f(?x)} is not Terms associated with a variable in a substitution are called bindings Unification produces a substitution Substitutions and applying them A substitution can be applied to a formula to produce a new formula done by replacing all variables in the expression by their bindings (unbound variables are not replaced) Example: P(?x,?x,?y,?v){?x/A,?y/F(B),?z/?w} = P(A,A,F(B),?v) The composition of 2 substitutions τ and σ (written as στ) is the substitution obtained by applying τ to the terms of σ, then adding the bindings of τ Example: {?w/g(?x,?y)}{?x/a,?y/b,?z/c} = {?w/g(a,b),?x/a,?y/b,?z/c} Two expressions are unifiable iff there exists a substitution σ which makes the expressions identical The most general unifier: the substitution which unifies two expressions such that its application gets rid of the fewest variables unify(human(?x),human(?y)) {?x/?y} (mgu) {?x/pat0,?y/pat0} (not mgu) unify(likes(?x,?y),likes(pat0,chris2)) {?x/pat0,?y/chris2} Example Unifications 13

14 unify(likes(?x,?x),likes(pat0,chris2)) False (they don t unify) unify(likes(?x,?x),likes(?y,pat0)) {?x/pat0,?y/pat0} unify(likes(?x,?x),likes(?z,mother-of(?z))) False unify(likes(?x,friend-of(?x)),likes(friend-of(?y),?y)) False Resolution Principle with Unification Φ Ψ ((Φ ϕ) (Ψ ψ))γ with ϕ Φ with ψ Ψ where ϕγ = ψγ Example: 1. P(?x) Q(?x,?y) 2. P(A) R(B,?z) 3. Q(A,?y) R(B,?z) 1,2 Resolution is sound and refutation complete!!! But, you can never be sure if something cannot be proven Resolution and Proof by Contradiction 1. Add to the negation of what is to be proven 2. Apply resolution until contradiction is reached (null clause) Given: 1. inst(?x,student) inst(?x,person) 2. inst(?x,dog) inst(?x,person) 3. inst(fido2,dog) Example Prove inst(fido2,student) Step Reason 4. inst(fido2,student) neg of thm 5. inst(?x,dog) inst(?x,student) 1,2 6. inst(fido2,student) 3,5 7. 4,6 Another example 1. DePaul s president is always a Vincentian priest. 1. inst(?x1,dpu-pres) inst(?x1,vpriest) 14

15 2. Vincentian priests are not married to anyone. 2. inst(?x2,vpriest) married(?x2,?x3) 3. If two people file joint tax returns, they are married to each other. 3. file-jointly(?x4,?x5) married-to(?x4,?x5) 4. Bill and Hillary file joint tax returns. 4. file-jointly(b0,h0) Prove that Bill is not DePaul s president. 5. inst(b0,dpu-pres) neg of thm 6. inst(b0,vpriest) 1,5 7. married(b0,?x3) 2,6 8. file-jointly(b0,?x3) 3,7 9. 4,8 15

16 Another Example of Proof by Resolution 1. Every student who took 480 bought a copy of Russell and Norvig. x y inst(x,student) & took(x,csc480) inst(y,rusnor) & bought(x,y) 1. inst(?x1,student) took(?x1,csc480) inst(skgn(?x1),rusnor) 2. inst(?x2,student) took(?x2,csc480) bought(?x2,skgn(?x2)) 2. If you bought something and haven t sold it, you own it. x y bought(x,y) & sold(x,y) owns(x,y) 3. bought(?x3,?x4) sold(?x3,?x4) owns(?x3,?x4) 3. Pat doesn t own a copy of Russell and Norvig. x inst(x,rusnor) & owns(pat0,x) 4. inst(?x5,rusnor) owns(pat0,?x5) 4. Pat is a student and took 480. inst(pat0,student) & took(pat0,csc480) 5. inst(pat0,student) 6. took(pat0,csc480) Prove that Pat sold a copy of Russell and Norvig. Step Reason 7. inst(?z,rusnor) sold(pat0,?z) neg of thm 8. took(pat0,csc480) inst(skgn(pat0),rusnor) 1,5 9. took(pat0,csc480) bought(pat0,skgn(pat0)) 2,5 10. inst(skgn(pat0),rusnor) 6,8 11. bought(pat0,skgn(pat0)) 6,9 12. owns(pat0,skgn(pat0)) 4, sold(pat0,skgn(pat0)) 7, sold(pat0,skgn(pat0)) owns(pat0,skgn(pat0)) 3, sold(pat0,skgn(pat0)) 12, ,15 16

17 Problems with Logic Deduction is conservative. It allows the computer to infer what must be true. People make risky inferences. Example: Tweety is a bird. Tweety can fly. Tweety is an ostrich. Tweety can t fly. In predicate calculus: x inst(x,bird) can-fly(x) x inst(x,ostrich) can-fly(x) x inst(x,ostrich) inst(x,bird) This is a contradiction. In clause form: 1. inst(?x,bird) can-fly(?x) 2. inst(?y,ostrich) can-fly(?y) 3. inst(?z,ostrich) inst(?z,bird) Can prove anything: Resolve 1 with 3, result with 2. Fixing the contradiction x inst(x,bird) & inst(x,ostrich) can-fly(x) x inst(x,ostrich) can-fly(x) x inst(x,ostrich) inst(x,bird) Now, it s impossible to prove can-fly(tweety0) from inst(tweety0,bird) Individual exceptions inst(?x,employee) marital-status(?x,single)?x=emp12 exemptions(?x,1) marital-status(emp12,single) exemptions(emp12,3) Doesn t work, unless you explicitly put in the database: emp0 emp1 emp0 emp2... emp1 emp0 emp1 emp2... Situation Variables 17

18 Facts about the world change over time. This causes problems: location(?x,?y) & move(?x,?y,?z) location(?x,?z) & location(?x,?y) The solution: situation variables add additional argument to predicates that change; this is the situation in which the formula is true location(?x,?y,?s) & move(?x,?y,?z,?s) location(?x,?z,result-of(move,?x,?y,?z,?s)) Situation variables prevent contradictions They also prevent many deductions: The Frame Problem paint(?x,?c,?s) color(?x,?c,result-of(paint,?x,?c,?s)) location(block0,loc0,s0) paint(block0,blue,s0) Where is block0 after it s been painted? Need frame axioms to tell us what hasn t changed: paint(?x,?c,?s) & location(?x,?y,?s) location(?x,?y,result-of(paint,?x,?c,?s)) 18

Propositional Logic Not Enough

Section 1.4 Propositional Logic Not Enough If we have: All men are mortal. Socrates is a man. Does it follow that Socrates is mortal? Can t be represented in propositional logic. Need a language that talks