CS544: Logic, Lecture 1

CS544: Logic, Lecture 1 February 22, 2011 Jerry R. Hobbs USC/ISI Marina del Rey, CA

Outline of My Four Lectures Feb 22: Logic for Representing Natural Language Feb 24: Syntax and Compositional Semantics of Clauses Mar 1: Syntax and Compositional Semantics of NPs Mar 3: Coordination and Comparison OR Inference, Coreference, and Metonymy

Logic and Information Logic and natural language are the best two ways we know of representing information / knowledge Natural language is too variable to compute with Logic has traditionally been too narrow in what can be represented A goal of knowledge representation research: Develop logics with expressivity closer to natural language

Propositional Logic Propositional constants: P, Q, R, ... Have vaues of either True or False. Logical connectives: and: & or  not: ~ or ¬ equivalent, iff: <--> or  or: v imply: --> or  Defined by truth tables: Latin word for “or” is “vel” & Q: T F P: T T F F F F v Q: T F P: T T T F T F ~ P: T F F T “Material implication”: either P is false or Q is true Definitions of <--> and -->: [P <-->Q] <--> [[P --> Q] & [Q --> P]] [P --> Q] <--> [~P v Q]

Properties of Logical Connectives Properties of logical connectives: Modus ponens: [P & [P-->Q]] --> Q & and v are associative and commutative: [[P & Q] & R] <--> [P & [Q & R]] [P & Q] <--> [Q & P] [[P v Q] v R] <--> [P v [Q v R]] [P v Q] <--> [Q v P] so we can write [P & Q & R & ...] and [P v Q v R v ...] (What about [[P --> Q] --> R] <-?-> [P --> [Q --> R]]?) Relating &, v and ~: ~[P & Q] <--> ~P v ~Q ~[P v Q] <--> ~P & ~Q [P & [Q v R]] <--> [[P & Q] v [P & R]] [P v [Q & R]] <--> [[P v Q] & [P v R]] Double negation: ~~P <--> P

Clause Form Clause form: [P1 v P2 v ~P3 v ....] & [Q1 v ~Q2 v ...] Negation applies only to propositional constants, not larger expressions. Disjunctions (v) appear at the midlevel, outscoping negation and outscoped by conjunctions. Conjunction at the highest level. Eliminate --> and <--> with the rules [P <--> Q] ==> [[P --> Q] & [Q --> P]] [P --> Q] ==> [~P v Q] Push ~ all the way inside with the rules: ~[P & Q] ==> ~P v ~Q ~[P v Q] ==> ~P & ~Q Push v inside & with the rule: [P v [Q & R]] ==> [[P v Q] & [P v R]] Eliminate double negations with the rule: ~~P ==> P It is always possible to reduce an expression to clause form. (Conjunctive Normal Form) clause

Example ~[P & [Q --> R]] ~[P & [~Q v R]] Eliminate --> ~P v ~[~Q v R] Move ~ inside ~P v [~~Q & ~R] Move ~ inside ~P v [Q & ~R] Cancel double negation [~P v Q] & [~P v ~R] Distribute v through & Clause form with two clauses We can rewrite this as two rules: P --> Q and P & R --> False Literals: P, ~Q, R, ... Positive literals: P, R, ... Horn clause: A clause with at most one positive literal. ~P v ~Q v R equivalent to [P & Q] --> R ~P v ~Q equivalent to [P & Q] --> False Definite Horn clause: A clause with exactly one positive literal.

First-Order Logic Propositional logic: Don’t look inside propositions: P, Q, R, ... First-order logic: Look inside propositions: p(x,y), like(J,M), ... Constants: John1, Sam1, ..., Chair-46, ..., 0, 1, 2, ... Variables: x, y, z, .... Predicate symbols: p, q, r, ..., like, hate, ... Function symbols: motherOf, sumOf, ... All the logical connectives of propositional logic. Predicates and functions apply to a fixed number of arguments: Predicates: like(John1,Mary1), hate(Mary1,George1), tall(Sue3), ... Functions: motherOf(Sam1) = Mary1, sumOf(2,3) = 5, ... In the expression: 3 + 2 > 4 function predicate Predicates applied to arguments are propositions and yield True or False. Functions applied to arguments yield entities in the domain.

Quantifiers Two different roles for variables: Recall from high-school algebra: (x + y)(x - y) = x2 - y2 x2 -7x + 12 = 0 universal statement: (A x,y)[(x + y)(x - y) = x2 - y2] existential statement: (E x)[x2 -7x + 12 = 0] Universal quantifier: A or : statement is true for all values of variable Existential quantifier: E or  : statement is true for some value of variable In (A x)[p(x) & q(y)] x is bound by the quantifier; y is not. Both are in the scope of the quantifier. We’ll only use variables that are bound by a quantifier. The quantifier tells how the variable is being used. Relation between A and E: ~(A x) p(x) <--> (E x)~p(x) Negation can be moved inside (A x) p(x) <--> (A y) p(y) The variable doesn’t matter (A x)[p(x)] & Q <--> (A x)[p(x) & Q] No harm scoping over what where no x in Q doesn’t involve the variable

Clause Form for 1st Order Logic Eliminate --> and <--> Move negation to the inside Give differently quantified variables different names: (A x)p(x) & (E x)q(x) ==> (A x)p(x) & (E y)q(y) Eliminate existential quantifiers with Skolem constants and functions: (E x)p(x) ==> p(A) (A x)(E y)p(x,y) ==> (A x)p(x,f(x)) Move universal quantifiers to outside: (A x)p(x) & (A y)[q(y) v r(y,f(y))] ==> (A x)(Ay)[p(x) & [q(y) v r(y,f(y))] Put matrix into clause form Skolem constant Skolem function prenex form: prefix matrix

Example (A x)[(E y)[p(x,y) --> q(x,y)] --> (E y)[r(x,y)]] (A x)[~(E y)[~p(x,y) v q(x,y)] v (E y)[r(x,y)]] Eliminate implication (A x)[(A y)~[~p(x,y)v q(x,y)] v (E y)[r(x,y)]] Move negation inside (A x)[(A y)[p(x,y) & ~q(x,y)] v (E y)[r(x,y)]] Move negation inside (A x)[(A y)[p(x,y) & ~q(x,y)] v (E z)[r(x,z)]] Rename variables (A x)[(A y)[p(x,y) & ~q(x,y)] v r(x,f(x))] Skolem function (A x)(A y)[[p(x,y) & ~q(x,y)] v r(x,f(x))] Prenex form (A x)(A y)[[p(x,y) v r(x,f(x))] & [~q(x,y) v r(x,f(x))]] Distribute v inside & p(x,y) v r(x,f(x)), ~q(x,y) v r(x,f(x)) Break into clauses

Horn Clauses Horn clause: A clause with one positive literal. ~p(x,y) v ~q(x) v r(x,y) is equivalent to [p(x,y) & q(x)] --> r(x,y) procedure name procedure body The key idea in Prolog Implicative normal form: (A x,y)[[p1(x,y) & p2(x,y) & ...] --> (E z)[q1(x,z) & q2(x,z) & ...]] Useful for commonsense knowledge: (A x)[[car(x) & intact(x) --> (E z)[engine(z) & in(z,x)]] Every intact car has an engine in it.

Logical Theories and Rules of Inference Logical theory: The logic as we have defined it so far + A set of logical expressions that are taken to be true (axioms) Rules of inference: Modus Ponens: From P, P --> Q infer Q Universal instantiation: From (A x)p(x) infer p(A) Theorems: Expressions that can be derived from the axioms and the rules of inference.

Models What do the logical symbols mean? What do the axioms mean? A logical theory is used to describe some domain. We assign an individual or entity in the domain to each constant (the denotation of that constant. To each unary predicate we assign a set of entities in the domain, those entities for which the predicate is true (the denotation or extension of p). To each binary predicate we assign a set of ordered pairs of entities, etc. ~P: true when P is not true. P & Q: true when P is true and Q is true P v Q: true when P is true or when Q is true p(A): true when the denotation of A is in the set assigned to p (A x)p(x): true when for every assignment of x, x is in the set assigned to p If all the axioms of the logical theory are true, then the domain is a model of the theory.

Examples Logical theory: Predicate: sum(x,y,z) (x is the “sum” of y and z) Axiom 1: (A x,y,z,w)[(E u)[sum(u,x,y) & sum(w,u,z)] <--> (E v)[sum(v,y,z) & sum(w,x,v)]] (associativity) Some models: addition of numbers, multiplication of numbers concatenation of strings Add Axiom 2: (A x,y,w)[sum(w,x,y) <--> sum(w,y,x)] (commutativity) Some models: addition of numbers, multiplication of numbers concatenation of strings In general, adding axioms eliminates models.

Some Uses of Models Consistency: A theory is consistent if you can’t conclude a contradiction. If a logical theory has a model, it is consistent. Independence: Two axioms are independent if you can’t prove one from the other. To show two axioms are independent, show that there is a model in which one is true and the other isn’t true. Soundness: All the theorems of the logical theory are true in the model. Completeness: All the true statements in the model are theorems in the logical theory. The logical theory should tell the whole truth (complete) and nothing but the truth (sound) Precision = 100% Recall = 100%

Extension vs. Intension (not “intention”) .... Clinton Bush Obama Extension: “president” predicate --> set of entities W1: .... Clinton Bush Obama W2: .... Clinton Gore Obama W3: .... Clinton Bush McCain Intension: “president” predicate --> set of possible world-entity pairs Frees meaning of predicate from accidents of how the world is

Back to Language Logic is about representing information. Language conveys information. Logic is a good way to represent the information conveyed by language. A man builds a boat. (E x,y)[man(x) & build(x,y) & boat(y)] A tall man builds a small boat. (E x,y)[tall(x) & man(x) & build(x,y) & small(y) & boat(y)] Seems simple enough, but problems arise. (e.g., the determiner “a”, the present tense, tall/small for what) Two ways to deal with these problems: Complicate the logic. Complicate our conceptualization of the underlying domain. Much computational semantics Me +

Reifying Events (Davidson) Events can be modified: John ran slowly. Events can be placed in space and time: On Tuesday, John ran in Chicago. Events can be causes and effects: John ran, because Sam was chasing him. Because John ran, he was tired. Events can be objects of propositional attitudes: Sam believes John ran. Events can be nominalized: John’s running tired him out. Events can be referred to by pronouns: John ran, and Sam saw it. To represent these, we need some kind of “handle” on the event. We need constants and variables to be able to denote events. We need to treat events as “things” -- reify events (from Latin “re(s)” - thing) Let e1 be John’s running. Then slow(e1) believe(Sam,e1) onDay(e1, ...), in(e1, Chicago) tiredOut(e1, John) cause(..., e1), cause(e1, ...) see(Sam, e1)

Representing Reifications Why not this? slow( run(John) ) This evaluates to True or False Then slow would describe not John’s running, but True or False This is easily understood, but it takes us out of logic. e1: run(John) run’(e1,John) This means “e1 is the event of John’s running” I’ll use this when I need to; run(John) otherwise.

Reifying Everything Not just events, but states, conditions, properties: John fell because the floor was slippery. cause(e1,e2) & fall’(e2, j) & slippery’(e1, f) The contract was invalid because John failed to sign it. cause(e1,e2) & invalid’(e2,c) & fail’(e1,j, e3) & sign’(e3,j,c) I will use the word “eventuality” to describe all these things -- events, states, conditions, etc. Controversial

Representing Case Relations Jenny pushed the chair from the living room to the dining room for Sam yesterday Case: Agent Theme Source Goal Benefactor Time Could represent this like push(Jenny, Chair1, LR, DR, Sam, 21Feb11, ...) Or like push’(e) & Agent(Jenny,e) & Theme(Chair1,e) & Source(LR,e) & Goal(DR,e) & Benefactor(Sam,e) & atTime(e, 21Feb11) Or like push’(e, Jenny, Chair1) & from(e, LR) & to(e, DR) & for(e, Sam) & yesterday(e, ...) from complements from adjuncts Equivalence of these: (A e,x,y)[push’(e,x,y) --> Agent(x,e) & Theme(y,e)]

Space, Time, Tense, and Manner tense John ran. run’(e,J) & Past(e) John ran on Tuesday. run’(e,J) & Past(e) & onDay(e,d) & Tuesday(d) John ran in Chicago. run’(e,J) & Past(e) & in(e,Chicago) John ran slowly. run’(e,J) & Past(e) & slow(e) John ran reluctantly. run’(e,J) & Past(e) & reluctant(J,e)

Attributives Some attributive adjectives have an implicit comparison set or scale: A small elephant is bigger than a big mosquito. That mosquito is big. mosquito(x) & big(x, s) The implicit comparison set or scale, which must be determined from context

Proper Names Proper names: Could treat them as constants: Springfield is the capital of Illinois. ==> capital(Springfield, Illinois) But there are many Springfields; we could treat it as a predicate true of any town named Springfield: capital(x,y) & Springfield(x) & Illinois(y) Or we could treat the name as a string, related to the entity by the predicate name: capital(x,y) & name(“Springfield”, x) & name(“Illinois”, y)

Indexicals An indexical or deictic is a word or phrase that requires knowledge of the situation of utterance for its interpretation. “I”, “you”, “we”, “here”, “now”, some uses of “this”, “that”, ... The property of being “I” is being the speaker of the current utterance Indexicals require an argument for the utterance or the speech situation. I(x,u): x is the speaker of utterance u you(x,u): x is the intended hearer of utterance u we(s,u): s is a set of people containing the speaker of utterance u here(x,u): x is the place of utterance u now(t,u): t is the time of utterance u Chris said, “I see you now.” ==> say(Chris,u) & content(e,u) & see’(e,x,y) & I(x,u) & you(y,u) & atTime(e,t) & now(t,u) from the quotation marks

Next Time ... How natural language conveys logical structure = Syntax + Compositional Semantics

CS544: Logic, Lecture 1