Computational Linguisticsnlpcl.kaist.ac.kr/~cs579_2020/slides/579-fall-2020-8-hw2.pdf · 2020. 9....

Computational LinguisticsCS579: Fall Semester 2020

School of ComputingKorea Advanced Institute of Science and Technology

Jong C. Park

© All rights reserved.

We reviewed the following aspects of First-Order Logic.• Three extensions: function symbols; equality;

sorted first-order logic We also reviewed three inference tasks,

along with related notions.• Querying; Consistency checking;

Informativity checking• Satisfiability and unsatisfiability; Validity

(invalidity), Valid and invalid arguments

Fall 2020 KAIST CS579: Computational Linguistics 2

Review of the Last Lecture

First-Order Logic Three Inference Tasks A First-Order Model Checker First-Order Logic and Natural Language


First-Order Logic

Goals Today• We review the syntax and semantics of first

order logic as the representation formalism.• We define the three inference tasks: querying,

consistency checking, and informativity checking.

• We present a first-order model checker to carry out querying.

• We see how good first-order logic is as a tool for computing natural language semantics.


First-Order Logic

Tasks• We decide how to represent models (in

Prolog).• We decide how to represent first-order

formulas (in Prolog).• We specify how (Prolog representations of)

first-order formulas are to be evaluated in (Prolog representations of) models with respect to (Prolog representations of) variable assignments.


A First-Order Model Checker

A Vocabulary{(LOVE,2), (CUSTOMER,1), (ROBBER,1),(JULES,0), (VINCENT,0), (PUMPKIN,0),(HONEY-BUNNY,0), (YOLANDA,0) }


Representing Models in Prolog

A Model in Prologmodel([d1,d2,d3,d4,d5],

[f(0,jules,d1), f(0,vincent,d2),f(0,pumpkin,d3), f(0,honey_bunny,d4),f(0,yolanda,d5), f(1,customer,[d1,d2]), f(1,robber,[d3,d4]),f(2,love,[(d3,d4)])]).


{(LOVE,2), (CUSTOMER,1), (ROBBER,1),(JULES,0), (VINCENT,0), (PUMPKIN,0),(HONEY-BUNNY,0), (YOLANDA,0) }

Another Model in Prologmodel([d1,d2,d3,d4,d5,d6],

[f(0,jules,d1), f(0,vincent,d2),f(0,pumpkin,d3), f(0,honey_bunny,d4),f(0,yolanda,d5), f(1,customer,[d1,d2,d5,d6]), f(1,robber,[d3,d4]),f(2,love,[])]).


First-order variables are represented by Prolog variables.• Use Prolog’s inbuilt unification mechanism to

handle variables.

A first-order constant c is represented by the Prolog atom and a first-order relation symbol R is represented by the Prolog atom .

The special two-place relation symbol = will be represented by the Prolog term .


Representing Formulas in Prolog

The connectives ∧, ∨, →, and ¬ are represented by the Prolog terms ,

, , and , respectively. Suppose is a first-order formula, and

is its representation as a Prolog term. Then will be represented as and as .


The task of evaluating (Prolog representations of) first-order formulas in (Prolog representations of) models with respect to (Prolog representations of) variable assignments is handled by the predicate

, whose four arguments are:1. the formula to be evaluated2. the model3. a list of assignments of members of the model’s

domain to any free variables the formula contains

4. a polarity feature ( or )


The Satisfaction Definition (in Prolog)

We use the Prolog terms of the form to indicate that a

variable has been assigned the element of the model’s domain.

The polarity feature is a flag that records whether we are trying to see if a particular subformula is true or false in a model.


for the boolean connectives• Negation

satisfy(not(Formula),Model,G,pos):-satisfy(Formula,Model,G,neg).

satisfy(not(Formula),Model,G,neg):-satisfy(Formula,Model,G,pos).


• Conjunction satisfy(and(Formula1,Formula2),Model,G,pos)

:-satisfy(Formula1,Model,G,pos),satisfy(Formula2,Model,G,pos).

satisfy(and(Formula1,Formula2),Model,G,neg):-satisfy(Formula1,Model,G,neg);satisfy(Formula2,Model,G,neg).


• Disjunction satisfy(or(Formula1,Formula2),Model,G,pos)

:-satisfy(Formula1,Model,G,pos);satisfy(Formula2,Model,G,pos).

satisfy(or(Formula1,Formula2),Model,G,neg):-satisfy(Formula1,Model,G,neg),satisfy(Formula2,Model,G,neg).


• Implicationsatisfy(imp(Formula1,Formula2),

Model,G,Pol) :-satisfy(or(not(Formula1),Formula2),

Model,G,Pol).


for existential quantificationsatisfy(some(X,Formula),model(D,F),G,pos) :-

memberList(V,D),satisfy(Formula,model(D,F),[g(X,V)|G],pos).

satisfy(some(X,Formula),model(D,F),G,neg) :-setof(V,(memberList(V,D),

satisfy(Formula,model(D,F),[g(X,V)|G],neg)), Dom),

setof(V,memberList(V,D),Dom).


model([d1,d2,d3,d4,d5],[f(0,jules,d1), f(0,vincent,d2),

f(0,pumpkin,d3), f(0,honey_bunny,d4),f(0,yolanda,d5), f(1,customer,[d1,d2]), f(1,robber,[d3,d4]), f(2,love,[(d3,d4)])]).

for universal quantificationsatisfy(all(X,Formula),Model,G,Pol) :-

satisfy(not(some(X,not(Formula))),Model,G,Pol).




for atomic formulas• One-place predicate symbols

satisfy(Formula,model(D,F),G,pos) :-compose(Formula,Symbol,[Argument]),i(Argument,model(D,F),G,Value),memberList(f(1,Symbol,Values),F),memberList(Value,Values).


compose(Term,Symbol,ArgList) :-Term =.. [Symbol|ArgList].



satisfy(Formula,model(D,F),G,neg) :-compose(Formula,Symbol,[Argument]),i(Argument,model(D,F),G,Value),memberList(f(1,Symbol,Values),F),\+ memberList(Value,Values).


Interpretation function in i(X,model(_,F),G,Value) :-

(var(X),memberList(g(Y,Value),G),Y==X, !

;atom(X),memberList(f(0,X,Value),F)

).


for atomic formulas• Two-place predicate symbols

satisfy(Formula,model(D,F),G,pos) :-compose(Formula,Symbol,[Arg1,Arg2]),i(Arg1,model(D,F),G,Value1),i(Arg2,model(D,F),G,Value2),memberList(f(2,Symbol,Values),F),memberList((Value1,Value2),Values).




satisfy(Formula,model(D,F),G,neg) :-compose(Formula,Symbol,[Arg1,Arg2]),i(Arg1,model(D,F),G,Value1),i(Arg2,model(D,F),G,Value2),memberList(f(2,Symbol,Values),F),\+ memberList((Value1,Value2),Values).




Equality in satisfy(eq(X,Y),Model,G,pos) :-

i(X,Model,G,Value1),i(Y,Model,G,Value2),Value1=Value2.

satisfy(eq(X,Y),Model,G,neg) :-i(X,Model,G,Value1),i(Y,Model,G,Value2),\+ Value1=Value2.


Example modelsexample(3,

model([d1,d2,d3,d4,d5,d6,d7,d8],[f(0,mia,d1), f(0,jody,d2),f(0,jules,d3), f(0,vincent,d4),f(1,woman,[d1,d2]),f(1,man,[d3,d4]), f(1,joke,[d5,d6]),f(1,episode,[d7,d8]),f(2,in,[(d5,d7),(d5,d8)]),f(2,tell,[(d1,d5),(d2,d6)])])).


Using the example modelsevaluate(Formula,Example,Assignment) :-

example(Example,Model),satisfy(Formula,Model,Assignment,Result),printStatus(Result).

evaluate(Formula,Example) :-evaluate(Formula,Example,[]).


example(3,model([d1,d2,d3,d4,d5,d6,d7,d8],

[f(0,mia,d1), f(0,jody,d2),f(0,jules,d3), f(0,vincent,d4),f(1,woman,[d1,d2]),f(1,man,[d3,d4]), f(1,joke,[d5,d6]),f(1,episode,[d7,d8]),f(2,in,[(d5,d7),(d5,d8)]),f(2,tell,[(d1,d5),(d2,d6)])])).

Test suite file (modelCheckerTestSuite.pl)test(some(X,robber(X)),1,[],pos).test(some(X,some(Y,love(X,Y))),1,[],pos).test(some(X,some(Y,love(X,Y))),2,[],neg).test(all(X,all(Y,love(X,Y))),2,[],neg).test(not(all(X,all(Y,love(X,Y)))),2,[],pos).…

Refer to modelChecker1.pl .


We can use the Model Checker to handle the querying task, but also to find elements in a model’s domain that satisfy certain descriptions. ?- satisfy(robber(X),

model([d1,d2,d3], [f(0,pumpkin,d1), f(o,jules,d2),f(1,customer,[d2]), f(1,robber,[d1,d3])]),[g(X,Value)], pos).

Value = d1 ;Value = d3


Refining the Model Checker

Problems • First Problem

?- evaluate(X,1).?- evaluate(imp(X,Y),4).

• Second Problem?- evaluate(mia,3).?- evaluate(all(mia,vincent),2).


• Third Problem?- evaluate(tasty(royale_with_cheese), 1).

• Fourth Problem?- evaluate(customer(X),1). Not satisfied in model.[ Cannot be evaluated.]

Refer to modelChecker2.pl .


First-order logic as a tool for computing natural language semantics• Inferential Capabilities• Resolution and tableau-based theorem provers• Model builders• Background knowledge

• Mia is married. She has a husband.• married mia → hashusband mia• ∀x woman x ∧ married x → hashusband x


First-Order Logic and Natural Language

• Representational Capabilities• Time (intervals, events, necessity and belief)• Higher-order entities

• Butch has every property that a good boxer has.• ∀P∀x goodboxer x → P x → P butch

• Partiality• FOL offers an elegant handle on partiality.

• Generalized quantifiers• Two robbers, most robbers, many robbers


First-Order Logic and Natural Language

SUMMARY

CS579: Computational Linguistics 33Fall 2020 KAIST

We reviewed a first-order model checker.• Represent models and first-order formulas in Prolog;

Specify the evaluation process in Prolog The Satisfaction Definition in Prolog

• satisfy/4 for boolean connectives (negation, conjunction, disjunction, implication), existential and universal quantification, one-place and two-place predicates, equality

• Interpretation function We discussed the need to refine the Model Checker. We examined the relationship between First-Order

Logic and Natural Language.• Inferential and Representational Capabilities


Summary

Make a bidding to the provided list of papers and, when the bidding is resolved, write a review of the assigned paper. • Select at least two “eager to review” papers and at

least five “willing to review” papers so that the final assignment is in your favor and evenly distributed as much as possible. (Bidding due: 29 September, 2020)

• The specific assignments will be announced as soon as the bids are fully resolved.

• Details of what are expected in your review will be provided.

Due: 28 October, 2020 Upload your review at KLMS.


Homework #2

Propose a topic for your term project.• The topic should lead to an implemented system with

documentation.• Give an appropriate scenario for user interaction, i.e.,

expected inputs and outputs, justify why this scenario is interesting; and explain the kind of work that should be done.

• Use a maximum of two A4 pages, single column, with appropriate margins and font sizes.

• Write your proposal in English. • Your proposal will be assessed with the metrics below:

novelty, substance, relevance, and clarity. Due: 14 October, 2020 (before 15 October, 2020) Upload your proposal at KLMS.


Term Project: Proposal

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Computational Linguisticsnlpcl.kaist.ac.kr/~cs579_2020/slides/579-fall-2020-8-hw2.pdf · 2020. 9....

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