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Working with Discourse Representation Theory Patrick Blackburn & Johan Bos Lecture 3 DRT and Inference

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This lecture Now that we know how to build DRSs for English sentences, what do we do with them? Well, we can use DRSs to draw inferences. In this lecture we show how to do that, both in theory and in practice.

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Overview Inference tasks Why FOL? From model theory to proof theory Inference engines From DRT to FOL Adding world knowledge Doing it locally

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The inference tasks The consistency checking task The informativity checking task

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Why First-Order Logic? Why not use higher-order logic? Better match with formal semantics But: Undecidable/no fast provers available Why not use weaker logics? Modal/description logics (decidable fragments) But: Can’t cope with all of natural language Why use first-order logic? Undecidable, but good inference tools available DRS translation to first-order logic Easy to add world knowledge

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Axioms encode world knowledge We can write down axioms about the information that we find fundamental For example, lexical knowledge, world knowledge, information about the structure of time, events, etc. By the Deduction Theorem 1 … n |= iff |= 1 & … & n That is, inference reduces to validity of formulas.

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From model theory to proof theory The inference tasks were defined semantically For computational purposes, we need symbolic definitions We need to move from the concept of |= to |-- In other words, from validity to provability

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Soundness If provable then valid: If |-- then |= Soundness is a `no garbage` condition

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Completeness If valid then provable If |= then |-- Completeness means that proof theory has captured model theory

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Decidability A problem is decidable, if a computer is guaranteed to halt in finite time on any input and give you a correct answer A problem that is not decidable, is undecidable

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First-order logic is undecidable What does this mean? It is not possible, to write a program that is guaranteed to halt when given any first-order formula and correctly tell you whether or not that formula is valid. Sounds pretty bad!

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Good news FOL is semi-decidable What does that mean? If in fact a formula is valid, it is always possible, to symbolically verify this fact in finite time That is, things are only going wrong for FOL when it is asked to tackle something that is not valid On some non-valid input, any algorithm is bound not to terminate

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Put differently Half the task, namely determining validity, is fairly reasonable. The other half of the task, showing non- validity, or equivalenty, satisfiability, is harder. This duality is reflected in the fact that there are two fundamental computational inference tools for FOL: theorem provers and model builders

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Theorem provers Basic thing they do is show that a formula is provable/valid. There are many efficient off-the-shelf provers available for FOL Theorem proving technology is now nearly 40 years old and extremely sophisticated Examples: Vampire, Spass, Bliksem, Otter

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Theorem provers and informativity Given a formula , a theorem prover will try to prove , that is, to show that it is valid/uninformative If is valid/uninformative, in theory, the theorem prover will always succeed So theorem provers are a negative test for informativity If the formula is not valid/uninformative, all bets are off.

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Theorem provers and consistency Given a formula , a theorem prover will try to prove , that is, to show that is inconsistent If is inconsistent, in theory, the theorem prover will always succeed So theorem provers are also a negative test for consistency If the formula is not inconsistent, all bets are off.

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Model builders Basic thing that model builders do is try to generate a [usually] finite model for a formula. They do so by iteration over model size. Model building for FOL is a rather new field, and there are not many model builders available. It is also an intrinsically hard task; harder than theorem proving. Examples: Mace, Paradox, Sem.

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Model builders and consistency Given a formula , a model builder will try to build a model for , that is, to show that is consistent If is consistent, and satisfiable on a finite model, then, in theory, the model builder will always succeed So model builders are a partial positive test for consistency If the formula is not consistent, or it is not satisfiable on a finite model, all bets are off.

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Finite model property A logic has the finite model property, if every satisfiable formula is satisfiable on a finite model. Many decidable logics have this property. But it is easy to see that FOL lacks this property.

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Model builders and informativity Given a formula , a model builder will try to build a model for , that is, to show that is informative If is satisfiable on a finite model, then, in theory, the model builder will always succeed So model builders are a partial positive test for informativity If the formula is not satisfiable on a finite model all bets are off.

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Yin and Yang of Inference Theorem Proving and Model Building function as opposite forces

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Doing it in parallel We have general negative tests [theorem provers], and partial positive tests [model builders] Why not try to get of both worlds, by running these tests in parallel? That is, given a formula we wish to test for informativity/consistency, we hand it to both a theorem prover and model builder at once When one succeeds, we halt the other

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Parallel Consistency Checking Suppose we want to test [representing the latest sentence] for consistency wrto the previous discourse Then: If a theorem prover succeeds in finding a proof for PREV , then it is inconsistent If a model builder succeeds to construct a model for PREV & , then it is consistent

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Why is this relevant to natural language? Testing a discourse for consistency DiscourseTheorem proverModel builder

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Why is this relevant to natural language? Testing a discourse for consistency DiscourseTheorem proverModel builder Vincent is a man. ??Model

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Why is this relevant to natural language? Testing a discourse for consistency DiscourseTheorem proverModel builder Vincent is a man. ??Model Mia loves every man. ??Model

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Why is this relevant to natural language? Testing a discourse for consistency DiscourseTheorem proverModel builder Vincent is a man. ??Model Mia loves every man. ??Model Mia does not love Vincent. Proof??

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Parallel informativity checking Suppose we want to test the formula [representing the latest sentence] for informativity wrto the previous discourse Then: If a theorem prover succeeds in finding a proof for PREV , then it is not informative If a model builder succeeds to construct a model for PREV & , then it is informative

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Why is this relevant to natural language? Testing a discourse for informativity DiscourseTheorem proverModel builder

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Why is this relevant to natural language? Testing a discourse for informativity DiscourseTheorem proverModel builder Vincent is a man. ??Model

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Why is this relevant to natural language? Testing a discourse for informativity DiscourseTheorem proverModel builder Vincent is a man. ??Model Mia loves every man. ??Model

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Why is this relevant to natural language? Testing a discourse for informativity DiscourseTheorem proverModel builder Vincent is a man. ??Model Mia loves every man. ??Model Mia loves Vincent. Proof??

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Let`s apply this to DRT Pretty clear what we need to do: Find efficient theorem provers for DRT Find efficient model builders for DRT Run them in parallel And Bob`s your uncle! Recall that theorem provers are more established technology than model builders So let`s start by finding an efficient theorem prover for DRT…

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Googling theorem provers for DRT

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Theorem proving in DRT Oh no! Nothing there, efficient or otherwise. Let`s build our own one. One phone call to Voronkov later: Oops --- does it take that long to build one from scratch? Oh dear.

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Googling theorem provers for FOL

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Use FOL inference technology for DRT There are a lot FOL provers available and they are extremely efficient There are also some interesting freely available model builders for FOL We have said several times, that DRT is FOL in disguise, so lets get precise about this and put this observation to work

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From DRT to FOL Compile DRS into standard FOL syntax Use off-the-shelf inference engines for FOL Okay --- how do we do this? Translation function (…) fo

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Translating DRT to FOL: DRSs x 1 …x n C1...CnC1...Cn ( ) fo = x 1 … x n ((C 1 ) fo &…&(C n ) fo )

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Translating DRT to FOL: Conditions (R(x 1 …x n )) fo = R(x 1 …x n ) (x 1 =x 2 ) fo = x 1 =x 2 (B) fo = (B) fo (B 1 B 2 ) fo = (B 1 ) fo (B 2 ) fo

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Translating DRT to FOL: Implicative DRS-conditions x 1 …x m C1...CnC1...Cn ( B ) fo = x 1 …x m (((C 1 ) fo &…&(C n ) fo )(B) fo )

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Two example translations Example 1 Example 2 x man(x) walk(x) y woman(y) x man(x) e adore(e) agent(e,x) theme(e,y)

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Example 1 x man(x) walk(x)

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Example 1 x man(x) walk(x) ) fo (

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Example 1 x ( ( man(x) ) fo & ( walk(x) ) fo )

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Example 1 x ( man(x) & ( walk(x) ) fo )

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Example 1 x ( man(x) & walk(x) )

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Example 2 y woman(y) x man(x) e adore(e) agent(e,x) theme(e,y)

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Example 2 y woman(y) x man(x) e adore(e) agent(e,x) theme(e,y) ) fo (

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Example 2 x man(x) e adore(e) agent(e,x) theme(e,y) y ( ) ( woman(y) ) fo & ( ) fo

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Example 2 x man(x) e adore(e) agent(e,x) theme(e,y) y ( ) woman(y) & ( ) fo

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Example 2 e adore(e) agent(e,x) theme(e,y) ) y (woman(y) &x ( ( man(x) ) fo ( ) fo )

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Example 2 e adore(e) agent(e,x) theme(e,y) ) y (woman(y) &x (man(x) ( ) fo )

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Example 2 y (woman(y) &x (man(x) e ( ( adore(e) ) fo & ( agent(e,x) ) fo & ( theme(e,y) ) fo )))

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Example 2 y (woman(y) &x (man(x) e (adore(e) & ( agent(e,x) ) fo & ( theme(e,y) ) fo )))

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Example 2 y (woman(y) &x (man(x) e (adore(e) & agent(e,x) & ( theme(e,y) ) fo )))

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Example 2 y (woman(y) &x (man(x) e (adore(e) & agent(e,x) & theme(e,y))))

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Basic setup DRS: x y vincent(x) mia(y) love(x,y)

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Basic setup DRS: FOL: x y(vincent(x) & mia(y) & love(x,y)) x y vincent(x) mia(y) love(x,y)

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Basic setup DRS: FOL: x y(vincent(x) & mia(y) & love(x,y)) Model: D = {d1} F(vincent)={d1} F(mia)={d1} F(love)={(d1,d1)} x y vincent(x) mia(y) love(x,y)

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Background Knowledge (BK) Need to incorporate BK Formulate BK in terms of first-order axioms Rather than just giving to the theorem prover (or model builder), we give it: BK & or BK

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Basic setup DRS: x y vincent(x) mia(y) love(x,y)

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Basic setup DRS: FOL: x y(vincent(x) & mia(y) & love(x,y)) x y vincent(x) mia(y) love(x,y)

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Basic setup DRS: FOL: x y(vincent(x) & mia(y) & love(x,y)) BK: x (vincent(x) man(x)) x (mia(x) woman(x)) x (man(x) woman(x)) x y vincent(x) mia(y) love(x,y)

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Basic setup DRS: FOL: x y(vincent(x) & mia(y) & love(x,y)) BK: x (vincent(x) man(x)) x (mia(x) woman(x)) x (man(x) woman(x)) Model: D = {d1,d2} F(vincent)={d1} F(mia)={d2} F(love)={(d1,d2)} x y vincent(x) mia(y) love(x,y)

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Local informativity Example: Mia is the wife of Marsellus. If Mia is the wife of Marsellus, Vincent will be disappointed. The second sentence is informative with respect to the first. But…

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x y mia(x) marsellus(y) wife-of(x,y) Local informativity

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x y z mia(x) marsellus(y) wife-of(x,y) vincent(z) wife-of(x,y) disappointed(z) Local informativity

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Local consistency Example: Jules likes big kahuna burgers. If Jules does not like big kahuna burgers, Vincent will order a whopper. The second sentence is consistent with respect to the first. But…

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x y jules(x) big-kahuna-burgers(y) like(x,y) Local consistency

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x y z jules(x) big-kahuna-burgers(y) like(x,y) vincent(z) u order(z,u) whopper(u) Local consistency like(x,y)

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DRT and local inference Because DRS groups information into contexts, we now have natural means to check not only global, but also local consistency and informativity. Important for dealing with presupposition. Presupposition is not about strange logic. But about using classical logic in new ways.

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Tomorrow Presupposition and Anaphora in DRT

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