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

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.

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

The inference tasks  The consistency checking task  The informativity checking task

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

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.

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

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

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

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

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!

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

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

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

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.

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.

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.

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.

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.

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.

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

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

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

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

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

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

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??

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

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

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

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

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??

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…

Googling theorem provers for DRT

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.

Googling theorem provers for FOL

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

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

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

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

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 )

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)

Example 1 x man(x) walk(x)

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

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

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

Example 1  x ( man(x) & walk(x) )

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

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

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

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

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

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

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

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

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

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

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

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

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)

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  

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

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

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)

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)

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…

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

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

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…

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

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)

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.

Tomorrow  Presupposition and Anaphora in DRT