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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.

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  1. Working with Discourse Representation TheoryPatrick Blackburn & Johan BosLecture 3DRT and Inference

  2. 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.

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

  4. The inference tasks • The consistency checking task • The informativity checking task

  5. 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

  6. 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.

  7. 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

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

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

  10. 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

  11. 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!

  12. 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

  13. 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

  14. 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

  15. 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 succeedSo theorem provers are a negative test for informativity • If the formula  is not valid/uninformative, all bets are off.

  16. 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 succeedSo theorem provers are also a negative test for consistency • If the formula  is not inconsistent, all bets are off.

  17. 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.

  18. 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 succeedSo 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.

  19. 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.

  20. 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 succeedSo model builders are a partial positive test for informativity • If the formula  is not satisfiable on a finite model all bets are off.

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

  22. 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

  23. 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

  24. Why is this relevant to natural language? • Testing a discourse for consistency

  25. Why is this relevant to natural language? • Testing a discourse for consistency

  26. Why is this relevant to natural language? • Testing a discourse for consistency

  27. Why is this relevant to natural language? • Testing a discourse for consistency

  28. 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

  29. Why is this relevant to natural language? • Testing a discourse for informativity

  30. Why is this relevant to natural language? • Testing a discourse for informativity

  31. Why is this relevant to natural language? • Testing a discourse for informativity

  32. Why is this relevant to natural language? • Testing a discourse for informativity

  33. 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…

  34. Googling theorem provers for DRT

  35. 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.

  36. Googling theorem provers for FOL

  37. 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

  38. 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

  39. Translating DRT to FOL: DRSs ( )fo = x1… xn((C1)fo&…&(Cn)fo)

  40. Translating DRT to FOL: Conditions (R(x1…xn))fo = R(x1…xn) (x1=x2)fo = x1=x2 (B)fo = (B)fo (B1B2)fo = (B1)fo (B2)fo

  41. Translating DRT to FOL:Implicative DRS-conditions ( B)fo = x1…xm(((C1)fo&…&(Cn)fo)(B)fo)

  42. Two example translations • Example 1 • Example 2

  43. Example 1

  44. Example 1 ( )fo

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

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

  47. Example 1 x(man(x) & walk(x))

  48. Example 2

  49. Example 2 ( )fo

  50. Example 2 (woman(y))fo & ( )fo ) y ( 

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