26. Januar Introduction to Computational Semantics

Transcription

1 1 Lehrstuhl für Künstliche Intelligenz Institut für Informatik Friedrich-Alexander-Universität Erlangen-Nürnberg 26. Januar Slides are mainly due to J. Bos and P. Blackburn course on An Introduction to Semantics

2 Overview

3 learn so far? What do we do in Construction of a Semantic Representation Semantic Resolution, Interpretation Semantic today s topic

4 FOL for Semantic Resolution, Interpretation Sample Interpretation of an Utterance: Bill likes a cartoon (2) FOL: x(cartoon(x) like(bill, x)) Elements of a domain: Bill, Simpsons Concepts of a domain: like, cartoon (cartoon(simpsons) like(bill, Simpsons)) is satisfied interpretation

5 Ingredients of FOL A Vocabulary Determines what we can talk about Syntax Uses vocabulary and rules to define the set of well-formed formulae (WFFs) Determines how we can talk about things Semantics Compositional (uses recursion) Truth, Satisfaction, Entailment. Notion of a, Interpretation

6 FOL A model is a pair (D; F) (Domian; Interpretation Function) Domain D contains the set of entities we want to talk about. D is called the domain, or universe of discourse, and must be non-empty. F is the interpretation function. It specifies what each symbol in the vocabulary stands for. It does so by associating each symbol in the vocabulary with an appropriate entity built from items in domain D. Sample Vocabulary: {(mia, 0), (robber, 1)} Sample Domain D: {d1} Sample Interpretation Function F: F(mia) = d1, F(rober) = {d1}

7 What can FOL do for NLP To perform inferences over sentence semantic representations (if we take a representation to be FOL formula), by evaluating certain kinds of descriptions (formulas) against certain kinds of situations (models) Now lets proceed to the details of it!

8 Task of Semntics: Automating Language Comprehension Today we work through the following three inference tasks:

9 Maxims Of Conversation via Why is Important? Maxims Of Conversation (1) each contribution of the discourse should be consistent with it, and (2) each contribution should bring new information Possible use of three inference strategies: Queston-Answering (1) Maxim of Conversation (2) Maxim of Conversation

10 task within FOL consists of a question: Given a model M and a formula φ, is φ true in M or not? Think of the model M as a picture of (a little part of) the world Then querying φ is asking whether or not the information φ is true in our little picture of the world A FOL tool that carries out this inference task is called a model checker. This is a relatively simple form of inference to implement for finite models and is useful for example for question answering and natural language generation.

11 I task within FOL consists of: A formula is consistent if it is satised in at least one model. So consistent formulas describe conceivable or possible states of afairs. For example, rober(mia) is consistent if there exists a model as ({d1,... };F(mia) = d1, F(rober) = {d1,...}) A formula that is not consistent is called inconsistent. So inconsistent describe inconceivable or impossible states of aairs. For example, robber(mia) robber(mia) is inconsistent. A finite set of formulas {φ 1,..., φ n } is consistent if φ 1 φ n. A finite set of formulas that is not consistent is called inconsistent.

12 II We would like to check whether the information given to us in natural language discourses is consistent or not. For if we are given inconsistent information, something may well be going wrong with communicative process... However consistency is a much harder inference task than querying. In fact, for FOL, consistency is undecidable. However there are two computational tools that can help us carry it out: theorem provers and model builders.

13 I task within FOL consists of: A valid sentence is a sentence that is true in all models (for example: robber(mia) robber(mia)). A sentence that is not valid is called invalid. Suppose φ 1,..., φ n, and ψ are a finite collection of first-order sentences. Then we say that the argument with premises φ 1,..., φ n and conclusion ψ is a valid argument if whenever all the premises are true in some model, the conclusion is true in that model also. The notation φ 1,..., φ n = ψ indicates a valid argument. The notation φ 1,..., φ n = ψ indicates a invalid argument.

14 II So what does validity have to do with informativity? This: We often call valid sentences uninformative sentences. (After all, if they re true in all models, they don t give us any specic information!) We often call a sentence that is not valid informative. If φ 1,..., φ n = ψ, then we say that ψ is uninformative with respect to φ 1,..., φ n. If φ 1,..., φ n = ψ, then we say that ψ is informative with respect to φ 1,..., φ n.

15 III We would like to check whether the information given to us in natural language discourses is informative with respect to the information we already have. For if it is not, something may well be going wrong with communicative process... is a much harder inference task than querying. In fact, for first order logic, consistency is undecidable. However there are two computational tools that can help us carry it out: theorem provers and model builders.

16 Relationships between and φ is informative (that is, not valid) if and only if φ is consistent. That is, informativity means the opposite really was an option. φ 1,..., φ n = ψ (that is, φ 1,..., φ n ) iff {φ 1,..., φ n, ψ} is consistent. φ is uninformative (that is, valid) if and only if φ is inconsistent. That is, uninformativity means that the opposite simply was not an option. φ 1,..., φ n = ψ (that is, φ is uninformative with respect to φ 1,..., φ n ) iff {φ 1,..., φ n, ψ} is inconsistent.

17 Properties We said that consistency and informativity are much harder tasks than querying. Why is this? Because both tasks are defined in terms of all models, and there are lots of models, and most are infinite. In short, both task are defined semantically and in a very abstract way. (By way of contrast, querying is dened semantically, but in a very concrete way.) Nevertheless the system that work with these tasks exist theorem provers, model builders and we shall learn more about them.

18 Proving I proving is the proving of mathematical theorems. A theorem is a proposition that has been or is to be proved on the basis of explicit assumptions. provers check whether a formula (or a set of formulas) is valid (true in all possible models)

19 Proving II Fact: First-order logic is not decidable! Now Suppose we give a first-order problem to a theorem prover. This means that there is a chance that, in theory, our theorem prover never comes back with an answer... Practical consequence: if no proof for φ is found, this does strictly speaking not mean that is not a theorem. (In practice, you will have to work with time-outs. ) On the other hand, if your theorem prover finds a proof, you can be sure φ is a theorem (is valid).

20 provers check whether a formula (or a set of formulas) is valid (true in all possible models) Builders attempt to construct a model for a formula (or a set of formulas) and thereby show that this formula is satisable (true in at least one model) There are some unpleasant restrictions for model building: You need to specify a domain size. This means that if your model builder cannot find a model with domain size n for input φ, it does not mean that φ is not satisable. Perhaps there is a model with domain size n + 1! On the other hand, if it finds a model, you can be sure that is satisable. Restricted to finite models. (Everybody has a mother)

21 Proving vs. I Suppose you want to check whether φ is consistent: Give φ to a theorem prover. If it succeeds in finding a proof, φ is not consistent Give φ to a model builder. If it finds a model, φ is consistent.

22 Proving vs. II Suppose you want to check whether φ is informative wrt ψ: Give ψ φ to a theorem prover. If it succeeds in finding a proof, φ is not informative. Give ψ φ and ψ φ to a model builder. If it finds a model in both cases, φ is informative.

23 Off-the-shelf Provers Provers that accept first-order formula syntax (as opposed to special CNF notation): OTTER PARDOX VAMPIRE There are many more.

24 Off-the-shelf Builders Accepting full first-order formula syntax (as opposed to special CNF notation): MACE KIMBA konrad/kimba.html There are some more.

25 What did We now have some answers to the two fundamental questions with which this course began: We know how to build first-order representations for natural language expressions with the help of DRT + the lambda calculus. We have learned about the tools that perform inference with first-order representations. In particular, we have learned something about theorem provers, model builder and model. We have learnd how these tools can be used for consistency and informativity check.

26 What will Next Class: It s time to have a little fun, bring the various bits and pieces together, and see what they can do for us in practice within NLP... Demo of a Curt-diaologue system Description of real-world applications