12. Dezember

Transcription

1 1 Lehrstuhl für Künstliche Intelligenz Institut für Informatik Friedrich-Alexander-Universität Erlangen-Nürnberg 12. Dezember Alex Lascarides SPNLP Course

2 Overview

3 do in Construction of a Semantic Representation Semantic Resolution Semantic Inference What is the task of Lexical Capture word meaning Capture lexical generalizations Yes Yes to incorporating taxonomic knowledge into lexicon, i.e. adding structured and relational view to the collection of words

4 Why FOL? Task of Semntics: Automating Comprehension (1) Every child likes cartoons. Bill is a child. (2) Bill would like to watch TV. Which chanel he prefers? By now we know how to parse sentences syntactically How do we parse them semantically?? How do we present information given by the sentence? FOL formula How do we work with information? FOL inference tools

5 Workflow of Semantic Analysis Construction of Semantic Representation Sentence Syntactic Parse Tree encoding of sentence s meaning in the language that: Unambiguous, Canonical Notion of a Model, Verifiability, Inference, Variables

6 Requirements Unambiguous Representation Are Simpsons shown daily? Canonical Simpsons, The Simpsons Interpretable Simpsons are my favourite. Inference and Variables When are Simpsons shown? Verifiability Are Simpsons shown every day? Expressiveness A, The, Some, Many, Although, And, Or...

7 Constructing Assumptions: Meaning of a sentence is combination of Words meaning Lexical Semantics Syntax of a sentence, i.e. how words are bind together FOL can be used as the of meaning

8 Construction Bill likes Simpsons (1) Lexicon provides words meaning: - Bill contributes constant Bill - Simpsons contributes constant Simpsons - Likes contributes predicate like(?,?) that takes two arguments: 1st argument agent 2d argument patient Syntax provides relations between words. Normally: subject agent object patient of (1): like(bill, Simpsons)

9 Semantic Construction Workflow Sentence Syntactic Parse Tree S:Bill likes Simpsons like(bill, Simpsons) NP:Bill Bill VP:likes Simpsons like(?, Simpsons) V:likes like(?,?) NP:Simpsons Simpsons Lexicon provides leaf informations (Lexical Semantic) Syntactic tree provides?-substitution

10 of s characterise the interpretation of utterance in terms of elements from the domain that the utterance describes we intuitively understand not a general meaning utterance but rather its individual specific meaning interpretation is bounded by domain individuals and concepts

11 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

12 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 against certain kinds of situations

13 Ingredients of FOL A Vocabulary (c.a. lexicon) 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 Unambiguous, Canonical Semantics Compositional (uses recursion) Truth, Satisfaction, Entailment. Notion of a Model, Verifiability, Inference,

14 FOL Vocabulary FOL designed to talk about various relationships and properties that hold among individuals (MIA,0) (HONEY-BUNNY,0) (VINCENT,0) (PUMPKIN,0) (STORE,1) (ROBBER,1) (LOVE,2) to determine the class of models (that is, the kinds of situation we want to describe)

15 Models Models Consists of Domain: The collection D of entities we can talk about; Function: A mapping F from each symbol in the vocabulary to its semantic value. The arity of a symbol s determines what kind of value F(s) should be e.g., F(MIA) D, F(ROBBER) D, F(LOVE) D x D Our Understanding of : characterise the interpretation of utterance in terms of elements from the domain that the utterance describes interpretation is bounded by domain individuals and concepts

16 model D = {d1; d2; d3; d4} F(MIA) = d1 F(HONEY-BUNNY) = d2 F(VINCENT) = d3 F(PUMPKIN) = d4 F(CUSTOMER) = {d1; d3} F(ROBBER) = {d2; d4} F(LOVE) = {(d4; d2); (d3; d1)}

17 The Consists of: Symbols in the vocabulary Variables: x, y,... Boolean connectives: (negation) (and) (or) (if... then) Quantiers: (all) (some) Equality: = Punctuation: brackets and comma

18 The Syntax All constants and variables are terms. Where R is an n-place relation symbol and τ 1,..., τ n are terms, R(τ 1,..., τ n ) is an atomic WFF. τ 1 = τ 2 is a WFF, where τ 1 and τ 2 are terms. If φ and ψ are WFFs, then so are: φ, φ ψ, φ ψ, φ ψ. If φ is a WFF and x is a variable, then xφ and xφ are WFFs.

19 Free and Bound Variables: (CUSTOMER(x) x(robber(x) y(person(y)))) First occurrence of x is free Second occurrence of x is bound Occurrence of y is bound Free variable pronouns She loves Vincent Context needed to interpret it; More than models needed to interpret free variables. A WFF with no free variables is a sentence FOL sentences WFFs

20 Some Examples (1) xlove(mia, x) Mia doesn t love anyone (2) x(robber(x) y(store(y) LOVE(x, y))) All robbers love a (perhaps diferent) store (3) y(store(y) x(robber(x) LOVE(x, y))) All robbers love the same store

21 Interpreting FOL Sentences Task: Compute whether a sentence is true or false with respect to a model Is the sentence an accurate description of the situation?

22 Satisfaction: M, g = φ The model M and variable assignment function g satisfy the formula φ: g defined for all variables g(x) D; I F g (τ) = g(τ) if τ is a variable I F g (τ) = F(τ) if τ is a constant Remember that F is an Function a mapping from each symbol in the vocabulary to its semantic value. M, g = xφ iff M, g = φ for some x-variant g of g Try to find some value for x that makes φ true Sentence s truth value defined in terms of satisfaction.

23 Full Denition of Satisfaction Where M = D, F : M, g = R(τ 1,..., τ n ) iff (I F g (τ 1 ),..., I F g (τ n ) F(R) M, g = τ 1 = τ n iff I F g (τ 1 ) = I F g (τ 2 ) M, g = φ iff M, g = φ M, g = φ ψ iff M, g = φ and M, g = ψ M, g = φ ψ iff M, g = φ or M, g = ψ M, g = φ ψ iff M, g = φ or M, g = ψ M, g = xφ iff M, g = φ for some x-variant g of g Try to find some value for x that makes φ true M, g = xφ iff M, g = φ for every x-variant g of g Verify that every value for x makes φ true

24 Truth (in terms of Satisfaction) It doesn t matter which variable assignment function g we use for sentences: Truth: A sentence φ is true in a model M (written M = φ) iff for any M, g = φ Validity: A sentence φ is valid (written = φ ) iff for any M, M = φ Entailment: φ 1,..., φ n = ψ iif if M, g = φ i for all i, then M, g = ψ

25 FOL Tools Theorem Provers (uninformative,inconsistency) Verifies if the FOL formula is valid (uninformative) Otter, Blicksem, Vampire Model Builder (consistency) Takes a FOL formula Looks for a model that satisfies it Takes a description and Makes a little picture of the world based on description Paradox, Mace Model Checkers (querying task) Takes a model and FOL formula Tests whether or not the formula is satisfied by the model

26 in Semantics Want to construct fol-representations of NL expressions so that the fol inferences capture what the NL expressions mean. Want to automate these inferences, so that once we can construct the logical forms of NL expressions, we can apply it in: question answering, information retrieval, tutorial dialogue systems...

27 Bill likes a cartoon (2) FOL: x(cartoon(x) like(bill, x)) Elements of a domain: Bill, Simpsons Concepts of a domain: like, cartoon Domain Knowledge: Bill may not be a patient of like, Simpsons is a cartoon, Bill is not a cartoon

28 Finding by Means of Mace Mace4 finite model builder for FOL formulas: x(cartoon(x) like(bill, x)) Mace4 Program: assign(domain_size, 2). % Two entities in the domain: 0, 1 formulas(theory). exists x(cartoon(x) & Like(Bill,x)). end_of_list. clauses(theory). Cartoon(Simpsons). Cartoon(Bill). Like(x,Bill). end_of_list.

29 Mace Output Only Model Found by Mace4: Bill : 0 $ c1 : 1 Simpsons : 1 Cartoon : Like : (cartoon(simpsons) like(bill, Simpsons)) is satisfied interpretation

30 learn today? What can FOL be for NLP?

31 What will we learn during the rest of semester? First Order (FOL) as a (LF) Discourse Representation Theory (DRT) as a al Form Why DRT as instead of FOL?