Semantic Inference in Linguistics: Models and Logical Relations

1. Computational Semanticshttp://www.coli.uni-sb.de/cl/projects/milca/esslli/ Day 5: Inference Aljoscha Burchardt, Alexander Koller, Stephan Walter, Universität des Saarlandes, Saarbrücken, Germany ESSLLI 2004, Nancy, France

2. lexicon Syntax Semantic construction Where are we by now? So far: John loves Mary. Sentence: Linguistic Analysis… Why??? Why meaning? Why logic? Formula: love(john, mary)

3. Motivations Why meaning? • The big question in the background of semantics: How do linguistic expressions relate to the world? • The need for inference in a broad sense is omnipresent in linguistic processing: Getting some piece of information out of another. This process is meaning based. Why logic? Using logic helps us in answering both problems at once.

4. Meaning based linguistic Inferences Peter loves Mary and she doesn't love him. No one is happy if he isn't loved by the one he loves.  Peter is not happy • Answering questions: A: "Is Peter happy" B: • Discourse „There is my car. The roof is red.“ => The roof of this particular car. • Pragmatics A: „Shall we watch Athens?“, B: „Oh, I hate Sports“ • Answer is "no." • ... "No"

5. Logical Inferences • Argumentation: Classical field => Answering questions „Every human is mortal“, „Socrates is a human“ => Socrates is mortal. x.human(x) -> mortal(x), human(soc) |= mortal(soc) • Discourse, Pragmatics, ... Inference problems during processing: • logical relations between readings (equivalence, implication, contradiction) yx.love(x,y)  xy.love(x,y) xy.love(x,y)  yx.love(x,y) • discourse maxims: utterance consistent? informative?… • "lexical" inference: "Brussels lowers taxes" • presuppositions

6. Next… How do linguistic expressions relate to the world? Building logical representations is a step towards a scientific theory of this relation! They're a way of replacing something we don't understand by something we understand (at least better). Why? Because we have a formal way of saying what they mean: Models.

7. The big question of semantics John loves Mary and Peter doesn't. Semantic construction love(john,mary) love(peter,mary) ??? "Understanding language" Logics {man(john), man(peter), woman(mary), love(john,mary)} ??? Cognition / Ontology ???  

8. Plan for Today • What's the advantage of FOL-formulae? • Interpretations and models • Doing things with semantic representations • Logical Inference and Proof Theory • A calculus • Automated Theorem Proving (first steps) • An implementation of propositional tableaux • A sample application

9. FOL-semantics What does a FO-formula mean? • It may be true or false (that's all) • Whether it is true or false is calculated given a model. • So: A formula is true or false in a model. • But what is a model?

10. Models A model can be thought of as a set of basic facts that describe a part of the world. E.g., talking about John, Mary, Peter, love, man and woman: • John loves Mary. • John is a man. • Mary doesn't love John. • Peter is a man. • Mary isn't a man. • Mary is a woman. • … • In this listing: • Who is there? • Which properties do (or don't) they have?

11. Formally This intuition is formalized as follows: A model is an ordered pair of a set and a Function: M = (D, F) The interpretation function: Which properties do these things have? (and more…) The domain: What is there.

12. Example model • D = {John, Mary, Peter} • F = {(John, John), (Mary, Mary), (Peter, Peter) (man, {John, Peter}), (woman, {Mary}), (love, {(John, Mary)}) }

13. iff iff and iff iff or iff or iff for some x-variant g' of g iff for all x-variants g' of g Truth in a model g: Assignment function, assigning values from D to variables

14. Models as Sets of Formulae • For our purposes, models are simply sets of literals (i.e. positive or negative atomic formulae). • Set contains all literals that are true in the model. • Our example: {man(john), man(peter), woman(mary), love(john,mary),love(mary,john),…} • Truth of atomic formulae without variables: R(t1,…,tn)  M

15. From theory to practice • Models define the semantics of logical languages… • …and are an interesting concept for relating language and the world. • But they're also of practical importance: They're the key to a formalization of inference. Now: some further important logical notions.

16. Inference and Entailment • Valid inference: Truth of premises guarantees truth of conclusion. • Entailment: Talking about all models. • Concept directly captures syllogistic reasoning. P, Q, … |=R For all M, g such that: we have: and and …

17. Validity • A related notion: Truth of a formula in all models: Validity |= A iff for all M,g: • Validity formalizes the notion of tautology, e.g.: Sylvester is either a cat or not. |= cat(s) v  cat(s) • Relation to entailment via the deduction theorem: A|= B iff |= AB

18. Where are we now? • Why meaning?  • Why logic?  • Relation to the world: Models  • Inferences: Entailment and validity  • How to compute with these notions?

19. How to work with all models? Entailment and validity are both defined with respect to all models. Problem: There are infinitely many models. How can we work with these notions then? Idea: Tell whether a formula is valid or not just by looking at it! The answer: A calculus.

20. Calculi • Calculi are rule-based systems for manipulating formulae according to their structure. • Some of the resulting configurations are called proofs. • Formulas with proofs are called theorems. • A good calculus produces a proof iff its input formula is valid.

21. "Good" Calculi Good Calculi are: • Sound: Only valid formulae get a proof. • Complete: All valid formulae get a proof. In other words: All and only theorems are valid. |- ≡ |= To achieve this, one has to give the right rules. Let's try…

22. Tableaux: The intuition I Truth conditions tell us what would have to hold in a model for a given formula, e.g.: • A and B hold in all models for A  B • For A  B, there are two kinds of models: Those for A and those for B. • … If we go on decomposing a formulas that way, we end up with sets of literals  models Example: smoke(john)  ( love(mary,john)   love(john,mary))  {smoke(john), love(john, mary)}  {smoke(john), love(mary, john)}

23. Tableaux: The intuition II We know: If a formula is valid, it's always true. I.e.: No model makes it false. Making formulae false: (smoke(john)  walk(john))F  {smoke(john), walk(john)} (smoke(john)  smoke(john)) F  {smoke(john), smoke(john)} "sign"  

24. Tableaux If we want to know whether a formula is valid, we systematically try to find a model that would make it false… … hoping that we find none. That is, all attempts should lead to contradictions. Next: A look at: ( ((pq) (pq)) )F

31. A simple fragment Next: The rules for a tableaux calculus for predicate logic without variables and quantifiers. • Actually propositional logic • Advantage 1: Decidable • Advantage 2: Rules are easy • Disadvantage: Boring and restricted More is possible – but not here and now.

32. Preprocessing • Reduce the number of connectives by translating  and  to  and . • Use logical equivalences: • A  B  (A  B) „De Morgan“ • A  B  (A  B)

33. Tableaux Inference Rules

34. “Mary loves Bill or John loves Mary'' |= ``John loves Mary“ ???

35. Summing up • Using predicate logic as representation language seemed to be a design decision on Monday. • Now we're happy we did it: • Models tell us when sentences are true. • Models give us a concept of logical inference. • This concept can be mechanized by calculi. • After the break: Calculi can be implemented in provers. And provers are useful!

36. More logics - Changing the language and/or the semantics. • Different phenomena, different logics: • Intensional logic (John seeks a unicorn) • Temporal logics (tense) • Dynamic logics (anaphora) • Higher Order (quantifiers) • Different tasks – different tools • Decidability and complexity • From propositional over first order to higher order • In between. E.g. Description logics.