Inference Tasks and Computational Semantics

Key Concepts

• Inference tasks

• Syntactic versus semantic approach to logic

• Soundness & completeness

• Decidability and undecidability

• Technologies:

• Theorem proving versus model building

INFERENCETASKS

QUERYINGCONSISTENCY

CHECKINGINFORMATIVITY

CHECKING

QUERYING

• Definition:– Given: Model M and formula P– Does M satisfy P?

• P is not necessarily a sentence, so have to handle assignments to free variables.

• Computability: yes if models are finite

Consistency Checking

• Definition: Given a formula P, is P consistent?

• Idea: consistent iff satisfiable in a model M, so task becomes discovering whether a model exists.

• This is a search problem.

• Computationally undecidable for arbitrary P.

Informativity Checking

• Definition: given P, is P informative or uninformative?

• Idea (which runs counter to logician's view)– informative = invalid– uninformative = valid (true in all possible models)

• Informativity: is genuinely new information being conveyed? Useful concept from PoV of communication

• Computability: validity worse than consistency checking since all models need to be checked for satisfiability.

Relations between Concepts

• P is consistent (satisfiable) iff –P is informative (invalid)

• P is inconsistent (unsatisfiable) iff –P uninformative (valid).

• P is informative (invalid) if –P is consistent (satisfiable).

• P is uninformative (valid) if –P is inconsistent (unsatisfiable).

Consistency within Discourse

Mia smokes.

Mia does not smoke.

• Should be possible to detect the inconsistency in such discourses

• To avoid detecting inconsistency in superficially similar discourses such as

Mia smokes.

Mia did not smoke

Consistency of Discourse w.r.t. Background Knowledge

Discourse:Mia is a beautiful woman.Mia is a tree

Background Knowledge:All women are humanAll trees are plants-Ex: human(x) and plant(x)

Consistency Checking for Resolving Scope Ambiguity

Every boxer has a broken nose

1. Ax(boxer(x) -Ey(broken-nose(y) & has(x,y)))

2. Ey(broken-nose(y) & Ax(boxer(x) → has(x,y)))

Second reading is inconsistent with world knowledge

• What world knowledge?• How represented and used?

Informativity Checking

• Make your contribution as informative as is required (for the current purposes of the exchange). H. P. Grice.Mia smokes.Mia smokes.Mia smokes

• Is not informative• Informativity checking also wrt background

knowledge

Informativity a `soft' signal

Mia is married

She has a husband

• Superficially uninformative wrt background knowledge.

• But nevertheless we can imagine contexts when such a discourse makes sense.

• Technically uninformative utterances can be used to “make a point”

Consistency Checking Task(CCT) in FOL

• Let Φ be the FOL semantic representation of the latest sentence in some ongoing discourse

• Suppose that the relevant lexical knowledge L, world knowledge W, natural language metaphysical assumption M, and the information from the previous discourse D has been represented in FOL

• CCT can be expressed:

L U W U M U D |= ¬Φ

To put it another way…

All-Our-Background-Stuff |= ¬Φ

hence

|= All-Our-Background-Stuff → ¬Φ

(Deduction Theorem)

Consequence: we can reduce CCT to deciding the validity of a single formula.

Informativity Checking Task(ICT) in FOL

• Let Φ be the FOL semantic representation of the latest sentence in some ongoing discourse

• Suppose that the relevant lexical knowledge L, world knowledge W, natural language metaphysical assumption M, and the information from the previous discourse D has been represented in FOL

• ICT can be expressed:

L U W U M U D |= Φ

To put it another way…

All-Our-Background-Stuff |= Φ

hence

|= All-Our-Background-Stu → Φ(Deduction Theorem)

Consequence: we can also reduce ICT to deciding the validity of a single formula.

Yes but …

• This definition is semantic, i.e. given in terms of models.

• This is very abstract, and

• defined in terms of all models.

• There are a lot of models, and most of them are very large.

• So is it of any computational interest whosoever?

Proof Theory

• Proof theory is the syntactic approach to logic.

• It attempts to define collections of rules and/or axioms that enable us to generate new formulas from old

• That is, it attempts to pin down the notion of inference syntactically.

• P |- Q versus P |= Q

Examples of Proof Systems

• Natural deduction• Hilbert-style system (often called axiomatic

systems)• Sequent calculus• Tableaux systems• Resolution• Some systems (notably tableau and

resolution) are particularly suitable for computational purposes.

Connecting Proof Theory toModel Theory

• Nothing we have said so far makes any connection between the proof theoretic and the model theoretic ideas previously introduced.

• We must insist on working with proof systems with two special properties

• Soundness

• Completeness.

Soundness

• Proof Theoretic Q is provable in proof theoretic system|- Q.

• Model Theoretic Q is valid in model theoretic system|= Q

• A PT system is sound iff|- Q implies |= Q

• Every theorem is valid

Remark on Soundness

• Soundness is typically an easy property to prove.

• Proofs typically have some kind of inductive structure.

• One shows that if the first part of proof is true in a model then the rules only let us generate formulas that are also true in a model.

• Proof follows by induction

Completeness

• Proof Theoretic Q is provable in proof theoretic system|- Q.

• Model Theoretic Q is valid in model theoretic system|= Q

• A PT system is sound iff|= Q implies |- Q

• Every valid formula is also a theorem

Remark on Completeness

• Completeness is a much deeper property that soundness,and is a lot more difficult to prove.

• It is typically proved by contraposition. We show that if some formula P is not provable then is not valid.

• This is done by building a model for ¬P• The 1st completeness proof for a 1st-order proof

system was given by Kurt Godel in his 1930 PhD thesis.

Sound and Complete Systems

• So if a proof system is both sound and complete (which is what we want) we have that:

|=Φ if and only if |-Φ

• That is, syntactic provability and semantic validity coincide.

• Sound and complete proof system, really capture the our semantic reality.

• Working with such systems is not just playing with symbols.

Blackburn’s Proposal

• Deciding validity (in 1st-order logic) is undecidable, i.e. no algorithm exists for solving 1st-order validity.

• Implementing our proof methods for 1st-order logic (that is, writing a theorem prover only gives us a semi-decision procedure.

• If a formulas is valid, the prover will be able to prove it, but if is not valid, the prover may never halt!

• Proposal – Implement theorem provers, – but also implement a partial converse tool: model builders.

Computational Tools

• Theorem prover: A tool that, when given a 1st-order formula Φ attempts to prove it.

• If Φ is in fact provable a (sound and complete) 1st-order prover can (in principle) prove it.

• Model builder: a tool that, when given a 1st-order formula Φ, attempts to build a model for it.

• It cannot (even in principle) always succeed in this task, but it can be very useful.

Theorem Provers and Model Checkers

• Theorem provers: a mature technology which provides a negative check on consistency and informativity

• Theorem provers can tell us when something is not consistent, or not informative.

• Model builders: a newer technology which provides a (partial) positive check on consistency and informativity

• That is, model builders can tell us when something is consistent or informative.

A Possible System

Let B be all our background knowledge, and Φ the representation of the latest sentence:

• Partial positive test for consistency: give MB B & Φ

• Partial positive test for informativity: give MB B & ¬Φ

• Negative test for consistency: give TP B → Φ

• Negative test for informativity: give TP B → ¬Φ

• And do this in parallel using the best available software!

Clever Use of Reasoning Tools(CURT)

• Baby Curt No inference capabilities• Rugrat Curt: negative consistency checks (naive prover)• Clever Curt: negative consistency checks (sophisticated

prover)• Sensitive Curt: negative and positive informativity checks• Scrupulous Curt: eliminating superfluuous readings• Knowledgeable Curt: adding background knowledge• Helpful Curt: question answering

Baby Curt computes semantic representations

Curt: 'Want to tell me something?'

> every boxer loves a woman

Curt: 'OK.'

> readings

1 forall A (boxer(A) > exists B (woman(B) & love(A, B)))

2 exists A (woman(A) & forall B (boxer(B) > love(B, A)))

Baby Curt accumulates information

> mia walks

Curt: 'OK.'

> vincent dances

Curt: 'OK.'

> readings

1 (walk(mia) & dance(vincent))

But Baby Curt is stupid

> mia walks

Curt: 'OK.'

> mia does not walk

Curt: 'OK.'

> ?- readings 1 (walk(mia) & - walk(mia))

Add Inference Component

• Key idea: use sophisticated theorem provers and model builders in parallel.

• The theorem prover provides negative check for consistency and informativity.

• The model builder provides positive check for consistency and informativity.

• The 1st to find a result, reports back, and stops the other

Example

> Vincent is a man

Message (consistency checking): mace found a result.

Curt: OK.

> ?- models

1 model([d1], [f(1, man, [d1]), f(0, vincent, d1)])

Example continued

> Mia likes every man.

Message (consistency checking): mace found a result.

Curt: OK.

> Mia does not like Vincent.

Message (consistency checking): bliksem found a

result.

Curt: No! I do not believe that!

Example 2

> ?- every car has a radioMessage (consistency checking): mace found a

result.Message (consistency checking): bliksem found aresult.Curt: 'OK.'> ?- readings1 forall A (car(A) > exists B (radio(B) & have(A,B)))

Issues

• Is a logic-based approach to feasible? How far can it be pushed?

• Is 1st-order logic essential?• Are there other interesting inference tasks?• Is any of this relevant to current trends in

computational linguistics, where shallow processing and statistical approaches rule?

• Are there applications?