Inference Tasks and Computational Semantics

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Inference Tasks and Computational Semantics
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Key Concepts

• Inference tasks
• Syntactic versus semantic approach to logic
• Soundness completeness
• Decidability and undecidability
• Technologies
• Theorem proving versus model building

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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

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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.

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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.

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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).

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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

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Consistency of Discourse w.r.t. Background
Knowledge

• Discourse
• Mia is a beautiful woman.
• Mia is a tree
• Background Knowledge
• All women are human
• All trees are plants
• -Ex human(x) and plant(x)

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Consistency Checking for Resolving Scope Ambiguity

• Every boxer has a broken nose
• Ax(boxer(x) -Ey(broken-nose(y) has(x,y)))
• Ey(broken-nose(y) Ax(boxer(x) ? has(x,y)))
• Second reading is inconsistent with world
  knowledge
• What world knowledge?
• How represented and used?

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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

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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

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Consistency Checking Task(CCT) in FOL

• Let F 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 F

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To put it another way

• All-Our-Background-Stuff F
• hence
• All-Our-Background-Stuff ? F
• (Deduction Theorem)
• Consequence we can reduce CCT to deciding the
  validity of a single formula.

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Informativity Checking Task(ICT) in FOL

• Let F 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 F

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To put it another way

• All-Our-Background-Stuff F
• hence
• All-Our-Background-Stu ? F
• (Deduction Theorem)
• Consequence we can also reduce ICT to deciding
  the validity of a single formula.

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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?

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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

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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.

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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.

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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

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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

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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

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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.

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Sound and Complete Systems

• So if a proof system is both sound and complete
  (which is what we want) we have that
• F if and only if -F
• 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.

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Blackburns 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.

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Computational Tools

• Theorem prover A tool that, when given a
  1st-order formula F attempts to prove it.
• If F 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 F, attempts to build a model
  for it.
• It cannot (even in principle) always succeed in
  this task, but it can be very useful.

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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.

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A Possible System

• Let B be all our background knowledge, and F
  the representation of the latest sentence
• Partial positive test for consistency give MB B
  F
• Partial positive test for informativity give MB
  B F
• Negative test for consistency give TP B ? F
• Negative test for informativity give TP B ? F
• And do this in parallel using the best available
  software!

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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

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Baby Curt computes semantic representations

• Curt 'Want to tell me something?'
• gt every boxer loves a woman
• Curt 'OK.'
• gt readings
• 1 forall A (boxer(A) gt exists B (woman(B)
  love(A, B)))
• 2 exists A (woman(A) forall B (boxer(B) gt
  love(B, A)))

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Baby Curt accumulates information

• gt mia walks
• Curt 'OK.'
• gt vincent dances
• Curt 'OK.'
• gt readings
• 1 (walk(mia) dance(vincent))

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But Baby Curt is stupid

• gt mia walks
• Curt 'OK.'
• gt mia does not walk
• Curt 'OK.'
• gt ?- readings 1 (walk(mia) - walk(mia))

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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

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Example

• gt Vincent is a man
• Message (consistency checking) mace found a
  result.
• Curt OK.
• gt ?- models
• 1 model(d1, f(1, man, d1), f(0, vincent,
  d1))

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Example continued

• gt Mia likes every man.
• Message (consistency checking) mace found a
  result.
• Curt OK.
• gt Mia does not like Vincent.
• Message (consistency checking) bliksem found a
• result.
• Curt No! I do not believe that!

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Example 2

• gt ?- every car has a radio
• Message (consistency checking) mace found a
  result.
• Message (consistency checking) bliksem found a
• result.
• Curt 'OK.'
• gt ?- readings
• 1 forall A (car(A) gt exists B (radio(B) have(A,
• B)))

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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?