Download Inference Tasks and Computational Semantics

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
INFERENCE
TASKS
QUERYING
CONSISTENCY
CHECKING
INFORMATIVITY
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 human
All trees are plants
-Ex: human(x) and plant(x)
Consistency Checking for
Resolving Scope Ambiguity
Every boxer has a broken nose
1.
2.
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?
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 to
Model 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 semidecision 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 1storder 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 1storder 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 radio
Message (consistency checking): mace found a
result.
Message (consistency checking): bliksem found a
result.
Curt: 'OK.'
> ?- readings
1 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?