Finite Model Building and Computational Semantics

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Finite Model Building and Computational Semantics

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Title: Finite Model Building and Computational Semantics

1
Finite Model Building and Computational Semantics
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...
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Johan Bos
University of Rome "La Sapienza
Dipartimento di Informatica

2
Aim of this talk

Give an introduction to model building
Show how model building can be used in
computational semantics
Convince you that model building is a very
interesting deduction technique

3
Models
4
Models
MltD,Fgt Dd1,d2 F(mia)d2 F(man)d1 F(woman)
d2 F(love)(d2,d1)
5
Models
MltD,Fgt Dd5
6
Models
MltD,Fgt Dd1,d2,d3,d4,d5, .. F(man)d1,d4,
d12 F(woman)d2,d3 F(car)d14,d13
F(love)(d2,d1), (d4,d4), . F(hate)(d5,d1)
, (d1,d4), .
7
Model Building

The task of constructing a model that satisfies a
particular first-order theory

8
Model Building

Also known as
Model Generation
Model Finding
Model Searching
Model Construction
Do not confuse with
Model Checking

9
Inference Tools (FOL)

Model Building
Useful for checking consistency and building a
discourse model
Model Checking
Useful for querying properties of the constructed
discourse model
Theorem Proving
Useful for drawing inferences, such as checking
for inconsistencies or uninformativity

10
Mathematicians vs. Linguists

Suppose we got a theory ?

11
Mathematicians vs. Linguists

Suppose we got a theory ?

12
Mathematicians vs. Linguists

Suppose we got a theory ?

13
Mathematicians vs. Linguists

Suppose we got a theory ?

14
Mathematicians vs. Linguists

Suppose we got a theory ?

15
Mathematicians vs. Linguists

Suppose we got a theory ?

16
Mathematicians vs. Linguists

Summing up
The mathematician thinks in terms of proofs and
counter-models
The linguist thinks in terms of models and
counter-proofs

17
Talk Outline

More about models and model building
Why models are useful in semantic interpretation
1 Linguistic interpretation
2 interface to application
3 robust inference
Limitations, Future, Conclusion

18
Finite Model Building

Model builders construct finite models for
logical theories
Minimal (no redundant information)
Flat (no recursion)
Deal naturally with quantification, disjunction,
conditionals, negation
Practical benefits
There are powerful off-the-shelf model builders
available
Model Checking tools available for complex queries

19
Off-the-shelf model builders

Model builders for First-order Logic
Finfimo Bry Torge
Finder Slaney
Sem Zhang
Kimba Konrad
Mace2 and Mace4 McCune
Paradox Claessen

20
Non-finite models

The following theory doesnt have a finite
modelperson(butch)?x(person(x) ? ?y(person(y)
parent(x,y)))?x?y?z(parent(x,y)parent(y,z)?par
ent(x,z))?x parent(x,x)
Everyone has a parent

21
Minimal Models

Models that only contain what is required to make
? true
The smallest models that satisfy ?
Small with respect to individuals
Small with respect to predicates

22
Domain-minimal model

Let M be a model (M ltD,Fgt) for a set of
first-order formulas ?
DefinitionM is an domain-minimal model for ?
if no model with a smaller domain exists that
satisfy ?

23
Example domain minimal

? ?x woman(x)
Example Models
Model 1 Model 2 Model
3
Model 3 is domain-minimal wrt ?,Models 2 and 3
are not

MltD,Fgt Dd1,d2F(woman)d1,d2
MltD,Fgt Dd1,d2F(woman)d2
MltD,Fgt Dd1F(woman)d1
24
Predicate-minimal model

Let M be a model (M ltD,Fgt) for a set of
first-order formulas ?, and P a predicate symbol
DefinitionM is a P-minimal model for ? if no
model exists that only differs from M in the
assignment of a smaller extension F(P) satisfying
M

25
Example predicate minimal

? ?x woman(x), Pwoman
Example Models
Model 1 Dd1,d2
F(woman)d1,d2
Model 2 Dd1,d2 F(woman)d2
Model 3 Dd1 F(woman)d1
Models 2 and 3 are P-minimal wrt ?, Model 1 is not

26
Example predicate minimal

? ?x woman(x), Pwoman
Example Models
Model 1 Model 2 Model
3
Models 2 and 3 are P-minimal wrt ?, Model 1 is not

MltD,Fgt Dd1,d2F(woman)d1,d2
MltD,Fgt Dd1,d2F(woman)d2
MltD,Fgt Dd1F(woman)d1
27
Local minimal models

Given a set of domain-minimal models satisfying a
theory ?, MltD,Fgt is locally minimal if there is
no other model M'ltD',F'gt such that F' ? F
See Gardent Konrad 2000

28
Minimal models in practice

Model builders normally generate models by
iteration over the domain size
As a side-effect, the output is a model with a
minimal domain size
Not always predicate minimal

29
Herbrand Models

Individual constants have a unique interpretation
Claimed to be more suitable for natural language
interpretion
E.g. Konrad, Gardent, Webber, Kohlhase
Not sure this is necessarily the case nicknames,
aliases, pseudonyms, etc.
However, might be more efficient

30
Mental Models

Cognitive Psychology
Human beings construct a model in the process of
interpreting language
Very similar to theorem proving vs. model
building
But incremental, non-monotonic
Johnson-Laird

31
Why models are good

Flat structures
No recursion
No implicit quantification
Easy and efficient to process
Something abstract made concrete
Able to quantify over objects

32
Model Building in Semantics

Three main uses 1 Linguistic Interpretation
2 Interface to Applications
3 Robust Inference

33
Model Building in Semantics

Three main uses 1 Linguistic Interpretation
2 Interface to Applications
3 Robust Inference

34
Linguistic Interpretation

When ambiguities arise, there are often clearly
preferred interpretations
Minimal models often correspond to preferred
interpretation
Type Coercion
Noun-noun compounds
Reciprocals
Definite descriptions
Accommodation

35
Type Coercion Gardent Webber 2001

Example 1
Mary enjoyed California.
Model

MltD,Fgt Dm,c,e1,e2F(enjoy) (e1,m,e2)
F(visit) (e2,m,c)
36
Type Coercion Gardent Webber 2001

Example 2
Mary read books about all Western states. She
enjoyed California.
Model 1
Model 2

MltD,Fgt Dm,c,e1,e2,e3, .F(visit)
(e2,m,c) F(enjoy) (e1,m,e2) F(read)
(e3,m,c)
MltD,Fgt Dm,c,e1,e2, .F(enjoy) (e1,m,e2)
F(read) (e2,m,c)
37
Type Coercion Gardent Webber 2001

Example 2
Mary read books about all Western states. She
enjoyed California.
Model 1
Model 2

MltD,Fgt Dm,c,e1,e2,e3, .F(visit)
(e2,m,c) F(enjoy) (e1,m,e2) F(read)
(e3,m,c)
MltD,Fgt Dm,c,e1,e2, .F(enjoy) (e1,m,e2)
F(read) (e2,m,c)
38
Noun-Noun Compounds Gardent Webber 2001

Example 1
The California student produced an excellent
report.
?x(Sx ?x(Cy FROMxy

39
Noun-Noun Compounds Gardent Webber 2001

Example 1
The California student produced an excellent
report.
?x(Sx ?x(Cy FROMxy
Example 2
Each student was assigned a state to study. The
California student produced an excellent report.
?x(Sx ?x(Cy ASSIGNEDxy

40
Reciprocals Gardent Konrad 2000

Examples
The students infected each other. ?x(Sx??y(Sy
x?y (Ixy ? Iyx)))
The students stared at each other. ?x(Sx??y(Sy
x?y Sxy))
The students like each other. ?x(Sx??y((Sy
x?y) ? Lxy))

41
Reciprocals Gardent Konrad 2000

Examples
The students infected each other. ?x(Sx??y((Sy
x?y) ? Lxy ? Lyx))
The students stared at each other. ?x(Sx??y((Sy
x?y) ? Lxy))
The students like each other. ?x(Sx??y((Sy
x?y) ? Lxy))
Introduce costs
?x(Sx??y((Sy x?y ? Lxy) ? Lxy))

42
Definite Descriptions

Van der Sandt binding preferred to accommodation
Binding leads to smaller models
Gardent Webber bridging preferred to
accommodation
Bridging leads to smaller models

43
Accommodation

Van der Sandt global accommodation is preferred
to local accommodation
However, local accommodation yields smaller models

44
Accommodation

Van der Sandt global accommodation is preferred
to local accommodation
However, local accommodation yields smaller
models
Example The king of France is not bald.
Global ?x (Kx ? Bx)
Local ? ?x (Kx Bx)

45
Accommodation

Van der Sandt global accommodation is preferred
to local accommodation
However, local accommodation yields smaller
models
Example The king of France is not bald.
Global ?x (Kx ? Bx)
Local ? ?x (Kx Bx)
Take world knowledge into account

46
Model Building in Semantics

Three main uses 1 Linguistic Interpretation
2 Interface to Applications
3 Robust Inference

47
Model Building in Semantics

Three main uses 1 Linguistic Interpretation
2 Interface to Applications
3 Robust Inference

48
Human-Computer Dialogue

Focus on small domains
Grammatical coverage ensured
Background knowledge already encoded
Spoken dialogue system
Godot the robot
Speech recognition and synthesis
People could give Godot directions, ask it
questions, tell it new information
Godot was a REAL robot

49
Godot the Robot
Godot with Tetsushi Oka
50
Spoken Dialogue System Architecture
Automatic Speech Recognition
Semantic Interpretation
Dialogue Manager
Inference
Speech Synthesis
Generation
Application Interface
Home Automation
Service Robots
51
Typical conversation with Godot

Simple dialogues in beginning
Human Robot?
Robot Yes?
Human Where are you?
Robot I am in the hallway.
Human OK. Go to the rest room!

52
Adding inference to Godot

We added semantics and inference
Using DRT
Inference tasks
Consistency checking
Informativity checking
Using FOL theorem provers and finite model
builders
Mace, Spass

53
Advanced conversation with Godot

Dealing with inconsistency and informativity
Human Robot, where are you?
Robot I am in the hallway.
Human You are in my office.
Robot No, that is not true.
Human You are in the hallway.
Robot Yes I know!

54
The Yin and Yang of Inference

Theorem Proving and Model Building function as
opposite forces
Assume ?, a logical formula, representing a
certain discourse ?
If a theorem prover succeeds in finding a proof
for ??, then ? is inconsistent
If a model builder succeeds to construct a model
for ?, then ? is consistent

55
Using minimal models

ExamplesTurn on a light.Turn on every
light.Turn on everything except the radio. Turn
off the red light or the blue light.Turn on
another light.

56
Using minimal models

Three Steps
Construct First-Order Theory
Build Model
Get relevant information from model

57
Step 1 Create First-Order Theory

Translate instruction into FOL
?w?s?k(possible-world(w) system(w,s)
kitchen(w,k) ?v?a(action(w,a,v)
?x(light(a,x) in(a,x,k) ? ?e(switch(a,e,s,x)
on(v,x)))))
Background Knowledge
?w?x(light(w,x) ? device(w,x))
?w?x(kitchen(w,x) ? ?device(w,x))
?w?v?x?y?z(switch(w,x,y,z)on(v,z)?poweron(w,z))
...
Situational Knowledge
?w(actual-world(w) light(w,c1) on(w,c1)
light(w,c2) off(w,c2) ...)

58
Step 2 Build First-Order Model

Give First-Order Theory to Model Builder
Model Builder will return minimal finite model

Example switch every light in the kitchen on
Use model checker
Get application primitives poweron(c1),poweron(c
2)

Dd1,d2,d3,d4,d5,d6,d7,d8 F(c1)d7 F(c2)d6 F(ac
tual_world)d1 F(possible_world)d1,d2,d3 F(sy
stem)(d1,d4),(d2,d4),(d3,d4) F(kitchen)(d1,d5
),(d2,d5),(d3,d5) F(action)(d1,d2,d3) F(light)
(d1,d6),(d2,d6),(d3,d6),(d1,d7),(d2,d7),(d3,d7)
F(in)(d1,d6,d5),(d2,d6,d5),(d3,d6,d5),(d1,d7,d5
),(d2,d7,d5),(d3,d7,d5) F(poweron)(d2,d6),(d2,d
7) F(poweroff) F(off)(d1,d6),(d1,d7) F(on)
(d3,d6),(d3,d7)
60
ASR
??
?
SEM
Theorem Prover
Model Builder
If proof found, discourse is inconsistent If
model found, use it for further actions
Model Checker
query
61
Videos of Godot
Video 1 Godot in the basement of Bucceuch Place
Video 2 Screenshot of dialogue manager with
DRSs and camera view of Godot
62
What can we say about

Demonstrated that FOL can play an interesting
role in dialogue systems
Also showed a new application of finite model
building
Limitations
Scalability, only small dialogues
Lack of incremental inference
Minimal models required

63
Model Building in Semantics

Three main uses 1 Linguistic Interpretation
2 Interface to Applications
3 Robust Inference

64
Model Building in Semantics

Three main uses 1 Linguistic Interpretation
2 Interface to Applications
3 Robust Inference

65
Robust Inference

We got excellent theorem provers at our disposal
But we need robust inference
Why theorem proving is bad for robust inference
How model building can help us to approximate
inferences

66
Theorem Provers

Theorem provers for first-order logic
developed over the last four decades
witnessed tremendous progress
We got at our disposal very efficient theorem
provers such as
OTTER
SPASS
BLIKSEM
VAMPIRE

67
We need robust inference for real applications

Even if are linguistic processing tools are
perfect, inferences are often made on the basis
of background knowledge
Appropriate background knowledge is notoriously
hard to compute
An example of a real application is the RTE
exercise
Recognising Textual Entailment

68
Example (Vampire proof)
RTE-2 489 (TRUE)
Initially, the Bundesbank opposed the
introduction of the euro but was compelled to
accept it in light of the political pressure of
the capitalist politicians who supportedits
introduction. ------------------------------------
----------------- The introduction of the euro
has been opposed.
?
69
Trouble with theorem proving

Theorem provers are extremely precise.
They wont tell you when there is almost a
proof.
Even if there is a little background knowledge
missing, even Vampire must give in and admit
dont know

70
Vampire no proof
RTE 1049 (TRUE)
Four Venezuelan firefighters who were traveling
to a training course in Texas were killed when
their sport utility vehicle drifted onto the
shoulder of a highway and struck a parked
truck. -------------------------------------------
--------------------- Four firefighters were
killed in a car accident.
?
71
Using Model Building

Basic Idea
Build models of T and TH
If the models are similar, then it is likely
that T entails H
Measure similarity between models
Define a threshold using machine learning
techniques

72
Model Similarity

When are two models similar?
Small difference in domain size
Small difference in predicate extensions
Quantify model similarity
Entailment Distance, ?

73
Semantic Distance 1/2

Suppose we want to measure the semantic distance
between p and q.
What do we do?
Give p to a model builder
Give pq to a model builder
Give ?q to a model builder
Give ?(qp) to a model builder
Assume that gives us four minimal models Mp,
Mpq, M?q, and M?(qp)

74
Semantic Distance 2/2

The Semantic Distance ? is computed as
follows ? (Mpq - Mp) (M?(qp) -
M?q)
The closer ? is to 0, the closer p and q are in
an entailment relation
? measures the amount of new information
introduced by q wrt p

75
Example 1

p John met Mary in Romeq John met Mary
Model p 3 entitiesModel pq
3 entitiesModel ?q 1 entityModel
?(qp) 1 entity
? (3-3)(1-1) 0
Prediction entailment

76
Example 2

p John met Mary q John met Mary in Rome
Model p 2 entitiesModel pq
3 entitiesModel ?q 1 entityModel
?(qp) 1 entity
? (3-2)(1-1) 1
Prediction almost entailment

77
Example 3

p John has no bike q John has no blue bike
Model p 1 predicateModel pq
1 predicate Model ?q 2
predicatesModel ?(qp) 2 predicates
? (1-1)(2-2) 0
Prediction entailment

78
Example 4

p John has no blue bike q John has no bike
Model p 1 predicateModel pq
1 predicate Model ?q 1
predicateModel ?(qp) 2 predicates
? (1-1)(2-1) 1
Prediction almost entailment

79
Model size differences