Reasoning with Dependency Structures and Lexicographic Definitions using Unit Graphs

undefined
Reasoning with Dependency Structures
and Lexicographic Definitions
using Unit Graphs
Maxime Lefrançois, Fabien Gandon
[ maxime.lefrancois | fabien.gandon ] @inria.fr
 
Depling’13, August 30
th
 2013, Prague
Knowledge Representation
•
answers recurrent needs
▫
represent
▫
manipulate
▫
query
▫
reason
▫
share
▫
...
•
here: applied to the linguistic domain
▫
Meaning-Text Theory
2
Lefrançois, Gandon, Reasoning  with Dependency  Structures and Lex. Definitions using Unit Graphs
 
3
Choose Formalism
Populate
Applications
t
Lefrançois, Gandon, Reasoning  with Dependency  Structures and Lex. Definitions using Unit Graphs
 
4
Choose Formalism
Populate
Applications
t
Lefrançois, Gandon, Reasoning  with Dependency  Structures and Lex. Definitions using Unit Graphs
5
Semantic Actant Slots (SemASlots)
= Participants of the linguistic situation denoted by L
that have a favoured position in sentences constructed with L
Linguistic Predicates
Need 
1
: Theory of Semantic Actants
(Mel’cuk, 2004)
Lefrançois, Gandon, Reasoning  with Dependency  Structures and Lex. Definitions using Unit Graphs
6
Semantic Actant Slots (SemASlots)
= Participants of the linguistic situation denoted by L
that have a favoured position in sentences constructed with L
Linguistic Predicates
Need 1: Theory of Semantic Actants
(Mel’cuk, 2004)
+ linguistic criterions
Lefrançois, Gandon, Reasoning  with Dependency  Structures and Lex. Definitions using Unit Graphs
7
Linguistic Predicates
Need 1: Theory of Semantic Actants
(Mel’cuk, 2004)
Lefrançois, Gandon, Reasoning  with Dependency  Structures and Lex. Definitions using Unit Graphs
(
to eat
)
(Paul ; eggs ; 
plate
)
logical
predicate
≠
 
8
Linguistic Predicates
Need 1: Theory of Semantic Actants
(Mel’cuk, 2004)
Lefrançois, Gandon, Reasoning  with Dependency  Structures and Lex. Definitions using Unit Graphs
(
outil
)
(Paul ; Untangle)
(
outil
)
(Paul ; Carpenter)
(
to eat
)
(Paul ; eggs ; 
plate
)
logical
predicate
≠
 
9
Linguistic Predicates
Need 1: Theory of Semantic Actants
(Mel’cuk, 2004)
Lefrançois, Gandon, Reasoning  with Dependency  Structures and Lex. Definitions using Unit Graphs
(
outil
)
(Paul ; Untangle)
(
outil
)
(Paul ; Carpenter)
(
to eat
)
(Paul ; eggs ; 
plate
)
logical
predicate
≠
 
Existing KR Formalisms
•
Semantic Web Formalisms
▫
RDF  - 
oriented labelled graphs
   
▫
OWL - 
Description Logics
▫
SPARQL - Query, ...
10
Lefrançois, Gandon, Reasoning  with Dependency  Structures and Lex. Definitions using Unit Graphs
Existing KR Formalisms
•
Semantic Web Formalisms
▫
RDF  - 
oriented labelled graphs
   
No Semantics
▫
OWL - 
Description Logics
   
Only binary relations
  
Reify ? -> No Semantics
11
Lefrançois, Gandon, Reasoning  with Dependency  Structures and Lex. Definitions using Unit Graphs
Existing KR Formalisms
•
Conceptual Graphs
▫
Sowa
Oriented labelled Graphs
▫
Sowa drew his inspiration from Tesnière
▫
Rules, reasoning,
 
(
Baget, Mugnier, Chein
, ...)
▫
Concepts and Relations definitions, 
(Sowa, Leclère, ...)
12
Lefrançois, Gandon, Reasoning  with Dependency  Structures and Lex. Definitions using Unit Graphs
Existing KR Formalisms
•
Conceptual Graphs
   
Alternation Concept – Relation
13
Lefrançois, Gandon, Reasoning  with Dependency  Structures and Lex. Definitions using Unit Graphs
14
t
The Unit Graphs Formalism
•
a graph-based formalism,
•
to represent linguistic units
Choose Formalism
Populate
Applications
Lefrançois, Gandon, Reasoning  with Dependency  Structures and Lex. Definitions using Unit Graphs
15
t
Draw inspiration from GC
and
Develop a RDF syntax
to exchange knowledge
The Unit Graphs Formalism
Choose Formalism
Populate
Applications
Lefrançois, Gandon, Reasoning  with Dependency  Structures and Lex. Definitions using Unit Graphs
16
t
The Unit Graphs Formalism
Choose Formalism
Populate
Applications
Lefrançois, Gandon, Reasoning  with Dependency  Structures and Lex. Definitions using Unit Graphs
Draw inspiration from GC
and
Develop a RDF syntax
to exchange knowledge
17
t
The Unit Graphs Formalism
What reasoning capabilities ?
Choose Formalism
Populate
Applications
Lefrançois, Gandon, Reasoning  with Dependency  Structures and Lex. Definitions using Unit Graphs
1. Choose Formalism
18
•
Needs, problems
•
Existing Knowledge Representation Formalisms
•
The Unit Graphs formalism
▫
Unit Graphs
▫
Rules and Definitions
•
Reasoning with Unit Graphs
▫
Logical Semantics
▫
Entailment problem
Lefrançois, Gandon, The Unit Graphs Framework: Linguistic Knowledge Representation
19
Units 
–
 Representations
(c.f., Mel’čuk, 2004)
Unit Types 
– Lexicon
Lefrançois, Gandon, Reasoning  with Dependency  Structures and Lex. Definitions using Unit Graphs
20
Unit Types
Lefrançois, Gandon, Reasoning  with Dependency  Structures and Lex. Definitions using Unit Graphs
Specify How their units are to be linked
to other units in Unit Graphs
through 
Actantial Structure
:
Actant Slots (ASlots)
Signatures
 
 
21
Unit Graphs
are defined over a Support
Lefrançois, Gandon, Reasoning  with Dependency  Structures and Lex. Definitions using Unit Graphs
22
Unit Graphs
are defined over a Support
Lefrançois, Gandon, Reasoning  with Dependency  Structures and Lex. Definitions using Unit Graphs
▫
Primitive Unit Types (PUTs)
    Lexical unit type ANIMAL
Grammatical unit type 
Verb
, 
Noun
, 
plur
Surface Semantic unit type 
(
animal
)
 
▫
C
o
n
j
u
n
c
t
i
v
e
 
U
n
i
t
 
T
y
p
e
s
 
(
P
U
T
s
)
A
 
U
n
i
t
 
m
a
y
 
c
o
n
s
i
s
t
 
o
f
 
s
e
v
e
r
a
l
 
c
o
n
j
o
i
n
t
 
P
U
T
s
   { 
def
, 
plur
, ANIMAL } (
(
the animals
)
)
23
Unit Graphs
are defined over a Support
Lefrançois, Gandon, Reasoning  with Dependency  Structures and Lex. Definitions using Unit Graphs
▫
Primitive Unit Types (PUTs)
    Lexical unit type ANIMAL
Grammatical unit type 
Verb
, 
Noun
, 
plur
Surface Semantic unit type 
(
animal
)
 
Are pre-ordered
 
plur     pluralizable
       
 
/
animal
\         /
dog
\
24
Unit Graphs
are defined over a Support
Lefrançois, Gandon, Reasoning  with Dependency  Structures and Lex. Definitions using Unit Graphs
Unit Types specify how their units
are to be linked through
Actantial Structure
Actant Slots (ASlots)
Signatures
 
 
25
Unit Graphs
are defined over a Support
Lefrançois, Gandon, Reasoning  with Dependency  Structures and Lex. Definitions using Unit Graphs
•
Actant Slots
▫
Have Symbols
    
 
Deep Semantics: lexicalized semantic roles
    Surface Semantics: Numbers
 
Deep Syntax: Roman numerals
▫
May be Obligatory,
 Optional, Prohibited
 
  
26
Organization of the Unit Types Hierarchy 
with respect to a unique ASymbol s
The complete Unit Types Hierarchy 
is an intricated superposition of such figures
Lefrançois, Gandon, Reasoning  with Dependency  Structures and Lex. Definitions using Unit Graphs
27
Unit Graphs
are defined over a Support
Lefrançois, Gandon, Reasoning  with Dependency  Structures and Lex. Definitions using Unit Graphs
•
Signatures
▫
denote the type of units
that fill an ASlot 
s
 of a unit of a type 
t
28
As we go down the hierarchy,
 
signatures may only be specialized
Lefrançois, Gandon, Reasoning  with Dependency  Structures and Lex. Definitions using Unit Graphs
29
Unit Graphs
are defined over a Support
Lefrançois, Gandon, Reasoning  with Dependency  Structures and Lex. Definitions using Unit Graphs
•
Circumstantial Symbols:
•
Deep and Surface Semantics: 
none
•
Deep Syntax: 
ATTR, COORD, APPEND
•
...
•
Are organized in hierarchy
•
ex: 
ATTR       
deep syntactic CSymbol
•
Are signed
30
Unit Graphs
are defined over a Support
Lefrançois, Gandon, Reasoning  with Dependency  Structures and Lex. Definitions using Unit Graphs
31
Unit Graphs
are defined over a Support
Unit nodes
Unit nodes labels : a type + a marker
Actantial triples
Circumstantial triples
Declared equivalences of unit nodes
Lefrançois, Gandon, Reasoning  with Dependency  Structures and Lex. Definitions using Unit Graphs
32
Unit Graphs
are defined over a Support
Lefrançois, Gandon, Reasoning  with Dependency  Structures and Lex. Definitions using Unit Graphs
33
Unit Graphs
are defined over a Support
Lefrançois, Gandon, Reasoning  with Dependency  Structures and Lex. Definitions using Unit Graphs
1. Choose Formalism
34
•
Needs, problems
•
Existing Knowledge Representation Formalisms
•
The Unit Graphs formalism
▫
Unit Graphs
▫
Rules and Definitions
•
Reasoning with Unit Graphs
▫
Logical Semantics
▫
Entailment problem
Lefrançois, Gandon, The Unit Graphs Framework: Linguistic Knowledge Representation
35
Lefrançois, Gandon, Reasoning  with Dependency  Structures and Lex. Definitions using Unit Graphs
Unit Graphs Homomorphism
•
« Is UG 
G
 contained in UG 
H
 ? »
•
Oriented Labelled Graphs Homomorphism
▫
arcs must be preserved
▫
labels may get specialized
36
Lefrançois, Gandon, Reasoning  with Dependency  Structures and Lex. Definitions using Unit Graphs
Unit Graphs Homomorphism
37
Lefrançois, Gandon, Reasoning  with Dependency  Structures and Lex. Definitions using Unit Graphs
Unit Graphs Rules
•
A Hypothesis 
H
•
A Conclusion 
G
•
A partial mapping from nodes of 
H
 to nodes of 
C
38
Lefrançois, Gandon, Reasoning  with Dependency  Structures and Lex. Definitions using Unit Graphs
Unit Graphs Rules
•
A rule 
R
 is Applicable to 
G
iif there is a homomorphism 
H
->
G
•
Apply rule 
R
: add 
C
 to 
G
 and merge nodes
39
Lefrançois, Gandon, Reasoning  with Dependency  Structures and Lex. Definitions using Unit Graphs
PUT Definitions
1. Choose Formalism
40
•
Needs, problems
•
Existing Knowledge Representation Formalisms
•
The Unit Graphs formalism
▫
Unit Graphs
▫
Rules and Definitions
•
Reasoning with Unit Graphs
▫
Logical Semantics
▫
Entailment problem
Lefrançois, Gandon, The Unit Graphs Framework: Linguistic Knowledge Representation
41
Lefrançois, Gandon, Reasoning  with Dependency  Structures and Lex. Definitions using Unit Graphs
Reasoning with Unit Graphs
42
Lefrançois, Gandon, Reasoning  with Dependency  Structures and Lex. Definitions using Unit Graphs
Logical Semantics of UGs: Closure
43
Lefrançois, Gandon, Reasoning  with Dependency  Structures and Lex. Definitions using Unit Graphs
Logical Semantics of 
UGs: Closure
44
Lefrançois, Gandon, Reasoning  with Dependency  Structures and Lex. Definitions using Unit Graphs
Logical Semantics of 
UGs: Closure
45
Lefrançois, Gandon, Reasoning  with Dependency  Structures and Lex. Definitions using Unit Graphs
Logical Semantics of 
UGs: Closure
46
Lefrançois, Gandon, Reasoning  with Dependency  Structures and Lex. Definitions using Unit Graphs
Logical Semantics of 
UGs: Closure
47
Lefrançois, Gandon, Reasoning  with Dependency  Structures and Lex. Definitions using Unit Graphs
Logical Semantics of 
UGs: Closure
48
Lefrançois, Gandon, Reasoning  with Dependency  Structures and Lex. Definitions using Unit Graphs
Logical Semantics of 
UGs: Closure
49
Lefrançois, Gandon, Reasoning  with Dependency  Structures and Lex. Definitions using Unit Graphs
Logical Semantics of 
UGs: Closure
50
Lefrançois, Gandon, Reasoning  with Dependency  Structures and Lex. Definitions using Unit Graphs
Logical Semantics of 
UGs: Closure
51
Lefrançois, Gandon, Reasoning  with Dependency  Structures and Lex. Definitions using Unit Graphs
Logical Semantics of 
UGs: Closure
1. Choose Formalism
52
•
Needs, problems
•
Existing Knowledge Representation Formalisms
•
The Unit Graphs formalism
▫
Unit Graphs
▫
Rules and Definitions
•
Reasoning with Unit Graphs
▫
Logical Semantics
▫
Entailment problem
Lefrançois, Gandon, The Unit Graphs Framework: Linguistic Knowledge Representation
53
Lefrançois, Gandon, Reasoning  with Dependency  Structures and Lex. Definitions using Unit Graphs
Reasoning with Unit Graphs
G
 entails 
H
if and only if
there is a homomorphism from 
H
 to 
cl
(
G
)
54
Lefrançois, Gandon, Reasoning  with Dependency  Structures and Lex. Definitions using Unit Graphs
Future Work
•
Problem: risk of infinite closure
•
need to find conditions :
▫
on the hierarchy of Unit Types
▫
on the set of definitions
Conclusions
55
•
Linguistic Knowledge Representation
•
The Unit Graphs Formalism
▫
Hierarchy of Unit Types
▫
Hierarchy of Circumstantial Symbols
▫
Unit Graphs
▫
Rules and Definitions
•
Reasoning with UGs
▫
Logical Semantics: Unit Graph Closure
▫
Entailment equivalent to the homomorphism problem
Lefrançois, Gandon, Reasoning  with Dependency  Structures and Lex. Definitions using Unit Graphs
undefined
Reasoning with Dependency Structures
and Lexicographic Definitions
using Unit Graphs
 
Depling’13, August 30
th
 2013, Prague
Thank you

								

								

								

										

												
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This piece discusses the application of knowledge representation in addressing recurrent needs related to manipulating, querying, reasoning, and sharing information, particularly in the linguistic domain of Meaning-Text Theory. Maxime Lefrançois and Fabien Gandon explore formalisms, such as the theory of semantic actants and linguistic predicates, to represent participants and semantic roles in linguistic situations. The use of unit graphs provides a structured approach to reasoning with dependency structures and lexicons to enhance understanding and analysis.


								
    
																		
  •   Knowledge representation
    • 
																	
  •    Dependency structures
    • 
																	
  •    Lexicographic definitions
    • 
																	
  •    Unit graphs
    • 
																	
  •    Meaning-Text Theory
    • 
																

								

								

									

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Presentation Transcript

								

								
  1. Depling13, August 30th 2013, Prague Reasoning with Dependency Structures  and Lexicographic Definitions using Unit Graphs Maxime Lefran  ois, Fabien Gandon [ maxime.lefrancois | fabien.gandon ] @inria.fr
    2. 
            
            
            
  2. Lefranois, Gandon, Reasoning  with Dependency  Structures and Lex. Definitions using Unit Graphs 2 Knowledge Representation     answers recurrent needs     represent     manipulate     query     reason     share     ...     here: applied to the linguistic domain     Meaning-Text Theory
    3. 
            
            
            
  3. Lefranois, Gandon, Reasoning  with Dependency  Structures and Lex. Definitions using Unit Graphs 3 Choose Formalism Populate Applications t
    4. 
            
            
            
  4. Lefranois, Gandon, Reasoning  with Dependency  Structures and Lex. Definitions using Unit Graphs 4 Choose Formalism Populate Applications t 1. Choose Formalism
    5. 
            
            
            
  5. Lefranois, Gandon, Reasoning  with Dependency  Structures and Lex. Definitions using Unit Graphs 5 Need 1: Theory of Semantic Actants (Mel   cuk, 2004) Linguistic Predicates Semantic Actant Slots (SemASlots) = Participants of the linguistic situation denoted by L that have a favoured position in sentences constructed with L
    6. 
            
            
            
  6. Lefranois, Gandon, Reasoning  with Dependency  Structures and Lex. Definitions using Unit Graphs 6 Need 1: Theory of Semantic Actants (Mel   cuk, 2004) Linguistic Predicates Semantic Actant Slots (SemASlots) = Participants of the linguistic situation denoted by L that have a favoured position in sentences constructed with L
    7. 
            
            
            
  7. Lefranois, Gandon, Reasoning  with Dependency  Structures and Lex. Definitions using Unit Graphs 7 Need 1: Theory of Semantic Actants (Mel   cuk, 2004) logical predicate     Linguistic Predicates optional (to eat)(Paul ; eggs ; plate)
    8. 
            
            
            
  8. Lefranois, Gandon, Reasoning  with Dependency  Structures and Lex. Definitions using Unit Graphs 8 Need 1: Theory of Semantic Actants (Mel   cuk, 2004) logical predicate     Linguistic Predicates optional (to eat)(Paul ; eggs ; plate) split (outil)(Paul ; Untangle) (outil)(Paul ; Carpenter)
    9. 
            
            
            
  9. Lefranois, Gandon, Reasoning  with Dependency  Structures and Lex. Definitions using Unit Graphs 9 Need 1: Theory of Semantic Actants (Mel   cuk, 2004) logical predicate     Linguistic Predicates optional (to eat)(Paul ; eggs ; plate) split (outil)(Paul ; Untangle) (outil)(Paul ; Carpenter) an actant may be a predicate
    10. 
            
            
            
  10. Lefranois, Gandon, Reasoning  with Dependency  Structures and Lex. Definitions using Unit Graphs 10 Existing KR Formalisms     Semantic Web Formalisms     RDF  - oriented labelled graphs      OWL - Description Logics     SPARQL - Query, ...
    11. 
            
            
            
  11. Lefranois, Gandon, Reasoning  with Dependency  Structures and Lex. Definitions using Unit Graphs 11 Existing KR Formalisms     Semantic Web Formalisms     RDF  - oriented labelled graphs  No Semantics     OWL - Description Logics Only binary relations Reify ? -> No Semantics
    12. 
            
            
            
  12. Lefranois, Gandon, Reasoning  with Dependency  Structures and Lex. Definitions using Unit Graphs 12 Existing KR Formalisms     Conceptual Graphs     SowaOriented labelled Graphs     Sowa drew his inspiration from Tesni  re     Rules, reasoning,(Baget, Mugnier, Chein, ...)     Concepts and Relations definitions, (Sowa, Lecl  re, ...)
    13. 
            
            
            
  13. Lefranois, Gandon, Reasoning  with Dependency  Structures and Lex. Definitions using Unit Graphs 13 Existing KR Formalisms     Conceptual Graphs Alternation Concept     Relation 
    14. 
            
            
            
  14. Lefranois, Gandon, Reasoning  with Dependency  Structures and Lex. Definitions using Unit Graphs 14 Choose Formalism Populate Applications t The Unit Graphs Formalism     a graph-based formalism,     to represent linguistic units
    15. 
            
            
            
  15. Lefranois, Gandon, Reasoning  with Dependency  Structures and Lex. Definitions using Unit Graphs 15 Choose Formalism Populate Applications t The Unit Graphs Formalism Draw inspiration from GC and Develop a RDF syntax to exchange knowledge
    16. 
            
            
            
  16. Lefranois, Gandon, Reasoning  with Dependency  Structures and Lex. Definitions using Unit Graphs 16 Choose Formalism Populate Applications t The Unit Graphs Formalism Draw inspiration from GC and Develop a RDF syntax to exchange knowledge
    17. 
            
            
            
  17. Lefranois, Gandon, Reasoning  with Dependency  Structures and Lex. Definitions using Unit Graphs 17 Choose Formalism Populate Applications t The Unit Graphs Formalism What reasoning capabilities ?
    18. 
            
            
            
  18. Lefranois, Gandon, The Unit Graphs Framework: Linguistic Knowledge Representation 18 1. Choose Formalism     Needs, problems     Existing Knowledge Representation Formalisms     The Unit Graphs formalism     Unit Graphs     Rules and Definitions     Reasoning with Unit Graphs     Logical Semantics     Entailment problem
    19. 
            
            
            
  19. Lefranois, Gandon, Reasoning  with Dependency  Structures and Lex. Definitions using Unit Graphs 19 Unit Types     Lexicon Units     Representations  (c.f., Mel     uk, 2004)
    20. 
            
            
            
  20. Lefranois, Gandon, Reasoning  with Dependency  Structures and Lex. Definitions using Unit Graphs 20 Unit Types Specify How their units are to be linked to other units in Unit Graphs through Actantial Structure:    Actant Slots (ASlots)    Signatures
    21. 
            
            
            
  21. Lefranois, Gandon, Reasoning  with Dependency  Structures and Lex. Definitions using Unit Graphs 21 Unit Graphs are defined over a Support
    22. 
            
            
            
  22. Lefranois, Gandon, Reasoning  with Dependency  Structures and Lex. Definitions using Unit Graphs 22 Unit Graphs are defined over a Support Hierarchy of Unit Types     Primitive Unit Types (PUTs) Lexical unit type ANIMAL Grammatical unit type Verb, Noun, plur Surface Semantic unit type (animal)     Conjunctive Unit Types (PUTs) A Unit may consist of several conjoint PUTs  { def, plur, ANIMAL } ((the animals))
    23. 
            
            
            
  23. Lefranois, Gandon, Reasoning  with Dependency  Structures and Lex. Definitions using Unit Graphs 23 Unit Graphs are defined over a Support Hierarchy of Unit Types     Primitive Unit Types (PUTs) Lexical unit type ANIMAL Grammatical unit type Verb, Noun, plur Surface Semantic unit type (animal) Are pre-ordered plur /animal\ pluralizable /dog\
    24. 
            
            
            
  24. Lefranois, Gandon, Reasoning  with Dependency  Structures and Lex. Definitions using Unit Graphs 24 Unit Graphs are defined over a Support Hierarchy of Unit Types Unit Types specify how their units  are to be linked through  Actantial Structure    Actant Slots (ASlots)    Signatures
    25. 
            
            
            
  25. Lefranois, Gandon, Reasoning  with Dependency  Structures and Lex. Definitions using Unit Graphs 25 Unit Graphs are defined over a Support Hierarchy of Unit Types     Actant Slots      Have Symbols Deep Semantics: lexicalized semantic roles Surface Semantics: Numbers Deep Syntax: Roman numerals     May be Obligatory, Optional, Prohibited
    26. 
            
            
            
  26. Lefranois, Gandon, Reasoning  with Dependency  Structures and Lex. Definitions using Unit Graphs 26 radix obligat prohibet absurd     Organization of the Unit Types Hierarchy  with respect to a unique ASymbol s The complete Unit Types Hierarchy  is an intricated superposition of such figures
    27. 
            
            
            
  27. Lefranois, Gandon, Reasoning  with Dependency  Structures and Lex. Definitions using Unit Graphs 27 Unit Graphs are defined over a Support Hierarchy of Unit Types     Signatures      denote the type of units  that fill an ASlot s of a unit of a type t
    28. 
            
            
            
  28. Lefranois, Gandon, Reasoning  with Dependency  Structures and Lex. Definitions using Unit Graphs 28 As we go down the hierarchy, signatures may only be specialized
    29. 
            
            
            
  29. Lefranois, Gandon, Reasoning  with Dependency  Structures and Lex. Definitions using Unit Graphs 29 Unit Graphs are defined over a Support Hierarchy of Circumstantial Symbols     Circumstantial Symbols:     Deep and Surface Semantics: none     Deep Syntax: ATTR, COORD, APPEND     ...     Are organized in hierarchy     ex: ATTR       deep syntactic CSymbol     Are signed
    30. 
            
            
            
  30. Lefranois, Gandon, Reasoning  with Dependency  Structures and Lex. Definitions using Unit Graphs 30 Unit Graphs are defined over a Support Unit Node Markers Arbitrary Symbols Every Element of M identifies a specific unit; Multiple elements of M may identify the same unit.
    31. 
            
            
            
  31. Lefranois, Gandon, Reasoning  with Dependency  Structures and Lex. Definitions using Unit Graphs 31 Unit Graphs are defined over a Support    Unit nodes    Unit nodes labels : a type + a marker    Actantial triples    Circumstantial triples    Declared equivalences of unit nodes
    32. 
            
            
            
  32. Lefranois, Gandon, Reasoning  with Dependency  Structures and Lex. Definitions using Unit Graphs 32 Unit Graphs are defined over a Support
    33. 
            
            
            
  33. Lefranois, Gandon, Reasoning  with Dependency  Structures and Lex. Definitions using Unit Graphs 33 Unit Graphs are defined over a Support
    34. 
            
            
            
  34. Lefranois, Gandon, The Unit Graphs Framework: Linguistic Knowledge Representation 34 1. Choose Formalism     Needs, problems     Existing Knowledge Representation Formalisms     The Unit Graphs formalism     Unit Graphs     Rules and Definitions     Reasoning with Unit Graphs     Logical Semantics     Entailment problem
    35. 
            
            
            
  35. Lefranois, Gandon, Reasoning  with Dependency  Structures and Lex. Definitions using Unit Graphs 35 Unit Graphs Homomorphism        Is UG G contained in UG H ?        Oriented Labelled Graphs Homomorphism     arcs must be preserved      labels may get specialized 
    36. 
            
            
            
  36. Lefranois, Gandon, Reasoning  with Dependency  Structures and Lex. Definitions using Unit Graphs 36 Unit Graphs Homomorphism
    37. 
            
            
            
  37. Lefranois, Gandon, Reasoning  with Dependency  Structures and Lex. Definitions using Unit Graphs 37 Unit Graphs Rules     A Hypothesis H     A Conclusion G     A partial mapping from nodes of H to nodes of C
    38. 
            
            
            
  38. Lefranois, Gandon, Reasoning  with Dependency  Structures and Lex. Definitions using Unit Graphs 38 Unit Graphs Rules     A rule R is Applicable to G iif there is a homomorphism H->G     Apply rule R: add C to G and merge nodes  
    39. 
            
            
            
  39. Lefranois, Gandon, Reasoning  with Dependency  Structures and Lex. Definitions using Unit Graphs 39 PUT Definitions
    40. 
            
            
            
  40. Lefranois, Gandon, The Unit Graphs Framework: Linguistic Knowledge Representation 40 1. Choose Formalism     Needs, problems     Existing Knowledge Representation Formalisms     The Unit Graphs formalism     Unit Graphs     Rules and Definitions     Reasoning with Unit Graphs     Logical Semantics     Entailment problem
    41. 
            
            
            
  41. Lefranois, Gandon, Reasoning  with Dependency  Structures and Lex. Definitions using Unit Graphs 41 Reasoning with Unit Graphs
    42. 
            
            
            
  42. Lefranois, Gandon, Reasoning  with Dependency  Structures and Lex. Definitions using Unit Graphs 42 Logical Semantics of UGs: Closure
    43. 
            
            
            
  43. Lefranois, Gandon, Reasoning  with Dependency  Structures and Lex. Definitions using Unit Graphs 43 Logical Semantics of UGs: Closure
    44. 
            
            
            
  44. Lefranois, Gandon, Reasoning  with Dependency  Structures and Lex. Definitions using Unit Graphs 44 Logical Semantics of UGs: Closure
    45. 
            
            
            
  45. Lefranois, Gandon, Reasoning  with Dependency  Structures and Lex. Definitions using Unit Graphs 45 Logical Semantics of UGs: Closure
    46. 
            
            
            
  46. Lefranois, Gandon, Reasoning  with Dependency  Structures and Lex. Definitions using Unit Graphs 46 Logical Semantics of UGs: Closure
    47. 
            
            
            
  47. Lefranois, Gandon, Reasoning  with Dependency  Structures and Lex. Definitions using Unit Graphs 47 Logical Semantics of UGs: Closure
    48. 
            
            
            
  48. Lefranois, Gandon, Reasoning  with Dependency  Structures and Lex. Definitions using Unit Graphs 48 Logical Semantics of UGs: Closure
    49. 
            
            
            
  49. Lefranois, Gandon, Reasoning  with Dependency  Structures and Lex. Definitions using Unit Graphs 49 Logical Semantics of UGs: Closure
    50. 
            
            
            
  50. Lefranois, Gandon, Reasoning  with Dependency  Structures and Lex. Definitions using Unit Graphs 50 Logical Semantics of UGs: Closure
    51. 
            
            
            
							

							

					

					
					
					

						
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