INVERSION FUNCTION OF MDS FOR SENTENCES ANALYSIS
Erqing Xu
Department of Mathematics, Cornell University, Ithaca, New York 14853, U.S.A.
Keywords: Natural language understanding, MDS, Inversion function, Proof tree.
Abstract: Traditional sentence analysis refers to finding the sentence structure for a given sentence. A question
different from this is: given a sentence Curry-Horwad isomorphic with a type, can we establish the proof
tree representing the sentence? Therefore, this paper combines the extensional Kripke interpretation and
MDS (Minimalist Deductive System); derives the Kripke model of MDS; provides the applicable inversion
function such that we are able to obtain the proof tree of typed λ-terms which represents sentence structure;
and demonstrates that the product-free proof trees obtained with inversion function of MDS enjoy the
property of Church-Rosser equality. Application examples demonstrate that our work is valid. The main
difference between our work and traditional sentence analysis approach is that the objects of analysis are
different. The object of our work is: Kripke model of MDS and type of sentence satisfied by assignment.
But the object of traditional sentence analysis approach is sentence. This paper enlarges the range of
application of sentence analysis, improves sentence analysis approach, enhances natural language
understanding, and thus is meaningful. Our work has not been seen in literature.
1 INTRODUCTION
In natural language understanding, parsing as logic
deduction has become one of the hot topics of
research. Minimalist Deductive System is a late
approach (Lecomte, 2004). In MDS calculus, a
sentence is Curry-Horwad isomorphic with a type.
The feature of sentence analysis with MDS is that
the establishment of proof tree is type-driven. Then
we may naturally have the question: for a given type
of sentence, can we establish the proof tree
representing the sentence? This question is
meaningful for the improvement of sentence
analysis and natural language understanding.
Coquand (2002) forwards inversion function of
simple type λ-calculus. This inversion function is
able to return typed λ-terms according to the given
type. However, inversion function relies on specific
Kripke model. The Kripke model of MDS has not
been seen. Therefore, in order to obtain the inversion
function of MDS, first we have to obtain the Kripke
model of MDS. Now we already have Kripke model
of intuitionnistic logic, and MDS is a fragment of
partially commutative linear logic. Since the
difference between linear logic and intuitionistic
logic is the absence of contraction and weakening
(Morrill, 1994), it is hopeful that Kripke model of
intuitionnistic logic becomes the Kripke model of
MDS.
The work of this paper is: 1. combining the
extensional Kripke interpretation and MDS to derive
the Kripke model of MDS; providing the applicable
inversion function for MDS calculus of types. 2.
forwarding the method of representing the result of
inversion function, i.e. typed λ-terms as a proof tree.
3. demonstrating product-free proofs obtained by
inversion function enjoys the property of strong
normalization. For MDS, the above-mentioned work
has not been seen in literature.
Comparison between the work of this paper and
related work is as follows:
The main difference between our work and
traditional sentence analysis approach is that the
objects of analysis are different. The object of our
work is: Kripke model of MDS and type of sentence
satisfied by assignment. But the object of traditional
sentence analysis approach is sentence.
The difference between our work and inversion
function of simple type λ-calculus is: 1. The calculus
is different. MDS calculus in this paper is linear
logic calculus embodying the minimalist grammar,
which is resource sensitive. Simple type λ-calculus
is pure typed λ-calculus, which is intuitionnistic
logic. Our work is applicable to Kripke model of
151
Xu E. (2010).
INVERSION FUNCTION OF MDS FOR SENTENCES ANALYSIS .
In Proceedings of the 2nd International Conference on Agents and Artificial Intelligence - Artificial Intelligence, pages 151-156
DOI: 10.5220/0002590401510156
Copyright
c
 SciTePress

MDS and sentences satisfied by assignment, while
the latter is applicable to pure typed semantic objects.
The organization of the rest of this paper is: 2.
Preliminaries, 3. Kripke model and Inversion
function for MDS, 4. Representing the result of
inversion function as proof tree, 5. Church-Rosser
equality of the result of inversion function, and 6.
Conclusion.
2 PRELIMINARIES
Definition 1. (Type) (Hindley & Seldin 1986)
Assume that we have been given some symbols
called atomic types; then we define types as follows:
(a)each atomic type is a type;
(b)if α and β are types, then (α →β) is a type.
Each type (α →β) is called a compound type.
Definition 2. (Typed λ-terms) (Hindley & Seldin
1986) For each type α, assume that we have
infinitely: many variables v:α of type α, and perhaps
some constants c:α of type α; then we define typed
λ-terms as follows:
(a) each v: α and c: α is a typed λ-term of
type α;
(b) if N: α →β and N: α are typed λ-terms of
types α →β and α respectively, then MN:β is a
typed λ-term of type β;
(c) if x: α is a variable of type α and M: β is a
typed λ-term of type β, then (λx.M): α →β is a λ-
term of type α →β.
Definition 3.
(MDS) (Lecomte, 2004) MDS is
composed of lexical entries and rules.
Generally speaking, a lexical entry consists
in an axiom
├ w: T
where T is of the following type:
((F
2
\ (F
3
\…(F
n
\( G
1
⊗ G
2
⊗…⊗ G
m
⊗A)/))))/F
1
)
where,
m and n can be any number greater than or equal to
0,
F
1
, …, F
n
are attractors,
G
1
, …, G
n
are features,
A is the resulting category type. (Lecomte, 2004)
There are nine rules in MDS, which are
illustrated in Figure 1.
Figure 1: Rules of MDS.
Definition 4. (Ranta 1994) A context, in the
technical sense of type theory, is a sequence of
hypotheses of the form
x
1
:A
1
, x
2
:A
2
(x
1
),…, x
n
:A
n
(x
1
,…,x
n-1
).
where the judgment x:A which introduces a
variable, is a hypothesis.
Definition 5. (Coquand 2002) The set of semantic
objects is defined as usual in Kripke semantics:
Force(ω, A)∈Set is written ω⊩ A, where T∈Set is
the set of types and W is the set of possible
worlds.
Note that Kripke interpretation is sometimes
called Kripke model. (Wang 1997)
Definition 6. (Simpson 1992) The extensional
Kripke interpretation is a sextuple:
where
•W is a set of possible worlds with a partial
ordering, ≤.
•{〚A〛
ω
} is a family of sets, with〚A〛
ω
,
indexed by types, A, and possible worlds, ω.
•{〚P〛
ω
} is a family of relations, 〚P〛
ω
⊆〚
A
1
〛
ω
× …× 〚 A
n
〛
ω
, indexed by predicate
symbols, P, with decorations, P: <A
1
,…, A
n
> and
possible worlds, ω.
•
is a family of functions, : 〚
A→B〛
ω
×〚A〛
ω
→〚B〛
ω
.
•
is a family of functions, :〚A〛
ω
→ 〚 A 〛
ω’
, indexed by types, A, and pairs of
possible worlds, ω′≥ω.
The extensional Kripke interpretation is simply
denoted as W.
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152

An inversion function, given a semantic object
in a particular Kripke model, returns a proof tree.
The function is defined together with function val
that intuitively takes a proof tree of the form of an
variable applied to zero or more arguments. The
definition is as follows.
In this paper, the extensional Kripke
interpretation is taken as the particular Kripke
model.
Definition 7. (Coquand 2002) Let a proof of type
M∈[ ]├A
1
→A
2
…→A
n
→o
where A
i,
i =1,2,…,n, is types. An inversion
function, denoted as reify, is defined as
reify([[M]])
≡ λ(z
1
:A
1
)…λ(z
n
:A
n
).{[[M]]val(z
1
)…val
(z
n
)}
where z
1
…z
n
are fresh names, i.e.
z
1
≡ gensym(Γ),
…,
z
n
≡ gensym([Γ,z
1
:A
1
,…, z
n-1
:A
n-1
])
here gensym∈(Γ∈C)Name, C is set of contexts,
Name is a countably infinite set, and
[[M]]val(z
1
)…val(z
n
)
is a proof tree of type o, atomic type. If x is a
variable of type A
1
→ …→A
k
→B, then
val(x)=Λ([v
1
]…Λ([v
k
](x reify (v
1
)…(reify(v
n
)))
where Λ is the simplified interpretation of
abstraction.
Note. The set of semantics objects ω ⊩ A in
Definition 2 is the same as {〚A〛
ω
} in Definition
3. It is denoted as [[A]] in inverse function.
Next, Definitions 8-10 define sentences
satisfied by the extensional Kripke model and
assignment.
Definition 8. (Coquand 2002) Suppose C is the
set of contexts. The set of environments is defined
as
where each variable in a context is associated with a
semantic object. Force_env(ω, Γ)∈Set is written ω
⊩ Γ.
Note. Environment is sometimes called
assignment.
Definition 9. (Coquand 2002) The interpretation for
proof tree of types in a given environment is defined
as:
[[ ]]
term
∈(Γ├A; ω⊩Γ)ω⊩A.
Definition 10.
(Wang 1997) It is inductively defined
as follows that in the extensional Kripke
interpretation of a formula of type, α, is satisfied by
the environment ω⊩Γ at possible ω∈W (denoted by
ω⊨α):
(1)when α is P(A
1
,…, A
n
) , where P is predicate
variable, A
i
, i =1,…,n, is a type, and [[P]]
ω
is
defined in Definition 2, for all ω′≥ω,
ω⊨P(A
1
, …, A
n
) iff <[[A
1
]]
term
,…, [[A
n
]]
term
>∈[[P]]
ω′
(2)when α is α
1
∧α
2
,
ω⊨α
1
∧α
2
iff for all ω′≥ω, ω⊨α
1
and ω⊨α
2
.
(3)when α is α
1
→α
2
,
ω⊨α
1
→α
2
iff for all ω′≥ω, if ω′⊨α
1
, then
ω′⊨α
2
.
Definition 11.
(Wang 1997; Coquand 2002) Suppose
α be a type lambda formula.
If ω⊨α, then α is
called K-satisfiable.
3 KRIPKE MODEL AND
INVERSION FUNCTION FOR
MDS
3.1 Kripke Model for MDS
It is composed of the following six components.
(1) Possible worlds. The world of mind can be seen
as a possible world. (Jiang & Pan 1998) The
possible world is denoted as w, and W={ w}.
Context is denoted as Γ. Contexts in Definition 4 is
taken as possible worlds in Definition 6, that is,
w=Γ. The possible world includes the set of typed λ-
terms representing words and sentences.
(2) If Γ⊆Γ′, then w≤w′.
(3) The set of semantic objects, 〚 A 〛
ω
, in
Definition 6 is λ-terms for lexical entries at w. And
each variable in a context is associated with a
semantic object. {〚A〛
ω
} is the set of 〚A〛
ω
in
all possible worlds.
(4)〚 P〛
ω
is the products of types in the possible
world w, and they occur in rule 7-9 of MDS. {
〚P〛
ω
} is the set of 〚P〛
ω
in all possible
worlds.
(5)
:〚A→B〛
ω
×〚A〛
ω
→〚B〛
ω
means
Definition 2 (c ) at w.
(6)
:〚A〛
ω
→〚A〛
ω’
means that if 〚A〛
holds at w, then 〚A〛holds for all w′≥w.
INVERSION FUNCTION OF MDS FOR SENTENCES ANALYSIS
153

3.2 Inversion Function for MDS
Definition 7 including symbols with the following
denotations leads to the inversion function for MDS.
1.[[M]].
Given the type of sentence is M. Let
M ∈[ ]├A
1
→A
2
…→A
n
→o.
For each A
i
, i =1,…,n, its semantics object, [[A
i
]] ,
is a variable z
i
which is a λ-term with type in any
context w′≥w. At any w′≥w, type M is mapped to its
semantics object, [[M]], which is λ-expression with
type for sentence.
2. reify([[M]])
reify([[M]])
≡ λ(z
1
:A
1
)…λ(z
n
:A
n
).{[[M]]val(z
1
)…val
(z
n
)}
is λ-expression with n bound variables z
1
,…, z
n
: of
types A
1
, …, A
n
, respectively where val(x) ia as
follows. If x is a variable of type A
1
→ …→A
k
→B,
then there are k bound variables in x, v
i
=[[A
i
]], i
=1,…,k at any w′≥w . Thus,
val(x)=Λ([v
1
]…Λ([v
k
](x reify (v
1
)…(reify(v
n
)))
is an variable x applied to k arguments v
1
,…,v
k
such
that val(x) is semantics object at any w′≥w.
3.3 Application of Inverse Function for
MDS
We take an example to show how the inverse
function returns λ - expression with type
representing a sentence.
Let M∈[ ]├(t→t)→t→t be a type of sentence.
From lexical entries of MDS, a list of lexical entries
appeared in the example is as follows.
α
1
=λv.seem(v): t→t (1)
α
2
=α
3
α
4
= approach(mary): t (2)
α
3
=λu.u(mary): (e→t) →t (3)
α
4
=λy.approach(y): e→t (4)
α
5
=x: e (5)
α
4
α
5
= approach(x): t (6)
λα
5
. α
4
α
5
= λx.approach(x): e→t (7)
α
3
λα
5
. α
4
α
5
=approach(mary) (8)
α
1
α
4
α
5
= seem(approach(x)): t (9)
λα
5
.α
1
α
4
α
5
=λx.seem(approach(x)): e→t (10)
α
3
(λα
5
.α
1
α
4
α
5
)=seem(approach(mary)): t (11)
λα
5
. α
4
α
5
=λx.approach(x): e→t (12)
From 3.2,
reify([[M]])
≡ λ(α
1
:t→t).λ(α
2
:t).[[α
1
α
2
]](α
1
=va
l(α
1
)α
2
=val(α
2
)) [[α
1
α
2
]]{α
1
=val(α
1
)α
2
=val(α
2
)}
≡ app(val(α
1
), val(α
2
))
≡ app(Λ[v](α
1
reify(v))), val(α
2
))
≡ α
1
reify (val(α
2
)) (13)
where ‘app’ is for application of λ-calculu, and last
equation above is due to that Λ[v](α
1
reify(v))) is
applied to val(α
2
). Because (2),in α
2
,there are two
arguments, α
3
and α
4
, therefore
reify(val(α
2
))
≡
λ(α
3
: (e→t)→t).λ(α
4
:
e→t).([[α
3
α
4
]] val(α
3
).val(α
4
))=[[ α
3
α
4
]]{
α
3
=val(α
3
), α
4
=val(α
4
)}
≡
app(val(α
3
), val(α
4
))
≡
app(Λ[v](α
3
reify(v))), val(α
4
))
≡
α
3
reify(val(α
4
)) (14)
Because α
4
, e→t, in (4) is a compound type, its
range is t and its domain is one argument, e. ∴
reify(val(α
4
))
≡
λ(α
5
: e).app(val(α
4
), val(α
5
))
≡
λ( α
5
:e)app(Λ([v] α
4
reify(v))), α
5
)
≡
λ(α
5
: e). α
4
reify(α
5
)
≡
λ(α
5
: e)( α
4
α
5
) (15)
‘reify(val(α
4
))’ in the result of (14) is replaced by
(15), and ‘reify (val(α
2
))’ in the result of (13) is
replaced by (14), it is obtained that
Reify([[M]])= λ(α
1
: t→t).λ(α
2
: t).(α
1
α
3
λ(α
5
:e).α
4
α
5
) (16)
α
1
to α
5
in (16) are replaced by (1)-(5),respectively,
it is obtained that
reify([[M]])
=(λv.seem(v): t→t) λ(approach(mary): t)
λv.seem(v) λu.u(mary) λ(x:e) λy.approach(y) x
=λ(λv.seem(v): t→t). λ(approach(mary): t)
(λv.seem(v). approach(mary)) (16′)
The inverse function results in (16′), the typed λ-
expression representing a sentence. (16′) is
equivalent to proof tree of sentence. (16′) can take as
the form of proof tree shown in the next section.
4 REPRESENTING THE RESULT
OF INVERSION FUNCTION AS
PROOF TREE
The method of representing the λ-terms obtained
with inversion function as a proof tree is as follows:
The λ-terms obtained in respective steps of the
application of the inversion function are transformed
into sub-proof trees. If the λ-terms obtained in a
certain step are juxtaposition, then transform the
result into a deductive sub-proof tree of application
illustrated by Definition 2(b). If a certain step
introduces a new variable, then transform the result
ICAART 2010 - 2nd International Conference on Agents and Artificial Intelligence
154

into a deductive sub-proof tree of abstraction
illustrated by Definition 2(c). Combine all the sub-
proof trees and we have the final proof tree.
We take（16）as an example to illustrate the
process of the derivation of the proof tree.
(13)/(16):
reify([[M]])
≡ (α
1
) reify (val(α
2
))
Its type is t. Since α
1
and reify (val(α
2
)) are
juxtaposition, then transform them into a deductive
sub-proof tree of application. We have deduction
(17)
α
1
: t → t reify (val(α
2
)): t
α
1
α
3
( λα
5
. α
4
α
5
): t (17)
From (14),
reify(val(α
2
))= (α
3
)reify(val(α
4
)
Its type is e→t. Since α
3
and reify (val(α
4
)) are
juxtaposition, we have similar situation. Then we
have deduction (18)
α
3
: (e→t) →t reify(val(α
4
): e→t
reify (val(α
2
)): t (18)
Use reify(val(α
4
))= λ(α
5
: e)( α
4
α
5
) in (18), and
we have deduction (19)
α
3
: (e→t) →t λ(α
5
: e)( α
4
α
5
) : e
→
t
reify (val(α
2
)): t (19)
Use (19) to substitute reify(val(α
2
)) in (17), and
we have (20).
α
3
: (e → t) → t λα
5
. α
4
α
5
: e → t
α
1
: t
→
t α
3
(λα
5
. α
4
α
5
): t
α
1
α
3
( λα
5
. α
4
α
5
): t (20)
λα
5
. α
4
α
5
: e → t in （15） and （20） can be
obtained in proof (21). In （ 21 ） , the upper
deduction is application, and the lower deduction is
λ-abstraction.
[α
4
: e → t]
2
[α
5
:e]
1
α
4
α
5
: t
λα
5
. α
4
α
5
: e → t (21)
Combine（17）-（21）, and we have (22）
[α
4
: e → t]
2
[α
5
:e]
1
α
4
α
5
: t
α
3
: (e → t) → t λα
5
. α
4
α
5
: e → t
α
1
: t → t α
3
(λα
5
. α
4
α
5
): t
α
1
α
3
( λα
5
. α
4
α
5
): t (22)
Replace α
1
through α
5
with the actual λ -terms
representing the lexical items, and we have:
which can be represented as the following proof
tree:
Figure 2: “It seems that Mary approaches”.
5 CHURCH-ROSSER EQUALITY
OF THE RESULT OF
INVERSION FUNCTION
Now refer to another proof which is equivalent in
the sense of Church-Rosser equality.
β
1
=λu.u(mary): (e→t) →t
β
2
=λβ
6
. β
3
β
5
β
6
= seem(approach(x)): e→t
β
3
=λv.seem(v): t→t
β
4
=β
5
β
6
= approach(x): t
β
5
=λy.approach(y): e→t
β
6
=x: e
β
3
β
5
β
6
= seem(approach(x)): t
β
1
(λβ
6
. β
3
β
5
β
6
)=seem(approach(mary)): t
Let M∈[ ]├((e→t) →t) →(e→t)→t be a proof
tree of types.
Then,
Reify([[M]])
≡
λ(β
1
: (e→t) →t) λ(β
2
: e→t) ([[β
1
β
2
]]
val(β
1
).val(β
2
))
≡
[[ β
1
β
2
]]{ β
1
=val(β
1
), β
2
=val(β
2
)}
≡
app(val(β
1
), val(β
2
))
≡
app(Λ[v]( β
1
reify(v))), val(β
2
))= β
1
.reify(val(β
2
)
reify(val(β
2
))
≡
λ(β
6
:e).app(val(β
3
),val(β
4
))
≡
λ(β
6
:e)app(Λ([v]β
3
reify(v))), β
4
)
≡
λ( β
6
: e). β
3
reify(β
4
)
∴reify(val(β
4
))
≡
λ (β
5
: e→t) λ(β
6
: e) [[β
5
. β
6
]] {β
5
=val(β
5
),
β
6
=val(β
6
)}
≡
app(val(β
5
), val(β
6
))=app(Λ[v]( β
5
reify(v))),
val(β
6
))
≡
β
5
reify (val(β
6
))= β
5
β
6
INVERSION FUNCTION OF MDS FOR SENTENCES ANALYSIS
155

∴reify[[M]]
≡ λ(β
1
: (e→t) →t) λ(β
2
: e→t) β
1
λ(β
6
:e) β
3
β
5
β
6
Replace β
1
through β
6
with lexical items, and
we have:
reify([[M]])
≡ λ(λu.u(mary):(e→t)→t).λ(seem(work(x)):e→t).
(λu.u(mary)λ(x:e) λv.seem(v) λy.work(y) x)
Now we transform the computational result of
the inversion function into a proof tree.
From reify[[M]]
≡ β
1
.(reify(val(β
2
)): e→t), we
have deduction (23)
β
1
: (e→t) →t reify(val(β
2
)): e→t
β
1
(λβ
6
. β
3
β
5
β
6
): t (23)
From reify(val(β
2
))= λ( β
6
: e). β
3
reify(val(β
4
)),
we have deduction (26)
β
1
: (e→t) →t λβ
6
. β
3
reify(val(β
4
)): e → t
β
1
(λβ
6
. β
3
β
5
β
6
): t (24)
From reify(val(β
4
))= β
5
β
6
: t and (8), we have
(25)
β
1
: (e → t) → t λ( β
6
:e). β
3
β
5
β
6
: e → t
β
1
(λβ
6
. β
3
β
5
β
6
): t (25)
where λ( β
6
:e). β
3
β
5
β
6
: e
→
t can be derived
from (26):
[β
5
: e → t]
2
[β
6
:e]
1
β
3
: t → t β
5
β
6
: t
β
3
β
5
β
6
: t
λβ
6
. β
3
β
5
β
6
: e → t (26)
∴From（25） and （26） we have （27）
[β
5
: e → t]
2
[β
6
:e]
1
β
3
: t
→
t β
5
β
6
: t
β
3
β
5
β
6
: t
β
1
: (e → t) → t λβ
6
. β
3
β
5
β
6
: e → t
β
1
(λβ
6
. β
3
β
5
β
6
): t (27)
Replace β
1
through β
6
with the actual λ -terms
representing the lexical items, and we have:
Figure 3: seem(approach(mary)): t.
which can be represented as the following proof
tree:
Figure 4: “Mary seems to approach”.
6 CONCLUSIONS
This paper realizes the establishment of the proof
tree representing the sentence according to the
sentence type with inversion function. Our work is
applicable to Kripke model of MDS and types of
sentences satisfied by assignment. Application
examples demonstrate that our work is valid. This
paper enlarges the range of application of sentence
analysis, improves the approach of sentence
analysis, and enhances natural language
understanding. Our work is meaningful.
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