Download chapter 5 conversion of imperative sentences from active to passive

CHAPTER 5
CONVERSION OF IMPERATIVE
SENTENCES FROM ACTIVE TO PASSIVE
ar
VOICE BY TOPOLOGICAL
HOMEOMORPHISM
In order to mathematical applications that successfully process in
el
grammar one needs good models of the algebra and grammar of as many
languages as possible. According to standard linguistic theory, each
sentence consist of parts of speeches, which are the smallest individually
Es
t
meaningful elements in a language.
In this present chapter we have applied properties of topological
spaces, bases, sub-bases, continuous mappings and homeomorphism of
abstract algebra for transformation of imperative sentences from active to
passivthesise voice by taking elements of the English language.
Part of this chapter has been presented in Uttakhand Congress of Science
and technology held at GBPUAT, Pantnagar, 2009.
Part of this chapter has appeared in Journal of Language in India, Volume
9:11– as follows research paper
Rakesh Pandey and H. S. Dhami (2009), Conversion of imperative
sentences from active to passive voice by topological homeomorphism,
Language in India, Volume 9 : 11 November 2009, pp 545-552.
5.1 Introduction
Mathematical tools and concepts have found find many applications
within the linguistic sciences, for example- the fast Fourier transform is
crucial tool for acoustic phoneticians working on speech analysis,
contemporary computational linguists make heavy use of probability
ar
theory for stochastic modeling, experimental psycholinguists and sociolinguists use statistics, patterns and tests are utilized in several linguistic
sub-disciplines
like
phonetics,
psycholinguistics,
sociolinguistics,
historical Linguistics, and syntax and so on. Marcus Kracht’s in his book
el
entitled Mathematics of Language (2003) has mentioned that for the last
fifty years a large amount of research within pure linguistics, particularly
in the United States, has been based on separating natural languages from
the physical, social, and cultural matrix in which their use is embedded,
Es
t
and regarding them in idealized form as collections of structured
mathematical objects. These approaches have led to the application of
algebraic methods —the discrete mathematics underlying formal
language theory, computational complexity theory, and mathematical
logic —to the description of natural languages and the study of their
abstract properties. Kornai in his recent book (2008) has observed that
many parts of linguistics field like formal syntax, logical semantics,
phonetics, phonology, philosophy of language etc. would fit nicely in
algebra and logic; but there are many others for which methods
belonging to other fields of mathematics are more appropriate.
The notion of a grammar is central to most work in computational
linguistics and natural language processing. In recent years, there has
been growing interest across a number of theoretical frameworks in
defining grammar formalisms for natural language which make available
stronger forms of psychological interpretation of the formalism than is
standard, giving rise to new ways of articulating the relationship between
grammar formalism and natural-language data.
The motivation behind preparation of this chapter underlies in the
fact that whether set of rules for conversion of sentences written in active
ar
voice can be done by the application of mathematics? Whether
mathematical abstraction can be an answer to the transformation rules in
strict adherence of the syntax and grammatical rules?
el
Initiating with the assumption that the epistemological problem
that all linguistics must deal is of a topological nature, a topological
model has been developed for internal linguistics by Lo’pez Garsia
(1994). Gorman and Curran (2007) have found that distributional models
Es
t
of language acquisition display topological properties to synonymy and
homonymy networks. In the process of determining the antecedent of an
anaphora, Sepassi and Marzban (2006) have examined the processing of
two types of causality structures in the active and passive voices.
Motivated by introducing a mathematical foundation of qualitative and
quantitative fuzziness of linguistic terms, Nguyen Cat Ho (2007) has
shown that a formal model of fuzziness of linguistic terms and hedges
can be defined reasonably in the semantic structure and there is a closed
relation between fuzziness of terms and semantic-based topology of these
algebras.
The present chapter is in continuation of chapter 4. With the aim to
cater the need of those students who face difficulty in applying
grammatical techniques in syntactic theory like that of transformations
from active voice to passive voice, we had developed a mathematical
model and consequently a computer program for affirmative sentences.
Here we have formed topological spaces for imperative sentences with
elements being the different parts of speeches. The property of
ar
homeomorphism of topological spaces has been used for defining the
mapping between active voice to passive voice sentence.
Generation of Topological Space for Imperative Sentences
el
5.2
Let E be a set comprising of all English words together with all
proper nouns and if we denote any word by a then
Es
t
a∈E
------------ (5.2.1)
It is a grammar phenomenon that an imperative sentence in active
voice contains verb, object and other part of speeches. Mathematical
sequence for this rule can be expressed asS a = {S v , S a , S e }
------------ (5.2.2)
Where S v , S a , S e respectively denote verb term, object term and other
terms after object.
We can consider a set X of parts of speeches as
X=
{noun,
pronoun,
verb,
conjunction, article, interjection} or
adjective,
adverb,
preposition,
X = {a1 , a2 , a3 , a4 , a5 , a6 , a7 , a8 , a9 }
------------ (5.2.3)
s
Where a i ' stand for different parts of speeches mentioned
above then by (1.1) each
a
will be in X as it will be any parts of
speech.
Now let us consider a power set P (X) of all subsets of X, then the
sequence of all parts of speeches in all English sentences (of cardinal
P (X). If in a sentence any number
ar
number 29) shall be the elements of
of parts of speech appears more than once then also it will be a subset of
X and ultimately the element of P(X).
el
In equation (1.2), the most possible elements of Sv shall be
{auxiliary verb, verb} and {art, noun} of So while in Se shall comprise
of parts of speeches.
S a = {{a 3 }, {a8 , aα }, {a i ,......, a j }}, α = 1,2
Es
t
Therefore
--------- (5.2.4)
Let us assume that finite intersections of the elements of Sa form
another set B, then
B = {{a3}, {a8, aα }, { a β }, X, φ}.
------------ (5.2.5)
where a β = {ai ,........, a j } .
The union of members of B shall be given by
Ta ={X, φ, {a3}, {a8, aα }, { a β }, {a3, a8, aα , ae}},
------------ (5.2.6)
Here it is obvious that Ta is a subset of P (X), and we can claim
that Ta is a topology on X on the basis of following properties-
X, φ∈ Ta
Arbitrary union of members of Ta belongs to Ta.
Finite intersection of members of Ta is again in Ta.
such that (Ta, X) is a topological space.
In order to facilitate the mapping of bases in active and passive
ar
voices, we shall now prove that Sa is a base for topology T on X.
Since (X, Ta) is a topological space, so
Sa ⊂ Ta such that Sa ≠ φ
------------ (5.2.7)
el
Sa will be said to be a base or basis for topology Ta on X if for any given
non- empty set
G ∈ Ta ⇒ ∃ S ⊂ Sa s.t. G = ∪{s : s ∈ S }
Es
t
It can also be defined as
x ∈ G ∈ T ⇒ S ∈ Sa s.t. x ∈ S ⊂ G
By equation (1.4) and (1.6) it is obvious that Sa satisfies the properties of
base of topological space, hence Sa is a base for Ta.
Now let we define another topology Tp over X as
Tm = {X, φ, {b, a3}, {l, a8, aα }, { a β }, {l, a8, aα , b, a3} --------- (5.2.8)
where l and b are two mathematical constants which shall correspond to
the words ‘let’ and ‘be’ respectively, in English language.
5.3 Application of Topological Homeomorphism
Let us define a map f from (X, Ta) to (X, Tm), that is,
f: (X, Ta) (X, Tm)
where
f(a3)=( b, a3)
------------ (5.3.1)
f(a8, aα ) =(l, a8, aα )
and


−1
({l , a8 , aα }) = {a8 , aα }


−1
({b, a3 }) = {b, a3 )

−1
({l , a8 , aα , b, a 3 , a β }) = {a8 , aα , a3 , a β }

−1

{a β } = {a β }

f
f
f
f
f −1 (φ ) = φ
------------ (5.3.2)
ar
f −1 ( X ) = X ,
el
In order to exhibit homeomorphism of the two topological spaces,
we require the mapping to be continuous mathematically.
We know that in two topological spaces, (X, Ta) and (X, Tm), a
mapping f of X into X is a Ta - Tm continuous if the inverse image under
Es
t
f of every Tm-open set is a Ta - open set. The mappings defined in (2.2)
satisfy this property and thus establish the fact that
f is a Ta-Tm
continuous.
The bases of the two topological spaces shall be mapped into each
other if we are able to establish the continuity of topological spaces. This
we can do with the assist of following explanation-:
Since (X, Ta) and (X, Tm) are two topological spaces [defined in
(5.2.6) and (5.2.8)] and f is an open mapping of (X, Ta) on to (X, Tm)
therefore the family of all images under f of the members of a base Sa for
Ta forms a base Sm for Tm.
Thus
{X, {b, a3}, {l, a8, ao}, {ae}} is a base of Tm and by the definition
of sub-base, given in the book of Allen Hatcher (2002).
{{b, a3}, {l, a8, aα }, {ae}} will be a sub-base for Tm.
We have established above that f is a continuous mapping from
(X, Ta) on to (X, Tm) and it is also a one-one mapping, therefore f will
be homeomorphism from (X, Ta) on to (X, Tm) , that is,
ar
f: (X, Ta) (X, Tm), and ( X , Ta ) ≈ ( X , Tm ) and Sa ≈ Sm ------- (5.3.3)
The transformation of verb forms can be justified with the help of
another mapping g from (X, Tm) into (X, Tp), where X contains only thirds
el
form of verbs in verb term av , defined as
g: (X, Tm) (X, Tp)
------------ (5.3.4)
such that Tp= {X, φ, {l, a8, aα }, {b, a3′′ }, { a β },{l, a8, aα , b, a3′′ ,, a β }}
Es
t
and
g (X)= X,
g (φ)=φ ,
Hence we can have


g -1 ({b, a 3′′′}) = {b, a 3 }

-1
g ({l , a8 , aα , b, a3′′′, a β }) = {l , a8 , aα , b, a3 , a β } ----------- (5.3.5)

g -1 ({a β }) = {a β }

g -1 ( X ) = X , g -1 (φ ) = φ
Now if we want to find the base of this topology we immediately get
Bp = {X, {b, a3′′ }, {l, a8, aα }, { a β }}
and its sub-base shall be
Sp = {b, a3′′ }, {l, a8, aα }, { a β }}
------------ (5.3.6)
Again from the definition of continuous mapping g is continuous from Tm
into Tp.
ar
Since f is continuous from Ta into Tm and g is continuous from Tm
into Tp hence composite map gof is continuous from Ta in to Tp also this
mapping is one-one hence it will also be a homeomorphism, that is,
------------ (5.3.7)
el
(X, Ta) ≈ (X, Tp)
Also there is a continuous mapping gof from (X, Ta) into (X, Tp) hence
their sub-bases shall map one another by the same mapping, such that
Es
t
gof(Sa)= (Sp)
gof{{a3}, {a8, aα }, { a β }} = {{b, a3}, {l, a8, aα }, { a β }}
Sa ≈ Sp, hence we can say Sa and Sp have the same meaning.
According to permutation law the mapped elements Sp = {{b, a3},
{l, a9, a1}, { a β }} can be putted in 6 ways. We can generate as many
sentences as we like for all the combinations but we should select only
those sentences, which follow grammatical rules and are meaningful. We
observe that the only permutation {{l, a9, aα }, {b, a3′′ }, { a β }} can give a
correct semantic meaning. Hence finally we have a mapping gof such
that
gof {{a3}, {a8, aα }, { a β }} = {{l, a8, aα }, {b, a3}, { a β }} ------- (5.3.8)
Following sentence of English language can do the justification of this
mathematical exercise.
Active- Open the door carefully.
Here ‘open’ represents verb, ‘the door’ object (the- article, door-noun)
and ‘carefully’ adverb
We can represent this sentence as
ar
(Sv, So, Se) [by equation (5.2.2)],
where Se (carefully) is the term after object element.
By topological homeomorphism, it will be mapped into
(l+So, b+ Sv′′′ , Se) [From equation (5.3.8)],
el
Where Sv′′′ is the third form of verbs as explained in the set of equations
(5.3.4) and (5.3.5).
In above transformed mapping l stands for ‘let’ and b for ‘be’, as defined
Es
t
in (5.2.8).
Hence the passive voice sentence is – Let the door be opened carefully.
5.4
Conclusion