Introduction
The writing systems of early civilizations, such as the Egyptian and the Hittite, were based on the use of symbols (hieroglyphs), representing a syllable. Egyptian hieroglyphics eventually included about 700 symbols (Anonymous, 1991). Realizing the difficulty encountered by the Hieroglyphic (or Hieratic, a cursive form of the Hieroglyphic) system in rapid processing of documents relating to business transactions, Phoenicians, who were primarily merchants, were the first to introduce the alphabet system of writing between 1700 and 1500 BC. While Phoenicians introduced twentytwo consonants in their writing system, Greeks complemented them with four vowels between 800 and 700 BC. A detailed discussion on this topic is available in Encyclopedia Britannica (1993).
Language evolution modeling has attracted increased attention from linguists,
cognitive scientists, neurobiologists, mathematicians and computer scientists
in the twentieth century. For example, Nowak and Komarova (2003) have remarked
that language is a biological trait that radically changed the performance of
one species and the appearance of the planet. Understanding how human language
came about is one of the most interesting tasks for evolutionary biology. These
authors have discussed how natural selection can guide the emergence of some
basic features of human language, including arbitrary signs, words, syntactic
communication and grammar. They further demonstrate how natural selection can
lead to the duality of patterning of human language: sequences of phonemes forming
words, sequences of which, in turn, forming sentences. Cangelosi (2001) has
discussed different types of models for the evolution of communication and language.
In particular, this study shows how evolutionary computation techniques, such
as Artificial Life, can be employed to investigate the emergence of syntax and
symbols from simple communication signals. Additionally, computational models
of syntax acquisition and evolution is also discussed in the researcch.
Redford et al. (2001) have developed a computational model of emergent syllable systems, based on a set of functional constraints on syllable systems and the assumption that language structure emerged through a cumulative change over time. The constraints have been derived from general communicative factors as well as from the phonetic principles of perceptual distinctiveness and articulatory ease. Their model has generated mock vocabularies optimized for the given constraints. More information on this class of papers is available in the Language Evolution Modeling newsletter (Zuidana, 2001).
According to Chomsky and Halle (1968), the phonology, the description of sound change, in a spoken language system can be compared with links in a chain. The chain can be smoothed out as one continuous piece. The effects of this phonetic system would create some lowpass filtering of signal variables and represent them as time functions. The filter of the time functions would be dependent on the language or speaker. The C/D model due to Fujimura (1992) depicts how a mixture of symbols and numbers can be assessed within the phonetic implementation process. More importantly, Fujimura’s (1992) C/D model shows “where and how the continuous nature of phonetic variables” can result from “phonologically significant articulatory variables” such as tongue movement. Many papers on experimental phonetics for linguistics, the phoneticsphonology interface, speech recognition, digital signal processing of speech and aeroacoustic modeling of speech are available on the web and will not be discussed further here in the interest of brevity (Fujimura, 2003; Anonymous, 2003).
The above review of the linguistic literature suggests that although substantial
progress has been reported on the topic of empirical modeling of language and
speech, a rigorous set theoretic basis for such modeling does not exist. The
primary objective of the present study is to bridge this longstanding mathematical
gap. To the authors’ knowledge, the aforementioned linguistic literature
reveals an absence of advanced mathematical techniques for mathematical representation
of syllable and alphabetbased writing, which will form the primary focus of
the present research. In this study, the syllable is approximated to be ideally
sampled version of morpheme (represented by a symbol in the case of Hieroglyphics),
while the phonetic alphabet is the physically sampled version of the corresponding
phoneme. In regards to the relationship between morphemes and syllables, the
above assumption is justified on the ground that in ancient Egyptians' writings,
morphemes were represented as syllables and vice versa. The assumption pertaining
to the relationship between phonetic alphabets and phonemes is justified on
the ground of ancient Phoenicians' pioneering usage to the same effect. The
classical language prevalent in ancient Northern India, Sanskrit and its derivatives,
Hindi, Bengali, Marathi, Punjabi, etc., also constitute a good example, where
this assumption is justified. In Sanskrit based languages, the letters are categorized
according to the primary organ giving rise to the phoneme. For example, the
Sanskritic counterparts for /k/, /kh/, /g/, /gh/ and throaty /ng/ are called
throat letters (kantha varna), softer (like French) versions of /t/, /th/, /d/,
/dh/ and /n/ are called dental letters (danta varna), while /p/, /ph/, /b/,
/bh/ and /m/ are called lip letters (oshtha varna) and so on. The notation '/
/' denotes a phoneme. In many (probably most) languages, there are more phonetic
alphabets than written alphabets or letters, a...z. For example, in English
26 letters are used to represent 40 phonetic alphabets (i.e., phonemes). The
fundamental steps to solving this problem are concerned with the (i) concept
of wave function as a solution to the wave equation subjected to initial and
boundary conditions for phonetic/linguistic modeling and (ii) derivation of
these solutions via Fourier series and Sinc series expansion techniques. A rigorous
set theoretic basis for mathematical modeling of a generic alphabet based written
language is provided through application of the reverse Cantor set based rational
deltafunction, first introduced by Chaudhuri (2005). Details are given in Appendix
A1A3.
Appendix: Reverse Cantor Based Rational Delta Function
In what follows, the concept of wave function is employed to mathematically
model the transformation of a natural language from its spoken to both written
forms–syllable or symbol based written language, e.g., Hieroglyphic (or
Hieratic) and alphabet based written language. The fundamental steps to solving
this problem are concerned with the solution to the wave equation subjected
to initial and boundary conditions and derivation of these solutions for syllables
and phonemes via Fourier series and Sinc series expansion techniques for symbol
and alphabetbased writings, respectively. For example, the name, Tutankhamon
(also known as Tutankhamen) of the boy king, who ruled Egypt from 1333 to 1323
BC, is comprised of three symbols in the Hieroglyphic (or Hieratic) system–Tut,
Ankh and Amon.

Fig. 1: 
Alphabet sampler inside a morphemic well 

Fig. 2: 
A graphical representation of the reverse Cantor set based
delta function (sequence) δ_{n}(x) vs. x 
The syllable based written language can be represented by a Fourier series,
which has a discrete frequency spectrum. In the other extreme, a spoken language
can be represented by a Fourier transform, characterized by a continuous frequency
spectrum. The alphabetical form of the written language lies somewhere in between,
characterized by a frequency spectrum neither entirely discrete nor completely
continuous. This will be represented by a Sinc series. The Fourier and Sinc
series, in turn, through ideal and physical sampling, respectively, reduce to
discrete time (space) Fourier series (DTFT or DSFT) for symbol based writing
and discrete Fourier transform (DFT) for alphabet based writing.
Materials and Methods
Wave Equation for Alphabetization and Syllabication of a Natural Language
Since speech is generated by the vibration of the human vocal chord, movement
of tongue and its touching various parts of the mouth and involvement of lips
and nose and the generated sound propagates in the air in the form of a wave
with velocity c, it is logical to start with the wave equation of the form:
in which u is the displacement, t is time and x is spatial coordinate. The angular frequency, ω = 2πf, f being the frequency, while the wave number, k = 2π/λ, λ being the wavelength. The wave velocity is given by
The initial condition is given by
In addition, the boundary conditions, which depend on the form of a written language, must be specified.
On applying the separation of variables,
Eq. 1 yields the following:
in which C is a constant. In order to obtain the desired solution in the form of e^{i(kx  ωt) }and e^{i(kx  ωt)} (or equivalently sine and cosine), C must be negative, which is taken to be equal to 1, without any loss of generality. This operation yields the following two ordinary differential equations (ODE) in x and t, respectively:
It may be noted that the separated wave equation in x, given by Eq. 6, is consistent with the underlying hypothesis of the present research that a spoken natural language (speech, word form) can be mathematically represented in the syllabified and alphabetized written forms of the same. This justifies the concept of the spatial wave function, Ψ(x), wherein x is given in terms of phoneme/alphabet units. The wave number k represents the corresponding eigenvalue.
Syllabic Form of Writing
In the syllabic form of writing, a symbol (representing a syllable) is abruptly
terminated before a new syllable starts. This is idealized by one dimensional
well (in analogy to quantum mechanics). The boundary conditions are specified
at the start (x = 0) and end (x = L) of the syllable, where L is given in terms
of phoneme units. These are given by
The solution to the ODE (6) subjected to the boundary condition (8) can then be easily obtained as follows:
which is the solution for a standing wave. The solution to the ODE (7) can also be easily obtained as follows:
The complete solution is given by
in which
After applying the initial conditions, the constant coefficients can be obtained as follows:
D_{n }can easily be determined by expanding g(x) in the form of Fourier sine series in x. The syllabic form of the written language needs to capture the initial condition (14) of the spoken language. Here the computed wave function is localized within the box (well), while in a speech, the wave function is extended all the way to "infinity".
Phonemic Analysis of Speech Sounds and Alphabetic Form of Writing
Morpheme is the smallest part of a word that has meaning of its own. Morphemes
may be words, prefixes, suffixes or endings that show inflection (Barnhart and
Barnhart, 1991). In the word carelessness, the morphemes are care, less and
ness. A morpheme, consisting of alphabets, "does not necessarily consist of
phonemes, but all morphemes are statable in terms of phonemes" (H.A. Gleason,
Jr., quoted by Barnhart and Barnhart, 1991). In the present work, morpheme/syllable
is assumed to be comprised of a number of phonemes (approximated by phonetic
alphabets). A phoneme acts as a lowpass filter in the frequency domain. It
follows that the corresponding alphabet acts as a longwave filter or low wave
number (k) filter/sampler in the wave number domain.
The characteristics of an ideal lowpass filter, in the frequency and time domains is discussed in standard texts e.g., Gajic (2003). An ideal lowpass filter transfer function is given by Gajic (2003).
The phase of the ideal filter is assumed to change linearly in frequency, which corresponds to the time shift of the filter input signals by t_{d}, known as the time delay. The impulse response of the ideal lowpass filter is obtained as follows. Using the time domain Fourier equivalent of a rectangular frequency domain pulse of unit height given by the Fourier transform pair
the ideal lowpass filter transfer function is given by Gajic (2003)
in which the time shift property of the Fourier transform has been utilized and the Sinc function is defined as:
The characteristics of the same ideal lowpass filter, in the wave number and space domains can be described in a similar manner. An ideal lowpass filter space transfer function can be written as follows:
The phase of the ideal filter is again assumed to change linearly in wave number, which corresponds to the space shift of the filter input signals by x_{d}, to be termed here as the space shift (delay). The spatial impulse response of the ideal lowpass filter is then obtained as follows. Using the space domain Fourier equivalent of a rectangular wave number domain pulse of height, g_{0}, given by the Fourier transform pair
the ideal lowpass filter spatial transfer function can now be written follows:
in which the spatial shift property of the Fourier transform has been utilized and the Sinc function is defined as before. Generalizing the above, g(x) can be expressed in standard sinc series (Stenger, 1993; Lund and Bowers, 1992) as follows:
in which
and g_{0} = g(0) corresponding to j = 0.
Sampling with an Ideal/Physical Sampler and the Discrete Time Fourier Transform
(DTFT)/Discrete Fourier Transform (DFT)
Shannon's Sampling Theorem (Gajic, 2003) states that a continuoustime bandlimited
signal, such as the Sinc function, sinc(t), with bandwidth frequency f_{max}
can be uniquely reconstructed from its sampled values at sinc (jT_{s}),
j = 0, ±1, ±2, ±3,..., if the sampling frequency f_{s
}= 1/ T_{s }satisfies
The frequency 2f_{max} is called the Nyquist frequency and the frequency interval [f_{max, }f_{max}] is called the Nyquist interval. This theorem can be easily extended to the space and wave number domains.
Standard texts on digital signal processing consider two types of samplers:
ideal sampler and the physical sampler (Gajic, 2003). It is shown that an ideal
sampler gives rise to discrete time Fourier transform (DTFT), while a real or
physical sampler produces discrete Fourier transform (DFT). Applying ideal sampling
operation to syllable (sound) based written language, such as Hieroglyphic or
Hieratic, these written symbols can be mathematically represented by DSFT (discrete
space Fourier transform), the spatial counterpart of the DTFT. This is expressed
by the spatial counterpart of Eq. (9.16) of Gajic (2003), which suggests that
the wave number spectrum of the ideal sampled "signal" (here the symbols) is
periodic.
As has been commented by Gajic (2003), the DTFT or in our case DSFT is the
Fourier transform of a discrete time or space "signal", useful for representing
a syllable based written language and obtained by sampling a continuoustime
or space "signal" by an ideal sampler. It has been shown (Gajic, 2003) that
DTFT and its inverse as defined in Eq. 9.23 and 9.27 of that book can be derived
using the discretetime or space Fourier series. By analogy to the continuoustime
or space Fourier series, the discretetime or space Fourier series are applicable
to discretetime or space periodic signals. A discretetime or space aperiodic
signal can be considered in the limit as a discretetime or space periodic
"signal" whose period tends to infinity and thus DTFT or DSFT tends to the corresponding
DFT (discrete Fourier transform), discussed in standard treatises (Gajic, 2003;
Press et al., 1992).
Application of the physical sampling to the phonemic signal analysis produces
an alphabetbased written language. Alphabet serves as the physical or real
sampler of narrow width, 'a'. The alphabet sampler inside the morphemic "well"
model is shown in Fig. 1. Assuming the number of alphabets
inside a morpheme to be N, the length of the syllable, L = Na, where a is the
width of the phoneme/alphabet. The boundary conditions are given by Eq.
(8) at x = ± L.
A rectangular pulse is defined as follows:
Since in the present alphabetic physical sampler inside the morphemic box (well)
problem, the wave function is permitted neither to extend all the way to "infinity",
nor to be localized in length scale of the order of "sampler width", a (<<
L), the computed eigenfunctions cannot be orthonormalized with respect to either
the Dirac or Kronecker δ functions. Consequently, these eigenfunctions
need to be orthonormalized with respect to the reverse Cantor set based rational
δ function, discussed in the Appendix. Therefore, the correct solution
must be of the form of discrete Fourier transform, as given by Eq.
(A10) and (A11).
where Ψ_{j} is the physically sampled spatial wave function. A_{k}, with k being the eigenvalue, need to be numerically evaluated by using the DFT (or FFT, DWT, etc.) technique and must satisfy the various boundary conditions of the problem. This scheme is currently being implemented and numerical results will be reported in future.
The fast Fourier transform (FFT) is an algorithm that computes in O (N log_{2}N) operations as compared to the DFT's O (N^{2}). The discrete wavelet transform (DWT) is a more recently developed fast computational tool that linearly operates on a data vector, the length of which is an integral power of 2 and transforms it into a numerically different vector without altering the length (Daubechies, 1992). Like the FFT, the DWT is invertible and orthogonal, its inverse transform, when viewed as a large matrix, being the transpose of the transform. Both the FFT and DWT can, therefore, be viewed as rotations in function space, from the input space to a new domain. In the case of FFT, the rotated domain has basis functions in the form of standard sines and cosines, while in the wavelet domain the basis functions are somewhat novel, with names like "mother functions" and "wavelets". The details of the FFT and DWT algorithms are available in Press et al. (1992) and will not be repeated here.
Conclusions
The concept of wave function is employed to mathematically model the transformation of a natural language from its spoken to both written forms – syllable (symbol) based and alphabet based writings. The fundamental steps to solving this problem are concerned with the solution to the wave equation subjected to initial and boundary conditions and derivation of these solutions for syllables and phonemes via Fourier series and Sinc series expansion techniques for symbol and alphabetbased writings, respectively. These, in turn, through ideal and physical sampling, reduce to discrete time (space) Fourier series (DTFT or DSFT) for symbol based writing and discrete Fourier transform (DFT) for alphabet based writing.
In this work, the syllable is approximated to be ideally sampled version of morpheme (represented by a symbol in the case of Hieroglyphics), while the phonetic alphabet is the physically sampled version of the corresponding phoneme. In regards to the relationship between morphemes (symbols) and syllables, the above assumption is justified on the ground that in ancient Egyptians' writings, morphemes were represented as syllables and vice versa. The assumption pertaining to the relationship between phonetic alphabets and phonemes is justified on the ground of ancient Phoenicians' pioneering usage to the same effect.
A rigorous set theoretic basis for mathematical modeling of a generic alphabet based written language is provided through application of the reverse Cantor set based rational delta function, first introduced by Chaudhuri (2006). Alphabetization bridges the gap between the two situations that arise in phonetic/linguistic research, namely the syllable based written language with discrete eigenvalues and the spoken language with a continuous spectrum of eigenvalues.
Most important, this novel rational delta function, δ_{n}(x), provides a rigorous set theoretic basis for permitting the resulting computed wave function to be expressed in the form of the discrete Fourier transform (DFT) in the Fourier domain and recover the sampled wave function in the physical domain by employing the inverse discrete Fourier transform (IDFT). The relatively straightforward transition to faster techniques, such as the fast Fourier transform (FFT), Sinc and discrete wavelet transform (DWT), will be the subject of future research.
This research helps bridge the longstanding gap between phonetic/linguistic research and modern development in computational spectral and pseudospectral methods, such as DFT, FFT, Sinc and DWT. The ease of the present method and its relative accuracy will help make challenging language problems numerically solvable.
Acknowledgement
The authors wish to thank Dr. J.A. Laursen and an anonymous reviewer for their
helpful suggestions on an earlier version of the manuscript.