Grammar Based Pre-Processing for PPM

International Journal of Computer Science & Information Technology (IJCSIT) Vol 9, No 1, February 2017
DOI:10.5121/ijcsit.2017.9101 1
GRAMMAR-BASED PRE-PROCESSING FOR PPM
William J. Teahan1
and Nojood O. Aljehane2
1
Department of Computer Science,University of Bangor Bangor, UK
2
Department of Computer Science,University of Tabuk,Tabuk, Saudi Arabia
ABSTRACT
In this paper, we apply grammar-based pre-processing prior to using the Prediction by Partial Matching
(PPM) compression algorithm. This achieves significantly better compression for different natural
language texts compared to other well-known compression methods. Our method first generates a grammar
based on the most common two-character sequences (bigraphs) or three-character sequences (trigraphs) in
the text being compressed and then substitutes these sequences using the respective non-terminal symbols
defined by the grammar in a pre-processing phase prior to the compression. This leads to significantly
improved results in compression for various natural languages (a 5% improvement for American English,
10% for British English, 29% for Welsh, 10% for Arabic, 3% for Persian and 35% for Chinese). We
describe further improvements using a two pass scheme where the grammar-based pre-processing is
applied again in a second pass through the text. We then apply the algorithms to the files in the Calgary
Corpus and also achieve significantly improved results in compression, between 11% and 20%, when
compared with other compression algorithms, including a grammar-based approach, the Sequitur
algorithm.
KEYWORDS
CFG, Grammar-based,Preprocessing, PPM, Encoding.
1. INTRODUCTION
1.1. PREDICTION BY PARTIAL MATCHING
The Prediction by Partial Matching (PPM) compression algorithm is one of the most effective
kinds of statistical compression. First described by Cleary and Witten in 1984 [1], there are many
variants of the basic algorithm, such as PPMA and PPMB [1], PPMC [2], PPMD [3], PPM* [4],
PPMZ [5] and PPMii [6]. The prediction of PPM depends on the bounded number of previous
characters or symbols. In PPM, to predict the next character or symbol, different orders of models
are used, starting from the highest orders down to the lowest orders. An escape probability
estimates if a new symbol appears in the context [1, 2]. Despite the cost of the terms of memory
and the speed of execution, PPM usually attains better compression rates compared with other
well-known compression methods.
One of the primary motivations for our research is the application of PPM to natural language
processing. Therefore, we report in this paper results on the use of PPM on natural language texts
as well as results on the Calgary Corpus, a standard corpus used to compare text compression
algorithms. PPM has achieved excellent results in various natural language processing
applications such as language identification and segmentation, text categorisation, cryptology,
and optical character recognition (OCR) [7].

2
Each symbol or character in PPM is encoded by using arithmetic coding using the probabilities
that are estimated by different models [8].To discuss the operation of PPM, Table 1 shows the
state of different orders for PPM, where k=2, 1, 0 and -1 after input string “abcdbc” has been
handled. In this exampleto estimate the probability for symbols in the contexts, the PPM
algorithm starts from the highest order k=2. To estimate the probability of the upcoming symbol
or character, if the context predicts the next symbol or character successfully, the associated
probability for this symbol or character will be used to encode it. Otherwise, the probability of the
escape will be estimated to let the encoder move down to the next highest order which is in this
case isk=1until the encoder reaches (if needed) the lowest order which is k=-1. Then the
probability for all symbols or characters will be estimated and encoded by| |
whereA is the size of
alphabets in the contexts. The experiments show the maximum order that usually gets good
compression rates for English is five [1][8][7]. For Arabic text, the experiments show that order
seven the PPM algorithm gives a good compression rate [9].
TABLE 1: PPMC model after processing the string “abcdbc”.
For example, if “c” followed the string “abcdbc”, the probability for an order 2 model with the
input symbol “abcdbc” would be because a successful prediction for the clause “ab→c”can be
made. Suppose the string “a” follows the input string “abcdbc” instead.The probability of for
an escape for the order 2 model would be encoded arithmetically, and the process of encoding
downgrades from order 2 model to the next order 1 model. Moreover, in order 1, the encoder does
not predict string “a”, so another escape (with probability ) is going to be encoded, and the
process of encoding downgrades to order 0. In order 0, the probability is willbe for string
“a”.Therefore, to encode string “a”, the total probability is * * = , which is 5.2 bits.
If a previously unseen character new string “n” follows the input string “abcdbc”, the process
encodes down to the lowest order which is k=-1 model. In this model k=-1, all strings are

3
encoded with the same probabilities which is | |
where A is size of alphabet. Supposing the size of
the alphabet is 256, so the probability for the new string “n” is for the order -1 model.
Therefore, the character “n” is encoded using the probability * * * which is 11.4 bits.
As our method uses both grammar-based and text pre-processing techniques prior to compression,
we will now briefly discuss related work in these two areas.
1.2.GRAMMAR-BASED COMPRESSION ALGORITHMS
Grammar-based compression algorithms depend on using a context-free grammar (CFG) to help
compress the input text. The grammar and the text is compressed by arithmetic coding or by
different statistical encoders [10]. Two examples of grammar-based compression schemes are the
Sequitur algorithm [11] and the Re-Pair algorithm [12].
Sequitur is an on-line algorithm that was developed by Nevill-Manning and Witten in 1997 [11].
Sequitur uses hierarchical structure as specified by a recursive grammar that is generated
iteratively in order to compress the text. Sequitur depends on repeatedly adding rules into the
grammar for the most frequent digram sequence (which may consist of terminal symbols that
appear in the text or non-terminal symbols that have been added to the grammar previously). The
rule for the start symbol S shows the current state of the corrected text sequence as it is being
processed. For instance, a text “abcdbcabc” is converted into three rules, which are S as the start
symbol and A and B as nonterminal symbols: S→BdAB, A→bc, B→aA.
In contrast, the Re-Pair algorithm proposed by Larsson and Moffat [12] in 2000 is off-line. Like
Sequitur, Re-Pair replaces the most frequent pair of symbols with a new symbol in the source
message essentially extending the alphabet. The frequencies of symbol pairs are then re-evaluated
and the process repeats until there are no longer any pair of symbols that occur twice. Through
this off-line process, what can be considered to be a dictionary has been generated, and then an
explicit representation of this dictionary is encoded as part of the compressed message.
1.3.TEXT PRE-PROCESSING FOR DATA COMPRESSION
Abel and Teahan [13] discuss text pre-processing techniques that have been found useful at
improving the overall compression for the Gzip, Burrows-Wheeler Transform (BWT) and PPM
schemes. The techniques work in such a way that they can easily be reversed while decoding in a
post-processing stage that follows the decompression stage. The methods discussed include a long
list of prior work in this area and various new techniques, and presents experimental results that
show significant improvement in overall compression for the Calgary Corpus. The methods most
similar to our method described in this paper are the bigraph replacement scheme described by
Teahan [7] and the token replacement scheme described by Abel and Teahan [13].
The main contribution of the work described in this paper is the improved pre-processing method
for PPM. This is due to the discovery that instead of using a fixed set of bigraphs for replacement
from a standard reference source (such as the Brown corpus),significantlybetter compression
results can be achieved by using bigraphs obtained from the text being compressed itself.
The rest of the paper is organised as follows. Our new approach is discussed in the next section.
Then we discuss experimental results on natural language texts and the Calgary Corpus by
comparing how well the new scheme performs compared to other well-known methods. The
summary and conclusions are presented in the final section.

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2. GRAMMAR BASED PRE-PROCESSING FOR PPM (GR-PPM)
In this section, a new off-line approach based on Context Free Grammars (CFG) is presented for
compressing text files. This algorithm, which we call GR-PPM (which is short for Grammar
based pre-processing for PPM), uses both CFGs and PPM as a general-purpose adaptive
compression method for text files.
In our approach, we essentially use the N most frequent n-graphs (e.g. bigraphs when n=2) in the
source files to first generate a grammar with one rule for each n-graph. We then substitute every
time one of these n-graphs occurs in the text with the single unique non-terminal symbol as
specified by its rule in the grammar in a single pass through the file. (Further schemes described
below may use multiple passes to repeatedly substitute commonly occurring sequences of n-
graphs and non-terminal symbols). This is done during the pre-processing phase prior to the
compression phase in a manner that allows the text to be easily regenerated during the
postprocessing stage. For bigraphs, for example, we call the variant of our scheme GRB- PPM
(Grammar Bigraphs for PPM). Our new method shows good results when replacing the N most
frequent symbols for bigraphs (for example, when N=100) but also using trigraphs (when n=3) in
a variant which we have called GRT-PPM (Grammar Trigraphs for PPM).
Each natural language text contains a high percentage of common n-graphs which comprise a
significant proportion of the text [7]. Substitution of these n-graphs using our context-free
grammar scheme and standard PPM can significantly improve overall compression as shown
below. For example, natural languages contain common sequences of two characters (bigraphs)
that often repeat in the same order in many different words, such as in the English “th”, “ed”and
“in”, and for the Arabic language, such as “ ‫,”ﺍﻝ‬ “‫ﻓﻲ‬” and “‫ﻻ‬ ” and so on.
The frequencies of common sequences of characters in reference corpora (such as the Brown
Corpus for American English [14] and the LOB Corpus for British English [15]) can be used to
define the n-graphs that will be substituted (without the need to encode the grammar separately,
making it possible to have the algorithm work in an online manner rather than offline). However,
although this method can be quite effective, what we have found to be most effective for our
scheme is to determine the list of n-graphs that define the grammar in a single or double pass
through the text being compressed prior to the compression phase, and then encoding the
grammar separately along with the corrected text which is encoded using PPM.
Our method replaces common sequences similar to both Sequitur and Re-Pair, but unlike them,
this is not done iteratively on the current most common digram sequence or phrase, but is done by
replacing the most common sequences as the text is processed from beginning to end in a single
pass (although further passes may occur later). Also, the PPM algorithm is used as the encoder
once the common sequences have been replaced whereas Re-Pair uses a dictionary based
approach for the encoding stage. Like Re-Pair, our method is only off-line during the phase which
generates the grammar.
Our approach adapts the bigraph replacement text pre-processing approach of Teahan [7] by
using an offline technique to generate the list of bigraphs first from the source file being
compressed. This approach is considered within a grammar-based context, and the approach is
further extended by considering multiple passes and recursive grammars.
Figure 1 shows the whole process of GR-PPM. First, the CFG will be generated from the original
source files by taking the N most frequent n-graphs and replacing them with the non-terminal

5
symbols as defined by their rules in the grammar. After rules are produced, the sender will do
grammar-based pre-processing to correct the text. Then, the corrected text is encoded by using
PPMD, and the resulting compressed text is then sent to the receiver. The receiver then decodes
the text by using PPMD to decompress the compressed file that was sent. Grammar-based post-
processing then facilitates the reverse mapping by replacing the encoded non-terminal symbols
with the original n characters or n-graphs.
Figure 1: The complete process of grammar based pre-processing for Prediction by Partial Matching (GR-
PPM).
One final step is the need to encode the grammar so that it is also known by the receiver since, as
stated, we have found we can achieve better compression by encoding separately a grammar
generated from the source file itself rather than using a general grammar (known to both sender
and receiver) that was generated from a reference corpus. We choose to use a very simple method
for encoding the grammar—simply transmitting the lists of bigraphs or trigraphs directly. (So
encoding a grammar containing N = 100 rules for the GRB-PPM scheme where n = 2 and
character size is 1 byte (for ASCII text files, say), this will incur an overhead of N × n bytes or
200 bytes for each file being encoded).
Table 2 illustrates the process of GRB-PPM using a line taken from the song by Manfred Mann
(one published paper uses this song as a good example of repeating characters [12]). The
sequence is “do wah diddy diddy dum diddy do”. For GRB-PPM, for example, there are five
bigraphs which are repeated more than once in the first pass through the text: in, do, di and dd .
Thus, these bigraphs will be substituted with new non-terminal symbols, say A, B, C and D,
respectively, and included in the grammar (for example, if we choose N = 4). Note that
substitution is performed as the text is processed from left to right. If a bigraph is substituted, the
process will move to the character following the bigraph before continuing.
Original
Text
Generate
Grammar
Original Text and
Grammar
Grammar-based
Pre-processing
Corrected Text and
Grammar
Encode
Compressed Text and
GrammarDecode
Corrected Text and
Grammar
Grammar-based
Post-processing

6
Table 2: An example of how GR-PPM works for a sample text.
Pass
 
Grammar  String & Corrected Strings
singing.do.wah.diddy.diddy.dum.diddy.do
1st A→in, B→do, C→di, D→dd sAgAg.B.wah.CDy.CDy.dum.CDy.B
2nd E→Ag, F→CD sEE.B.wah.Fy.Fy.dum.Fy.B
In GRBB-PPM (Grammar Bigraph to Bigraph for PPM), new rules are formed in a second pass
through the text in the corrected text that resulted from the first pass for the further bigraphs, Ag,
CD. These will then be represented with non-terminals E and F that are added to the expanded
grammar. After all bigraphs have been substituted, the message is reduced to the new sequence
sEE.B.wah.Fy.Fy.dum.Fy.B.
Note that we ignore spaces and any punctuation characters because based on our experiments,
including these symbols decreases the compression rate. Moreover, the grammar will be
transmitted to the receiver with the original text after all bigraphs are substituted in the original
text with their non-terminal symbols. In the above example, it is clear that the number of symbols
is reduced from 31 symbols in the original text to 23 symbols in the first pass (GRB-PPM) and to
20 symbols in the next pass (GRBB-PPM).
The grammar in both GRB-PPM and GRT-PPM share the same characteristic, which is that no
pair of characters appears in the grammar more than once. This property ensures that every
bigraph in the grammar is unique, a property called non-terminal uniqueness using the same
terminology proposed by Neville-Manning and Witten [11]. To make sure that each rule in the
grammar is useful, the second property, referred to as rule utility, is that every rule in the
grammar is used more than once in the corrected text sequence. These two features are in the
grammar that GR-PPM generates and are discussed in more detail next.
2.1. NON-TERMINAL UNIQUENESS
For the more general case (i.e. considering n-graphs, not just bigraphs), each n-graph has to
appear only once in the grammar and is also substituted once by a non-terminal symbol. To
prevent the same n-graph from occurring elsewhere in the corrected text sequence, each n-graph
is substituted based on the n-graph that will be generated by the algorithm. In the example of
Table 2, the list of most frequent bigraphs that form the grammar are added at the same time as
the text is processed in each pass. For example, when di appears in the text more than once, the
new non-terminal symbol A is substituted. On the other hand, in the Sequitur algorithm [11], only
the most frequent digram (i.e. bigraph using our terminology) are added incrementally to the
grammar.
2.2.RULE UTILITY
Every rule in the grammar should be used more than one time to ensure that the rule utility
constraint is applied. When di appears in the text more than once, the new non-terminal symbol C
is substituted. In our approach, rule utility does not require the creating and deleting of rules,
which makes the rules more stable. This method retains the tracking of long files, thus avoiding

the requirement of exterior data structures. However, in the Sequitur algorithm, when the new
letter in the input appears in first rule
digram [11]. The process of deleting and creating of the rules in the grammar each time when the
new symbol appears is unstable and inefficient. In the Sequitur approach, the grammar
dynamically created. To avoid separate data structures, we apply a multi
PPM and add multiple symbols to the grammar at the same time and this causes a greater stability
and efficiency.
2.3.HIERARCHICAL GRAMMATICAL
In order to further illustrate our method, Figures 2a and 2b show the hierarchical grammatical
structures that are generated by our
The hierarchical grammatical structures are formed based on the most frequent two characters or
bigraph in each text. For example, in Figure 2a, the word
bigraph in GRB-PPM, and the non
bigraph for the next pass for GRBB
Figure 2: Hierarchical structure in the grammars generated by our algorithm for sample sequences in two
The same algorithm generates the Arabic version in Figure 2b, where the word
‫ﻛﻒ‬ and ‫ﻩ‬in a similar way (‫ﻛﻒ‬
spaces to make them more visible,
algorithms and structures. On the other hand, in the Sequitur algorithm, spaces are part of the
algorithm, which means it is also part of the grammatical structures generated by the algorithm.
Figure 3: Pseudo
Figure 3 summarises the algorithm using pseudo
passes the algorithm performs (from 1 up to a maximum of
to find the N most frequent bigraphs and substitute them with
through 5 implement the bigraphs utility constraints. Line 7 compresses the final text file by using
PPMD after substituting N bigraphs.
ears in first rule S, the grammar creates a new rule and deletes the old
]. The process of deleting and creating of the rules in the grammar each time when the
new symbol appears is unstable and inefficient. In the Sequitur approach, the grammar
dynamically created. To avoid separate data structures, we apply a multi-pass approach for GR
RAMMATICAL STRUCTURES
structures that are generated by our approach for two different languages, English and Arabic.
bigraph in each text. For example, in Figure 2a, the word the is split into th and
PPM, and the non-terminal that represents it and the letter e forms the second
bigraph for the next pass for GRBB-PPM), and so on for other words in the texts.
languages: (a) English (b) Arabic.
The same algorithm generates the Arabic version in Figure 2b, where the word ‫ﻛﻔﻪ‬is also split into
is the bigraph and ‫ﻩ‬is the second bigraph). We use bullets for
more visible, but nevertheless, spaces will not be considered in our
ure 3: Pseudo-code for the GR-PPM algorithm.
Figure 3 summarises the algorithm using pseudo-code. Line 1 is for a loop to define how many
passes the algorithm performs (from 1 up to a maximum of P passes). Lines 2 to 4 are for a loop
requent bigraphs and substitute them with non-terminal symbols. Lines 3
bigraphs.
7
e and deletes the old
]. The process of deleting and creating of the rules in the grammar each time when the
new symbol appears is unstable and inefficient. In the Sequitur approach, the grammar is being
pass approach for GR-
guages, English and Arabic.
and e (th is the
forms the second
is also split into
is the second bigraph). We use bullets for
but nevertheless, spaces will not be considered in our
code. Line 1 is for a loop to define how many
passes). Lines 2 to 4 are for a loop
terminal symbols. Lines 3

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3. EXPERIMENTAL RESULTS
This section discusses experimental results for variants of our method GR-PPM de-scribed above
for compression of various text files compared with other well-known schemes.
We have found that variants of the GR-PPM algorithm achieve the best compression ratio for
texts in different languages, such as English, Arabic, Chinese, Welsh and Persian. Also, we have
compared the results with those of different compression methods that are known to obtain good
results, including Gzip and Bzip2. BS-PPM uses order 4 PPMD to compress UTF-8 text files and
is a recently published variant of PPM that achieves excellent results for natural language texts
[9]. For both GRB-PPM and GRT-PPM, we use the 100 most frequent bigraphs or trigraphs and
order 4 PPMD for the encoding stage.
Table 3 compares the results of using GRB-PPM and GRT-PPM (using N = 100 and order 4
PPMD) with different versions of well-known compression methods, such as Gzip, Bzip2 and
BS-PPM order 4. (PPMD has become the de facto standard for comparing variants of the PPM
algorithm, and so is included with our results listed here.) We do these experiments with different
text files in different languages, including American English, British English, Arabic, Chinese,
Welsh and Persian. We use the Brown corpus for American English and LOB for British English.
For Arabic, we use the BACC [16]. For Persian, the Hamshahri corpus is used [17]. The LCMC
corpus is used for Chinese [18] and the CEG corpus is used for Welsh [19].
It is clear that GRB-PPM achieves the best compression rate (shown in bold font) in bits per
character (bpc) for all cases in different languages. Also, GRB-PPM is significantly better than
various other compression methods. For instance, for Arabic text, GRB-PPM shows a nearly 45%
improvement over Gzip and approximately 15% improvement over Bzip2. For Chinese, GRB-
PPM shows a 36% improvement over BS-PPM and 38% improvement over Gzip. For the Brown
corpus, GRB-PPM shows nearly a 35% improvement over Gzip and approximately 15%
improvement over Bzip2. For the LOB corpus, GRB-PPM shows a 36% improvement over Gzip
and 15% improvement over Bzip2. For Welsh, GRB-PPM shows a 31% improvement over BS-
PPM and 48% improvement over Gzip. For Persian, GRB-PPM shows nearly a 50%
improvement over Gzip and approximately 22% improvement over Bzip2.
GRBB-PPM and GRTT-PPM, which are the second passes of GRB-PPM and GRT-PPM
respectively, achieve better compression ratios than their single pass variants for all the different
language texts, such as English, Arabic, Welsh and Persian. In GRBB-PPM and GRTT-PPM, we
use the 100 most frequent bigraphs for both passes and order 4 PPMD for the encoding stage as
before.

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Table 3: GRB-PPM and GRT-PPM compared with other compression methods for different natural
language texts.
File Language Size Gzip Bzip2 PPMD4 BS-
PPM
GRT-
PPM
GRB-
PPM
(bpc) (bpc) (bpc) (bpc) (bpc) (bpc)
Brown American
English
5968707 3.05 2.33 2.14 2.11 2.03 1.99
LOB British
English
6085270 2.95 2.22 2.03 2.10 1.92 1.88
LCMC Chinese   5379203 2.58 1.64 1.61 2.49 1.61 1.59
BACC Arabic  69497469 2.14 1.41 1.68 1.32 1.29 1.21
Hamshahri Persian  53472934 2.58 1.64 1.68 1.25 1.31 1.22
CEG Welsh 6753317 2.91 1.95 1.69 2.20 1.57 1.51
Average 2.14 1.86 1.80 1.91 1.62 1.56
Table 4 shows the results of the different single pass and double pass variants of GR-PPM. The
results for the single pass variants GRB-PPM and GRT-PPM have been included from Table 3 for
ease of comparison as well as the results of the grammar-based compression algorithm Sequitur.
It is clear that GRBB-PPM achieves the best compression rate (bpc) for almost all cases in the
different language texts with only the single result on the Chinese text, the LCMC corpus, being
better for the Sequitur algorithm. For Arabic, GRBB-PPM shows 29% improvement over PPMD
and a nearly 19% improvement over the Sequitur algorithm. For American English, Brown
GRBB-PPM shows 8% improvement over PPMD and a nearly 23% improvement over Sequitur.
For British English, LOB GRBB-PPM shows a 20% improvement over the Sequitur algorithm.
For Welsh, GRBB-PPM shows 12% improvement over PPMD and 27% improvement over the
Sequitur algorithm. For Persian, GRBB-PPM shows 27% improvement over PPMD and a nearly
14% improvement over Sequitur.
Table 4: Variants of GR-PPM compared with other compression methods for different natural language
texts.
File PPMD
Order4
Sequitur GRT-
PPM
GRTT-
PPM
GRB-
PPM
GRBB-
PPM
(bpc) (bpc) (bpc) (bpc) (bpc) (bpc)
Brown 2.14 2.55 2.03 2.00 1.99 1.97
LOB 2.03 2.34 1.92 1.90 1.88 1.88
LCMC 1.61 1.45 1.61 1.61 1.59 1.59
BACC 1.68 1.47 1.29 1.29 1.21 1.20
Hamshahri 1.68 1.42 1.31 1.31 1.22 1.22
CEG 1.69 2.04 1.57 1.54 1.51 1.49
Average 1.80 1.87 1.62 1.60 1.56 1.55
Table 5 shows the compression rate for PPMC, Sequitur [20], Gzip, GRB-PPM and GRBB-PPM
on the Calgary corpus. Overall, the GRBB-PPM algorithm outperforms all the well-known
compression methods. For the Sequitur algorithm, GRBB-PPM shows on average a nearly 19%
improvement and on average a 17%, 12% and 1% improvement over Gzip, PPMC and GRB-
PPM, respectively. Although GRBB-PPM achieves similar results on the book1, book2, news and
pic files compared to GRB-PPM, GRBB-PPM is better than GRB-PPM for the other files.

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Table 5: Performance of various compression schemes on the Calgary Corpus.
File Size PPMC Gzip Sequitur GRB- PPM GRBB-
PPM
(bytes) (bpc) (bpc) (bpc) (bpc) (bpc)
bib 111261 2.12 2.53 2.48 1.87 1.85
book1 768771 2.52 3.26 2.82 2.25 2.25
book2 610856 2.28 2.70 2.46 1.91 1.91
news 377109 2.77 3.07 2.85 2.32 2.32
paper1 53161 2.48 2.79 2.89 2.34 2.32
paper2 82199 2.46 2.89 2.87 2.29 2.26
pic 513216 0.98 0.82 0.90 0.81 0.81
progc 39611 2.49 2.69 2.83 2.36 2.33
progl 71646 1.87 1.81 1.95 1.66 1.61
progp 49379 1.82 1.82 1.87 1.70 1.64
trans 93695 1.75 1.62 1.69 1.48 1.45
Average 2.14 2.29 2.32 1.91 1.88
4. SUMMARY AND CONCLUSIONS
In this paper, we have described new algorithms for improving the compression for different
natural language texts. These algorithms work by substituting a repeated symbol (bigraph or
trigraph) with a non-terminal symbol from a grammatical rule in a CFG before using PPM to
compress the text files. These algorithms are maintained by two constraints, which are non-
terminal uniqueness and rule utility. These techniques also work well as good compression
methods for general text files.
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