Grammatical Complexity Of Symbolic Sequences: A Brief Introducton

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Grammatical ComplexityOf Symbolic SequencesA
Brief Introducton
Bailin Hao T-Life Research Center, Fudan
University Institute of Theoretical Physics,
Academia Sinica The Santa Fe Institute, New
Mexico, USA http//www.itp.ac.cn/hao/
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Three Paradigms in TheoreticalDescription of
Nature

• Deterministic based on periodicities and
  recurrences, from Kepler to Yang-Mills
• Stochastic based on randomness, from Brownian
  motion to MSR field theory of hydrodynamics and
  molecular motors
• Fractal, self-similar, scale invariant from
  phase transitions and critical phenomena to
  chaotic dynamics
• Finiteness is the unifying Physics languages

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???(language ?? philology)??

• ?????
• ???????
• Zipf ??
• ???????????????
• ????Chomsky??
• ????Lindenmayer ??(???????)
• ??????(Factorizable language)

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?????????

• ???
• ???
• ???
• ?????
• ????
• ????????????
• ?????,???????
• ????????
• ????????
• ??????????
• ????????

??? ????????? ????????? ?????????? ??????????
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An Observation

• u d c s b t
• charge, mass, flavor, charm,
• p n e
• charge, mass, spin, magnetic momentum,
• H C N O P
• atomic number, ion radius, valence, affinity,
• H2O NO CO2
• molecular weight, polarity,
• a c g t
• A D E F G H W Y V
• BRCA1 PDGF

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A PROGRAMME

• Coarse-Grained Description of Nature
• Use of Symbols and Symbolic Strings
• Language
• Grammar and Complexity
• (Chomsky, Lindenmayer, etc.)
• So far this programme has been best realized in
  the study of dynamics by using Symbolic Dynamics.
• There have been preliminary attempts in analyzing
  biological sequences.

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• It may not be a coincidence that the two systems
  in the universe that most impress us with their
  open-ended complex design life and mind are
  based on discrete combinatorial systems. Many
  biologists believe that if inheritance were not
  discrete, evolution as we know it could not have
  taken place.
• S. Pinker, The Language Instinct (1995)

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Simple Examples

• At the level of words
• DOG GOD
• At sentence level
• Dog bites Man
• Man bites Dog

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• N C
  EGF (Epidermal GF)
• N C
  Chymotrypsin (??????)
• N C
  Urokinase (UK) (???)
• N
  C Factor IX
• 
  (????IX, X-mas??????)
• N
  C Plasminogen
• 
  (???????)
• ?????????domain??
• B.Alberts ?,Mol.Biology of the Cell
  ??? 1994. P.123

Ca ????
?3?-s-s-
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• GC ?????
• ??? ?
• ?1. ? a, c, g, t
• ?2. ? A, C, D
  W, Y
• ?3. ? a, z, A,
  Z, , ,
• ????????????????
• (????) ?
• ???????????????
• ?? ???,????,????
• ????????

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Classification of Formal Languages

• Chomsky Hierarchy
• Sequential production rules
• Lindenmayer Systems
• Parallel production rules

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Generative Grammar

• S ? Sentence
• NP ? Noun Phrase
• VP ? Verb Phrase
• Adj ? Adjective
• Art ? Article

S NP VP VP V NP NP (Art) Adj N
S if S then S S either S or S
N boy girl scientist V sees
believes loves eats Adj young good
beautiful Art a one the
Non-Terminal and Terminal Symbols
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• Chomsky ????
• N ??????(?????)
• T ?????
• S ? N ????
• P ????(x y)???
• x, y ???? ?? x, y ???????????
• ?? G (N, T, P, S)
• 0 ???
• x ? (N?T) N(N?T)
• y ? (N?T)

???????????
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• 1 ??? ???????
• x t1 a t2
• t1, t2 ? T
• a ? N
• 2 ??? ???????
• x a ? N
• 3 ??? ????
• x a y b ? bc
• a, c ? N b ? ? b ? T

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?????Chomsky??
? ?? ??? ????
0 ???? REL ??? (?????) ??
1 ????? CSL ??????? ???????
2 ????? CFL ????? ??? (??)
3 ?? RGL ????? ???
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R L R R
R L R R

a
b
(i)
(ii)
? (a, R) b A Finite State Automaton (FSA)
R L
a b c
b
c
d
A transfer function
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FSA Finite State Automata

• Deterministic FSA
• Non-Deterministic SFA
• Equivalence of DFSA and NDFSA subset
  construction
• Minimal DFSA
• Myhill-Nerode theorem (1958) number of nodes in
  minDFSA

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A Pushdown Automaton

• Pushdown list
• Stack
• First In Last Out (FILO)

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A Turing MachineAlan M. Turing (1912-1954)

• FSA ? R/W tape
• Church-Turing Thesis (1936)
• Any effective (mechanical) computation can
• be carried out by a Turing machine

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Example ai b ici igt0 CSL

• Terminals a, b, c
• Non-terminal A, B
• Sequential rules B aBAc abc
• bA bb
• cA Ac
• B abc
• B aBAc aabcAc
  aabAcc
• B abAc aaBAcAc
• 
  aaBAAc
• 
  aaabcAAc
• 
  aaabAcAc aaabbAcc

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Rules to Generate Gene-Like Sequences( according
to David Searls )

• gene upstream transcript downstream
• transcript 5-untranslated-region
  start-codon coding-region
• 3-untranslated-region
• coding-region codon coding-region
  stop-codon splice
• coding region
• codon lys asn thr met glu his
  pro asp ala gly tyr
• trp phe leu ile ser
  arg gln val cys
• start-codon met
• stop-codon taa tag tga

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• leu tt purine ct base (6)
• ser ag pyrimidine tc base (6)
• arg ag purine cg base (6)
• val gt base
  pro cc base (4)
• ala gc base
  gly gg base (4)
• thr ac base
  (4)
• ile at pyrimidine ata (3)
• lys aa purine
  asn aa pyrimidine (2)
• gln ca purine
  his ca pyrimidine (2)
• glu ga purine
  cys tg pyrimidine (2)
• phe tt pyrimidine tyr
  ta pyrimidine (2)
• asp ga pyrimidine (2)
• met atg
  trp tgg
• base m a c g t
  purine a g
• primidine c t

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• splice intron intron
  gt intron-body ag
• splice a a intron splice c c
  intron
• splice t t intron splice g g
  intron
• a splice intron a c splice
  intron c
• t splice intron t g splice
  intron g
• upstream enhancer promotor enhancer
• enhancer
• promotor
• silencer
• isolator

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• These rules are capable to generate an
  unlimited
• set of gene-like sequences, mostly biological
  nonsense.
• They may be used to recognize gene-like segments
• in long DNA sequences.
• Syntax versus Semantics texts vs. grammar.
• Physics behind this coarse-grained
  description
• stereochemistry, interaction between proteins and
• DNA chains, metallic ions etc.

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Symbolic Dynamics Languages
1991
1999
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????????

• ?????????????????????????????????
• ?????????????????????
• ????????????????????????

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Subintervals determined by the periodic
kneading Sequence (RLRRC)8
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Order of visits in the periodic kneading Sequence
(RLLRC)8
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Transformations of subintervals

• a ? c d (on reading L)
• b ? d (on reading R)
• c ? b c (on reading R)
• d ? a (on reading R)

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Input L R R R
q a b c d
d 1 1 0 0
c 1 0 1 0
b 0 0 1 0
a 0 0 0 1
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Transfer Functions
R L
a c, d
b d
c b, c
d a
R L
a,b,c,d a,b,c,d c,d
c,d a,b,c
a,b,c b,c,d c,d
b,c,d a,b,c,d
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(No Transcript)
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Stefan matrix for 256P in Feigenbaum cascade
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Stefan matrix for F13233 Case (a)
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Stefan matrix for F13233. Case (b)
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Stefan matrix for F13233. Case (c)
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Stefan matrix for F13233. Case (d)
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Symbolic Dynamics Languages
1991
1999
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Development of Anabaena catenula (????????)

• br
  bl
• ar
  al
• albr
  blar
• Alphabet S ar, al, br, bl
• Production rules
• Initial symbol (axiom) ? ar
• Grammar G (S, P, ?)
• Language L (G) ? S

br ar ar albr bl al al
blar
P
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• Lindenmayer Systems
• Parallel production rules. Finer classification
• D0L Deterministic, no interaction, i.e.,
  context-free
• 0L non-deterministic, no interaction
• IL non-deterministic, with Interaction, i.e.,
  context
• sensitive
• T0L with Table of production rules
• TIL
• E0L Extended to non-terminal symbols
• ET0L
• EIL ? REL of Chomsky

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CSL
CFL
RGL
FIN
DOL
REL

• RGL Regular CFL
  Context-Free
• CSL Context-Sensitive REL Recursively
  Enumerable

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0REL
EIL

• Chomsky
• Lindenmayer
• Indexed

1CSL
IND
ET0L
IL
E0L
2CFL
T0L
3RGL
0L
D0L
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Example a la Lindenmayer

• L aibici i gt 0 CSL
• G (S, T, ?)
• ? abc
• S a, b, c
• T t1, t2
• T1 a aa, b bb, c cc
• T2 a ?, b ?, c ?
  
• 
  T0L

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Dyck language A language of nested parentheses

• Many types of parentheses
• Finite depth of nesting
• Context-free language
• Our case
• Only 3 types of parentheses
• Shallow nesting
• Conjecture (Xie) may be regular language

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• ?????
• ??????Z .G .Yu (???2001)
• ???????????
• Consensus ???????
• ????
• ????? ??????
• ?????????

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Factorizable Languages

• Symbolic dynamics leads to factorizable languages
• A complete genome defines a factorizable langauge
• An amino acid sequence with unique reconstruction
  (at certain K) defines a factorizable language

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Modeling in Biology

• Cells
• Tissues
• Organs
• Systems circulation, respiration,
  reproduction, neural, sensory, musclular, etc.
• Organisms, population, ecosystems
• Animals versus plants
• Plant development, morphology, physiology and
  pathology

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Modeling of Plant MorphologyBy using L-System

• P. Prusinkiewicz, J. Hanan, Lindenmayer Systems,
  Fractals, and Plants, LN in Biomath., vol. 79,
  Springer, 1989
• P. Prusinkiewicz, A. Lindenmayer, The Algorithmic
  Beauty of Plants, Springer, 1990
• P. Prusinkiewicz, M. Hammel, J. Hanan, R. Mech,
  Visual models of plant development, Chap.9 in
  Handbook of Formal Languages, Vol.3, Springer,
  1997

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Consistency of Macro-and Micro-Description of
Nature

• Molecular phylogeny versus phylogeny based on
  morphological features
• Modeling plant development without getting into
  molecular and cellular description
• No need to model protein folding by invoking
  quarks!

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Some Useful URLs

• www.grogra.org (Growth Grammar)
• http//www.computableplant.org
• http//algorithmicbotany.org

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• Huimin Xie ???
• Grammatical Complexity and
• 1D dynamical Systems
• Vol.6 in Directions in
  Chaos
• WSPC, 1996.
• ??? ????????
• ?????????, 1994
• Bailin Hao, Weimou Zheng, Applied Symbolic
  Dynamics and Chaos (WSPC, 1998), Chap. 8
• J.Hopcroft, J.Ullman, Introduction to Automata
  Theory, Languages and Computation,
  Addison-Wesley, 1979.

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Thanks!