STOCHASTIC CONTEXT FREE GRAMMAR

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STOCHASTIC CONTEXT FREE GRAMMAR. PARSING &amp; USE. OUTLINE. Introduction to Stochastic Context Free Grammar(SCFG) Parsing of SCFG Use to RNA secondary structure prediction. SCFG. Chomsky hierarchy:. CONTEXT FREE GRAMMAR It’s a triple where: ∑ = set of terminal symbols(alphabet)

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### STOCHASTIC CONTEXT FREE GRAMMAR

PARSING & USE

OUTLINE
• Introduction to Stochastic Context Free Grammar(SCFG)
• Parsing of SCFG
• Use to RNA secondary structure prediction
SCFG

Chomsky hierarchy:

• CONTEXT FREE GRAMMAR
• It’s a triple where:
• ∑ = set of terminal symbols(alphabet)
• V = set of non terminal symbols
• R = set of production rules in the form:
• S=special start symbol and ∑ П V=Φ

A string can be derived from another string ( ) if:

and the production is a production of the grammar.

SCFG

A Stochastic Context Free Grammar is a quadruple G=(∑,V,R,P):

Probability function

constraint

Def.: Let G=(∑,V,R,P) a SCFG and a derivation sequence d,

where is a string of non terminal symbols, the probability of the derivation d is:

,

SCFG
• Grammar can be ambiguous
• Def. : The probability of SCFG G that produce the string s, is: , where are the derivation sequences that produces s.
SCFG
• Chomsky Normal Form(CNF)
• Def.: A CFG(or SCFG) is in CNF if all the rules are in this form:

B and C non terminal symbol

αis a single terminal symbol

SCFG - Parsing
• Parsing process

sequence

Parser

(synctacticanalyzer)

Parse tree

Give a sequence and a grammar, which is the best parse tree that generate the sequence, wath is which is the parse tree with the highest probability?

CYK algorithm

SCFG - Parsing
• CYK algorithm (Cocke-Younger-Kasami)
• High usedfor NLP(NaturalLanguage Processing)
• Dynamicprogramming
• Work with SCFG in CNF
SCFG – Parsing
• Input: SCFG G in CNF and word s.
• Data Structure: dynamic programming 3-D arrray holds the maximum probability for a constituent with non terminal a spanning words i…j. Back-pointers to construct the parse tree.
• Output: maximum probability parse.
SCFG - Parsing
• Initialization: n = length of ,R = number of nonterminals in G.

Table P[n,n,R] = 0 // set all values in table to 0.

Triples G[n,n,R] = triples of (position,nonterminal1,nonterminal2). //traceback pointers

For j = 1 to n do

for all unit productions of type do

if s[j] == then

set P[j,1,V] = Pv() // the probability of the production

set G[j,1,V] = new Triple(0,0,0) // indicates no further traceback - i.e. a child node

end if

end for

end for

SCFG - Parsing
• Mainloop:

//i is the length of the span, j the start and k where to split into two subspans

for i = 2 to n do

for j = 1 to n-i+1 do

for k = 1 to i-1 do

for all productions of type do

set newprob = P[j, k, X] *P[j + k, i – k, Y ]*pv(XY )

if newprob > P[j, i, V ] then

set P[j, i, V ] = newprob

set G[j, i, V] = new Triple(k,X,Y) // new traceback // point

end if

end for

end for

end for

end for

P[1][n][Start symbol in G] holds the probability of the most likely parse.

SCFG - Parsing
• Memory cost: O(n^2*M)
• Time cost: O(n^3*T)

n=length of the input string

M=number of non terminal symbols

T=number of production rules in the type V-->XY

SCFG - Use
• RNA primary structure: a nucleotide sequence constituent the mulecule, represented with a single string of the {a,c,g,u} alphabet
• RNA secondary structure: refer to the retreat of the sequence(that is the primary structure) to her-self, due to the action of hydrogen link.
SCFG - Use

Stem & loop

SCFG - Use
• The secondarystructureof RNA isimportantbecause:
• RNA “preserve” thisstructureduring the time
• It’s common findsimilar RNA thathave the similarsecondarystructure, butdifferntsequenceofnucleotides
• Evolutionof the RNA “follow” hisstructure

Sequenceanalysisof RNA is more difficultthan DNA and otherproteins

SCFG - Use
• Problem:

- Prediction of RNA secondary structure for a single sequence?

Analogy with SCFG

Calculate the most likely “parse tree” that derive a string

SCFG - Use
• Simple grammar for RNA:
• S -> gSc | cSg | aSu | uSa | ε (complementary couples)
• S -> aS | cS | gS | uS (left single basis)
• S -> Sa | Sc | Sg | Su (right single basis)
• S -> a | c | g | u (single basis)
• S -> SS (fork)
SCFG - Use

Nucleotides sequence

String

RNA secondary structure

Parse tree