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STOCHASTIC CONTEXT FREE GRAMMARPowerPoint Presentation

STOCHASTIC CONTEXT FREE GRAMMAR

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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)

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