Download Horn Clause Computation with DNA Molecules

Horn Clause Computation with
DNA Molecules
Satoshi Kobayashi
Journal of Combinatorial Optimization, v.3
pp.277-299, 1999
Summarized by In-hee Lee
1.Introduction

Horn program
 Subclass of the formulas of first-order logic
 Have cloas relation to PROLOG language.
 Finite set of Horn clauses is computationally equivalent
to Turing machine.

Its parallel implementation with huge number of
molecules might have possibility to overcome the
computational power of conventional computers.
 Propose an experimental method for implementing
deduction with a subclass of Horn programs.
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2.Simple Horn Program

Simple Horn clause
 Three disjoint sets C, V, R
 C:
set of constants, V: set of variables, R: set of relation
symbols
 Each relation symbol A is associated with nonnegative integer,
arity, ary(A)
 Connectives , , and a quantifier 
 Simple atomic formula(atom): A( X 1 ,, X ary( A) )
 Simple ground atom(ground atom)
 Simple Horn clause(clause): X 1 ,, X m ( F  F1    Fn )
 Fact(n=0),
rule(n>0)
Head
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Body
2.Simple Horn Program

Herbrand base of H (BH)
 The set of all ground atoms with relation symbols and
constants used in H.

Substitution
 A mapping from V to VC.
 A ground instance is an expression which doesn’t
contain any variable.

Immediate consequence operator TH(I)
 The set of all ground atoms can be proved by one step
derivation using I and H.
TH ( I )  { A  BH | {B1 ,, Bk }  I 
A  B ,, B is a ground instance of a clause in H
k
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Lab, http://bi.snu.ac.kr/
 H is a simple Horn program  I is a set of simple ground atoms }
3.Deduction with DNA Molecules

Elementary form of simple Horn program
 Corresponding to graph G(H)=(V,E)
V  { Ai | A  R  1  i  ary ( A)}
E  {( Ai , B j ) | A and B are in the same body of a rule in H and
ith argument of A and jth argument of B are the same variable}





The arguments of every rule are only variables
Any relation symbol appear in the body of rule at most once.
Any variable appears in the body of rule at most two.
Every fact is ground
Every variable appeared in the head of a rule must appear in the
body of that rule
 Corresponding graph is bipartite.
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3.Deduction with DNA Molecules
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3.1
Overall
Procedure
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3.2 Representation of Atoms
A( X 1 ,, X k )
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3.2 Representation of Atoms

Since G(H) is bipartite, V can be divided two
disjoint set V1 and V2.
 V1: CCCAi'GGGXCCCAi GGG
 V2: GGGAi'CCCXGGGAi CCC
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3.3 Argument Equality Check

For every pair (Ai, Bj) in G(H)
1. Add Ai R by ligation and run whiplash PCR with {A,
C, T}
2. Destroy the binding to beads
R
3. Extract strands which have B j by beads.
4. Amplify extracted strands
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3.3 Argument Equality Check
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3.4 Argument Copy

For every variable x in head of a rule r and edge(x,
y) in G(H)
 Argument copy(x, y)
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3.4 Argument Copy
Copying ith
argument of A
to jth argument
of B
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3.4 Argument Copy

C2
Example: make A(d1, d3) from B(d1, d2)C(d3, d4)
D4
C2
C1
D3
C1
A2
D3
C1
B2
D2
B2
B1
A1
D1
D1
B1
B1
A2
D3
C1
C2D4C2C1 S3 C1B2D2B2B1D1B1C1D3A2B1D1A1
A2D3A2A1 D1 A1 A2 D3 A2
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4. Discussion

The feasibility of the proposed methods depends
on that of whiplash PCR.
 Too much human intervention.
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