Horn Clause Computation by Self-Assembly of DNA Molecules Hiroki Uejima Masami Hagiya Satoshi...
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![Page 1: Horn Clause Computation by Self-Assembly of DNA Molecules Hiroki Uejima Masami Hagiya Satoshi Kobayashi.](https://reader035.fdocuments.us/reader035/viewer/2022081603/56649f335503460f94c50071/html5/thumbnails/1.jpg)
Horn Clause Computation by Self-Assembly of DNA Molecules
Hiroki UejimaMasami Hagiya
Satoshi Kobayashi
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Previous Works(SIMD Type Computation) Solution to HPP by Adleman (1994)
For a 7-vertex directed graph Adleman-Lipton paradigm (1995)
Solution candidates are randomly generated. Real solutions are selected from among the genera
ted candidates. Applying a single operation to multiple molecul
es expressing data at once.
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Previous Works(Computational Power/Model) The correspondence between forms of DNA m
olecule and computational power based on formal languages.
Various computational models Branching program Turing machine Boolean circuit Random Access Memory Horn clause computation (Kobayashi)
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Horn Clause Computation Model by Kobayashi Each molecule corresponds to a
Horn clause. One step of derivation is realized
by one biological operation. SIMD type computation The number of operations is
proportional to the size of problem.
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Previous Works(Autonomous Computation) Computation proceeds
autonomously by self-assembly of DNA.
Possible to keep the number of operations constant.
Computation with DNA tiles A simulation of 1-D cellular automata String tiling
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Structure of DNA Tile
X
X
X
Y
Y
Z
Z
Z
Y
W
W
W
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cf. Winfree’s DNA Tile
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Contribution of This Work A Proposal and an analysis of a
new model of DNA computation Based on Horn clause computation Autonomous by self-assembly of DNA
molecules A theoretical research on a
possibility of molecular computation.
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Outline of The Algorithm To generate ground Horn clauses by vari
able substitution, using string tiles. The ground clauses are generated randomly
by self-assembly of DNA. This phase proceeds autonomously.
To make a deduction on the ground clauses. This phase also proceeds autonomously.
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Horn Clause Usedin This Algorithm A term in a rule is the form f1(…fn(X)…). The arity of a predicate is at most 2. The arity of a function is 1 The variable of the 1st argument of an at
om is X, the 2nd is Y. A fact contains no variables.
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Correspondence between DNA and Horn Clause DNA molecule expressing Horn
clause Fact molecule Rule molecule
~Q ~R
P
Q
~Q
P
P ← Q, RP ← QQ
sticky end
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The Resolution Principleby Self-Assembly of DNA
~Q
~R
P
Q
~S
~T
P ← Q, R
Q ← S, T
P ← Q, R Q ← S, T
P ← S, T, R
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Result Detection To put query molecules in To ligate molecules To detect a circular form
molecule~P P
The query molecule to
detect the fact P
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Start!
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Self-assembly
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Self-assembly
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Putting query molecules in
Query molecule
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Ligation
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Another example of circular molecule
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Computational Complexity Time complexity
(The number of operations): constant Space complexity
(The minimum number of molecules to derive a fact): O(2n)
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What’s String Tile Proposed by Winfree et al. (2000) String tiling is the collapse of multi-layer
assemblies into simpler superstructures. A string tile has a directed graph inside, t
he edges of the graph corresponds to DNA strands.
The graphs are connected with each other by hybridization of tiles.
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Variable Substitutionby Self-Assembly of String Tile
P(f(X), Y) ← Q(X, g(Y))a / Y g(X) / X b / X
P(f(g(b)), a) ← Q(g(b), g(a))
Substitution tile Substitution tileSeed tile
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A(f(X),Y) ← B(X, g(Y)), C(X, Y)
g(X) / X b / Xa / Y
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A(f(g(b)), a) ← B(g(b), g(a)), C(g(b), a)
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A(f(g(b)), a) ← B(g(b), g(a)), C(g(b), a)
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A(f(g(b)), a) ← B(g(b), g(a)), C(g(b), a)
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B(g(b), g(a))
C(g(b), a)
A(f(g(b)), a)
A(f(g(b)), a) ← B(g(b), g(a)), C(g(b), a)
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A(f(g(b)), a)
B(g(b), g(a))
C(g(b), a)
A(f(g(b)), a) ← B(g(b), g(a)), C(g(b), a)
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NTM Simulation by Horn Clause Computation
Configuration is expressed by fact. Ss(ft(-1)(ft(-2)(fb(a1))), ft(0)(ft(1)(fb(fb(a2)))))
Transition rule is expressed by rule. Ss’(X, ft(-1)(ft’(0)(Y))) ← Ss(ft(-1)(X), ft(0)(Y)) Ss’(ft’(0)(X), Y) ← Ss(X, ft(0)(Y))
b t(-2) t(-1) t(0) t(1) b b
s
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Features of Our Model Autonomous computation keeps
the number of operations constant. Our model is equivalent to non-
deterministic Turing machine. Variable substitution phase are
separated from deduction phase completely.
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Advantage of Our Model Close relation to high-level
programming language PROLOG (Horn clause computation)
More suitable for expressing complex algorithms than other models.
Small number of operations(Autonomous computation)
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Weak Point of Our Model Error-prone deduction
Term encoding has problem Too long sticky end Biased deduction
Estimation of complexity is not appropriate. Time complexity: Time to reach equilibrium is more
appropriate than the number of operations. Space complexity: More molecules will be required
because multiple proof trees are generated. 3-D conformation of proof tree molecule
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Future Works Thermodynamic/kinetic analysis of
autonomous DNA computation Optimization of parameters
according to the analysis Temperature Salt concentration
Analysis of DNA computation as probabilistic algorithm