Horn Clause Computation with DNA Molecules Satoshi Kobayashi Journal of Combinatorial Optimization, v.3 pp.277-299, 1999 Summarized by In-hee Lee

(C) 2001, SNU Biointelligence Lab, http://bi.snu.ac.kr/ 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. (C) 2001, SNU Biointelligence Lab, http://bi.snu.ac.kr/

(C) 2001, SNU Biointelligence Lab, http://bi.snu.ac.kr/ 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): Simple ground atom(ground atom) Simple Horn clause(clause): Fact(n=0), rule(n>0) Head Body (C) 2001, SNU Biointelligence Lab, http://bi.snu.ac.kr/

(C) 2001, SNU Biointelligence Lab, http://bi.snu.ac.kr/ 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. (C) 2001, SNU Biointelligence Lab, http://bi.snu.ac.kr/

3.Deduction with DNA Molecules Elementary form of simple Horn program Corresponding to graph G(H)=(V,E) 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. (C) 2001, SNU Biointelligence Lab, http://bi.snu.ac.kr/

3.Deduction with DNA Molecules (C) 2001, SNU Biointelligence Lab, http://bi.snu.ac.kr/

(C) 2001, SNU Biointelligence Lab, http://bi.snu.ac.kr/ 3.1 Overall Procedure (C) 2001, SNU Biointelligence Lab, http://bi.snu.ac.kr/

3.2 Representation of Atoms (C) 2001, SNU Biointelligence Lab, http://bi.snu.ac.kr/

3.2 Representation of Atoms Since G(H) is bipartite, V can be divided two disjoint set V1 and V2. V1: V2: (C) 2001, SNU Biointelligence Lab, http://bi.snu.ac.kr/

3.3 Argument Equality Check For every pair (Ai, Bj) in G(H) Add by ligation and run whiplash PCR with {A, C, T} Destroy the binding to beads Extract strands which have by beads. Amplify extracted strands (C) 2001, SNU Biointelligence Lab, http://bi.snu.ac.kr/

3.3 Argument Equality Check (C) 2001, SNU Biointelligence Lab, http://bi.snu.ac.kr/

(C) 2001, SNU Biointelligence Lab, http://bi.snu.ac.kr/ 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) (C) 2001, SNU Biointelligence Lab, http://bi.snu.ac.kr/

(C) 2001, SNU Biointelligence Lab, http://bi.snu.ac.kr/ 3.4 Argument Copy Copying ith argument of A to jth argument of B (C) 2001, SNU Biointelligence Lab, http://bi.snu.ac.kr/

(C) 2001, SNU Biointelligence Lab, http://bi.snu.ac.kr/ 3.4 Argument Copy Example: make A(d1, d3) from B(d1, d2)C(d3, d4) C2 D4 C1 D3 B2 D2 B1 D1 A1 D1 B1 A2 D3 C1 C1 D3 A2 C2 D4 C1 S3 B2 D2 B1 D1 D3 A2 A1 A2 D3 A2 A1 D1 A1 A2 D3 (C) 2001, SNU Biointelligence Lab, http://bi.snu.ac.kr/

(C) 2001, SNU Biointelligence Lab, http://bi.snu.ac.kr/ 4. Discussion The feasibility of the proposed methods depends on that of whiplash PCR. Too much human intervention. (C) 2001, SNU Biointelligence Lab, http://bi.snu.ac.kr/