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Circular Post Machines

Artiom Alhazov Università degli Studi di Milano-Bicocca, Milano Alexander Krassovitsky Rovira i Virgili University, Tarragona Yurii Rogozhin. Circular Post Machines. P systems with Exo- Insertion and Deletion. AND. bbbabcbbb. bbb bbb. abc. q. abbaabba baabbaab.

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Circular Post Machines

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  1. Artiom Alhazov Università degli Studi di Milano-Bicocca, Milano Alexander Krassovitsky Rovira i Virgili University, Tarragona Yurii Rogozhin CircularPost Machines P systems with Exo- Insertion and Deletion AND bbbabcbbb bbb bbb abc q abbaabba baabbaab Institute of Mathematics and Computer Science, Academy of Sciences of Moldova, Chişinău

  2. Insertion/Deletion • Are fundamental string operations in FLT • & motivated by mismatched annealing, point mutations etc. • Insert α between u and v: • like rewriting rule uvuαv • x1uvx2x1uαvx2 • Delete α between u and v: • like rewriting rule uαvuv • x1uαvx2x1uvx2 • In the main part of the presentation the contexts are always empty

  3. Insertion/Deletion Systems • (Alphabet,Axioms,Rules,Terminals) • Computationally complete without contexts • Size: 6-tuple of numbers bounding lengths of • (inserted strings, left insertion contexts, right insertion contexts; deleted strings, left deletion contexts, right deletion context) • E.g., IDSs of size (2,0,0;3,0,0) and (3,0,0;2,0,0) are universal, but not those of size (2,0,0;2,0,0).

  4. Insertion/Deletion PSystems • (V,T,µ,M1,…,Mk,R1,…,Rk) • Rules are ins/del rules with target indications • Without context: can be written as • ins(α,tar), del(α,tar) • Multiplicities are not tracked, evolution proceeds in all possible ways and the original string remains • This does not influence the result • P systems of size (2,0,0;2,0,0) still not universal

  5. Exo- Insertion/Deletion PSystems • Rules are applied at left or right, written as • insL(α,tar), insR(α,tar), delL(α,tar), delR(α,tar) • Ex.: L({a,b},{a,b},[ ]1,{babbab}, • {delL(b,here); insR(a,here)})= {λ,b}abbab{a}* • The class of such systems is denoted • SPk(e-insm0,0,e-deln0,0) • The red comma is replaced by < if deletion rules have priority over insertion rules

  6. Other Notations • P  tP: for tissue P systems • e-  r- or l-: for right-only or left-only rules • E prefix: result is collected modulo the terminal alphabet • RE: recursively enumerable languages • REG: regular languages

  7. Circular Post machines q abbaabba baabbaab • A variant of Turing machines • On a circular tape • One-way: right move rewriting the symbol • Can insert a symbol at left or right • Can delete a symbol read • Are computationally complete • E.g., accepting CPM0 simulate DTM Intro- duced in this paper

  8. Circular Post machines++ q abbaabba baabbaab • A final state qf is distinguished • The configuration [picture] can be written as • qbaabbaababbaabba • Ex: qbcra  raaabbaababbaabbac

  9. CPM5 example: Collatz problem • n=1:stop. n=2m:nm,repeat. n=2m+1:n3m+2,repeat. • Start in q1anM. q1aq2, q2Mqf, q2aq3, q3aq4, q4aq5, q5aq3, • q4Mq6, q6Mq1, q5Mq7, q7Mq8, • q8aq9, q9aq10, q10aq11,q11aq8, q10Mq12,q12Mq1. • q1aaaMq2aaMq3aMq4aMaq5Maq7aq8aMq9aMaq10aMaaq11Maaq8Maaa • q9Maaaaq10Maaaaaq12aaaaaq1aaaaaMq2aaaaMq3aaaMq4aaaMaq5aaMa • q3aMaq4aMaaq5Maaq7aaq8aaMq9aaMaq10aaMaaq11aMaaq8aMaaa • q9aMaaaaq10aMaaaaaq11Maaaaaq8Maaaaaaq8Maaaaaaq9Maaaaaaa • q10Maaaaaaaaq12aaaaaaaaq1aaaaaaaaMq2aaaaaaaMq3aaaaaaMq4aaaaaaMa • q5aaaaaMaq3aaaaMaq4aaaaMaaq5aaaMaaq3aaMaaq4aaMaaaq5aMaaa • q3Maaaq4Maaaaq6aaaaq1aaaaMq2aaaMq3aaMq4aaMaq5aMaq3Ma • q4Maaq6aaq1aaMq2aMq3Mq4Maq6aq1aM q2Mqf. q1:start q2:one q4:even q5:odd q8:triple

  10. CPM0 by CPM5 • Erasing rule: itself • Rewriting rule: pxpx, pxyq • Lengthening: p0p0, p0yp’0, p’0aqr’, raqq, • r’0r, rxrx, rxxr • Skipping some configurations allowed: • Faster simulation is possible • p0p0, p0yp’0 • If next instruction is q0s, set p’0=s • If next instruction is q0zs, add rule p’0zs • If next instruction is q0zs0, set p’0=q0

  11. Non-deterministic CPM5 • Generating RE is possible also under conditions: • Non-final states: disjoint union of Q1 and Q2. • For every state pQ1 and x, there is exactly 1 instruction pxq • For every state pQ2, there are 2 instructions pyq1, pyq2. • Indeed, use q11q1, q11q2 to yield q21+, and start a DCPM5 in q2 to simulate a DTM converting 1+ into any RE language.

  12. eIDP: Known results • Recall that SP(ins20,0,del20,0) are NOT universal • SP(e-ins20,0,e-del10,0) are universal • SP(e-ins10,0,e-del20,0) are universal • StP(e-ins10,0,e-del10,0) are universal • (*) SP(e-ins10,0,e-del10,0) --- still open • Main result of this paper: • SP(e-ins10,0<e-del10,0) are universal • Another result: a lower bound for (*)

  13. Main: LSP(e-ins10,0<l-del10,0)=RE delL(S) delL(zf) delL(zi) delL(qi) insL(zi) insL(S) qiajql delL(qj) delL(aj) insL(ql) delL(S) delL(zf) insR(ak) insL(zi) delL(zi) delL(qi) qiakql insL(ql) delL(qj)

  14. Low: ELSP(r-ins10,0,r-del10,0)REG delR(S) delR(qf) delR(qi) insR(S) qiajql m insR(aj) insR(ql)

  15. Conclusions bbbabcbbb bbbbbb abc • LSP(e-ins10,0<e-del10,0) =RE. • ELSP(e-ins10,0,e-del10,0) – open, REG. • Boxes: result valid in one-sided case (r or l). • We introduced an instrument: CPM5 • pxq, pyq • Thank you for your attention q abbaabba baabbaab

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