Uniformity is Weaker than Semi-Uniformity for Some Membrane Systems

We investigate computing models that are presented as families of finite computing devices with a uniformity condition on the entire family. Examples of such models include Boolean circuits, membrane systems, DNA computers, chemical reaction networks and tile assembly systems, and there are many others. However, in such models there are actually two distinct kinds of uniformity condition. The first is the most common and well-understood, where each input length is mapped to a single computing device (e.g. a Boolean circuit) that computes on the finite set of inputs of that length. The second, called semi-uniformity, is where each input is mapped to a computing device for that input (e.g. a circuit with the input encoded as constants). The former notion is well-known and used in Boolean circuit complexity, while the latter notion is frequently found in literature on nature-inspired computation from the past 20 years or so. Are these two notions distinct? For many models it has been found that these notions are in fact the same, in the sense that the choice of uniformity or semi-uniformity leads to characterisations of the same complexity classes. In other related work, we showed that these notions are actually distinct for certain classes of Boolean circuits. Here, we give analogous results for membrane systems by showing that certain classes of uniform membrane systems are strictly weaker than the analogous semi-uniform classes. This solves a known open problem in the theory of membrane systems. We then go on to present results towards characterising the power of these semi-uniform and uniform membrane models in terms of NL and languages reducible to the unary languages in NL, respectively.

[1]  Artiom Alhazov,et al.  Polarizationless P Systems with Active Membranes , 2004, Grammars.

[2]  Gheorghe Paun,et al.  The Oxford Handbook of Membrane Computing , 2010 .

[3]  Nancy A. Lynch,et al.  Comparison of polynomial-time reducibilities , 1974, STOC '74.

[4]  Lloyd M. Smith,et al.  DNA computing on surfaces , 2000, Nature.

[5]  Ho-Lin Chen,et al.  Active Self-Assembly of Simple Units Using an Insertion Primitive , 2013, SODA.

[6]  Mario J. Pérez-Jiménez,et al.  Characterizing Tractability by Tissue-Like P Systems , 2009, Workshop on Membrane Computing.

[7]  Ian Parberry,et al.  Circuit complexity and neural networks , 1994 .

[8]  Uzi Vishkin,et al.  Constant Depth Reducibility , 1984, SIAM J. Comput..

[9]  Eric Allender,et al.  Amplifying Lower Bounds by Means of Self-Reducibility , 2008, 2008 23rd Annual IEEE Conference on Computational Complexity.

[10]  Erik Winfree,et al.  An information-bearing seed for nucleating algorithmic self-assembly , 2009, Proceedings of the National Academy of Sciences.

[11]  Ker-I Ko.  Distinguishing Conjunctive and Disjunctive Reducibilities by Sparse Sets , 1989, Inf. Comput..

[12]  Erik Winfree,et al.  The program-size complexity of self-assembled squares (extended abstract) , 2000, STOC '00.

[13]  Neil Immerman.  Nondeterministic Space is Closed Under Complementation , 1988, SIAM J. Comput..

[14]  Niall Murphy.  Uniformity conditions for membrane systemsUncovering complexity below P , 2010 .

[15]  Artiom Alhazov,et al.  Uniform Solution to QSAT Using Polarizationless Active Membranes , 2006 .

[16]  Eric Allender,et al.  RUSPACE(log n) $\subseteq$ DSPACE(log2n/log log n) , 1998, Theory of Computing Systems.

[17]  Giancarlo Mauri,et al.  Recent complexity-theoretic results on P systems with active membranes , 2015, J. Log. Comput..

[18]  Jehoshua Bruck,et al.  Programmability of Chemical Reaction Networks , 2009, Algorithmic Bioprocesses.

[19]  Niall Murphy,et al.  A Characterisation of NL Using Membrane Systems without Charges and Dissolution , 2008, UC.

[20]  Erik Winfree,et al.  Complexity of Self-Assembled Shapes , 2004, SIAM J. Comput..

[21]  Anne Condon,et al.  Space and Energy Efficient Computation with DNA Strand Displacement Systems , 2012, DNA.

[22]  Christopher Joseph Thachuk.  Space and energy efficient molecular programming and space efficient text indexing methods for sequence alignment , 2013 .

[23]  Heribert Vollmer,et al.  Introduction to Circuit Complexity: A Uniform Approach , 2010 .

[24]  Petr Sosík.  The computational power of cell division in P systems: Beating down parallel computers? , 2004, Natural Computing.

[25]  Gheorghe Paun.  Further Twenty Six Open Problems in Membrane Computing , 2005 .

[26]  R.E. Ladner,et al.  A Comparison of Polynomial Time Reducibilities , 1975, Theor. Comput. Sci..

[27]  G Rozenberg,et al.  Computing with DNA by operating on plasmids. , 2000, Bio Systems.

[28]  N. Immerman,et al.  On uniformity within NC 1 . , 1988 .

[29]  Matthew Cook,et al.  Computation with finite stochastic chemical reaction networks , 2008, Natural Computing.

[30]  Andrew Winslow,et al.  Tight Bounds for Active Self-Assembly Using an Insertion Primitive , 2015, Algorithmica.

[31]  Artiom Alhazov,et al.  Uniform Solution of , 2007, MCU.

[32]  Lulu Qian,et al.  Supporting Online Material Materials and Methods Figs. S1 to S6 Tables S1 to S4 References and Notes Scaling up Digital Circuit Computation with Dna Strand Displacement Cascades , 2022 .

[33]  Giancarlo Mauri,et al.  Sublinear-Space P Systems with Active Membranes , 2012, Int. Conf. on Membrane Computing.

[34]  P D Kaplan,et al.  DNA solution of the maximal clique problem. , 1997, Science.

[35]  Jin-Yi Cai,et al.  Resolution of Hartmanis' conjecture for NL-hard sparse sets , 1997, Theor. Comput. Sci..

[36]  Niall Murphy,et al.  On acceptance conditions for membrane systems: characterisations of L and NL , 2008, CSP.

[37]  Erik Winfree,et al.  Active self-assembly of algorithmic shapes and patterns in polylogarithmic time , 2013, ITCS '13.

[38]  Róbert Szelepcsényi,et al.  The method of forced enumeration for nondeterministic automata , 1988, Acta Informatica.

[39]  Erik Winfree,et al.  The computational power of Benenson automata , 2005, Theor. Comput. Sci..

[40]  Agustín Riscos-Núñez,et al.  Computational efficiency of dissolution rules in membrane systems , 2006, Int. J. Comput. Math..

[41]  Neil Immerman,et al.  On Uniformity within NC¹ , 1990, J. Comput. Syst. Sci..

[42]  Erik Winfree,et al.  Two computational primitives for algorithmic self-assembly: copying and counting. , 2005, Nano letters.

[43]  Ho-Lin Chen,et al.  Fast Algorithmic Self-assembly of Simple Shapes Using Random Agitation , 2014, DNA.

[44]  Sanjeev Arora,et al.  Computational Complexity: A Modern Approach , 2009 .

[45]  Niall Murphy,et al.  AND and/or OR: Uniform Polynomial-Size Circuits , 2013, MCU.

[46]  L M Adleman,et al.  Molecular computation of solutions to combinatorial problems. , 1994, Science.

[47]  Mario J. Pérez-Jiménez,et al.  Complexity classes in models of cellular computing with membranes , 2003, Natural Computing.

[48]  Emil L. Post.  Recursively enumerable sets of positive integers and their decision problems , 1944 .

[49]  Salil P. Vadhan,et al.  Computational Complexity , 2005, Encyclopedia of Cryptography and Security.

[50]  Michael Sipser,et al.  Parity, circuits, and the polynomial-time hierarchy , 1981, 22nd Annual Symposium on Foundations of Computer Science (sfcs 1981).

[51]  Mario J. Pérez-Jiménez,et al.  Computational complexity of tissue-like P systems , 2010, J. Complex..

[52]  Eric Allender,et al.  Uniform constant-depth threshold circuits for division and iterated multiplication , 2002, J. Comput. Syst. Sci..

[53]  Ker-I Ko,et al.  On Sets Truth-Table Reducible to Sparse Sets , 1988, SIAM J. Comput..

[54]  Niall Murphy,et al.  Active Membrane Systems Without Charges and Using Only Symmetric Elementary Division Characterise P , 2007, Workshop on Membrane Computing.

[55]  Artiom Alhazov,et al.  Solving a PSPACE-Complete Problem by Recognizing P Systems with Restricted Active Membranes , 2003, Fundam. Informaticae.

[56]  A. Condon,et al.  Less haste, less waste: on recycling and its limits in strand displacement systems , 2011, Interface Focus.

[57]  Allan Borodin,et al.  On Relating Time and Space to Size and Depth , 1977, SIAM J. Comput..

[58]  Edith Hemaspaandra,et al.  SPARSE reduces conjunctively to TALLY , 1993, [1993] Proceedings of the Eigth Annual Structure in Complexity Theory Conference.

[59]  Ján Manuch,et al.  Less Haste, Less Waste: On Recycling and Its Limits in Strand Displacement Systems , 2011, DNA.

[60]  Neil Immerman,et al.  Descriptive Complexity , 1999, Graduate Texts in Computer Science.

[61]  R J Lipton,et al.  DNA solution of hard computational problems. , 1995, Science.

[62]  Niall Murphy,et al.  The computational power of membrane systems under tight uniformity conditions , 2010, Natural Computing.