Complexity of reachability problems for finite discrete dynamical systems

Sequential Dynamical Systems (SDSs) are a special type of finite discrete dynamical systems that can be used to model simulation systems. We focus on the computational complexity of testing several phase space properties of SDSs. Our main result is a sharp delineation between classes of SDSs whose behavior is easy to predict and those whose behavior is hard to predict. Specifically, we show the following. 1.Several state reachability problems for SDSs are PSPACE-complete, even when restricted to SDSs whose underlying graphs are of bounded bandwidth (and hence of bounded pathwidth and treewidth), and the function associated with each node is symmetric. Moreover, this result holds even when the underlying graph is d-regular for some constant d and all the nodes compute the same symmetric Boolean function. An immediate corollary of this result is a PSPACE-hard lower bound on the complexity of reachability problems for regular generalized 1D-Cellular Automata and undirected systolic networks with Boolean totalistic local transition functions. 2.In contrast, the above reachability problems are solvable in polynomial time for SDSs when the Boolean function associated with each node is symmetric and monotone. The PSPACE-completeness results follow as corollaries of simulation results which show for several classes of SDSs, how one class of SDSs can be efficiently simulated by another (more restricted) class of SDSs. We also prove several structural properties concerning the phase space of an SDS. SDSs are closely related to Cellular Automata (CA), concurrent transition systems, discrete Hopfield networks and systolic networks. This observation in conjunction with our lower bounds for SDSs, yields new PSPACE-hard lower bounds on the complexity of state reachability problems for these models, extending some of the earlier results in [K. Culik II, J. Karhumaki, On totalistic systolic networks, Inform. Process. Lett. 26 (5) (1988) 231-236; P. Floreen, E. Goles, G. Weisbuch, Transient length in sequential iterations of threshold functions, Discrete Appl. Math. 6 (1983) 95-98; P. Floreen, P. Orponen, Complexity issues in discrete Hopfield networks, Research Report No. A-1994-4, Department of Computer Science, University of Helsinki, 1994. Also appears in: I. Parberry (Ed.), Comp. and Learning Complexity of Neural Networks: Advanced Topics, 1999; D. Harel, O. Kupferman, M.Y. Vardi, On the complexity of verifying concurrent transition systems, Inform. and Comput. 173 (2002) 143-161; S.K. Shukla, H.B. Hunt III, D.J. Rosenkrantz, R.E. Stearns, On the complexity of relational problems for finite state processes, in: International Colloquium on Automata Programming and Languages, ICALP, 1996, pp. 466-477; A. Rabinovich, Complexity of equivalence problems for concurrent systems of finite agents, Inform. and Comput. 127 (2) (1997) 164-185].

[1]  Pekka Orponen.  An Overview Of The Computational Power Of Recurrent Neural Networks , 2000 .

[2]  Pekka Orponen,et al.  General-Purpose Computation with Neural Networks: A Survey of Complexity Theoretic Results , 2003, Neural Computation.

[3]  Moore,et al.  Unpredictability and undecidability in dynamical systems. , 1990, Physical review letters.

[4]  Wuxu Peng,et al.  Deadlock detection in communicating finite state machines by even reachability analysis , 1995, Proceedings of Fourth International Conference on Computer Communications and Networks - IC3N'95.

[5]  Amir Pnueli,et al.  Reachability Analysis of Dynamical Systems Having Piecewise-Constant Derivatives , 1995, Theor. Comput. Sci..

[6]  Madhav V. Marathe,et al.  Science and engineering of large scale socio-technical simulations. , 2001 .

[7]  H. Gutowitz.  Cellular automata: theory and experiment : proceedings of a workshop , 1991 .

[8]  Olivier Bournez,et al.  On the Computational Power of Dynamical Systems and Hybrid Systems , 1996, Theor. Comput. Sci..

[9]  Klaus Sutner,et al.  De Bruijn Graphs and Linear Cellular Automata , 1991, Complex Syst..

[10]  Harry B. Hunt,et al.  Gardens of Eden and Fixed Points in Sequential Dynamical Systems , 2001, DM-CCG.

[11]  Sampath Kannan,et al.  Communicating Hierarchical State Machines , 1999, ICALP.

[12]  David S. Johnson,et al.  Computers and Intractability: A Guide to the Theory of NP-Completeness , 1978 .

[13]  Max H. Garzon,et al.  Models of Massive Parallelism , 1995, Texts in Theoretical Computer Science. An EATCS Series.

[14]  Frederic Green,et al.  NP-Complete Problems in Cellular Automata , 1987, Complex Syst..

[15]  Davide Sangiorgi,et al.  Communicating and Mobile Systems: the π-calculus, , 2000 .

[16]  Cristopher Moore,et al.  Generalized shifts: unpredictability and undecidability in dynamical systems , 1991 .

[17]  Christian M. Reidys,et al.  On Acyclic Orientations and Sequential Dynamical Systems , 2001, Adv. Appl. Math..

[18]  Karel Culik,et al.  On the Limit Sets of Cellular Automata , 1989, SIAM J. Comput..

[19]  Karel Culik,et al.  Undecidability of CA Classification Schemes , 1988, Complex Syst..

[20]  Gustavo Deco,et al.  Finit Automata-Models for the Investigation of Dynamical Systems , 1997, Inf. Process. Lett..

[21]  Pekka Orponen,et al.  Computational complexity of neural networks: a survey , 1994 .

[22]  Yuval Rabani,et al.  Simulating quadratic dynamical systems is PSPACE-complete (preliminary version) , 1994, STOC '94.

[23]  Norman Margolus,et al.  An embedded DRAM architecture for large-scale spatial-lattice computations , 2000, Proceedings of 27th International Symposium on Computer Architecture (IEEE Cat. No.RS00201).

[24]  Giovanni Pighizzini,et al.  Asynchronous Automata Versus Asynchronous Cellular Automata , 1994, Theor. Comput. Sci..

[25]  Christian M. Reidys,et al.  Discrete, sequential dynamical systems , 2001, Discret. Math..

[26]  Reinhard Laubenbacher,et al.  Decomposition and simulation of sequential dynamical systems , 2003, Adv. Appl. Math..

[27]  Bruno Durand.  Inversion of 2D Cellular Automata: Some Complexity Results , 1994, Theor. Comput. Sci..

[28]  David Harel,et al.  On the Complexity of Verifying Concurrent Transition Systems , 1997, Inf. Comput..

[29]  J. Taylor,et al.  Switching and finite automata theory, 2nd ed. , 1980, Proceedings of the IEEE.

[30]  H. T. Kung.  Why systolic architectures? , 1982, Computer.

[31]  Madhav V. Marathe,et al.  Adversarial models in evolutionary game dynamics , 2001, SODA '01.

[32]  Edward A. Feigenbaum,et al.  Switching and Finite Automata Theory: Computer Science Series , 1990 .

[33]  Christian M. Reidys,et al.  Elements of a theory of simulation III: equivalence of SDS , 2001, Appl. Math. Comput..

[34]  Christian M. Reidys,et al.  Elements of a theory of computer simulation I: Sequential CA over random graphs , 1999, Appl. Math. Comput..

[35]  Michel Cosnard,et al.  Computability with Low-Dimensional Dynamical Systems , 1994, Theor. Comput. Sci..

[36]  Dan Gordon,et al.  On the computational power of totalistic cellular automata , 1987, Mathematical systems theory.

[37]  Hans L. Bodlaender,et al.  A Tourist Guide through Treewidth , 1993, Acta Cybern..

[38]  C. A. R. Hoare,et al.  Communicating sequential processes , 1978, CACM.

[39]  Christopher L. Barrett,et al.  Emergent Local Control Properties in Particle Hopping Traffic Simulations , 1995 .

[40]  Pekka Orponen,et al.  Attraction Radii in Binary Hopfield Nets are Hard to Compute , 1993, Neural Computation.

[41]  Karel Culik,et al.  A Simple Universal Cellular Automaton and its One-Way and Totalistic Version , 1987, Complex Syst..

[42]  Sandeep Kumar Shukla.  Uniform approaches to the verification of finite state systems , 1997 .

[43]  Alexander Moshe Rabinovich,et al.  Complexity of Equivalence Problems for Concurrent Systems of Finite Agents , 1997, Inf. Comput..

[44]  Harry B. Hunt,et al.  Reachability problems for sequential dynamical systems with threshold functions , 2003, Theor. Comput. Sci..

[45]  P. Gács.  Deterministic Computations Whose Hisrtory is Independent of the Order of Updating , 1995 .

[46]  Daniel Brand,et al.  On Communicating Finite-State Machines , 1983, JACM.

[47]  Harry B. Hunt,et al.  Analysis Problems for Sequential Dynamical Systems and Communicating State Machines , 2001, MFCS.

[48]  Sandeep K. Shukla,et al.  On the Complexity of Relational Problems for Finite State Processes (Extended Abstract) , 1996, ICALP.

[49]  Robin Milner,et al.  Communicating and mobile systems - the Pi-calculus , 1999 .

[50]  Péter Gács,et al.  Deterministic computations whose history is independent of the order of asynchronous updating , 2001, ArXiv.

[51]  Hava T. Siegelmann,et al.  On the Computational Power of Neural Nets , 1995, J. Comput. Syst. Sci..

[52]  Zsuzsanna Róka.  One-way Cellular Automata on Cayley Graphs , 1993, FCT.

[53]  Mohamed G. Gouda,et al.  Proving liveness for networks of communicating finite state machines , 1986, TOPL.

[54]  Pekka Orponen,et al.  Complexity Issues in Discrete Hopfield Networks , 1994 .

[55]  Max H. Garzon,et al.  Models of massive parallelism: analysis of cellular automata and neural networks , 1995 .

[56]  John N. Tsitsiklis,et al.  On the predictability of coupled automata: an allegory about chaos , 1990, Proceedings [1990] 31st Annual Symposium on Foundations of Computer Science.

[57]  Eugene Asarin,et al.  On some Relations between Dynamical Systems and Transition Systems , 1994, ICALP.

[58]  Reinhard Laubenbacher,et al.  Equivalence Relations on Finite Dynamical Systems , 2001, Adv. Appl. Math..

[59]  Klaus Sutner,et al.  On the Computational Complexity of Finite Cellular Automata , 1995, J. Comput. Syst. Sci..

[60]  Karel Culik,et al.  On Totalistic Systolic Networks , 1988, Inf. Process. Lett..

[61]  B A Huberman,et al.  Evolutionary games and computer simulations. , 1993, Proceedings of the National Academy of Sciences of the United States of America.

[62]  Eric Goles Ch.,et al.  Transient length in sequential iteration of threshold functions , 1983, Discret. Appl. Math..

[63]  Stephen Wolfram,et al.  Theory and Applications of Cellular Automata , 1986 .

[64]  Bruno Martin,et al.  A Geometrical Hierarchy on Graphs via Cellular Automata , 2002, Fundam. Informaticae.