## Computation, Automata, Languages

*31 Dec 2015 12:19*

Computers aren't made ofIdeal, theoretical computers are rather mathematical objects: they are, equivalently, algorithms, or effective procedures, or abstract automata, or functions which can be specified recursively, or formal languages.matter.

--- Greg Egan, Permutation City

*Things to learn more about:* Classifications of machines and
languages (beyond the classical, four-level Chomsky hierarchy). Hierarchies of
computational power. Abstract-algebraic treatment of automata. Effects of
making automata stochastic. Techniques for proving equivalence of automata; of
minimizing automata.
Bisimulation. Techniques for inferring
automata or grammars from their languages, especially when generation is
stochastic. Non-finite-state transducers. Stochastic context-free
grammars and their connections
with branching processes. "Logics of
time and computation".

Analog computation. What forms are structurally stable? Other forms of unconventional computation. DNA computation doesn't interest me very much, because that's just another kind of hardware, and slow, big and noisy at that. But quantum computation is very interesting, because it can do something new. So, possibly, is computation in dynamical systems.

Complexity classes --- in space (memory), time, other resources? Analog equivalents. "Phase transitions" between complexity classes, and the analogy to physical phase transitions.

See also: AI; Algorithmic Information Theory; Biological computers; Cellular Automata; Computational Mechanics (for exploring the intrinsic computation of physical processes); Computers; Dynamics; Gödel's Theorem (a consequence of the existence of uncomputable functions); Grammatical Inference; Math I Ought to Learn; Machine Learning, Statistical Inference and Induction; Parallel and Distributed Computing; Physics of Computation and Information; Programming; Quantum Mechanics (for quantum computers); Symbolic Dynamics; Transducers

- Recommended, big picture:
- Gary William Flake, The Computational Beauty of Nature: Computer Explorations of Fractals, Chaos, Complex Systems, and Adaptation [Review: A Garden of Bright Images]
- Harry R. Lewis and Christos H. Papadimitriou, Elements of the Theory of Computation [Very nice textbook; the proofs, for instance, are comprehensible and correct, which is not always the case with the competition.]
- Marvin Minsky, Computation: Finite and Infinite Machines
- Cristopher Moore and Stephan Mertens, The Nature of Computation [Cris and Stephan were kind enough to let me read this in manuscript; it's magnificent. Review: Intellects Vast and Warm and Sympathetic]

- Recommended, close-ups:
- Michael Arbib, Brains, Machines and Mathematics
- George S. Boolos and Richard C. Jeffrey, Computability and Logic
- Taylor L. Booth, Sequential Machines and Automata Theory [Fascinating material on probabilistic machines which has dropped out of later texts]
- Andrei Broder and Jorge Stolfi, Pessimal Algorithms and Simplexity Analysis
- J. Richard Büchi, Finite Automata, Their Algebras and Grammars: Towards a Theory of Formal Expressions
- Marco Giunti, Computation, Dynamics, and Cognition [The first two-thirds has a nice treatment of abstract computers as discrete dynamical systems, including some apparently new results about non-Turing computation; the stuff about cognition and scientific explanation seems, by contrast, strained and tacked-on. By Giunit's standards, no analog computation is "computation"!]
- Joseph Y. Halpern, "Beyond Nash Equilibrium: Solution Concepts for the 21st Century", arxiv:0806.2139
- J. Hartmanis and R. E. Stearns, Algebraic Structure Theory of Sequential Machines
- Eric Mjolsness, "Stochastic Process Semantics for Dynamical Grammar Syntax: An Overview", cs.AI/0511073
- Cristopher Moore,
"Recursion Theory on the Reals and Continuous-Time Computation,"
Theoretical Computer Science,
**162**(1999): 23--44 - Matthias Scheutz, "Computational versus Causal
Complexity", Minds and Machines
**11**(2001): 543--566 ["notions of implementation based on an isomorphic correspondence between physical and computational states are not tenable. Rather, `implementation' has to be based on the notion of `bisimulation' in order to ... incorporate intuitions from computational practice. A formal definition of implementation is suggested ... to make the functionalist notion of `physical realization' precise. The upshot of this new definition ... is that implementation cannot distinguish isomorphic bisimilar from non-isomporphic bisimilar systems anymore, thus driving a wedge between the notions of causal and computational complexity." PDF] - Claude E. Shannon and John McCarthy (eds.), Automata
Studies [Think of this as a collection of many of the ways computer
science
*could*have gone, including some of the ways it did, e.g., Kleene's paper introducing regular expressions and finite automata] - J. F. Traub and A. G. Werschulz, Complexity and Information
- E. Vidal, F. Thollard, C. de la Higuera, F. Casacuberta and R. C. Carrasco, "Probabilistic Finte-State Machines"

- To read:
- Andrew Adamatzky (ed.), Collison-Based Computing
- Jiri Adamek and Vera Trnkova, Automata and Algebras in Categories
- James Anderson, Automata Theory with Modern Applications
- Arbib (ed.), Algebraic Theory of Machines, Langauges, and Semigroups
- G. Ausiello, Complexity and Approximation: Combinatorial Optimization Problems and Their Approximability Properties
- Falk Bartels, Ana Sokolova and Erik de Vink, "A hierarchy of
probabilistic system types", Theoretical Computer
Science
**327**(2004): 3--22 - Asa Ben-Hur, Alexander Roitershtein and Hava T. Siegelmann,
"Probabilistic analog automata", Theoretical Computer
Science
**320**(2004): 449--464 - Udi Boker and Nachum Dershowitz, "Comparing Computational Power", cs.LO/0510069
- Paul Bohan Broderick, "On Communication and Computation",
Minds and Machines
**14**(2004): 1--19 ["The most famous models of computation and communication, Turing Machines and (Shannon-style) information sources, are considered. The most significant difference lies in the types of state-transitions allowed in each sort of model. This difference does not correspond to the difference that would be expected after considering the ordinary usage of these terms."] - C. S. Calude, J. Casti, and M. J. Dinneen, eds., Unconventional Models of Computation
- John Carroll and Darrell Long, Theory of Finite Automata: with an Introduction to Formal Languages [Good chapter on finite-state transducers; dunno about the rest yet]
- Manuel Lameiras Campagnolo, Cristopher Moore, and José Félix Costa, "Iteration, Inequailities, and Differentiability in Analog Computers" [On-line]
- Manuel Lameiras Campagnolo and Cristopher Moore, "Upper and lower bounds on continuous-time computation" [On-line]
- J. H. Conway, Regular Algebra and Finite Machines
- Jean-Charles Delvenne, Petr Kurka and Vincent Blondel, "Computational Universality in Symbolic Dynamical Systems", cs.CC/0404021
- Alan John Dix, Formal Methods for Interactive Systems
- E.-E. Doberkat, Stochastic Automata: Stability, Nondeterminism and Prediction [In our library...]
- David Doty and Jared Nichols, "Pushdown Dimension", cs.IT/0504047
- Stefano Galatolo, Mathieu Hoyrup, Cristóbal Rojas, "Dynamical systems, simulation, abstract computation", arxiv:1101.0833
- Joseph A. Goguen and Grant Malcolm, Algebraic Semantics of Imperative Programs
- Robert Goldblatt, Logics of Time and Computation
- Odd Goldreich, Computational Complexity: A Conceptual Perspective
- Gramss, Bornholdt, Gross, Mitchell and Pellizzari (eds.), Non-Standard Computation: Molecular Computation --- Cellular Automata --- Evolutionary Algorithms --- Quantum Computers
- David Harel
- The Science of Computing: Exploring the Nature and Power of Algorithms
- Computers Ltd: What They Really Can't Do

- Thomas A. Henzinger, Rupak Majumdar and Jean-Francois Raskin, "A Classification of Symbolic Transition Systems," cs.LO/0101013
- Dorit Hochbaum, Approximation Algorithms for NP-Hard Problems
- Marcus Hutter, "The Fastest and Shortest Algorithm for All Well-Defined Problems," cs.CC/0206022
- Giorgi Japaridze, "Computatbility Logic: A Formal Theory of Interaction", cs.LO/0404024
- Johannes Kobler and Rainer Schuler, "Average-case intractability
vs. worst-case intractability", Information and
Computation
**190**(2004): 1--17 - Eyal Kushilevitz and Noam Nisan, Communication Complexity
- A. A. Lorents, Stochastic automata: constructive theory
- Marc Mezard and Andrea Montanari, Information, Physics, and Computation
- Marcin Milkowski, Explaining The Computational Mind [For the material on what it means to explain phenomena computatinally]
- Rajeev Motwani, Randomized Algorithms
- Jorg Olschewski, Michael Ummels, "The Complexity of Finding Reset Words in Finite Automata", arxiv:1004.3246
- Leszek Plaskota, Noisy Information and Computational Complexity
- György E. Révész, Introduction to Formal Languages
- National Research Council (USA), Probability and Algorithms [online]
- Wolfgang Reisig, Petri Nets: An Introduction
- Hartley Rogers Jr., Theory of Recursive Functions and Effective Computability
- Arto Salomaa, Computation and Automata
- David Sankoff, "Branching Processes with Terminal Types:
Application to Context-Free Grammars", Journal of Applied
Probability
**8**(1971): 233--240 [JSTOR] - Yuzuru Sato, Logic and Computation in Dynamical Systems [Ph.D. thesis, University of Tokyo, 2000]
- Géraud Sénizergues, "Complete Formal Systems for
Equivalence Problems," Theoretical Computer Science
**231**(2000): 309--334 - Tanya Sienko, Andrew Adamatzky and Nicholas Rambidi (eds.), Molecular Computing
- Edward P. Stabler and Edward L. Keenan, "Structural similarity
within and among languages," Theoretical
Computer Science
**293**(2003): 345--363 - Wojciech Szpankowski, Average Case Analysis of Algorithms on Sequences [Preprint version]
- J. F. Traub and H. Wozniakowski, "Persepctives on Information-Based
Complexity," Bulletin of the American Mathematical
Society
**26**(1992): 29--52, math.NA/9201269 - J. V. Tucker and J. I. Zucker
- "Abstract Computability, Algebraic Specification and Initiality," cs.LO/0109001
- "Abstract versus Concrete Computation on Metric Partial Algebras," cs.LO/0108007

- Vijay V. Vazirani, Approximation Algorithms
- R. F. Walters, Categories and Computer Science
- Herbert S. Wilf, Algorithms and Complexity [online]
- Wlodek Zadrozny, "Minimum Description Length and Compositionality," cs.CL/0001002