A modelforalarge network of"neurons" withagraded response (orsigmoid input-output relation) is studied. Thisdeterministic system hascollective properties in veryclose correspondence withtheearlier stochastic model based onMcCulloch-Pitts neurons. Thecontent-addressable memoryandother emergent collective properties oftheorigi- nalmodelalso arepresent inthegraded response model. The idea that suchcollective properties areusedinbiological sys- temsisgiven addedcredence bythecontinued presence ofsuch properties formorenearly biological "neurons." Collective analog electrical circuits ofthekinddescribed will certainly function. Thecollective states ofthetwomodels haveasimple correspondence. Theoriginal modelwill continue tobeuseful forsimulations, because its connection tograded response sys- temsisestablished. Equations that include theeffect ofaction potentials inthegraded response system arealsodeveloped. Recent papers (1-3) haveexplored theability ofasystem of highly interconnected "neurons" tohaveuseful collective computational properties. Theseproperties emergesponta- neously inasystem having alarge numberofelementary "neurons." Content-addressable memory(CAM)isoneof thesimplest collective properties ofsucha system. The mathematical modeling hasbeenbasedon"neurons" that aredifferent bothfromreal biological neurons andfromthe realistic functioning ofsimple electronic circuits. Someof these differences aremajor enough thatneurobiologists and circuit engineers alike havequestioned whether real neural orelectrical circuits wouldactually exhibit thekindofbe- haviors foundinthemodelsystem evenifthe"neurons" wereconnected inthefashion envisioned. Twomajor divergences between themodelandbiological orphysical systems stand out.Realneurons (andreal physi- caldevices suchasoperational amplifiers that might mimic them) havecontinuous input-output relations. (Action po- tentials areomitted until Discussion.) Theoriginal modeling usedtwo-state McCulloch-Pitts (4)threshold devices having outputs of0or1only. Realneurons andreal physical circuits haveintegrative timedelays duetocapacitance, andthetime evolution ofthestate ofsuchsystems should berepresented byadifferential equation (perhaps withaddednoise). The original modeling usedastochastic algorithm involving sud- den0-1or1-0changes ofstates ofneurons atrandom times. Thispaper showsthat theimportant properties oftheorigi- nalmodelremain intact whenthese twosimplifications of themodeling areeliminated. Although itisuncertain wheth- ertheproperties ofthese newcontinuous "neurons" areyet close enough totheessential properties ofrealneurons (and/or their dendritic arborization) tobedirectly applicable toneurobiology, amajor conceptual obstacle hasbeenelimi- nated. Itiscertain that aCAM constructed onthebasic ideas

Neurons withgraded response havecollective computational properties likethoseoftwo-state neurons

Let T1,…,Tl:X→X be commuting measure-preserving transformations on a probability space . We show that the multiple ergodic averages are convergent in as for all ; this was previously established for l=2 by Conze and Lesigne [J. P. Conze and E. Lesigne. Theoremes ergodique por les mesures diagonales. Bull. Soc. Math. France112 (1984), 143–175] and for general l assuming some additional ergodicity hypotheses on the maps Ti and TiTj−1 by Frantzikinakis and Kra [N. Frantzikinakis and B. Kra. Convergence of multiple ergodic averages for some commuting transformations. Ergod. Th. & Dynam. Sys.25 (2005), 799–809] (with the l=3 case of this result established earlier by Zhang [Q. Zhang. On the convergence of the averages . Mh. Math.122 (1996), 275–300]). Our approach is combinatorial and finitary in nature, inspired by recent developments regarding the hypergraph regularity and removal lemmas, although we will not need the full strength of those lemmas. In particular, the l=2 case of our arguments is a finitary analogue of those by Conze and Lesigne.

Norm convergence of multiple ergodic averages for commuting transformations

Let $T_1, ..., T_l: X \to X$ be commuting measure-preserving transformations on a probability space $(X, \X, \mu)$. We show that the multiple ergodic averages $\frac{1}{N} \sum_{n=0}^{N-1} f_1(T_1^n x) ... f_l(T_l^n x)$ are convergent in $L^2(X,\X,\mu)$ as $N \to \infty$ for all $f_1,...,f_l \in L^\infty(X,\X,\mu)$; this was previously established for $l=2$ by Conze and Lesigne and for general $l$ assuming some additional ergodicity hypotheses on the maps $T_i$ and $T_i T_j^{-1}$ by Frantzikinakis and Kra (with the $l=3$ case of this result established earlier by Zhang). Our approach is combinatorial and finitary in nature, inspired by recent developments regarding the hypergraph regularity and removal lemmas, although we will not need the full strength of those lemmas. In particular, the $l=2$ case of our arguments are a finitary analogue of those of Conze and Lesigne.

Church's Thesis asserts that the only numeric functions that can be calculated by effective means are the recursive ones, which are the same, extensionally, as the Turing-computable numeric functions. The Abstract State Machine Theorem states that every classical algorithm is behaviorally equivalent to an abstract state machine. This theorem presupposes three natural postulates about algorithmic computation. Here, we show that augmenting those postulates with an additional requirement regarding basic operations gives a natural axiomatization of computability and a proof of Church's Thesis, as Godel and others suggested may be possible. In a similar way, but with a different set of basic operations, one can prove Turing's Thesis, characterizing the effective string functions, and—in particular—the effectively-computable functions on string representations of numbers.

/pdf/a-natural-axiomatization-of-computability-and-proof-of-2nph0ml8so.pdf

A natural axiomatization of computability and proof of church's thesis

http://www.people.cs.uchicago.edu/~soare/History/turing.pdf

Turing Oracle Machines, Online Computing, and Three Displacements in Computability Theory

In recent years, classical computability has expanded beyond its original scope to address issues related to computability and complexity in algebra, analysis, and physics. The deep interconnection between "computation" and "proof" has originated much of the most significant work in constructive mathematics and mathematical logic of the last 70 years. Moreover, the increasingly compelling necessity to deal with computability in the real world (such as computing on continuous data, biological computing, and physical models) has brought focus to new paradigms of computation that are based on biological and physical models. These models address questions of efficiency in a radically new way and even threaten to move the so-called Turing barrier, i.e. the line between the decidable and the un-decidable.

This book examines new developments in the theory and practice of computation from a mathematical perspective, with topics ranging from classical computability to complexity, from biocomputing to quantum computing. The book opens with an introduction by Andrew Hodges, the Turing biographer, who analyzes the pioneering work that anticipated recent developments concerning computations allegedly new paradigms. The remaining material covers traditional topics in computability theory such as relative computability, theory of numberings, and domain theory, in addition to topics on the relationships between proof theory, computability, and complexity theory. New paradigms of computation arising from biology and quantum physics are also discussed, as well as the computability of the real numbers and its related issues.

This book is suitable for researchers and graduate students in mathematics, philosophy, and computer science with a special interest in logic and foundational issues. Most useful to graduate students are the survey papers on computable analysis and biological computing. Logicians and theoretical physicists will also benefit from this book.

Benedikt Lwe

Papers

New Computational Paradigms: Changing Conceptions of What is Computable