Jiří Wiedermann

The Turing machine paradigm in contemporary computing

The importance of algorithms is now recognized in all mathematical sciences, thanks to the development of computability and computational complexity theory in the 20th century. The basic understanding of computability theory developed in the nineteen thirties with the pioneering work of mathematicians like Godel, Church, Turing and Post. Their work provided the mathematical basis for the study of algorithms as a formalized concept. The work of Hartmanis, Stearns, Karp, Cook and others in the nineteen sixties and seventies showed that the refinement of the theory to resource-bounded computations gave the means to explain the many intuitions concerning the complexity or ‘hardness’ of algorithmic problems in a precise and rigorous framework.

Characterizing the super-Turing computing power and efficiency of classical fuzzy Turing machines

The first attempts concerning formalization of the notion of fuzzy algorithms in terms of Turing machines are dated in late 1960s when this notion was introduced by Zadeh. Recently, it has been observed that corresponding so-called classical fuzzy Turing machines can solve undecidable problems. In this paper we will give exact recursion-theoretical characterization of the computational power of this kind of fuzzy Turing machines. Namely, we will show that fuzzy languages accepted by these machines with a computable t-norm correspond exactly to the union Σ10 ∪ Π10 of recursively enumerable languages and their complements. Moreover, we will show that the class of polynomially time-bounded computations of such machines coincides with the union NP ∪ co-NP of complexity classes from the first level of the polynomial hierarchy.

How We Think of Computing Today

Classical models of computation no longer fully correspond to the current notions of computing in modern systems. Even in the sciences, many natural systems are now viewed as systems that compute. Can one devise models of computation that capture the notion of computing as seen today and that could play the same role as Turing machines did for the classical case? We propose two models inspired from key mechanisms of current systems in both artificial and natural environments: evolving automata and interactive Turing machines with advice. The two models represent relevant adjustments in our apprehension of computing: the shift to potentially non-terminating interactive computations, the shift towards systems whose hardware and/or software can change over time, and the shift to computing systems that evolve in an unpredictable, non-uniform way. The two models are shown to be equivalent and both are provably computationally more powerful than the models covered by the old computing paradigm. The models also motivate the extension of classical complexity theory by non-uniform classes, using the computational resources that are natural to these models. Of course, the additional computational power of the models cannot in general be meaningfully exploited in concrete goal-oriented computations.

Computation as an unbounded process

We develop a model of computation as an unbounded process, measuring complexity by the number of observed behavioural changes during the computation. In a natural way, the model brings effective unbounded computation up to the second level of the Arithmetical Hierarchy, unifying several earlier concepts like trial-and-error predicates and relativistic computing. The roots of the model can be traced back to the circular a-machines already distinguished by Turing in 1936. The model allows one to introduce nondeterministic unbounded computations and to formulate an analogue of the P-versus-NP question. We show that under reasonable assumptions, the resource-bounded versions of deterministic and nondeterministic unbounded computation have equal computational power but that in general, the corresponding complexity classes are different (P^m^i^n^d@?NP^m^i^n^d).

A Theory of Interactive Computation

Many embedded systems behave very differently from classical machine models: they interact with an unpredictable environment, they are “always on”, and they change over time. This leads to the interesting question of what a computational theory of interactive, evolving programs should look like. While the behavior of such programs has been well-studied in concurrency theory, there has been much less emphasis on their computational aspects. A theory of interactive computation must necessarily lead beyond the classical, finitary models of computation.

On the Universal Computing Power of Amorphous Computing Systems

Amorphous computing differs from the classical ideas about computations almost in every aspect. The architecture of amorphous computers is random, since they consist of a plethora of identical computational units spread randomly over a given area. Within a limited radius the units can communicate wirelessly with their neighbors via a single-channel radio. We consider a model whose assumptions on the underlying computing and communication abilities are among the weakest possible: all computational units are finite state probabilistic automata working asynchronously, there is no broadcasting collision detection mechanism and no network addresses. We show that under reasonable probabilistic assumptions such amorphous computing systems can possess universal computing power with a high probability. The underlying theory makes use of properties of random graphs and that of probabilistic analysis of algorithms. To the best of our knowledge this is the first result showing the universality of such computing systems.

Complexity of Evolving Interactive Systems

We study a versatile model of evolving interactive computing: lineages of automata. A lineage consists of a sequence of interactive finite automata, with a mechanism of passing information from each automaton to its immediate successor. Lineages enable a definition of a suitable complexity measure for evolving systems. We show several complexity results, including a hierarchy result.

A Computability Argument Against Superintelligence

Using the contemporary view of computing exemplified by recent models and results from non-uniform complexity theory, we investigate the computational power of cognitive systems. We show that in accordance with the so-called extended Turing machine paradigm such systems can be modelled as non-uniform evolving interactive systems whose computational power surpasses that of the classical Turing machines. Our results show that there is an infinite hierarchy of cognitive systems. Within this hierarchy, there are systems achieving and surpassing the human intelligence level. Any intelligence level surpassing the human intelligence is called the superintelligence level. We will argue that, formally, from a computation viewpoint the human-level intelligence is upper-bounded by the

Computability in Amorphous Structures

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