Functioning and interaction of distributed devices and concurrent algorithms are analyzed in the context of the theory of algorithms. Our main concern here is how and under what conditions algorithmic interactive devices can be more powerful than the recursive models of computation, such as Turing machines. Realization of such a higher computing power makes these systems superrecursive. We find here five sources for superrecursiveness in interaction. In addition, we prove that when all of these sources are excluded, the algorithmic interactive system in question is able to perform only recursive computations. These results provide computer scientists with necessary and sufficient conditions for achieving superrecursiveness by algorithmic interactive devices.
Deep Dive into Superrecursive Features of Interactive Computation.
Functioning and interaction of distributed devices and concurrent algorithms are analyzed in the context of the theory of algorithms. Our main concern here is how and under what conditions algorithmic interactive devices can be more powerful than the recursive models of computation, such as Turing machines. Realization of such a higher computing power makes these systems superrecursive. We find here five sources for superrecursiveness in interaction. In addition, we prove that when all of these sources are excluded, the algorithmic interactive system in question is able to perform only recursive computations. These results provide computer scientists with necessary and sufficient conditions for achieving superrecursiveness by algorithmic interactive devices.
There is a tendency to oppose algorithms and interaction (cf., for example, [17]). This opposition is based on a very restricted understanding of algorithms, which is based on the Church-Turing Thesis that equates algorithms with Turing machines or other mathematical schemas that give rules for computation of a function. Some researchers claim that interactive computation is more powerful than Turing machines (cf., for example, [6,7,14,15,17]), while others insist that the Church-Turing Thesis still holds (cf., for example, [9]). However, contemporary understanding extends the concept of algorithm, making it closer to the general usage of the word "algorithm". Namely, algorithm is informally perceived as a (finite) structure (e.g., a system of rules) that contains for some performer (class of performers) exact information (e.g., instructions) that allows some performer(s) to pursue a definite goal (cf., for example, [3,4]).
People need information that is contained in algorithms to make their activity efficient and purposeful. Consequently, one main achievement of 20 th century scientific thought was elaboration of the theory of algorithms and computation. This theory studies abstract and real automata, computers and networks, computation and communication. In many ways, this theory is the central cornerstone for computer science. Many key accomplishments in the theory of algorithms and computation converge to the famous Church-Turing Thesis, a statement determining the boundaries of algorithmic computations. The Church-Turing Thesis has long been considered as the most fundamental law within computing. However, recent developments in the theory of algorithms allow overcoming limitations in the Church-Turing Thesis. New mathematical models for algorithms and computation have appeared that extend prior theory in a manner similar to the way relativity theory and quantum mechanics went beyond Newtonian mechanics. These new models are more powerful than the classical recursive algorithm models, i.e., Turing machines, partial recursive functions, Lambdacalculus, and cellular automata.
Algorithms and automata that are more powerful than Turing machines are called super-recursive.
At the first glance, it looks like interactive systems are essentially different from computers and cannot be represented by computing models, such as Turing machines.
Really, as Leeuwen and Wiedermann write [13], the purpose of an interactive system is usually not to compute some final result but to react to or interact with the environment in which the system is placed and to maintain a well-defined action-reaction behavior.
Interactive systems are always operating and thus, may be seen as machines on infinite strings, but differ in the sense that their inputs are not specified and may depend on intermediate outputs and external sources.
However, if we consider only systems that work with symbolic information, then reaction to or interaction with the environment or with another such system is information transformation and exchange, or communication. In other words, functioning of and interactive system that works with symbolic information consists of computation and communication processes.
In this paper, we analyze sources of an interactive recursive algorithm/machine ability to outperform (have more computing power than) conventional Turing machines, that is, to be able to compute recursively non-computable functions. We find five such sources of interactive superrecursiveness. Namely, interactive superrecursiveness is possible when: 1) the interactive algorithm is itself superrecursive; 2) (Proposition 1) the interactive algorithm is recursive but contains initial information about some recursively non-computable function (has a non-recursive oracle); 3) (Proposition 2) the interactive recursive algorithm interacts with a superrecursive algorithm (a non-recursive environment); 4) (Theorems 2, 3 and 4) time of interaction is not recursively coordinated; 5) (Theorems 5) communication space is not recursively coordinated. The first three cases are not interesting because either superrecursive power comes not from interaction (case 1) or as it is well known, if a recursive device have access to a super-recursive information, then this device can compute recursively non-computable functions.
However, after finding sources of interactive superrecursiveness, it is natural to ask a question if all sources have been found or there are other sources that we have not been able to see. To show that our analysis of interactive superrecursiveness is complete, we prove (Theorems 6) that if interacting algorithms/devices are recursive and their interaction is organized/controlled by a recursive device/algorithms, then computable functions are also recursive. Thus, we consider algorithms in the form of rules and devices that perform simple and constructive operations at each step and give a result after a finite number of steps (in finite t
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