scholarly journals Information Processing by Symmetric Inductive Turing Machines

Proceedings ◽  
2020 ◽  
Vol 47 (1) ◽  
pp. 28
Author(s):  
Mark Burgin
Keyword(s):  
Information Processing ◽  
Turing Machines ◽  
Recursive Functions ◽  
Partial Recursive Functions ◽  
Software Programs ◽  
Models Of Computations

Traditional models of computations, such as Turing machines or partial recursive functions, perform computations of functions using a definite program controlling these computations. This approach detaches data, which are processed, and the permanent program, which controls this processing. Physical computers often process not only data but also their software (programs). To reflect this peculiarity of physical computers, symmetric models of computations and automata were introduced. In this paper, we study information processing by symmetric models, which are called symmetric inductive Turing machines and reflexive inductive Turing machines.

scholarly journals Information Processing by Symmetric Inductive Turing Machines

Proceedings ◽  
2020 ◽  
Vol 47 (1) ◽  
pp. 28
Author(s):  
Mark Burgin
Keyword(s):  
Information Processing ◽  
Turing Machines ◽  
Recursive Functions ◽  
Partial Recursive Functions ◽  
Software Programs ◽  
Models Of Computations

Traditional models of computations, such as Turing machines or partial recursive functions, perform computations of functions using a definite program controlling these computations. This approach detaches data, which are processed, and the permanent program, which controls this processing. Physical computers often process not only data but also their software (programs). To reflect this peculiarity of physical computers, symmetric models of computations and automata were introduced. In this paper, we study information processing by symmetric models, which are called symmetric inductive Turing machines and reflexive inductive Turing machines.

scholarly journals Unsolid and fluid strong varieties of partial algebras

2006 ◽  
Vol 2006 ◽  
pp. 1-19
Author(s):  
S. Busaman ◽  
K. Denecke
Keyword(s):  
Partial Algebra ◽  
The Other ◽  
Turing Machines ◽  
Left Hand ◽  
Recursive Functions ◽  
Right Hand ◽  
Algebraic Description ◽  
Partial Recursive Functions ◽  
Partial Algebras ◽  
The Right

A partial algebra𝒜=(A;(fiA)i∈I)consists of a setAand an indexed set(fiA)i∈Iof partial operationsfiA:Ani⊸→A. Partial operations occur in the algebraic description of partial recursive functions and Turing machines. A pair of termsp≈qover the partial algebra𝒜is said to be a strong identity in𝒜if the right-hand side is defined whenever the left-hand side is defined and vice versa, and both are equal. A strong identityp≈qis called a strong hyperidentity if when the operation symbols occurring inpandqare replaced by terms of the same arity, the identity which arises is satisfied as a strong identity. If every strong identity in a strong variety of partial algebras is satisfied as a strong hyperidentity, the strong variety is called solid. In this paper, we consider the other extreme, the case when the set of all strong identities of a strong variety of partial algebras is invariant only under the identical replacement of operation symbols by terms. This leads to the concepts of unsolid and fluid varieties and some generalizations.

Mathematical Dynamical Systems and Computational Systems

1997 ◽  
Author(s):  
Marco Giunti
Keyword(s):  
Cellular Automata ◽  
Finite Automata ◽  
Turing Machines ◽  
Computational System ◽  
Canonical Systems ◽  
Recursive Functions ◽  
Computation Theory ◽  
Partial Recursive Functions ◽  
Push Down ◽  
Computational Systems

The main thesis of this chapter is that a dynamical viewpoint allows us to better understand some important foundational issues of computation theory. Effective procedures are traditionally studied from two different but complementary points of view. The first approach is concerned with individuating those numeric functions that are effectively calculable. This approach reached its systematization with the theory of the recursive functions (Gödel, Church Kleene).This theory is not directly concerned with computing devices or computations. Rather, the effective calculability of a recursive function is guaranteed by the algorithmic nature of its definition. In contrast, the second approach focuses on a family of abstract mechanisms, which are then typically used to compute or recognize numeric functions, sets of numbers, or numbers. These devices can be divided into two broad categories: automata or machines (Turing and Post), and systems of rules for symbol manipulation (Post). The mechanisms that have been studied include: a. Automata or Machines 1. gate-nets and McCulloch-Pitts nets 2. finite automata (Mealy and Moore machines) 3. push-down automata 4. stack automata 5. Turing machines 6. register machines 7. wang machines 8. cellular automata b. Systems of Rules 9. monogenic production systems in general 10. monogenic Post canonical systems 11. monogenic Post normal systems 12. tag systems. I call any device studied by computation theory a computational system. Computation theory is traditionally interested in studying the relations between each type of computational system and the others, and in establishing what class of numeric functions each type can compute. Accordingly one proves two kinds of theorem: (1) that systems of a given type emulate systems of another type (examples: Turing machines emulate register machines and cellular automata; cellular automata emulate Turing machines, etc.), and (2) that a certain type of system is complete relative to the class of the (partial) recursive functions or, in other words, that this type of system can compute all and only the (partial) recursive functions (examples of complete systems: Turing machines, register machines, cellular automata, tag systems, etc.). All different types of computational systems have much in common. Nevertheless, it is not at all clear exactly which properties these mechanisms share.

Preliminaries

2017 ◽  
Author(s):  
David J. Lobina
Keyword(s):  
Data Structures ◽  
Production Systems ◽  
Specific Data ◽  
Recursive Functions ◽  
Recursive Solution ◽  
Partial Recursive Functions ◽  
Recursive Structures ◽  
Recursive Data Structures ◽  
Recursive Data ◽  
The Relationship

Recursion, or the capacity of ‘self-reference’, has played a central role within mathematical approaches to understanding the nature of computation, from the general recursive functions of Alonzo Church to the partial recursive functions of Stephen C. Kleene and the production systems of Emil Post. Recursion has also played a significant role in the analysis and running of certain computational processes within computer science (viz., those with self-calls and deferred operations). Yet the relationship between the mathematical and computer versions of recursion is subtle and intricate. A recursively specified algorithm, for example, may well proceed iteratively if time and space constraints permit; but the nature of specific data structures—viz., recursive data structures—will also return a recursive solution as the most optimal process. In other words, the correspondence between recursive structures and recursive processes is not automatic; it needs to be demonstrated on a case-by-case basis.

scholarly journals Recursive density types and Nerode extensions of arithmetic

1975 ◽  
Vol 20 (2) ◽  
pp. 146-158 ◽  
Author(s):  
P. Aczel
Keyword(s):  
Equivalence Classes ◽  
Partial Functions ◽  
Recursive Functions ◽  
Cardinal Numbers ◽  
Partial Recursive Functions

The notion of a recursive density type (R.D.T.) was introduced by Medvedev and developed by Pavlova (1961). More recently the algebra of R.D.T.'s was initiated by Gonshor and Rice (1969). The R.D.T.'s are equivalence classes of sets of integers, similar in many respects to the R.E.T.'s. They may both be thought of as effective analogues of the cardinal numbers. While the equivalence relationfor R.E.T.'s is defined in terms of partial recursive functions, that for R.D.T.'s may be characterized in terms of recursively bounded partial functions (see 4.22a).

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