Recursive function theory studies functions that can be evaluated by an algorithm based on some computing model such as a Turing machine or computer program. These functions are called partial recursive (p.r.) and are defined on subsets of N, where N = {0, 1,... }, with values in N. (A good introduction to the theory may be found in [2].) This paper explains the Parameter Theorem, illustrates some unusual computer programs, and explains their connection to recursive function theory. Basing our computing model on a modern high level language, we can construct programs whose existence is known through more formal means. Our functions will operate on strings rather than numbers, however. There is no loss in generality since strings can be viewed as members of N, and vice-versa. From a practical aspect, this approach is somewhat more desirable since conventional programming languages often allow making use of long strings, but place more restrictive limits on the size of integers. We first pose a problem: Does there exist a program (written in some conventional programming language) which halts if and only if its input is identical to the program's text (i.e., source code)? For the remainder of this note X is the set of ASCII characters 'a,' 'b, +,' space, line feed, and so forth, needed to write programs in most programming languages. We denote by X * the set of all strings (finite sequences) over X. Let P1 be the set of all one variable partial recursive functions on X *. A function I belongs to P1 if and only if there is a subset of S of X* and some algorithm which computes 't by associating with each element of S a well-defined element of X *. Similarly, 't belongs to P2 if and only if there is a set S c X * X X *, and an algorithm for 't which associates with each element of S a well-defined element of X *. A partial recursive function I in P1 (or in P2) is recursive if the algorithm is defined on all of X * (or X * X X *). This is equivalent to saying that the algorithm which evaluates I always halts. Our computing model will be based on Adat; its high-level string-handling properties simplify later examples. For readability, however, we will express algorithms in a more familiar Pascal-like style. Readers familiar with either programming language should find forthcoming algorithms sufficiently clear. A formal presentation in Ada is given in the Appendix. A string of ASCII characters, such as in FIGURE 1, represents a function in our language which takes strings to strings. Of course, strings of ASCII characters such as The quick brown fox and %*!! represent nothing at all in our programming language. Let A1 be the set of all strings in X * representing functions in our language of one input STRING which outputs a STRING. Similarly let A2 be the set of all strings in
22Algorithmic Procedures, Generalized Turing Algorithms, and Elementary Recursion Theory
Algorithmic Procedures, Generalized Turing Algorithms, and Elementary Recursion Theory
30Church's thesis without tears
The modern theory of computability is based on the works of Church, Markov and Turing who, starting from quite different models of computation, arrived at the same class of computable functions. The purpose of this paper is the show how the main results of the Church-Markov-Turing theory of computable functions may quickly be derived and understood without recourse to the largely irrelevant theories of recursive functions, Markov algorithms, or Turing machines. We do this by ignoring the problem of what constitutes a computable function and concentrating on the central feature of the Church-Markov-Turing theory: that the set of computable partial functions can be effectively enumerated. In this manner we are led directly to the heart of the theory of computability without having to fuss about what a computable function is.The spirit of this approach is similar to that of [RGRS]. A major difference is that we operate in the context of constructive mathematics in the sense of Bishop [BSH1], so all functions are computable by definition, and the phrase “you can find” implies “by a finite calculation.” In particular ifPis some property, then the statement “for eachmthere isnsuch thatP(m, n)” means that we can construct a (computable) functionθsuch thatP(m, θ(m))for allm. Church's thesis has a different flavor in an environment like this where the notion of a computable function is primitive.One point of such a treatment of Church's thesis is to make available to Bishopstyle constructivists the Markovian counterexamples of Russian constructivism and recursive function theory. The lack of serious candidates for computable functions other than recursive functions makes it quite implausible that a Bishopstyle constructivist could refute Church's thesis, or any consequence of Church's thesis. Hence counterexamples such as Specker's bounded increasing sequence of rational numbers that is eventually bounded away from any given real number [SPEC] may be used, as Brouwerian counterexamples are, as evidence of the unprovability of certain assertions.
Read more10Continuous-time computation with restricted integration capabilities
Continuous-time computation with restricted integration capabilities
70On a theory of computation over the real numbers; NP completeness, recursive functions and universal machines
A model for computation over an arbitrary (ordered) ring R is presented. In this general setting, universal machines, partial recursive functions, and NP-complete problems are obtained. While the theory reflects of classical over Z (e.g. the computable functions are the recursive functions), it also reflects the special mathematical character of the underlying ring R (e.g. complements of Julia sets provide natural examples of recursively enumerable undecidable sets over the reals) and provides a natural setting for studying foundational issues concerning algorithms in numerical analysis.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">></ETX>
Read more43The Insufficiency of Formal Design Methods " The Necessity of an Experimental Approach - for the Understanding and Control of Complex MAS
We highlight the limitations of formal methods by exhibiting two results in recursive function theory: that there is no effective means of finding a program that satisfies a given formal specification; or checking that a program meets a specification. We exhibit a simple MAS which has all the power of a Turing machine. We argue that any pure design methodology will face insurmountable difficulties in today's open and complex MAS. We recommend instead a methodology based on experimental method - scientific foundations for MAS construction and control.
Read more11Tarski's theory of definability: common themes in descriptive set theory, recursive function theory, classical pure logic, and finite-universe logic
Tarski's theory of definability: common themes in descriptive set theory, recursive function theory, classical pure logic, and finite-universe logic
Read more64Indexings of subrecursive classes
Indexings of subrecursive classes
3Computation of recursive functionals using minimal initial segments
Computation of recursive functionals using minimal initial segments
4Lower radicals of certain classes of rings
i. S. A. Berezin, "An algebra of one-place primitive recursive functions with iteration operation of general form," Kibernetika, No. 3, 12-19 (1976). 2. G. P. Gavrilov, "Functional completeness in a countably valued logic," in: Problems of Cybernetics [in Russian], No. 15 (1965), pp. 5-64. 3. D. A. Zakharov, Recursive Functions [in Russian], Novosibirsk (1970). 4. V. V. Koz'minykh, "One-place primitive recursive functions," Algebra Logika, ~, 75-90 (1968). 5. A. I. ~l'tsev, "Iterative algebras and Post varieties," Algebra Logika, ~, 5-24 (1966). 6. A. I. Mal'tsev, "Constructive algebras, I," Usp. Mat. Nauk, 16, No. 3, 3-60 (1961). 7. S. S. Marchenkov, ,"B°unded. . recursions," Math. Balkanlca," 2, 124-142 (1972).~ 8. S S. ~rchenkov, Ellmlnationof recursion schemes in the--Grzegorczyk class ," Mat. Zametki, 5, No. 5, 561-568 (1969). 9. V, L. Mikheev, "A class of algebras of primitive recursive functions," Mat. Zametki, 14, No. 1 143-156 (1973). i0. S. V. Pakhomov, "A simple syntactical definition of all classes of the Grzegorczyk hierarchy," Zap. Nauch. Sem. Leningr. Otd. ~at. Inst. Akad. Nauk SSSR, 40, 127-130 (1974). Ii. H. Rogers, Jr., Theory of Recursive Functions and Effective Computability, McGraw-Hill (1967). 12. M. G. Rozinas, "An algebra of many-place primitive recursive functions," Uch. Zap. Ivanovsk. Gos. Pedagog. Inst., i17, 95-111 (1972). 13. A. Cobham, "The intrinsic computational difficulty of functions," in: Logic, Methodology and Philosophy of Science, Amsterdam (1965), pp. 24-30. 14. A. Grzegorczyk, "Some classes of recursive functions," in: Rozprawy Matematiczne, Vol. 4, Warsaw (1953). 15. R. W. Ritchie, "Classes of predictably computable functions," Trans. Am. Math. Soc., 106, 139-173 (1963).
Read more36The Mathematica programmer
Preface. About This Book. Part 1: Paradigms of Programming. Introduction. Logic Programming. Higher-Order Functions. Combinators. Turing Machines. Part 2: Visualization. Animated Algorithms. Function Iteration and Chaos. Fractional Brownian Motion. Uniform Polyhedra. The Stellated Icosahedra. Ray Tracing. Single-Image Stereograms. Appendixes. References. Index of Programs. Index. Preface. About This Book: Overview. About the Programs. Notation and Terminology. The Mathematica Programmer CD-ROM. The Mathematica Programmer WWW Archive. Colophon.Part 1: Paradigms of Programming: Introduction: Mathematica's Programming Language. Pattern Matching and Term Rewriting. Programming Styles. Program Organization. Logic Programming: The Ingredients of Logic Programs. A PROLOG Interpreter for Mathematica. Lists in PROLOG. Backtracking. Deduction. Higher-Order Functions: Introduction. The Functional Features of Mathematica. Functions as Data. Fixed Points of Higher-Order Functions. Combinators: Introduction. Combinatory Algebras. Combinatory Abstraction. Converting Functions to Combinators. Applications. Turing Machines: Introduction. A Turing Machine Simulator. Assembly Programming. Recursive Functions. Optimization. Conclusions. The Complete Code of TuringMacros.m. Part 2: Visualization: Animated Algorithms: Three Standard Sorting Algorithms. Asymptotic Behavior. Conclusions. Function Iteration and Chaos: Function Iteration. Bifurcations. The Final-State Diagram. The Ingredients of Chaos. Super-Attractive Orbits. Conclusions. Fractional Brownian Motion: Introduction. Random Additions. Fourier Synthesis. Random Faults. Analysis of fBm Data. Uniform Polyhedra: Introduction. Uniform Construction. Data Structures. Rendering. Auxiliary Programs. The Stellated Icosahedra: Introduction. Rendering. Discussion. The Complete Code of Icosahedra.m. Ray Tracing: A Data Type for Surfaces. Photorealistic Rendering. Converting Mathematica Graphics. Sample Images. Stereo Pairs. The Complete Code of POVray.m. Single-Image Stereograms: Introduction. The Classis SIRDS in Mathematica. Designing Good Images. Exact Stereograms. Interface to External SIS Generators. The Complete Code of SIS.m. Appendixes. References. Index of Programs. Index.
Read more21Simple computation-universal cellular spaces and self-reproduction
Cellular spaces computationally equivalent to any given Turing machine are exhibited which are simple in the sense that each cell has only a small number of states and a small neighborhood. Neighborhood reduction theorems are derived in this interest, and several simple computationuniversal cellular spaces are presented. Conditions for computation-universality of a cellular space are investigated, and, in particular, the conjecture that unbounded but boundable propagation in a space is a sufficient condition is refuted. Finally, the computation-universal spaces derived in the study are used to introduce, via recursive function theory, examples of simple self-reproducing universal Turing machine configurations in one and two dimensions.
Read more12Recursive functions of context free languages (II)
In this paper we proved that the function class CFRF and its proper subclass CFPRF are respectively the partial recursive functions and primitive recursive functions of context free languages (CFLs). Also we discussed the relation between them and recursive functions defined on other domains. It is indicated that the functions of natural numbers and/or symbol strings (words) are functions of CFLs. Several frequently used primitive recursive functions on words were given, including logical connectives, conditional expressions. Also the powerful operators (bounded maximization and minimization operators) for constructing primitive recursive functions were defined. Two important nontrivial algorithms, the characteristic function of arbitrary CFL and the parse function of CFL sentences were constructed. Based on them, the method for extending or restricting function domain was described.
Read moreMinimization and NP multifunctions
Minimization and NP multifunctions
17Toward A More Scalable End-User Scripting Language
End-user scripting languages are relatively easy to learn, but have limited expressive power. Tile-based scripting systems are particularly accessible to beginners, but usually are very limited in scope and usually lack extensibility, and for some tasks the tile idiom becomes cumbersome. Conventional programming languages used by computer professionals are far more powerful, but at the cost of additional complexity and limited environmental support, which place them out of the casual programmer's reach. This paper presents TileScript, an attempt to combine the accessibility of a tile-based programming interface with the leverage of a full textual programming language and with a simple means of extension, making it potentially an appealing tool for the novice programmer without sacrificing any expressiveness. All TileScript programs, whether built originally with tiles or textually, can always be edited both graphically via a drag-and-drop tile interface and textually, and the user can freely switch back and forth between tile and textual representations at any time. Additionally TileScript's simple yet powerful extensibility mechanisms allow the language to be used to tackle problems that would normally be out of the scope of an end-user scripting language.
Read moreHyperarithmetical Index Sets in Recursion Theory
We define a family of properties on hyperhypersimple sets and show that they yield index sets at each level of the hyperarithmetical hierarchy.An extension yields a nj-complete index set.We also classify the index set of quasimaximal sets, of coinfinite r.e.sets not having an atomless superset, and of r.e.sets major in a fixed nonrecursive r.e.set. Introduction.The present paper deals with index sets, i.e., sets of indices of partial recursive (p.r.) functions and recursively enumerable (r.e.) sets that are defined through the p.r. functions or r.e.sets they code.The early results in index sets used geometric arguments in one-or two-dimensional arrays: Rogers showed the 3 and n3-completeness of the index sets of recursive and simple sets, respectively, in a finite injury argument.Lachlan, D. A. Martin, R. W. Robinson, and Yates (1968, unpublished, later appearing in Tulloss [Tu71]) showed the ^-completeness of the index set of maximal sets in an infinite injury argument.Tulloss [ibid.]also mentions for the first time the question whether the index set of quasimaximal sets is E5-complete.However, the geometric method was too complex at higher levels of the arithmetical hierarchy.During the 1970's, progress in index sets was mainly made in other areas by several Russian mathematicians as well as L. Hay.Schwarz [Schta] was the first to introduce induction into index set proofs (in the r.e.degrees) and was able to show that the index sets of low" and highn r.e.sets are "+3 and En+4-complete, respectively.Solovay [JLSSta] then extended Schwarz's methods to show the w+i-completeness of the index sets of low<w (lown for some n) and of high<w (high" for some n) r.e.sets as well as the H^+i-completeness of the index set of intermediate degrees (degrees neither low<w nor high<w).In this paper, we exhibit a family of algebraically invariant properties LWliWdefinable in , that yields index sets at any level of the %perarithmetical hierarchy.The proof is based on induction and Lachlan's theorem [La68] that any E3-Boolean algebra is isomorphic to the lattice of r.e.supersets of some r.e.set (modulo finite sets).It uses tree arguments and the fact that the Cantor-Bendixson rank of a tree corresponds to certain properties of the lattice of r.e.supersets of the set constructed.An extension yields a nj-complete index set.A corollary shows the ^-completeness of the index set of quasimaximal sets, thereby settling this longopen question.Further results classify the index sets of atomic sets and of r.e.sets major in a fixed nonrecursive r.e.set.Our notation is fairly standard and generally follows Soare's forthcoming book Recursively Enumerable Sets and Degrees [Sota].
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