Subrecursive Classes Research Articles

On subrecursive complexity of integration

We consider the complexity of the integration operator on real functions with respect to the subrecursive class M2. We prove that the definite integral of a uniformly M2-computable analytic real function with M2-computable limits is itself M2-computable real number. We generalise this result to integrals with parameters and with varying limits. As an application, we show that the Euler-Mascheroni constant is M2-computable.

Categorical comprehensions and recursion

A new categorical setting is defined in order to characterize the subrecursive classes belonging to complexity hierarchies. This is achieved by means of coercion functors over a symmetric monoidal category endowed with certain recursion schemes that imitate the bounded recursion scheme. This gives a categorical counterpart of generalized safe composition and safe recursion.

Complexity-Theoretic Hierarchies Induced by Fragments of Gödel’s T

We introduce two hierarchies of unknown ordinal height. The hierarchies are induced by natural fragments of a calculus based on finite types and Godel’s T, and all the classes in the hierarchies are uniformly defined without referring to explicit bounds. Deterministic complexity classes like logspace, p, pspace, linspace and exp are captured by the hierarchies. Typical subrecursive classes are also captured, e.g. the small relational Grzegorczyk classes ℰ*0, ℰ*1 and ℰ*2.

Some subrecursive versions of Grzegorczyk's Uniformity Theorem

AbstractA theorem published by A. Grzegorczyk in 1955 states a certain kind of effective uniform continuity of computable functionals whose values are natural numbers and whose arguments range over the total functions in the set of the natural numbers and over the natural numbers. Namely, for any such functional a computable functional with one function‐argument and the same number‐arguments exists such that the values of the first of the functionals at functions dominated by a given one are completely determined by their restrictions to numbers not exceeding the corresponding values of the second functional at the given function. We prove versions of this theorem for the class of the primitive recursive functionals and for the class of the elementary recursive ones. Analogous results can be proved for many other subrecursive classes of functionals. (© 2004 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Continuous-time computation with restricted integration capabilities

Recursion theory on the reals, the analog counterpart of recursive function theory, is an approach to continuous-time computation inspired by the models of Classical Physics. In recursion theory on the reals, the discrete operations of standard recursion theory are replaced by operations on continuous functions such as composition and various forms of differential equations like indefinite integrals, linear differential equations and more general Cauchy problems. We define classes of real recursive functions in a manner similar to the standard recursion theory and we study their complexity. We prove both upper and lower bounds for several classes of real recursive functions, which lie inside the elementary functions, and can be characterized in terms of space complexity. In particular, we show that hierarchies of real recursive classes closed under restricted integration operations are related to the exponential space hierarchy. The results in this paper, combined with earlier results, suggest that there is a close connection between analog complexity classes and subrecursive classes, at least in the region between FLINSPACE and the primitive recursive functions.

Continuous-time computation with restricted integration capabilities

Recursion theory on the reals, the analog counterpart of recursive function theory, is an approach to continuous-time computation inspired by the models of Classical Physics. In recursion theory on the reals, the discrete operations of standard recursion theory are replaced by operations on continuous functions such as composition and various forms of differential equations like indefinite integrals, linear differential equations and more general Cauchy problems. We define classes of real recursive functions in a manner similar to the standard recursion theory and we study their complexity. We prove both upper and lower bounds for several classes of real recursive functions, which lie inside the elementary functions, and can be characterized in terms of space complexity. In particular, we show that hierarchies of real recursive classes closed under restricted integration operations are related to the exponential space hierarchy. The results in this paper, combined with earlier results, suggest that there is a close connection between analog complexity classes and subrecursive classes, at least in the region between FLINSPACE and the primitive recursive functions.

Accessible Recursive Functions

AbstractThe class of all recursive functions fails to possess a natural hierarchical structure, generated predicatively from “within”. On the other hand, many (proof-theoretically significant) sub-recursive classes do. This paper attempts to measure the limit of predicative generation in this context, by classifying and characterizing those (predictably terminating) recursive functions which can be successively defined according to an autonomy condition of the form: allow recursions only over well-orderings which have already been “coded” at previous levels. The question is: how can a recursion code a well-ordering? The answer lies in Girard's theory of dilators, but is reworked here in a quite different and simplified framework specific to our purpose. The “accessible” recursive functions thus generated turn out to be those provably recursive in ( –CA)0.

Intuitionistic formal theories with realizability in subrecursive classes

A family of formal intuitionistic theories is proposed (with each theory equivalent to the Peano arithmetic by expressive power) with realizability of proved formulas in several subrecursive classes, e.g. Grzegorczyk classes, polynomial-time computable functions class, etc. ∀ xA( x, f( x)) Algorithm extraction for∀ x∃ yA( x, y) is shown for various classes of bounded complexity. The results on polynomial computability are closely connected to work on the Bounded Arithmetic by S. Buss.

Equational derivation vs. computation

Subrecursive hierarchy classifications are used to compare the complexities of recursive functions according to (i) their derivations in a version of Kleene's equation calculus, and (ii) their computations by term-rewriting. In each case ordinal bounds are assigned, and it turns out that the respective complexity measures are given by (i) a version of the Fast Growing Hierarchy, and (ii) the Slow Growing Hierarchy. Known comparisons between the two hierarchies then provide ordinal trade-offs between (i) derivation and (ii) computation. Characteristics of some well-known subrecursive classes are also read off.

Diagonalization, uniformity, and fixed-point theorems

We derive new fixed-point theorems for subrecursive classes, together with a theorem on the uniformity of certain reductions, from a general formulation of the technique of delayed diagonalization. This formulation extends the main theorem of U. Schöning (Theoret. Comput. Sci. 18 (1982), 95–103) to cases which involve infinitely many diagonal classes Ck, and which allow each Ck to contain uncountably many members. The main technical work ties the familiar concept of a witness function directly to the often-studied Cantor-set topology on languages, and provides a “delay construction” which refines those due to Schöning, S. Breidtbart, and D. Schmidt. Our “a.e.” fixed-point theorems do not require that the “programming system” for the subrecursive class in question be well-behaved; we compare them to results which do. The other theorem is similar to the “uniform boundedness theorem” of classical analysis, and extends work of J. Grollmann and A. Selman.

Indexings of subrecursive classes

A subrecursive indexing is a programming language or Gödel numbering for a class of total recursive functions. Several properties of subrecursive indexings, such as effective composition and generation of constant functions, are investigated from an axiomatic point of view. The result is a theory akin to the axiomatic treatment of recursive function theory of Strong and Wagner. Using this formalism, we prove results relating the complexity of uniform simulation, diagonalization, deciding membership, and deciding halting; we give a subrecursive analog of Rice's theorem; we give a characterization of the combinatorial power of subrecursive indexings analogous to the combinatorial completeness of the lambda calculus; finally, we give a characterization the power of diagonalization over subrecursive classes and show that if P≠NP is provable at all, then it is provable by diagonalization.

Complexity of index sets and translating functions

The connection between index stes appearing in recursive enumerations of subrecursive classes and translating functions is investigated. Referring to a construction, which yields enumerations with low complexity for some index sets, we show that in such a case the complexity for some translating functions must be high. Furthermore a class of enumerations is discussed, which has elementary translating functions, but an important part of the index sets is not decidable by algorithms of the enumerated class (these index sets are of the form { x : ψ x ( a ) = f ( x )}).

On unsolvability in subrecursive classes of predicates.

On unsolvability in subrecursive classes of predicates.

Minimal pairs of polynomial degrees with subexponential complexity

The goal of extending work on relative polynomial time computability from computations relative to sets of natural numbers to computations relative to arbitrary functions of natural numbers is discussed. The principal techniques used to prove that the honest subrecursive classes are a lattice are then used to construct a minimal pair of polynomial degrees with subexponential complexity; that is two sets computable by Turing machines in subexponential time but not in polynomial time are constructed such that any set computable from both in polynomial time can be computed directly in polynomial time.

Polynomial and abstract subrecursive classes

The reducibility “polynomial time computable in” for arbitrary functions isintroduced. It generalizes Cook's definition from sets to arbitrary functions. A complexity-theoretic as well as a syntactic characterization is given and their equivalence is shown. This equivalence and the naturalness of both definitions give evidence that our notion is “correct.” The computable functions are classified into polynomial classes according to this reducibility. The ordering of these classes under set inclusion is studied. Honest classes are introduced and using them, the classification is related to the computational complexity of the functions classified. The algebraic structure of honest classes is also investigated. In Section II abstract subrecursive reducibilities are introduced. An axiomaticdefinition in purely recursion-theoretic terms is given; in particular, no reference to a particular machine model is needed. The definition is in the spirit of Strong's and Wagner's characterizations of basic recursive function theories. All known reducibilities are abstract reducibilities. The algebraic structure of abstract classes is explored.

Helping and the meet of pairs of honest subrecursive classes

A notion of one computable function helping the computation of another by an amount equal to a third is defined in terms of the lattices of honest subrecursive classes. It is said that two honest computable functions help each other's computation by an amount equal to a third honest function if the intersection (meet) of the subrecursive classes which the first two generate is equal to the class generated by the third; that is, two functions help each other's computation by an amount equal to a third if the information content they have in common is that given by the third. A technical characterization is given of those pairs of honest subrecursive classes for which there exist nontrivial third honest subrecursive classes whose information content in common with the first class is equal to the second class. Further, it is shown that for every pair of honest subrecursive classes with the first properly containing the second there is an effective, increasing sequence of honest subrecursive classes which have information content in common with the first class equal to the second class, and this sequence is cofinal upward with the set of all honest subrecursive classes which have information content in common with the first class equal to the second class. Although there is always an upper bound to such a sequence, the sequence may or may not be eventually constant (and therefore there may or may not be a maximal class whose information content in common with the first class is equal to the second).

On the density of honest subrecursive classes

The relation of honest subrecursive classes to the computational complexity of the functions they contain is briefly reviewed. It is shown that the honest subrecursive classes are dense under the partial ordering of set inclusion. In fact, any countable partial ordering can be embedded in the gap between an effective increasing sequence of honest subrecursive classes and an honest subrecursive class which is properly above the sequence (or in the gap between an eflective decreasing sequence and a class which is properly below the sequence). Information is obtained about the possible existence of least upper bounds (greater lower bounds) of increasing (decreasing) sequences of honest subrecursive classes. Finally it is shown that for any two honest subrecursive classes, one properly containing the other, there exists a pair of incomparable honest subrecursive classes such that the greatest lower bound of the pair is the smaller of the first two classes and the least upper bound of the pair is the larger of the first two classes.

The honest subrecursive classes are a lattice

The relation of honest subrecursive classes to the computational complexity of the functions they contain is reviewed. It is shown that the honest subrecursive classes are a distributive lattice under the partial ordering of set inclusion. The meet and join operations of the lattice are effective, and every honest subrecursive class is the greatest lower bound of two setwise incomparable honest subrecursive classes.

Subrecursive Programming Languages, Part I

The structural complexity of programming languages, and therefore of programs as well, can be measured by the subrecursive class of functions which characterize the language. Using such a measure of structural complexity, we examine the tradeoff relationship between structural and computational complexity. Since measures of structural complexity directly related to high level languages interest us most, we use abstract language models which approximate highly struc+, -,., +

Subrecursiveness: Machine-independent notions of computability in restricted time and storage

This paper deals with computability of recursive functions in restricted time and storage. Section 1 introduces some terminology, including the notion of an arithmetization of a computing machine. Section 2, extending previous work of Ritchie [11] and Cobham [2], characterizes classes of functions computable with limited rate of growth of storage, which are stable under variations in the type of computing machine allowed. Section 3 provides a similarly machine-independent characterization of the class of functions computable in linear bounded storage with polynomial bounded time (the bounds being in terms of the length of the argument), examines the conjecture that it is a proper subclass of the functions computable in linear bounded storage, and shows that it is the smallest subrecursive class with certain stability and adequacy properties. Attempts to deal with sharper bounds on computation time, in particular, to characterize computability in linear time and storage in a suitably machineindependent fashion, require more detailed considerations of machine structure and will be discussed in a future publication. Many of the results below appeared in Chapter 1 of the author's thesis [14]; some exceptions include Theorems 7 and 8, Corollary 8, and Examples 5 and 7.