This paper is a continuation of the authors' work , where the main problem considered was whether a given recursive structure is recursively isomorphic to a polynomial-time structure. In that paper, a recursive Abelian group was constructed which is not recursively isomorphic to any polynomial-time Abelian group. We now show that if every element of a recursive Abelian group has finite order, then the group is recursively isomorphic to a polynomial-time group. Furthermore, if the orders are bounded, then the group (...) is recursively isomorphic to a polynomial-time group with universe A being the set of tally representations of natural numbers Tal = s{;1s};* or the set of binary representations of the natural numbers Bin. We also construct a recursive Abelian group with all elements of finite order but which has elements of arbitrary large finite order which is not isomorphic to any polynomial-time group with universe Tal or Bin. Similar results are obtained for structures , where f is a permutation on the set A. (shrink)

The central problem considered in this paper is whether a given recursive structure is recursively isomorphic to a polynomial-time structure. Positive results are obtained for all relational structures, for all Boolean algebras and for the natural numbers with addition, multiplication and the unary function 2x. Counterexamples are constructed for recursive structures with one unary function and for Abelian groups and also for relational structures when the universe of the structure is fixed. Results are also given which distinguish primitive recursive structures, (...) exponential-time structures and structures with honest witnesses. (shrink)

We develop a theory of LOGSPACE structures and apply it to construct a number of examples of Abelian Groups which have LOGSPACE presentations. We show that all computable torsion Abelian groups have LOGSPACE presentations and we show that the groups ${\mathbb {Z}, Z(p^{\infty})}$ , and the additive group of the rationals have LOGSPACE presentations over a standard universe such as the tally representation and the binary representation of the natural numbers. We also study the effective categoricity of such groups. For (...) example, we give conditions are given under which two isomorphic LOGSPACE structures will have a linear space isomorphism. (shrink)

A Π01 class is an effectively closed set of reals. We study properties of these classes determined by cardinality, measure and category as well as by the complexity of the members of a class P. Given an effective enumeration {Pe:e < ω} of the Π01 classes, the index set I for a certain property is the set of indices e such that Pe has the property. For example, the index set of binary Π01 classes of positive measure is Σ02 complete. (...) Various notions of boundedness are discussed and classified. For example, the index set of the recursively bounded classes is Σ03 complete and the index set of the recursively bounded classes which have infinitely many recursive members is Π04 complete. Consideration of the Cantor-Bendixson derivative leads to index sets in the transfinite levels of the hyperarithmetic hierarchy. (shrink)

In this paper, we study the lattice of r.e. subspaces of a recursively presented vector space V ∞ with regard to the various complexity-theoretic speed-up properties such as speedable, effectively speedable, levelable, and effectively levelable introduced by Blum and Marques. In particular, we study the interplay between an r.e. basis A for a subspace V of V ∞ and V with regard to these properties. We show for example that if A or V is speedable , then V is levelable (...) . We also show that if A is an r.e. subset of a recursive basis for V ∞ , then A is levelable iff V is speedable while it is not the case that A is levelable iff V is speedable. We also provide several contrasts between the lattice of r.e. subspaces and the lattice of r.e. sets with respect to these speed-up properties. For example, every maximal set is levelable but we show that there exist supermaximal spaces which are nonspeedable in all possible r.e. degrees as well as supermaximal spaces which are levelable in all r.e. degrees which contain levelable sets. (shrink)

We examine the effective categoricity of equivalence structures via Ershov's difference hierarchy. We explore various kinds of categoricity available by distinguishing three different notions of isomorphism available in this hierarchy. We prove several results relating our notions of categoricity to computable equivalence relations: for example, we show that, for such relations, computable categoricity is equivalent to our notion of weak ω-c.e. categoricity, and that $\Delta _2^0 $ -categoricity is equivalent to our notion of graph-ω-c.e. categoricity.

The problem of when a recursive graph has a recursive k-coloring has been extensively studied by Bean, Schmerl, Kierstead, Remmel, and others. In this paper, we study the polynomial time analogue of that problem. We develop a number of negative and positive results about colorings of polynomial time graphs. For example, we show that for any recursive graph G and for any k, there is a polynomial time graph G′ whose vertex set is {0,1}* such that there is an effective (...) degree preserving correspondence between the set of k-colorings of G and the set of k-colorings of G′ and hence there are many examples of k-colorable polynomial time graphs with no recursive k-colorings. Moreover, even though every connected 2-colorable recursive graph is recursively 2-colorable, there are connected 2-colorable polynomial time graphs which have no primitive recursive 2-coloring. We also give some sufficient conditions which will guarantee that a polynomial time graph has a polynomial time or exponential time coloring. (shrink)

We give resource bounded versions of the Completeness Theorem for propositional and predicate logic. For example, it is well known that every computable consistent propositional theory has a computable complete consistent extension. We show that, when length is measured relative to the binary representation of natural numbers and formulas, every polynomial time decidable propositional theory has an exponential time (EXPTIME) complete consistent extension whereas there is a nondeterministic polynomial time (NP) decidable theory which has no polynomial time complete consistent extension (...) when length is measured relative to the binary representation of natural numbers and formulas. It is well known that a propositional theory is axiomatizable (respectively decidable) if and only if it may be represented as the set of infinite paths through a computable tree (respectively a computable tree with no dead ends). We show that any polynomial time decidable theory may be represented as the set of paths through a polynomial time decidable tree. On the other hand, the statement that every polynomial time decidable, relative to the tally representation of natural numbers and formulas, is equivalent to P = NP. For predicate logic, we develop a complexity theoretic version of the Henkin construction to prove a complexity theoretic version of the Completeness Theorem. Our results imply that that any polynomial space decidable theory △ possesses a polynomial space computable model which is exponential space decidable and thus △ has an exponential space complete consistent extension. Similar results are obtained for other notions of complexity. (shrink)

We study computability theoretic properties of and equivalence structures and how they differ from computable equivalence structures or equivalence structures that belong to the Ershov difference hierarchy. Our investigation includes the complexity of isomorphisms between equivalence structures and between equivalence structures.

Index sets are used to measure the complexity of properties associated with the differentiability of real functions and the existence of solutions to certain classic differential equations. The new notion of a locally computable real function is introduced and provides several examples of Σ04 complete sets.

An ω-language is a set of infinite sequences on a countable language, and corresponds to a set of real numbers in a natural way. Languages may be described by logical formulas in the arithmetical hierarchy and also may be described as the set of words accepted by some type of automata or Turing machine. Certain families of languages, such as the equation image languages, may enumerated as P0, P1, … and then an index set associated to a given property R (...) of languages is just the set of e such that Pe has the property. The complexity of index sets for 7 types of languages is determined for various properties related to the size of the language. (shrink)

The first-order theory of the lattice of recursively enumerable closed subsets of an effective topological space is proved undecidable using the undecidability of the first-order theory of the lattice of recursively enumerable sets. In particular, the first-order theory of the lattice of recursively enumerable closed subsets of Euclidean n -space, for all n , is undecidable. A more direct proof of the undecidability of the lattice of recursively enumerable closed subsets of Euclidean n -space, n ⩾ 2, is provided using (...) the method of reduction and the recursive inseparability of the set of all formulae satisfiable in every model of the theory of SIBs and the set of all formulae refutable in some finite model of the theory of SIBs. (shrink)

We investigate the complexity of finding solutions to infinite recursive constraint satisfaction problems. We show that, in general, the problem of finding a solution to an infinite recursive constraint satisfaction problem is equivalent to the problem of finding an infinite path through a recursive tree. We also identify natural classes of infinite recursive constraint satisfaction problems where the problem of finding a solution to the infinite recursive constraint satisfaction problem is equivalent to the problem of finding an infinite path through (...) finitely branching recursive trees or recursive binary trees. There are a large number of results in the literature on the complexity of the problem of finding an infinite path through a recursive tree. Our main result allows us to automatically transfer such results to give equivalent results about the complexity of the problem of finding a solution to a recursive constraint satisfaction problem. (shrink)

A computable graph is constructed which is not computably isomorphic to any polynomial-time graph with a standard universe . Conditions are given under which a computable graph is computably isomorphic to a polynomial-time graph with a standard universe — for example, if every vertex has finite degree. Two special types of graphs are studied. It is shown that any computable tree is recursively isomorphic to a p-time tree with standard universe and that any computable equivalence relation is computably isomorphic to (...) a p-time equivalence relation with standard universe. (shrink)

This paper discusses an extension of Answer Set Programming called Hybrid Answer Set Programming which allows the user to reason about dynamical systems that exhibit both discrete and continuous aspects. The unique feature of Hybrid ASP is that it allows the use of ASP type rules as controls for when to apply algorithms to advance the system to the next position. That is, if the prerequisites of a rule are satisfied and the constraints of the rule are not violated, then (...) the algorithm associated with the rule is invoked. (shrink)

We investigate notions of randomness in the space ${{\mathcal C}(2^{\mathbb N})}$ of continuous functions on ${2^{\mathbb N}}$ . A probability measure is given and a version of the Martin-Löf test for randomness is defined. Random ${\Delta^0_2}$ continuous functions exist, but no computable function can be random and no random function can map a computable real to a computable real. The image of a random continuous function is always a perfect set and hence uncountable. For any ${y \in 2^{\mathbb N}}$ , (...) there exists a random continuous function F with y in the image of F. Thus the image of a random continuous function need not be a random closed set. The set of zeroes of a random continuous function is always a random closed set. (shrink)