A hierarchy of functions with respect to their role as bounds in the Turing reducibility of functions is introduced and studied. This hierarchy leads to a certain notion of incompressibility of sets which is also investigated.

The project was partially supported by a NSF grant of China. The author was grateful to Professor S. Lempp for his encouragement and suggestion while the author was visiting the Department of Mathematics at the University of Wisconsin.

In [3], two different effective versions of Borel embedding are defined. The first, called computable embedding, is based on uniform enumeration reducibility, while the second, called Turing computable embedding, is based on uniform Turing reducibility. While [3] focused mainly on computable embeddings, the present paper considers Turing computable embeddings. Although the two notions are not equivalent, we can show that they behave alike on the mathematically interesting classes chosen for investigation in [3]. We give a (...) “Pull-back Theorem”, saying that if Φ is a Turing computable embedding of K into K’, then for any computable infinitary sentence φ in the language of K’, we can find a computable infinitary sentence φ* in the language of K such that for all. (shrink)

This article offers comprehensive criticism of the Turing test and develops quality criteria for new artificial general intelligence (AGI) assessment tests. It is shown that the prerequisites A. Turing drew upon when reducing personality and human consciousness to “suitable branches of thought” re-flected the engineering level of his time. In fact, the Turing “imitation game” employed only symbolic communication and ignored the physical world. This paper suggests that by restricting thinking ability to symbolic systems alone Turing (...) unknowingly constructed “the wall” that excludes any possi-bility of transition from a complex observable phenomenon to an abstract image or concept. It is, therefore, sensible to factor in new requirements for AI (artificial intelligence) maturity assessment when approaching the Tu-ring test. Such AI must support all forms of communication with a human being, and it should be able to comprehend abstract images and specify con-cepts as well as participate in social practices. (shrink)

What is the relation between intelligence and computation? Although the difficulty of defining `intelligence' is widely recognized, many are unaware that it is hard to give a satisfactory definition of `computational' if computation is supposed to provide a non-circular explanation for intelligent abilities. The only well-defined notion of `computation' is what can be generated by a Turing machine or a formally equivalent mechanism. This is not adequate for the key role in explaining the nature of mental processes, because it (...) is too general, as many computations involve nothing mental, nor even processes: they are simply abstract structures. We need to combine the notion of `computation' with that of `machine'. This may still be too restrictive, if some non-computational mechanisms prove to be useful for intelligence. We need a theory-based taxonomy of {\em architectures} and {\em mechanisms} and corresponding process types. Computational machines my turn out to be a sub-class of the machines available for implementing intelligent agents. The more general analysis starts with the notion of a system with independently variable, causally interacting sub-states that have different causal roles, including both `belief-like' and `desire-like' sub-states, and many others. There are many significantly different such architectures. For certain architectures (including simple computers), some sub-states have a semantic interpretation for the system. The relevant concept of semantics is defined partly in terms of a kind of Tarski-like structural correspondence (not to be confused with isomorphism). This always leaves some semantic indeterminacy, which can be reduced by causal loops involving the environment. But the causal links are complex, can share causal pathways, and always leave mental states to some extent semantically indeterminate. (shrink)

We begin with the history of the discovery of computability in the 1930’s, the roles of Gödel, Church, and Turing, and the formalisms of recursive functions and Turing automatic machines . To whom did Gödel credit the definition of a computable function? We present Turing’s notion [1939, §4] of an oracle machine and Post’s development of it in [1944, §11], [1948], and finally Kleene-Post [1954] into its present form. A number of topics arose from Turing functionals (...) including continuous functionals on Cantor space and online computations. Almost all the results in theoretical computability use relative reducibility and o-machines rather than a-machines and most computing processes in the real world are potentially online or interactive. Therefore, we argue that Turing o-machines, relative computability, and online computing are the most important concepts in the subject, more so than Turing a-machines and standard computable functions since they are special cases of the former and are presented first only for pedagogical clarity to beginning students. At the end in §10–§13 we consider three displacements in computability theory, and the historical reasons they occurred. Several brief conclusions are drawn in §14. (shrink)

The paper gives a soundness and completeness proof for the implicative fragment of intuitionistic calculus with respect to the semantics of computability logic, which understands intuitionistic implication as interactive algorithmic reduction. This concept — more precisely, the associated concept of reducibility — is a generalization of Turing reducibility from the traditional, input/output sorts of problems to computational tasks of arbitrary degrees of interactivity.

I propose to consider the question, "Can machines think?" This should begin with definitions of the meaning of the terms "machine" and "think." The definitions might be framed so as to reflect so far as possible the normal use of the words, but this attitude is dangerous, If the meaning of the words "machine" and "think" are to be found by examining how they are commonly used it is difficult to escape the conclusion that the meaning and the answer to (...) the question, "Can machines think?" is to be sought in a statistical survey such as a Gallup poll. But this is absurd. Instead of attempting such a definition I shall replace the question by another, which is closely related to it and is expressed in relatively unambiguous words. The new form of the problem can be described in terms of a game which we call the 'imitation game." It is played with three people, a man (A), a woman (B), and an interrogator (C) who may be of either sex. The interrogator stays in a room apart front the other two. The object of the game for the interrogator is to determine which of the other two is the man and which is the woman. He knows them by labels X and Y, and at the end of the game he says either "X is A and Y is B" or "X is B and Y is A." The interrogator is allowed to put questions to A and B. We now ask the question, "What will happen when a machine takes the part of A in this game?" Will the interrogator decide wrongly as often when the game is played like this as he does when the game is played between a man and a woman? These questions replace our original, "Can machines think?". (shrink)

In this paper we consider the we known method by E. Post of solving the problem of construction of recursively enumerable sets that have a degree intermediate between the degrees of recursive and complete sets with respect to a given reducibility. Post considered reducibilities ≤m, ≤btt, ≤tt and ≤T and solved the problem for al of them except ≤T. Here we extend Post's original method of construction of incomplete sets onto two wide classes of sub-Turing reducibilities what were (...) studying in [1, 2]. (shrink)

We aim at providing a philosophical analysis of the notion of “proof by Church’s Thesis”, which is – in a nutshell – the conceptual device that permits to rely on informal methods when working in Computability Theory. This notion allows, in most cases, to not specify the background model of computation in which a given algorithm – or a construction – is framed. In pursuing such analysis, we carefully reconstruct the development of this notion (from Post to Rogers, to the (...) present days), and we focus on some classical constructions of the field, such as the construction of a simple set. Then, we make use of this focus in order to support the following encompassing claim (which opposes to a somehow commonly received view): the informal side of Computability, consisting of the large class of methods typically employed in the proofs of the field, is not fully reducible to its formal counterpart. (shrink)

We consider questions related to the rigidity of the structure R, the PTIME-Turing degrees of recursive sets of strings together with PTIME-Turing reducibility, pT, and related structures; do these structures have nontrivial automorphisms? We prove that there is a nontrivial automorphism of an ideal of R. This can be rephrased in terms of partial relativizations. We consider the sets which are PTIME-Turing computable from a set A, and call this class PTIMEA. Our result can be stated (...) as follows: There is an oracle, A, relative to which the PTIME-Turing degrees are not rigid . Furthermore, the automorphism can be made to preserve the complexity classes DTIMEA for all k 1, or to move any DTIMEA for n 2. From the existence of such an automorphism we conclude as a corollary that there is an oracle A relative to which the classes DTIME are not definable over R. We carry out the corresponding partial relativization for the many-one degrees to construct an oracle, A, relative to which the PTIMA-many-one degrees relative to A have a nontrivial automorphism, and one relative to which the lattice of sets in PTIMEA under inclusion have a nontrivial automorphism. The proof is phrased as a forcing argument; we construct the set A to meet a particular collection of dense sets in our notion of forcing. Roughly, the dense sets will guarantee that if A meets these sets and we split A into two pieces, A0 and A1, in a simple way, and then switching the roles of A0 and A1 in all computations from A will produce an automorphism of the ideal of PTIMA-degrees below A. We force A0 and A1 to have different PTIME-Turing degree; this will then make the automorphism nontrivial. An appropriately generic set A is constructed using a priority argument. (shrink)

We say that A≤LRB if every B-random set is A-random with respect to Martin–Löf randomness. We study this relation and its interactions with Turing reducibility, classes, hyperimmunity and other recursion theoretic notions.

Polynomial clone reducibilities are generalizations of the truth-table reducibilities. A polynomial clone is a set of functions over a finite set X that is closed under composition and contains all the constant and projection functions. For a fixed polynomial clone ${\fancyscript{C}}$ , a sequence ${B\in X^{\omega}}$ is ${\fancyscript{C}}$ -reducible to ${A \in {X}^{\omega}}$ if there is an algorithm that computes B from A using only effectively selected functions from ${\fancyscript{C}}$ . We show that if A is Kurtz random and ${\fancyscript{C}_{1} (...) \nsubseteq \fancyscript{C}_{2}}$ are polynomial clones, then there is a B that is ${\fancyscript{C}_{1}}$ -reducible to A but not ${\fancyscript{C}_{2}}$ -reducible to A. This implies a generalization of a result first proved by Lachlan (Z Math Logik Grundlagen Math 11:17–44, 1965) for the case |X| = 2. We also show that the same result holds if Kurtz random is replaced by Kolmogorov–Loveland stochastic. (shrink)

We sketch the historical and conceptual context of Turing's analysis of algorithmic or mechanical computation. We then discuss two responses to that analysis, by Gödel and by Gandy, both of which raise, though in very different ways. The possibility of computation procedures that cannot be reduced to the basic procedures into which Turing decomposed computation. Along the way, we touch on some of Cleland's views.

Informal construction -- Formal construction -- Limiting results.

We consider the strongest forms of enumeration reducibility, those that occur between 1- and npm-reducibility inclusive. By defining two new reducibilities which are counterparts to 1- and i-reducibility, respectively, in the same way that nm- and npm-reducibility are counterparts to m- and pm-reducibility, respectively, we bring out the structure of the strong reducibilities. By further restricting n1- and nm-reducibility we are able to define infinite families of reducibilities which isomorphically embed the r. e. (...) class='Hi'>Turing degrees. Thus the many well-known results in the theory of the r. e. Turing degrees have counterparts in the theory of strong reducibilities. We are also able to positively answer the question of whether there exist distinct reducibilities ≤y and ≤a between ≤e and ≤m such that there exists a non-trivial ≤y-contiguous ≤z degree. (shrink)

A version of the Church-Turing Thesis states that every effectively realizable physical system can be simulated by Turing Machines (‘Thesis P’). In this formulation the Thesis appears to be an empirical hypothesis, subject to physical falsification. We review the main approaches to computation beyond Turing definability (‘hypercomputation’): supertask, non-well-founded, analog, quantum, and retrocausal computation. The conclusions are that these models reduce to supertasks, i.e. infinite computation, and that even supertasks are no solution for recursive incomputability. This yields (...) that the realization of hypercomputing devices is implausible, and that Thesis P is not essentially different from the standard Church-Turing Thesis. (shrink)

Subcreative sets, introduced by Blum, are known to coincide with the effectively speedable sets. Subcreative sets are shown to be the complete sets with respect to S-reducibility, a special case of Turing reducibility. Thus a set is effectively speedable exactly when it contains the solution to the halting problem in an easily decodable form. Several characterizations of subcreative sets are given, including the solution of an open problem of Blum, and are used to locate the subcreative sets (...) with respect to the complete sets of other reducibilities. It is shown that q-cylindrification is an order-preserving map from the r.e. T-degrees to the r.e. S-degrees. Consequently, T-complete sets are precisely the r.e. sets whose q-cylindrifications are S-complete. (shrink)

Subcreative sets, introduced by Blum, are known to coincide with the effectively speedable sets. Subcreative sets are shown to be the complete sets with respect to S-reducibility, a special case of Turing reducibility. Thus a set is effectively speedable exactly when it contains the solution to the halting problem in an easily decodable form. Several characterizations of subcreative sets are given, including the solution of an open problem of Blum, and are used to locate the subcreative sets (...) with respect to the complete sets of other reducibilities. It is shown that q-cylindrification is an order-preserving map from the r.e. T-degrees to the r.e. S-degrees. Consequently, T-complete sets are precisely the r.e. sets whose q-cylindrifications are S-complete. (shrink)

Though less well known than his other work, Turings 1938 Princeton Thesis, this title which includes his notion of an oracle machine, has had a lasting influence on computer science and mathematics. It presents a facsimile of the original typescript of the thesis along with essays by Appel and Feferman that explain its still-unfolding significance.

We show the decidability of the existential theory of the recursively enumerable degrees in the language of Turing reducibility, Turing reducibility of the Turing jumps, and least and greatest element.

Schnorr randomness and computable randomness are natural concepts of random sequences. However van Lambalgen’s Theorem fails for both randomnesses. In this paper we define truth-table Schnorr randomness and truth-table reducible randomness, for which we prove that van Lambalgen's Theorem holds. We also show that the classes of truth-table Schnorr random reals relative to a high set contain reals Turing equivalent to the high set. It follows that each high Schnorr random real is half of a real for which van (...) Lambalgen's Theorem fails. Moreover we establish the coincidence between triviality and lowness notions for truth-table Schnorr randomness. © 2011 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim. (shrink)

We consider the computably enumerable sets under the relation of Q-reducibility. We first give several results comparing the upper semilattice of c.e. Q-degrees, RQ, Q, under this reducibility with the more familiar structure of the c.e. Turing degrees. In our final section, we use coding methods to show that the elementary theory of RQ, Q is undecidable.

Let ≤r and ≤sbe two binary relations on 2ℕ which are meant as reducibilities. Let both relations be closed under finite variation and consider the uniform distribution on 2ℕ, which is obtained by choosing elements of 2ℕ by independent tosses of a fair coin.Then we might ask for the probability that the lower ≤r-cone of a randomly chosen set X, that is, the class of all sets A with A ≤rX, differs from the lower ≤s-cone of X. By c osure (...) under finite variation, the Kolmogorov 0-1 aw yields immediately that this probability is either 0 or 1; in case it is 1, the relations are said to be separable by random oracles.Again by closure under finite variation, for every given set A, the probability that a randomly chosen set X is in the upper ≤r-cone of A is either 0 or 1; let Almostr be the class of sets for which the upper ≤r-cone has measure 1. In the following, results about separations by random oracles and about Almost classes are obtained in the context of generalized reducibilities, that is, for binary relations on 2ℕ which can be defined by a countable set of total continuous functionals on 2ℕ in the same way as the usual resource-bounded reducibilities are defined by an enumeration of appropriate oracle Turing machines. The concept of generalized reducibility comprises a natura resource-bounded reducibilities, but is more general; in particular, it does not involve any kind of specific machine model or even effectivity. The results on generalized reducibilities yield corollaries about specific resource-bounded reducibilities, including several results which have been shown previously in the setting of time or space bounded Turing machine computations. (shrink)

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In our former works, for a given concept of reduction, we study the following hypothesis: “For a random oracle A, with probability one, the degree of the one-query tautologies with respect to A is strictly higher than the degree of A.” In our former works (Suzuki in Kobe J. Math. 15, 91–102, 1998; in Inf. Comput. 176, 66–87, 2002; in Arch. Math. Logic 44, 751–762), the following three results are shown: The hypothesis for p-T (polynomial-time Turing) reduction is equivalent (...) to the assertion that the probabilistic complexity class R is not equal to NP; The hypothesis for p-tt (polynomial-time truth-table) reduction implies that P is not NP; The hypothesis holds for each of the following: disjunctive reduction, conjunctive reduction, and p-btt (polynomial-time bounded-truth-table) reduction. In this paper, we show the following three results: (1) Let c be a positive real number. We consider a concept of truth-table reduction whose norm is at most c times size of input, where for a relativized propositional formula F, the size of F denotes the total number of occurrences of propositional variables, constants and propositional connectives. Then, our main result is that the hypothesis holds for such tt-reduction, provided that c is small enough. How small c can we take so that the above holds? It depends on our syntactic convention on one-query tautologies. In our setting, the statement holds for all c < 1. (2) The hypothesis holds for monotone truth-table reduction (also called positive reduction). (3) Dowd (in Inf. Comput. 96, 65–76, 1992) shows a polynomial upper bound for the minimum sizes of forcing conditions associated with a random oracle. We apply the above result (1), and get a linear lower bound for the sizes. (shrink)