Acceptable programming systems have many nice properties like s-m-n-Theorem, Composition and Kleene Recursion Theorem. Those properties are sometimes called control structures, to emphasize that they yield tools to implement programs in programming systems. It has been studied, among others by Riccardi and Royer, how these control structures influence or even characterize the notion of acceptable programming system. The following is an investigation, how these control structures behave in the more general setting of complete numberings as defined by Mal'cev and Eršov.

Acceptable programming systems have many nice properties like s-m-n-Theorem, Composition and Kleene Recursion Theorem. Those properties are sometimes called control structures, to emphasize that they yield tools to implement programs in programming systems. It has been studied, among others by Riccardi and Royer, how these control structures influence or even characterize the notion of acceptable programming system. The following is an investigation, how these control structures behave in the more general setting of complete numberings as defined by Mal'cev and Eršov.

Terry Horgan and John Tienson have suggested that connectionism might provide a framework within which to articulate a theory of cognition according to which there are mental representations without rules (RWR) (Horgan and Tienson 1988, 1989, 1991, 1992). In essence, RWR states that cognition involves representations in a language of thought, but that these representations are not manipulated by the sort of rules that have traditionally been posited. In the development of RWR, Horgan and Tienson attempt to forestall a particular (...) line of criticism, theSyntactic Argument, which would show RWR to be inconsistent with connectionism. In essence, the argument claims that the node-level rules of connectionist networks, along with the semantic interpretations assigned to patterns of activation, serve to determine a set of representation-level rules incompatible with the RWR conception of cognition. The present paper argues that the Syntactic Argument can be made to show that RWR is inconsistent with connectionism. (shrink)

This paper discusses refinements of the natural ordering of them-degrees (1-degrees) of strong recursive reducibility classes. Such refinements are obtained by posing complexity conditions on the reduction function. The discussion uses the axiomatic complexity theory and is hence very general. As the main result it is proved that if the complexity measure is required to be linearly bounded (and space-like), then a natural class of refinements forms a lattice with respect to a natural ordering upon them.

We provide and interpret a new measure independent characterization of the Gödel speed-up phenomenon. In particular, we prove a theorem that demonstrates the indifference of the concept of a measure independent Gödel speed-up to an apparent weakening of its definition that is obtained by requiring only those measures appearing in some fixed Blum complexity measure to participate in the speed-up, and by deleting the “for all r” condition from the definition so as to relax the required amount of speed-up. We (...) interpret our results as correlating the relative difficulty of mechanically recognizing theories with the relative power and the relative abstractness of the theories. We conclude by providing two open problems concerning possible similarities and relationships between the Blum speedability and Gödel speed-up phenomena. MSC: 03D15, 03F20. (shrink)

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” means that we can construct a functionθsuch thatP)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. (shrink)

The received view is that computational states are individuated at least in part by their semantic properties. I offer an alternative, according to which computational states are individuated by their functional properties. Functional properties are specified by a mechanistic explanation without appealing to any semantic properties. The primary purpose of this paper is to formulate the alternative view of computational individuation, point out that it supports a robust notion of computational explanation, and defend it on the grounds of how computational (...) states are individuated within computability theory and computer science. A secondary purpose is to show that existing arguments for the semantic view are defective. (shrink)

According to some philosophers, computational explanation is proprietary
to psychology—it does not belong in neuroscience. But neuroscientists routinely offer computational explanations of cognitive phenomena. In fact, computational explanation was initially imported from computability theory into the science of mind by neuroscientists, who justified this move on neurophysiological grounds. Establishing the legitimacy and importance of computational explanation in neuroscience is one thing; shedding light on it is another. I raise some philosophical questions pertaining to computational explanation and outline some promising answers that (...) are being developed by a number of authors. (shrink)

In this work a double exponential time inseparability result is proven for a finitely axiomatizable first order theory Q₊. The theory, subset of Presburger theory of addition S₊, is the additive fragment of Robinson system Q. We prove that every set that separates Q₊` from the logically false sentences of addition is not recognizable by any Turing machine working in double exponential time. The lower bound is given both in the non-deterministic and in the linear alternating time models. The result (...) implies also that any theory of addition that is consistent with Q₊—in particular any theory contained in S₊—is at least double exponential time difficult. Our inseparability result is an improvement on the known lower bounds for arithmetic theories. Our proof uses a refinement and adaptation of the technique that Fischer and Rabin used to prove the difficulty of S₊. Our version of the technique can be applied to any incomplete finitely axiomatizable system in which all of the necessary properties of addition are provable. (shrink)

We discuss Kunen’s algorithmic implementation of a proof for the Paris–Harrington theorem, and the author’s and da Costa’s proposed “exotic” formulation for the P = NP hypothesis. Out of those two examples we ponder the relation between mathematics within an axiomatic framework, and intuitive or informal mathematics.

We show that the P 2 0 sentence, and explore some of theconsequences of that fact. This paper summarizes recent workby the author with N. C. A. da Costa on the P

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A realizability notion that employs only Kalmar elementary functions is defined, and, relative to it, the soundness of EA-(Π₁⁰-IR), a fragment of Heyting Arithmetic (HA) with names and axioms for all elementary functions and induction rule restricted to Π₁⁰ formulae, is proved. As a corollary, it is proved that the provably recursive functions of EA-(Π₁⁰-IR) are precisely the elementary functions. Elementary realizability is proposed as a model of strict arithmetic constructivism, which allows only those constructive procedures for which the amount (...) of computational resources required can be bounded in advance. (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)

Hypercomputation—the hypothesis that Turing-incomputable objects can be computed through infinitary means—is ineffective, as the unsolvability of the halting problem for Turing machines depends just on the absence of a definite value for some paradoxical construction; nature and quantity of computing resources are immaterial. The assumption that the halting problem is solved by oracles of higher Turing degree amounts just to postulation; infinite-time oracles are not actually solving paradoxes, but simply assigning them conventional values. Special values for non-terminating processes are likewise (...) irrelevant, since diagonalization can cover any amount of value assignments. This should not be construed as a restriction of computing power: Turing’s uncomputability is not a ‘barrier’ to be broken, but simply an effect of the expressive power of consistent programming systems. (shrink)

New success criteria of inductive inference in computational learning theory are introduced which model learning total (not necessarily recursive) functions with (possibly everywhere) imprecise theories from (possibly always) inaccurate data. It is proved that for any level of error allowable by the new success criteria, there exists a class of recursive functions such that not all f are identifiable via the criterion at that level of error. Also, necessary and sufficient conditions on the error level are given for when more (...) classes of functions may be identified. (shrink)

Hay and, then, Johnson extended the classic Rice and Rice-Shapiro Theorems for computably enumerable sets, to analogs for all the higher levels in the finite Ershov Hierarchy. The present paper extends their work to analogs in the transfinite Ershov Hierarchy. Some of the transfinite cases are done for all transfinite notations in Kleene's important system of notations, equation image. Other cases are done for all transfinite notations in a very natural, proper subsystem equation image of equation image, where equation image (...) has at least one notation for each constructive ordinal. In these latter cases it is open as to what happens for the entire set of transfinite notations in equation image. (shrink)

We investigate a new paradigm in the context of learning in the limit, namely, learning correction grammars for classes of computably enumerable (c.e.) languages. Knowing a language may feature a representation of it in terms of two grammars. The second grammar is used to make corrections to the first grammar. Such a pair of grammars can be seen as a single description of (or grammar for) the language. We call such grammars correction grammars. Correction grammars capture the observable fact that (...) people do correct their linguistic utterances during their usual linguistic activities. We show that learning correction grammars for classes of c.e. languages in the TxtEx-model (i.e., converging to a single correct correction grammar in the limit) is sometimes more powerful than learning ordinary grammars even in the TxtBe-model (where the learner is allowed to converge to infinitely many syntactically distinct but correct conjectures in the limit). For each n ≥ 0, there is a similar learning advantage, again in learning correction grammars for classes of c.e. languages, but where we compare learning correction grammars that make n + 1 corrections to those that make n corrections. The concept of a correction grammar can be extended into the constructive transfinite, using the idea of counting-down from notations for transfinite constructive ordinals. This transfinite extension can also be conceptualized as being about learning Ershov-descriptions for c.e. languages. For u a notation in Kleene's general system $(O,\, < _o )$ of ordinal notations for constructive ordinals, we introduce the concept of an u-correction grammar, where u is used to bound the number of corrections that the grammar is allowed to make. We prove a general hierarchy result: if u and v are notations for constructive ordinals such that $u\, < _o \,v$ , then there are classes of c.e. languages that can be TxtEx-learned by conjecturing v-correction grammars but not by conjecturing u-correction grammars. Surprisingly, we show that— above "ω-many" corrections—it is not possible to strengthen the hierarchy: TxtEx-learning u-correction grammars of classes of c.e. languages, where u is a notation in O for any ordinal, can be simulated by TxtBe-learning ω-correction grammars, where ω is any notation for the smallest infinite ordinal ω. (shrink)

A generator program for a computable function (by definition) generates an infinite sequence of programs all but finitely many of which compute that function. Machine learning of generator programs for computable functions is studied. To motivate these studies partially, it is shown that, in some cases, interesting global properties for computable functions can be proved from suitable generator programs which cannot be proved from any ordinary programs for them. The power (for variants of various learning criteria from the literature) of (...) learning generator programs is compared with the power of learning ordinary programs. The learning power in these cases is also compared to that of learning limiting programs, i.e., programs allowed finitely many mind changes about their correct outputs. (shrink)

Freivalds defined an acceptable programming system independent criterion for learning programs for functions in which the final programs were required to be both correct and "nearly" minimal size, i.e., within a computable function of being purely minimal size. Kinber showed that this parsimony requirement on final programs limits learning power. However, in scientific inference, parsimony is considered highly desirable. A lim-computablefunction is (by definition) one calculable by a total procedure allowed to change its mind finitely many times about its output. (...) Investigated is the possibility of assuaging somewhat the limitation on learning power resulting from requiring parsimonious final programs by use of criteria which require the final, correct programs to be "not-so-nearly" minimal size, e.g., to be within a lim-computable function of actual minimal size. It is shown that some parsimony in the final program is thereby retained, yet learning power strictly increases. Considered, then, are lim-computable functions as above but for which notations for constructive ordinals are used to bound the number of mind changes allowed regarding the output. This is a variant of an idea introduced by Freivalds and Smith. For this ordinal notation complexity bounded version of lim-computability, the power of the resultant learning criteria form finely graded, infinitely ramifying, infinite hierarchies intermediate between the computable and the lim-computable cases. Some of these hierarchies, for the natural notations determining them, are shown to be optimally tight. (shrink)

links to current studies in the theory of scientific discovery. Our own theory is elaborated in four papers accessible in pdf format through the links on the left.