The Church-Turing Thesis has been the subject of many variations and interpretations over the years. Specifically, there are versions that refer only to functions over the natural numbers (as Church and Kleene did), while others refer to functions over arbitrary domains (as Turing intended). Our purpose is to formalize and analyze the thesis when referring to functions over arbitrary domains. First, we must handle the issue of domain representation. We show that, prima facie, the thesis is not well defined for arbitrary domains, since the choice of representation of the domain might have a non-trivial influence. We overcome this problem in two steps: (1) phrasing the thesis for entire computational models, rather than for a single function; and (2) proving a “completeness” property of the recursive functions and Turing machines with respect to domain representations. In the second part, we propose an axiomatization of an “effective model of computation” over an arbitrary countable domain. This axiomatization is based on Gurevich’s postulates for sequential algorithms. A proof is provided showing that all models satisfying these axioms, regardless of underlying data structure, are of equivalent computational power to, or weaker than, Turing machines.

For Yuri, profound thinker, esteemed expositor, and treasured friend. Abstract. Over the past two decades, Gurevich and his colleagues have developed axiomatic foundations for the notion of algorithm, be it classical, interactive, or parallel, and formalized them in a new framework of abstract state machines. Recently, this approach was extended to suggest axiomatic foundations for the notion of effective computation over arbitrary countable domains. This was accomplished in three different ways, leading to three, seemingly disparate, notions of effectiveness. We show that, though having taken different routes, they all actually lead to precisely the same concept. With this concept of effectiveness, we establish that there is – up to isomorphism – exactly one maximal effective model across all countable domains.

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We recently bought an IBM-BLOKBUSTR computer, after reading Osherson’s (1985) beguiling description of its ability to do the uncomputable. We had hoped to use it in our study of the human ability to generate music. But we ran into problems that we had not anticipated. Readers of this journal who plan to use this rather unusual machine in their research, might want to profit from our experience. What makes BLOKBUSTR different from other computers is that it can do more than compute. For example, it can run a program, K-BAR, that outputs all and only the integers in the set that mathematicians call “K-bar”. K-bar is uncomputable. It consists of the indexes of Turing machines that do not halt when run on their own indexes. Since we can prove-mathematically-that this set cannot be generated by a computation, BLOKBUSTR can do something no ordinary computer can. In our lab, we feel that some human cognitive abilities require more than computing. The ability to generate and recognize good music, for example.

After some preliminary grammatical considerations in this chapter, we collect the material on truth-functional logic in chapter 2; the material on quantifiers and identity in chapter 3 (proofs), chapter 5 (symbolization), and chapter 6 (semantics); some applications in chapter 7 (logical theory, arithmetic, set theory), and a discussion of definitions in chapter 8. Chapter 13 is an unfulfilled promise of a discussion of the chief theorems about modern logic.

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The present article describes a genome database reviewing gene-related knowledge of two model bacteria, Bacillus subtilis and Escherichia coli. The database, Indigo, is open through the World-Wide Web