Showing papers in "Studies in logic and the foundations of mathematics in 1980"
TL;DR: It is argued that Turing's analysis of computation by a human being does not apply directly to mechanical devices, and it is proved that if a device satisfies the principles then its successive states form a computable sequence.
Abstract: After a brief review of Church's thesis and Godel's objection to it, it is argued that Turing's analysis of computation by a human being does not apply directly to mechanical devices. A set-theoretic form of description for discrete deterministic machines is elaborated and four principles (or constraints) are enunciated, which, it is argued, any such machine must satisfy. The most important of these, called “the principle of local causality” rejects the possibility of instantaneous action at a distance. Although the principles are justified by an appeal to the geometry of space-time, the formulation is quite abstract, and can be applied to all kinds of automata and to algebraic systems. It is proved that if a device satisfies the principles then its successive states form a computable sequence. Counter-examples are constructed which show that if the principles be weakened in almost any way, then there will be devices which satisfy the weakened principles and which can calculate any number-theoretic function.
329 citations
Abstract: We prove some results in group theory in a model theoretic spirit (i) We construct Jonsson groups of cardinality χ, and other cardinalities as well This answers an old question of Kurosh (ii) Our group is simple with no maximal subgroup; so it follows that taking Frattini subgroups does not commute with direct products (ii) Assuming the continuum hypothesis, our group is not a topological group, except with the trivial topologies This answers a quite old question of AA Markov In the construction we use small cancellation theory We try to make the paper intelligible to both group theorists and model theorists Only a knowledge of naive set theory and group theory is needed
132 citations
TL;DR: In this article, it was shown that a finitely generated simple group has a solvable word problem if and only if there is an embedding of the group into a finite presented simple group that is a subgroup of the finitely presented group.
Abstract: Publisher Summary It is known that a finitely generated simple group is a subgroup of a finitely presented group if and only if it has a recursively enumerable set of defining relations, and that it has a recursively enumerable set of defining relations if and only if it has a solvable word problem. This chapter presents the results that a finitely generated group has a solvable word problem if and only if there is an embedding of the group into a finitely generated simple group that is a subgroup of a finitely presented group. It was shown that if a finitely generated group has a word problem that is solvable with respect to one presentation on a finite set of generators, then the word problem is solvable with respect to every presentation on a finite set of generators. Thus, it can be said that the word problem for a finitely generated group is solvable or unsolvable without referring to a specific presentation. The chapter also discusses the countable groups assuming that all the presentations referred to are on finite or countably infinite set of generators.
110 citations
TL;DR: The chapter describes self-application to recursion by the proof of David Park's theorem to the effect that the least fixed-point operator and the paradoxical combinator are the same in a wide class of well-behaved models.
Abstract: Publisher Summary The chapter presents an exposition of why the λ-calculus has models. The A-calculus was one of the first areas of research of Professor Kleene, in which the experience gained by him was surely beneficial in his later development of the recursive function theory. The chapter discusses a very short historical summary, and it is found that there is considerable overlap with CURRY. There is a review of the theory of functions and relations as sets leading up to the important notion of a continuous set mapping. The problem of the self-application of a function to itself as an argument is discussed in the chapter from a new angle. The model (essentially due to PLOTKIN) of the basic laws of λ -calculus thus results. The chapter describes self-application to recursion by the proof of David Park's theorem to the effect that the least fixed-point operator and the paradoxical combinator are the same in a wide class of well-behaved models. The connection thus engendered to recursion theory (r. e. sets) is outlined, and some remarks on recent results about ill-behaved models and on induction principles are discussed. The theme of type theory and a construction of an (η)-model with fewer -type distinctions is presented. There is a brief discussion of how to introduce more type distinctions into models via equivalence relations. The chapter also presents various points of philosophical disagreement with Professor Curry.
109 citations
TL;DR: This paper constitutes a first attempt to sistematize the present state of the development of para-consistent logic, as well as the main topics and open questions related to it, and to have mainly an expository character.
Abstract: This paper constitutes a first attempt to sistematize the present state of the development of para-consistent logic, as well as the main topics and open questions related to it. As we want this paper to have mainly an expository character, we will not in general be rigorous, especially when an intuitive presentation is better for a first understanding of the questions under consideration, as, for example, in Section 1. Section 6 is perhaps the only one where the reader will find some original results. The bibliography, though large, is of course not intended to be complete. A general idea of the content of this paper is given by the Index.
89 citations
TL;DR: In this article, it was shown that the first order theories of many reducibility orderings are recursively isomorphic to second order logic on countable sets (and so to true second order arithmetic).
Abstract: We show that the first order theories of many reducibility orderings are recursively isomorphic to second order logic on countable sets (and so to true second order arithmetic). The reduction procedure uses some initial segment results and Spector's theorem on countable ideals in the degrees to code quantification over symmetric irreflexive binary relations. This is known to be enough to obtain full second order logic. Applications to other theories are mentioned as are several to problems of definability in, and automorphisms of, the Turing degrees.
59 citations
55 citations
TL;DR: In this article, a characterization of principal congruences of De Morgan algebras is given and from it it is derived that the variety of finite De Morgan algebra has DPC and CEP and this characterization is then applied to give a new proof of Kalman's characterization of subdirectly irreducible in this variety and thus to obtain the representation theorem for DeMorgan algebra.
Abstract: In this paper a characterization of principal congruences of De Morgan algebras is given and from it we derive that the variety of De Morgan algebras has DPC and CEP The characterization is then applied to give a new proof of Kalman's characterization of subdirectly irreducibles in this variety and thus to obtain the representation theorem for DeMorgan algebras first proved by Kalman and independently, using topological methods, by Bialynicki-Birula and Rasiowa From this representation it is deduced that finite De Morgan algebras are not the only ones with Boolean congruence lattices Finally it is shown that the compact elements in the congruence lattice of a De Morgan algebra form a Boolean sublattice
40 citations
TL;DR: In this paper, it was shown that 2 ω is not the ω 1 union of meager sets, and the latter does not imply CH, while the former does imply CH.
Abstract: It is shown that 2 ω is the ω 1 union of meager sets does not imply 2 ω is the ω 1 , union of disjoint non-empty closed sets and the latter does not imply CH.
26 citations
TL;DR: In this article, the algebraically closed groups up to partial isomorphism are characterized by recursion-theoretic invariants and their properties determined by looking at these invariants, and the results surveyed are: (1) a non-trivial group M is algebraic closed if every finite system of equations with coefficients from M, which is solvable in a supergroup of M, has a solution in M.
Abstract: Publisher Summary This chapter characterizes the algebraically closed groups up to partial isomorphism by “recursion-theoretic” invariants and determines their properties by looking at these invariants. It reviews the known about results algebraically closed groups. The results surveyed are: (1) a non-trivial group M is algebraically closed if every finite system of equations with coefficients from M, which is solvable in a supergroup of M, has a solution in M. By the theorem of Higman, Neumann, and Neumann— that is, every isomorphism of subgroups is extendable to an inner automorphism of a supergroup, it can be concluded that: Algebraically closed groups are ω-homogeneous—that is, every isomorphism of finitely generated subgroups is extendable to an automorphism. ω-homogeneous groups M are determined up to partial isomorphism by their skeleton Sk(M) (the class of all finitely generated groups that are embeddable in M). Therefore, in the study of properties that are compatible with partial isomorphism, it is enough to study the skeletons of algebraically closed groups.
23 citations
TL;DR: The present paper takes steps toward developing a certain semantics to flesh out the formal bones of those computations of the generalized recursion theory of γ-functionals.
Abstract: In my June 13, 1977 Oslo address (Generalized recursion theory 11, 1978) I overcame the limitations in my 1959, 1963 theory on substitution of γ-functionals and on the use of the first recursion theorem by introducing a new list of schemata, with computation rules that do not require values of parts to be computed unless and until they are needed. Computation thereby became starkly formal. The present paper takes steps toward developing a certain semantics to flesh out the formal bones of those computations. The old types 0,1,2 are represented within new types where type 0 = {u, 0, 1,2,…} (u = “undefined” or “unknown”), and the higher types are composed of the “unimonotone” partial one-place functions from the preceding type into {0,1,2,…}. These are the functions which are monotone and can each be embodied in an oracle who, e.g. at type 2, has been programmed by Apollo to respond (if she does), to any question “ 2 ( 1 )?” we ask, on the basis of information about 1 which she obtains by asking in turn questions of an oracle embodying 1 and observing her responses.
TL;DR: In this article, the authors discuss Malcev's problem and groups with a normal form and provide a detailed definition of a group with standard basis, together with some examples and discusses Novikov groups.
Abstract: Publisher Summary This chapter discusses Malcev's problem and groups with a normal form. The problem known as Malcev's problem is “does there exist an associative ring R, without zero divisors that is not embeddable in a skew field whose multiplicative semigroup R* of non-zero elements is embeddable in a group.” The chapter also provides the detailed definition of a group with standard basis, together with some examples and discusses Novikov groups.
TL;DR: In this paper, the authors give back-and-forth systems characterizing elementary equivalence in those logics and their fragments of bounded quantifier rank and apply to higher order quantifiers also.
Abstract: L ∞Ω (K) is the logic obtained by adding a Lindstrom's quantifier K 1 … k ( ϕ 1 ( 1 )… ϕ k ( k )) to the logical operations of L Ω . The corresponding finitary logic is L ∞ω ( K ), and L Ω ( K i ) i∉⌉ is obtained by adjoining a family of quantifiers. In this paper, we give back-and-forth systems characterizing elementary equivalence in those logics and their fragments of bounded quantifier rank. This generalizes work of Fraisse and Ehrenfeucht for L ∞ω , Karp for L Ω , Brown, Lipner, and Vinner for cardinal quantifiers, Badger for Magidor-Malitz quantifiers, and others. Our systems apply to higher order quantifiers also.
TL;DR: In this article, a discussion of some philosophical criticisms of combinatory logic is presented, preceded by a brief survey to give background and a brief discussion of the main arguments of these criticisms.
Abstract: This is a discussion of some philosophical criticisms of combinatory logic, preceded by a brief survey to give background.
TL;DR: In this paper, it was shown that a non-trivial solution of a quadratic equation W = 1 in a group G=(X: R) induces a cancellation diagram on a compact surface S defined by an endomorphic image of W. If it is assumed that R satisfies a suitable small cancellation hypothesis, such a diagram cannot exist, and thus it can be conclude that all solutions of W=1 in G are trivial.
Abstract: Publisher Summary This chapter discusses cancellation diagrams that are a fundamental tool of combinatorial group theory. There is an intuitive connection between the question of commuting elements and cancellation diagrams on the torus, which has yet to be made precise. Studies suggest that the connection can be made precise for any quadratic word. In terms to be defined precisely, the chapter shows that a “non-trivial” solution of a “quadratic equation” W=1 in a group G=(X: R) induces a “non-trivial” cancellation diagram on a compact surface S defined by an endomorphic image of W. If it is assumed that R satisfies a suitable small cancellation hypothesis, such a diagram cannot exist, and thus it can be conclude that all solutions of W=1 in G are “trivial.” The consequences of this result are also discussed in the chapter. The present methods also yield a very simple proof of the important theorem of Nielsen that if G is the fundamental group of a compact surface then all the automorphisms of G are induced by automorphisms of the corresponding free group.
TL;DR: In this article, it was shown that there exist 2N 0 analytic subsets of the Cantor space having pairwise incomparable Kleene degrees, and that no two of these sets are Borel isomorphic.
Abstract: We show that the following assertion is true in a wide class of models of Zermelo-Fraenkel set theory. There exist 2N0 analytic subsets of the Cantor space having pairwise incomparable Kleene degrees. It follows by a lemma of Kuratowski that no two of these sets are Borel isomorphic.
TL;DR: This chapter defines modular machines that are related to Minsky machines, defined in such a way that Turing computable functions are computable by modular machines, which provides a new proof that Turing-computable functions are partial recursive.
Abstract: Publisher Summary This chapter defines modular machines that are related to Minsky machines. These machines act on N2, the set of pairs of natural numbers, with a very simple transition function. It will then be almost immediate that any function computable by a modular machine is partial recursive. On the other hand, modular machines are defined in such a way that Turing computable functions are computable by modular machines. This provides a new proof that Turing-computable functions are partial recursive. It also provides an easy proof of the normal form theorem for the partial recursive functions, since the data for a modular machine, being numerical in nature, can easily be encoded by a natural number with the decodings being primitive recursive. The chapter provides applications of modular machines in group theory.
TL;DR: In this paper, modular machines and embedding theorems are used to give simple proofs to various theorem, such as any finitely generated recursively presented group can be embedded in a finitely presented group.
Abstract: Publisher Summary This chapter discusses modular machines and embedding theorems. It uses modular machines to give simple proofs to various theorems, such as any finitely generated recursively presented group can be embedded in a finitely presented group. The chapter also discusses reduction lemmas, Turing machines, and embedding theorems.
TL;DR: A survey of algorithms that have been used to solve decision problems (mainly word problems) in various varieties of algebras, e.g., lattices, commutative semigroups, and quasigroups.
Abstract: Publisher Summary This chapter provides a survey of algorithms that have been used to solve decision problems (mainly word problems) in various varieties of algebras, e.g., lattices, commutative semigroups, and quasigroups. The interest is in the algebraic properties that imply the existence of such algorithms. The chapter discusses the connection between embedding of partial algebras in a variety and the solvability of the word problem for finitely presented (f.p.) algebras in the variety. It considers algorithms based on the finite separability properties. The chapter discusses the aspects of normal form theorems. The work is limited to finitely presented variety V— that is, a variety defined by a finite number of finitary operations and a finite set of identities, and by algebra.
TL;DR: A system ALPO of set theory is presented and proved to form a conservative extension of Peano arithmetic to allow a very substantial amount of analysis to be directly formalized, including Lebesgue integration theory, without resorting to wholesale padding of objects with extra information.
Abstract: A system ALPO of set theory is presented and proved to form a conservative extension of Peano arithmetic. The system has sufficient strength to allow a very substantial amount of analysis to be directly formalized, including Lebesgue integration theory, without resorting to wholesale padding of objects with extra information.
TL;DR: It was shown in this article that in intuitionistic logic unrestricted use of infinite disjunction will even with two propositional variables give rise to a proper class of equivalence classes of formulae.
Abstract: It is shown that in intuitionistic logic unrestricted use of infinite disjunction will even with two propositional variables give rise to a proper class of equivalence-classes of formulae.
TL;DR: In this article, the algebraic structure of algebraically closed groups is studied and it is shown that no a.c. group can be embedded into a finitely generated subgroup of itself.
Abstract: Publisher Summary This chapter presents several results on the algebraic structure of algebraically closed (a.c.) groups. It discusses that no a.c. group can be embedded into a finitely generated subgroup of itself.
TL;DR: In this article, the adequacy of constant domain relational world semantics for quantified relevant logics is investigated, and the main problem is solved, though in a disagreeably circuitous way, for many weaker relevant Logics, and an outline of how the solution may be extended to stronger Logics such as RQ is given.
Abstract: The main problem investigated is the adequacy of constant domain relational world semantics for quantified relevant logics. The problem is solved, though in a disagreeably circuitous way, for many weaker relevant logics, and an outline of how the solution may be extended to stronger logics such as RQ is given. Alternative necessity and intensional-conjunction style rules for the evaluation of quantifiers are studied and shown to simply force the main problems above with the usual (extensional conjunction style) quantifier - rule to reappear, unmitigated, at alternative outlets. Finally some philosophical problems allegedly engendered by constant domain world semantics are examined briefly: it is argued that the “problems” are no problems.
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TL;DR: This chapter discusses reducible braids and describes the results obtained by Garside, an algorithm to determine if a given braid is reducible or not.
Abstract: Publisher Summary This chapter discusses reducible braids. The aim is to develop an algorithm to determine if a given braid is reducible or not. In order to state the results precisely, the chapter describes the results obtained by Garside.
TL;DR: In this article, the authors discuss the algorithmic problems for solvable groups and consider restrictions having a natural group-theoretic character, such as nilpotency, polycyclicity, and solvability.
Abstract: Publisher Summary This chapter discusses the algorithmic problems for solvable groups. Classical algorithmic problems— the word problem, conjugacy problem, and isomorphism problem; arose in the theory of groups from topolocal considerations and were formulated for the first time at the beginning of this century by Dehn. Studies show that these problems have negative solutions in the class of all groups. However, for important classes of groups such as nilpotent and solvable groups, these problems remained unsolved. Recently, a number of results have been obtained in this field and this chapter is intended to survey them. The majority of algorithmic problems have a negative solution in the class of all groups. Therefore, it is natural to study algorithmical problems with additional restrictions on the groups considered. In a number of papers, such restrictions were on the form of the defining relations. However, the chapter considers restrictions having a natural group-theoretic character, such as nilpotency, polycyclicity, and solvability.
TL;DR: In this paper, an absolute version of Post's problem for 3 E is devised, and studied when certain recursively enumerable projecta are equal, with the aid of the notions of indexicality and ordinal recursiveness.
Abstract: The unresolved character of the power set operation stymies the solution of elementary problems arising in Kleene's theory of recursion in objects of finite type. E.g. Post's problem for 3 E has a positive solution if V=L (NORMANN, 1975), and a negative if AD holds. Let δ be the class of all sets R ⊆ 2ω such that R is recursive in 3 E, b for some real b . A forcing construction shows δ is not recursively enumerable in 3 E when there is a recursively regular well-ordering of 2″ recursive in 3 E. It follows that the concepts of Σ * , and weak Σ * , definability differ. With the aid of the notions of indexicality and ordinal recursiveness, an absolute version of Post's problem for 3 E is devised, and studied when certain recursively enumerable projecta are equal.
TL;DR: In this paper, Corner showed that for any positive integer n = 1, it is sufficient to construct a non-trivial finitely generated group isomorphic to its own direct square.
Abstract: Publisher Summary This chapter discusses the isomorphisms of direct powers. The construction presented is the simple groups constructed by Camm. The use of the structure theorems for amalgamated free products and HNN extensions is done. Corner showed that, given any positive integer n, there exists a countable abelian group H such that H r ≅ H s sif and only if r is congruent to s modulo n . (where H r denotes the direct sum of r isomorphic copies of H .) The chapter shows that similar results can be obtained for finitely generated non-abelian groups. For the case n =1, it is sufficient to construct a non-trivial finitely generated group isomorphic to its own direct square.
TL;DR: In this article, it was shown that for the class of stable, stable and ℵ 0 -cate gorical groups G for which [ G :Z(G )] is finite, the theory of { G ; [ G: Z(G)] ≤ n } is undecidable.
Abstract: Following R. Baer and B. H. Neumann a group G is called and F-C Group if for each g ɛ G the conjugacy class { χ -1 gχ;ɛ ɛ G } of g is finite. In her celebrated paper W. Szmielew solved the problems of decidability and internal characterization of elementary equivalence for the class of abelian groups. The model theory of abelian groups is by now well established. For the next slightly larger class, i.e. the class of nilpotent groups (even nil-2) only a few sporadic results are known. We found it more promising to look at another generalization of commutativity, namely the class of FC-groups. We have the following results: (1) For all nɛω : the theory of { G ; [ G : Z( G )] ≤ n } is decidable. (2) The theory of { G ; [ G : Z( G )] 0 } is undecidable. (3) The theory of all FC-groups and the theory of all BFC-groups are both undecidable. (4) We classified all stable, ω -stable and ℵ 0 - cate gorical groups G for which [ G :Z( G )] is finite. (5) We determined the Δ 2 - theory of all periodic FC-groups. (6) We have the surprising result, that the Hypercenter of every FC-groups of finite exponent is first-order definable without parameters. It seems to us that it is possible in the near future to get elementary invariants for periodic FC-groups.
TL;DR: The essence of the development of regular patterns described by finite automata is illustrated by following the mainstream of work flowing from Kleene's 1951 theorem on regular events to recent results in computational complexity and algorithmic logic.
Abstract: Regular patterns described by finite automata are evident in the behavior of computers, in the structure of programming languages and in the rules for reasoning about programs. The systematic study of these patterns has shaped computing theory, providing theorems, techniques and a paradigm with far reaching and subtle effects on many aspects of modem computing theory. The essence of the development is illustrated in this paper by following the mainstream of work flowing from Kleene's 1951 theorem on regular events to recent (circa 1976) results in computational complexity and algorithmic logic.
TL;DR: In this article, the basic ideas of a first-order model theory with variable binding term operators (vbto) introduced as new primitive symbols added to usual firstorder languages are presented.
Abstract: By a variable binding term operator (vbto) we understand an operator which binds variables of formulas to form terms Examples of vbtos: the description operator ι, Hilbert's symbols τ and e, the classifier {|}, and Russell's operator ⁁ Usual vbtos are sometimes introduced by contextual definition, and therefore can be eliminated Nonetheless, owing to the fact that the elimination of vbtos requires elaborate metatheorems and to the circunstance that there are relevant cases in which vbtos are employed as primitive, a systematic model-theoretical treatment of such operators is in order Here we present the basic ideas of a first-order model theory with vbtos introduced as new primitive symbols added to usual first-order languages