Showing papers on "Turing machine published in 1975"

Toward a mathematical theory of inductive inference

Abstract: Intelligence tests occasionally require the extrapolation of an effective sequence (e.g. 1661, 2552, 3663, …) that is produced by some easily discernible algorithm. In this paper, we investigate the theoretical capabilities and limitations of a computer to infer such sequences. We design Turing machines that in principle are extremely powerful for this purpose and place upper bounds on the capabilities of machines that would do better.

On some central problems in computational complexity.

Abstract: In this thesis we examine some of the central problems in the theory of computational complexity, like the trade-offs between time and memory, the power of nondeterminism and parallelism, and the speed gained by adding new operations to random access machines. Our main result is the cahracterization of the power of multiplication in random access acceptors: we show, in Chapter 3, that for such models nondeterministic and deterministic computations are polynomially related and that there is a polynomial relationship between the amount of time required for acceptance by random access machines with multiplication, and the amount of tape required by Turing machines. Thus, the additional power gained by using multiplication is the same as that of memory over time (if any). We derive similar results for some other interesting instruction sets. We also have some results for probabilistic and nondeterministic computations: we define threshold machines and show how probabilistic Turing machines may simulate them, and exhibit a set of complete problems for threshold machines. For nondeterministic computations, we present a hierarchy of the elementary recursive languages obtained by polynomially bounded quantification over objects of higher and higher type, which represent nondeterministic time bounded computations with larger and larger bounds. Finally, we discuss some deterministic computations, and conclude with a look at some open problems.

The complexity of negation-limited networks — A brief survey

Abstract: Proving lower bounds on the combinational complexity of concrete functions is a difficult and challenging problem. Previous successes in establishing lower bounds for monotone networks and the known gaps between the monotone and general combinational complexity indicate the key role that negations play in determining combinational complexity.

From automata theory to brain theory

Abstract: Although the brain modeler can gain many useful insights from such concepts of orthodox automata theory as finite automata, network complexity theory, and Turing machines, we here stress that the best neural modelling will bear little resemblance to a straight application of such techniques. This general perspective is complemented by a survey of eight levels of neural modelling, coupled with an extensive bibliography. The eight levels are: formfunction relations in single neurons; lateral inhibition; mode selection; statistical mechanics; adaptive neural networks; holography; control theory; and cognitive modelling.

Mind, Language and Reality: The refutation of conventionalism

Abstract: I shall discuss conventionalism in Quine's writing on the topic of radical translation and in the writings of Reichenbach and Grunbaum on the nature of geometry. One preliminary remark: in one respect, the situation is extremely complicated with respect to the views of both of these men. With respect to Quine, the situation is so confused that one perhaps should distinguish between two QuinesQuine 1 and Quine 2. Quine 1 is the Quine who everybody thinks wrote Word and Object, that is to say, the Quine whose supposed proof of the impossibility of radical translation, of the impossibility of there being a unique correct translation between radically different and unrelated languages, is discussed in journal article after journal article and is the topic of at least fifty percent of graduate student conversation nowadays. Quine 2 is the far more subtle and guarded Quine who defended his formulations in Word and Object recently at the Conference on Philosophy of Language at Storrs. In the light of what Quine said at Storrs, I am inclined to think that Word and Object may have been widely misinterpreted. At any rate, Quine seems to think that Word and Object has been widely misinterpreted, although he was charitable enough to take some of the blame himself for his own formulations. In what follows, then, I shall be criticizing the views of Quine 1, even if Quine 1 is a cultural figment not to be identified with the Willard Van Orman Quine who teaches philosophy at Harvard. It is the views of Quine1 that are generally attributed to Willard Van Orman Quine, and it is worthwhile showing what is wrong with those views. If I can have the help of Quine2-of Willard Van Orman Quine himself-in refuting' the views of Quine1, then so much the better. There is a similar problem with respect to the work of Reichenbach. The arguments for the conventionality of geome-

A survey of partial degrees

Abstract: Partial degrees are equivalence classes of partial natural number functions under some suitable extension of relative recursiveness to partial functions. The usual definitions of relative recursiveness, equivalent in the context of total functions, are distinct when extended to partial functions. The purpose of this paper is to compare the upper semilattice structures of the resulting degrees.Relative partial recursiveness of partial functions was first introduced in Kleene [2] as an extension of the definition by means of systems of equations of relative recursiveness of total functions. Kleene's relative partial recursiveness is equivalent to the relation between the graphs of partial functions induced by Rogers' [10] relation of relative enumerability (called enumeration reducibility) between sets. The resulting degrees are hence called enumeration degrees. In [2] Davis introduces completely computable or compact functionals of partial functions and uses these to define relative partial recursiveness of partial functions. Davis' functionals are equivalent to the recursive operators introduced in Rogers [10] where a theorem of Myhill and Shepherdson is used to show that the resulting reducibility, here called weak Turing reducibility, is stronger than (i.e., implies, but is not implied by) enumeration reducibility. As in Davis [2], relative recursiveness of total functions with range ⊆{0, 1} may be defined by means of Turing machines with oracles or equivalently as the closure of initial functions under composition, primitive re-cursion, and minimalization (i.e., relative μ-recursiveness). Extending either of these definitions yields a relation between partial functions, here called Turing reducibility, which is stronger still.

"Helping": Several Formalizations

Abstract: Much recent work in the theory of computational complexity ([Me], [FR]. [SI]) is concerned with establishing “the complexity” of various recursive functions, as measured by the time or space requirements of Turing machines which compute them. In the above work, we also observe another phenomenon: knowing the values of certain functions makes certain other functions easier to compute than they would be without this knowledge. We could say that the auxiliary functions “help” the computation of the other functions. For example, we may conjecture that the “polynomial-complete” problems of Cook [C] and Karp [K] and Stockmeyer [S2], such as satisfiability of propositional formulas or 3-colorability of planar graphs, in fact require time proportional to n log 2 n to be computed on a deterministic Turing machine. Then since the time required to decide if a planar graph with n nodes is 3-colorable can be lowered to a polynomial in n if we have a precomputed table of the satisfiable formulas in the propositional calculus, it is natural to say that the satisfiability problem “helps” the computation of the answers to the 3-coloring problem. Similar remarks may be made for any pair of polynomial-complete problems. As a further illustration, Meyer and Stockmeyer [MS] have shown that, for a certain alphabet Σ, recognition of the set of regular expressions with squaring which are equivalent to Σ* requires Turing machine space c n for some constant c , on an infinite set of arguments. We also know that this set of regular expressions, which we call RSQ, may actually be recognized in space d n for some other constant d. Theorem 6.2 in [LMF] implies that there is some problem (not necessarily an interesting one) of complexity approximately equal to that of RSQ, which does not reduce the complexity of RSQ below c n . It does not “help” the computation of RSQ.

Ten years of speedup

Abstract: The paper presents a survey on the speedup phenomenon in the machine-independent theory of recursive functions, the techniques used to prove its existence, its non-effectiveness, its generalizations, and the relations between the speedup in recursion theory, and similar phenomena in logic.

Describing automata in terms of languages associated with their peripheral devices.

Abstract: A unified approach is presented to deal with automata having different kinds of peripheral devices. This approach is applied to pushdown automata and Turing machines, leading to elementary proofs of several well-known theorems concerning transductions, relationship between pushdown automata and context-free languages, as well as homomorphic characterization and undecidability questions. In general, this approach leads to homomorphic characterization of language families generated by a single language by finite transduction.

On converting on-line algorithms into real-time and on real-time algorithms for string-matching and palindrome recognition

Abstract: We give a sufficient condition when an on-line algorithm can be transformed into a real-time algorithm. We use this condition to construct real-time algorithms for string-matching and palindrome recognition problems by random access machines and by Turing machines.

A convenient cryptomorphic version of recursive function theory

Abstract: When several axiom schemes can serve equally well in defining a theory, one generally seeks that axiomatic treatment which is most convenient for the intended applications. We present the results of such a study regarding the parallel theories of (partial) recursive functions and Turing machines.

On models of protection in operating systems

Abstract: In [ii], a model of protection systems was introduced. Much of that paper is devoted to an explanation of the model. There are examples of the use of the model in capturing aspects of real systems. A formulation of the safety question is presented and it is shown that the question of whether a given general protection system is safe or not is undecidable. That paper has provoked a number of questions as to why certain features were or were not included in the model. The present paper attempts to answer these questions as well as giving some new results.

How to make abstract ideas more concrete

Abstract: Games that students play, either against each other or against the computer, can help to make some of the abstract ideas from the theory of computing seem more concrete and more interesting than they now appear to many students. In this paper, I describe the general structure of some games based on such theoretical constructs as the Turing machine, the context-free grammar and the like. I indicate some of the ways that such games can be “built” either using pencil, paper and scrap materials or using computer programs. Some experiences with the use of such games, and some of their mathematical and pedagogical implications, are discussed.