Nonrecursive tilings of the plane. I

1974 ◽  
Vol 39 (2) ◽  
pp. 283-285 ◽  
Author(s):  
William Hanf
Keyword(s):  
Turing Machine ◽  
Recursive Function ◽  
Positive Answer ◽  
Additional Restriction ◽  
Detailed History ◽  
Finite Set

A finite set of tiles (unit squares with colored edges) is said to tile the plane if there exists an arrangement of translated (but not rotated or reflected) copies of the squares which fill the plane in such a way that abutting edges of the squares have the same color. The problem of whether there exists a finite set of tiles which can be used to tile the plane but not in any periodic fashion was proposed by Hao Wang [9] and solved by Robert Berger [1]. Raphael Robinson [7] gives a more detailed history and a very economical solution to this and related problems; we will assume that the reader is familiar with §4 of [7]. In 1971, Dale Myers asked whether there exists a finite set of tiles which can tile the plane but not in any recursive fashion. If we make an additional restriction (called the origin constraint) that a given tile must be used at least once, then the positive answer is given by the main theorem of this paper. Using the Turing machine constructed here and a more complicated version of Berger and Robinson's construction, Myers [5] has recently solved the problem without the origin constraint.Given a finite set of tiles T1, …, Tn, we can describe a tiling of the plane by a function f of two variables ranging over the integers. f(i, j) = k specifies that the tile Tk is to be placed at the position in the plane with coordinates (i, j). The tiling will be said to be recursive if f is a recursive function.

Locally finite minimal non-FC-groups

1989 ◽  
Vol 105 (3) ◽  
pp. 417-420 ◽  
Author(s):  
Mahmut Kuzucuoglu ◽  
Richard E. Phillips
Keyword(s):  
Simple Group ◽  
Proper Subgroup ◽  
Positive Answer ◽  
Locally Finite ◽  
Finite Set

We recall that a group G is an FC-group if for every x∈G the set of conjugates {xg|g∈G} is a finite set. Our interest here is with those groups G which are not FC groups while every proper subgroup of G is an FC-group: such groups are called minimal non-FC-groups. Locally finite minimal non-FC-groups with (G ≠ G′ are studied in [1] and the structure of these groups is reasonably well understood. In [2] Belyaev has shown that a perfect, locally finite, minimal non-FC-group is either a simple group or a p-group for some prime p. Here we make use of the results of [5] to refine the result of Belyaev and provide a positive answer to problem 5·1 of [11]; in particular, we prove the followingTheorem. There exists no simple, locally finite, minimal non-FC-group.

Number of variables is equivalent to space

2001 ◽  
Vol 66 (3) ◽  
pp. 1217-1230 ◽  
Author(s):  
Neil Immerman ◽  
Jonathan F. Buss ◽  
David A. Mix Barrington
Keyword(s):  
Turing Machine ◽  
Descriptive Complexity ◽  
First Order ◽  
Trade Offs ◽  
Finite Set ◽  
The Universe

AbstractWe prove that the set of properties describable by a uniform sequence of first-order sentences using at most k + 1 distinct variables is exactly equal to the set of properties checkable by a Turing machine in DSPACE[nk] (where n is the size of the universe). This set is also equal to the set of properties describable using an iterative definition for a finite set of relations of arity k. This is a refinement of the theorem PSPACE = VAR[O[1]] [8]. We suggest some directions for exploiting this result to derive trade-offs between the number of variables and the quantifier depth in descriptive complexity.

scholarly journals An Informal Arithmetical Approach to Computability and Computation

1961 ◽  
Vol 4 (3) ◽  
pp. 279-293 ◽  
Author(s):  
Z.A. Melzak
Keyword(s):  
Turing Machine ◽  
System Changes ◽  
Finite Set ◽  
Paper And Pencil ◽  
Effective Computability

In 1936 A. M. Turing published his analysis of the notion of effective computability. Very roughly speaking, its object was to distinguish between numbers defined by existential statements and those defined by purely constructive ones. Guided by his schematizing of what goes on when a person calculates using paper and pencil, Turing introduced the concept of an A-machine, eventually to be called a Turing machine. Such a machine consists of a box and a linear tape divided into squares, indefinitely long in both directions. The tape passes through the box and exactly one square of it is always within the box. The system changes only at certain discrete times: t = 0, 1, …. Each square can carry exactly one of the finite set of symbols: s1, s2, …, sn, one of which may be the blank.

Church’s thesis

2018 ◽  
Author(s):  
Stewart Shapiro
Keyword(s):  
Turing Machine ◽  
Recursive Function ◽  
Turing Computability ◽  
Church’S Thesis ◽  
The Third ◽  
Alan Turing ◽  
Rigorous Formulation ◽  
Person Following ◽  
Mechanical Procedure ◽  
Church's Thesis

An algorithm or mechanical procedure A is said to ‘compute’ a function f if, for any n in the domain of f, when given n as input, A eventually produces fn as output. A function is ‘computable’ if there is an algorithm that computes it. A set S is ‘decidable’ if there is an algorithm that decides membership in S: if, given any appropriate n as input, the algorithm would output ‘yes’ if n∈S, and ‘no’ if n∉S. The notions of ‘algorithm’, ‘computable’ and ‘decidable’ are informal (or pre-formal) in that they have meaning independently of, and prior to, attempts at rigorous formulation. Church’s thesis, first proposed by Alonzo Church in a paper published in 1936, is the assertion that a function is computable if and only if it is recursive: ‘We now define the notion…of an effectively calculable function…by identifying it with the notion of a recursive function….’ Independently, Alan Turing argued that a function is computable if and only if there is a Turing machine that computes it; and he showed that a function is Turing-computable if and only if it is recursive. Church’s thesis is widely accepted today. Since an algorithm can be ‘read off’ a recursive derivation, every recursive function is computable. Three types of ‘evidence’ have been cited for the converse. First, every algorithm that has been examined has been shown to compute a recursive function. Second, Turing, Church and others provided analyses of the moves available to a person following a mechanical procedure, arguing that everything can be simulated by a Turing machine, a recursive derivation, and so on. The third consideration is ‘confluence’. Several different characterizations, developed more or less independently, have been shown to be coextensive, suggesting that all of them are on target. The list includes recursiveness, Turing computability, Herbrand–Gödel derivability, λ-definability and Markov algorithm computability.

Undecidability and initial segments of the (r.e.) tt-degrees

1990 ◽  
Vol 55 (3) ◽  
pp. 987-1006 ◽  
Author(s):  
Christine Ann Haught ◽  
Richard A. Shore
Keyword(s):  
Turing Machine ◽  
Recursive Function ◽  
Equivalence Classes ◽  
Truth Table ◽  
Turing Degree ◽  
General Notion ◽  
Basic Information ◽  
Recursive Procedure ◽  
Tabular Type

A notion of reducibility ≤r between sets is specified by giving a set of procedures for computing one set from another. We say that a set A is r-reducible to a set B, A ≤rB, if one of the procedures applied to B gives A. Associated with any such reducibility notion is the structure of r-degrees, the equivalence classes of sets with respect to this reducibility, with the induced ordering. The most general notion of a computable reducibility is that of Turing, ≤T. Here we say that A ≤TB if there is a Turing machine φe which, when equipped with an oracle for B, computes A: φeB = A. Such Turing degree computations are characterized by the phenomenon that only during the computation itself do we discover which questions about B need to be answered to compute A(x). In contrast, for nearly all other computable reducibilities the set of questions needed is given in advance by a recursive procedure. Perhaps the most common example of such a procedure is many-one reducibility, ≤m: A ≤mB if there is a recursive function f such that x ∈ A ⇔ f(x) ∈ B.Reducibilities with the property that the output, A(x), is determined by the answers that B gives to a set of questions calculated recursively from x are said to be of tabular type. The most general tabular reducibility is called truth-table reducibility, ≤tt. The procedures [e] associated with this reducibility are specified by a recursive function f (= {e}) which, for each x, gives a set of n questions about the oracle and, for each of the possible 2n sets of answers, gives the corresponding output. As usual this defines A ≤ttB as “there is an e such that [e]B = A”. It is with this notion of reducibility and the associated tt-degrees that we shall be concerned in this paper. Basic information on several such strong reducibilities can be found in Rogers [28]. For more information we recommend the survey articles by Odifreddi [25] and Degtev [2] as well as Odifreddi's book [26].

Remarks on Complementation in the Lattice of all Topologies

1966 ◽  
Vol 18 ◽  
pp. 83-88 ◽  
Author(s):  
Haim Gaifman
Keyword(s):  
Positive Answer ◽  
Finite Set

Our aim is to prove that certain topologies have complements in the lattice of all the topologies on a given set. Lattices of topologies were studied in (1-8). In (7) Hartmanis points out that the lattice of all the topologies on a finite set is complemented and poses the question whether this is so if the set is infinite. A positive answer is given here for denumerable sets. This result was announced in (6). The case of higher powers remains unsettled, although quite a few topologies turn out to have complements. As far as the author knows, no one has proved the existence of a topology that has no complement.

On Generalized Quantum Turing Machine and Its Applications

2009 ◽  
Vol 16 (02n03) ◽  
pp. 195-204
Author(s):  
Satoshi Iriyama ◽  
Masanori Ohya
Keyword(s):  
Computational Complexity ◽  
Turing Machine ◽  
Polynomial Time ◽  
Recursive Function ◽  
Quantum Algorithm ◽  
Partial Recursive Function ◽  
Sat Problem ◽  
Input Size ◽  
Amplification Process ◽  
Quantum Turing Machine

Ohya and Volovich discussed a quantum algorithm for the SAT problem with a chaos amplification process (OMV SAT algorithm) and showed that the number of steps it performed was polynomial in input size. In this paper, we define a generalized quantum Turing machine (GQTM) and related computational complexity. Then we show that there exists a GQTM which recognizes the SAT problem in polynomial time. Moreover, we discuss the problem of finding the quantum algorithm for a partial recursive function.

Degree problems for modular machines

1980 ◽  
Vol 45 (3) ◽  
pp. 510-528 ◽  
Author(s):  
Daniel E. Cohen
Keyword(s):  
Group Theory ◽  
Turing Machine ◽  
Recursive Function ◽  
Related Result ◽  
Turing Degrees ◽  
General Sense ◽  
Halting Problem ◽  
The Right

Modular machines were introduced in [1] and [2], where they were used to give simple proofs of various unsolvability results in group theory. Here we discuss the degrees of the halting, word, and confluence problems for modular machines, both for their own sake and in the hope that the results may be useful in group theory (see [4] for an application of a related result to group theory).In the course of the analysis, I found it convenient to compare degrees of these problems for a Turing machine T and for a Turing machine T1 obtained from T by enlarging the alphabet but retaining the same quintuples (or quadruples). The results were surprising. The degree for a problem of T1 depends not just on the corresponding degree for T, but also on the degrees of the corresponding problems when T is restricted to a semi-infinite tape (both semi-infinite to the right and semi-infinite to the left). For the halting and confluence problems, the Turing degrees of the problems for these three machines can be any r.e. degrees. In particular the halting problem of T can be solvable, while that of T1 has any r.e. degree.A machine M (in the general sense) consists of a countable set of configurations (together with a numbering, which we usually take for granted), a recursive subset of configurations called the terminal configurations, and a recursive function, written C ⇒ C′, on the set of configurations. If, for some n ≥ 0, we have C = C0 ⇒ C1 ⇒ … ⇒ Cn = C′, we write C → C′. We say M halts from C if C → C′ for some terminal C′.

scholarly journals Retract Rationality and Algebraic Tori

2019 ◽  
Vol 63 (1) ◽  
pp. 173-186 ◽  
Author(s):  
Federico Scavia
Keyword(s):  
Prime Number ◽  
Affirmative Answer ◽  
Positive Answer ◽  
Algebraic Tori ◽  
Finite Set ◽  
Multiplicative Type

AbstractFor any prime number $p$ and field $k$, we characterize the $p$-retract rationality of an algebraic $k$-torus in terms of its character lattice. We show that a $k$-torus is retract rational if and only if it is $p$-retract rational for every prime $p$, and that the Noether problem for retract rationality for a group of multiplicative type $G$ has an affirmative answer for $G$ if and only if the Noether problem for $p$-retract rationality for $G$ has a positive answer for all $p$. For every finite set of primes $S$ we give examples of tori that are $p$-retract rational if and only if $p\notin S$.

Mars: was There an Ancient Eden?

1997 ◽  
Vol 161 ◽  
pp. 203-218 ◽  
Author(s):  
Tobias C. Owen
Keyword(s):  
Positive Answer ◽  
Biogenic Elements ◽  
Habitable Zones ◽  
The Face ◽  
The Origin Of Life ◽  
Early Mars ◽  
Favorable Conditions ◽  
Open Question ◽  
Rocky Planets ◽  
Impact Erosion

AbstractThe clear evidence of water erosion on the surface of Mars suggests an early climate much more clement than the present one. Using a model for the origin of inner planet atmospheres by icy planetesimal impact, it is possible to reconstruct the original volatile inventory on Mars, starting from the thin atmosphere we observe today. Evidence for cometary impact can be found in the present abundances and isotope ratios of gases in the atmosphere and in SNC meteorites. If we invoke impact erosion to account for the present excess of129Xe, we predict an early inventory equivalent to at least 7.5 bars of CO2. This reservoir of volatiles is adequate to produce a substantial greenhouse effect, provided there is some small addition of SO2(volcanoes) or reduced gases (cometary impact). Thus it seems likely that conditions on early Mars were suitable for the origin of life – biogenic elements and liquid water were present at favorable conditions of pressure and temperature. Whether life began on Mars remains an open question, receiving hints of a positive answer from recent work on one of the Martian meteorites. The implications for habitable zones around other stars include the need to have rocky planets with sufficient mass to preserve atmospheres in the face of intensive early bombardment.

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