The concept of a μ-recursive function

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.

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Counting to Infinity: Does learning the syntax of the count list predict knowledge that numbers are infinite?

10.31234/osf.io/5c6pb ◽

2020 ◽

Author(s):

Junyi Chu ◽

Pierina Cheung ◽

Rose M. Schneider ◽

Jess Sullivan ◽

David Barner

Keyword(s):

Natural Number ◽

Recursive Function ◽

Implicit Knowledge ◽

Successor Function ◽

Counting Rule ◽

Number Words ◽

The Us ◽

Counting Range ◽

Per Se

By around the age of 5½, many children in the US judge that numbers never end, and that it is always possible to add +1 to a set. These same children also generally perform well when asked to label the quantity of a set after 1 object is added (e.g., judging that a set labeled “five” should now be “six”). These findings suggest that children have implicit knowledge of the “successor function”: every natural number, n, has a successor, n+1. Here, we explored how children discover this recursive function, and whether it might be related to discovering productive morphological rules that govern language-specific counting routines (e.g., the rules in English that represent base 10 structure). We tested 4- and 5-year-old children’s knowledge of counting with three tasks, which we then related to (1) children’s belief that 1 can always be added to any number (the successor function), and (2) their belief that numbers never end (infinity). Children who exhibited knowledge of a productive counting rule were significantly more likely to believe that numbers are infinite (i.e., there is no largest number), though such counting knowledge wasn’t directly linked to knowledge of the successor function, per se. Also, our findings suggest that children as young as four years of age are able to implement rules defined over their verbal count list to generate number words beyond their spontaneous counting range, an insight which may support reasoning over their acquired verbal count sequence to infer that numbers never end.

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