Dale Myers

Nonrecursive Tilings of the Plane. II

We show that there is a finite set of tiles which can tile the plane but not in any recursive way. This answers a natural sequel to Hao Wangs problem of the existence of a finite set of tiles which can tile the plane but not in any periodic way. In the proof, an elaboration of Robinsons method of transforming origin-constrained problems into unconstrained problems is applied to Hanfs origin-constrained tiling of Part I. We will assume familiarity with §§2, 3, and 7 of [3]. Following Robinson, we will mark the edges of our tiles with symbols and configurations of arrow heads and tails as well as with colors. The matching condition for abutting tiles will be that the symbols on adjacent edges must be identical, every arrow head must match with a tail, every tail with a head, and the colors must be the same. This is, of course, mathematically equivalent to the original condition which involved only colors. A tiling of the plane by a set of tiles is a covering of the plane with translated copies of the tiles such that adjacent edges of abutting tiles satisfy the above matching condition. A set of tiles is consistent if the plane can be tiled by the set. A set of tiles with a designated origin tile is origin-consistent if there is a tiling of the plane with the origin tile at the center. A square block of tiles is a tiling if every pair of abutting tiles satisfies the above matching condition. If an origin tile has been designated, a block of tiles is an origin-constrained tiling if it is a tiling with the origin tile at the center. Two blocks have the “ same ” center row if the blocks are of the same size and have identical center rows or if the smaller blocks center row is a centered segment of the larger blocks center row.

On the solvability of domino snake problems

Abstract In this paper we present an extensive treatment of tile connectability problems, sometimes called domino snake problems. The interest in such problems stems from their relationship to classical tiling problems, which have been established as an important, simple and useful tool for obtaining basic lower bound results in complexity and computability theory. We concentrate on the following two contrasting results: The general snake problem is undecidable in a half-plane (due to Ebbinghaus), but is decidable in the whole plane. This surprising decidability result was announced without proof by Myers in 1979. We provide here the first full proof, and show that the problem is actually PSPACE-complete. We also prove many results concerning the difficulty of variants of these general snake problems and their extension to infinite snakes. In addition, we establish a resemblance between snake problems and classical tiling problems, considering the corresponding bounded, unbounded and recurring cases.

Boolean sentence algebras: Isomorphism constructions

Associated with each first-order theory is a Boolean algebra of sentences and a Boolean space of models. Homomorphisms between the sentence algebras correspond to continuous maps between the model spaces. To what do recursive homomorphisms correspond? We introduce axiomatizable maps as the appropriate dual. For these maps we prove a Cantor-Bernstein theorem. Duality and the Cantor-Bernstein theorem are used to show that the Boolean sentence algebras of any two undecidable languages or of any two functional languages are recursively isomorphic where a language is undecidable iff it has at least one operation or relation symbol of two or more places or at least two unary operation symbols, and a language is functional iff it has exactly one unary operation symbol and all other symbols are for unary relations, constants, or

An Interpretive Isomorphism Between Binary and Ternary Relations

It is well-known that in first-order logic, the theory of a binary relation and the theory of a ternary relation are mutually interpretable, i.e., each can be interpreted in the other. We establish the stronger result that they are interpretively isomorphic, i.e., they are mutually interpretable by a pair of interpretations each of which is the inverse of the other.

Elementary properties of free extensions

AbstractFor any partial groupoid

Subgoal Strategies for Solving Board Puzzles

\mathfrak{A}

Boolean spaces with countable base but uncountably many orbits

, let Fr