R70-9 A Graph-Theoretic Model for Periodic Discrete Structures

E.L. Lawler
1970 IEEE transactions on computers  
A. GRAPH THEORY R70-9 A Graph-Theoretic Model for Periodic Discrete Structures-James Turner (J. Franklin Institute, vol. 285, pp. 52-58, January 1968). Wiring schemes which exploit the potential of modern integrated circuit technology are likely to exhibit a high degree of "periodicity," as well as various other attributes. The first contribution of this paper is to formulate a precise definition of periodicity in terms of a graph-theoretic model originated by Yoeli. Roughly speaking, an
more » ... e graph is said to be periodic if it can be generated from a finite template which is moved (periodically) from point to point in the plane. This definition is interesting and useful, and generalizes nicely to infinite graphs in n-space. Now, suppose that in addition to periodicity we require planarity and point symmetry. (Roughly speaking, a graph is point symmetric if it "looks the same" from each of its vertices.) What types of infinite graphs satisfy all three of these requirements? The second contribution of this paper is to reveal that there are exactly eleven such graphs, and they correspond to the three regular and eight semi-regular tesselations of the plane. A planar tesselation is a set of polygons fitting together side by side which cover the plane simply and without gaps. A tesselation is regular if each face is the same regular polygon. The three regular tesselations are composed of triangles, squares, and hexagons, respectively. A semiregular tesselation permits the faces to be different, but the same regular polygons must surround each vertex. An example of a semi-regular tesselation is one in which each vertex is surrounded by a square and two octagons. It is interesting to compare these results with those of Eric Wagner, who has considered various wiring schemes that do not meet all ofTurner's conditions. For example, Wagner proposes an array which would appear to correspond to an (impossible) regular tesselation composed of pentagons. However, Wagner's scheme is not periodic under the Yoeli-Turner definition. The third problem considered in this paper is: In how many distinct ways can the eleven undirected point-symmetric planar arrays be oriented so that the resulting directed array is point-symmetric? This proves to be a rather difficult problem, in general. The author shows that there are exactly four such orientations in the case of the quadratic lattice (corresponding to the regular tesselation of squares). The paper concludes with the statement of a number of unsolved problems which would appear to be interesting topics for further research, e.g., find an algorithm to determine if two infinite periodic graphs are isomorphic, where each is defined according to Yoeli's model. This paper contributes to the literature on tree automata (also known as algebra automata), which is providing new insights at the interface between automata theory and formal language theory-an interface which hopefully will show increasing relevance both to linguists and compiler writers. Formal language theory has long employed pushdown automata, linear bounded automata, etc., to process, parse, and recognize the strings of a formal language. Lurking in the background have been the derivation trees which show how such strings can be derived by various rewriting rules from some initial symbol. The new approach (spurred by the linguists' interest in transformational grammars which actually transform derivation trees, and by Buchi's observation-built upon by such workers as Mezei, Thatcher, Eilenberg, Wright, and Giveon in the pages of Information and Control that ordinary automata may be viewed as a special kind of universal algebra) looks at automata that act directly on the trees, rather than on the strings they produce. Brainerd gives a neat self-contained proof of the expected result that the methods developed by Moore, Myhill, and Nerode to reduce the number of states of a finite-state string-processor carry over clearly to finitestate tree-processors which work from the "leaves" of a tree towards its "root," with the "input" labeling any node acting upon the "previous states" already computed at the nodes leading to it, to compute a new state at the given node. The ordinary theory is recovered when each node is limited to at most one input node-the tree then becomes a string of inputs which successively update the sate of a sequential machine. The late John von Neumann took an interest in a mechanical modeling of self-reproducing phenomena of living things, which are far more complex and purposeful than a mere regeneration of a facsimile of the structures found, for example, in the growth of a crystal in a solution of the constituent substance. After some initial work with some kinematic models of such phenomena, he found a two-dimensional array of identical finite-state machines to be a convenient medium in which to study selfreproducing phenomena represented as the successive state distributions offinite state machines [1 ]. A celebrated cellular space used by von Neumann for his modeling, and now by Mukhopadhyay, consists of an infinitely extended checkerboard, each of whose squares stands for a 29-state sequential machine of 563
doi:10.1109/t-c.1970.222980 fatcat:be3yk2xieba3nf3wtvy2ynstsy