Here, we give a full proof of Theorem 1. Recall that an equivalent construction of the PAGA graphs is obtained by collapsing the graph G 1 to G t m leading to t super nodes. Accordingly, based on G t m, we can directly generate G m by (temporarily) adding m more nodes, each linking with a single edge, and then collapsing these m nodes to the (t+ 1)th super node. Let dt,i(v) denote the degree of...

The programming language Pascal waH originally designed for tcaching introductory programming, currently, however, production systems use it as the primary implementation language. This paper describes extensions of Pascal intended to aid the large program developer. The extensions are implemented in a macro preprocessor MAP, which supports constant expression evaluation, source file inclusion,...

Pretty-printing a piece of Pascal code with TEX is often done via an external preprocessor. Actually, the job can be done entirely in TEX; this paper introduces PPP, a Pascal pretty-printer environment that allows you to typeset Pascal code by simply typing \Pascal {Pascal code) \endpascal. The same approach of "preprocessing w i t h TEX" namely two-token tail-recursion around a \FIND-like macr...

Since its introduction in 1971, PASCAL language has received rapid acceptance as a structured programming tool in a wide variety of applications ranging from computer science education to systems programming in production environments. It is, therefore, not surprising that numerous reports relating to implementation of PASCAL compilers on various computers continue to appear in the current lite...

We introduce a generalization of Pascal triangle based on binomial coefficients of finite words. These coefficients count the number of times a word appears as a subsequence of another finite word. Similarly to the Sierpiński gasket that can be built as the limit set, for the Hausdorff distance, of a convergent sequence of normalized compact blocks extracted from Pascal triangle modulo 2, we de...