The Combinatorics of Non-determinism

A deep connection exists between the interleaving semantics of concurrent processes and increasingly labelled combinatorial structures. In this paper we further explore this connection by studying the rich combinatorics of partially increasing structures underlying the operator of non-deterministic choice. Following the symbolic method of analytic combinatorics, we study the size of the computation trees induced by typical non-deterministic processes, providing a precise quantitative measure of the so-called "combinatorial explosion" phenomenon. Alternatively, we can see non-deterministic choice as encoding a family of tree-like partial orders. Measuring the (rather large) size of this family on average offers a key witness to the expressiveness of the choice operator. As a practical outcome of our quantitative study, we describe an efficient algorithm for generating computation paths uniformly at random.

[1]  Alain Denise,et al.  Coverage-biased Random Exploration of Models , 2008, Electron. Notes Theor. Comput. Sci..

[2]  Philippe Flajolet,et al.  Birthday Paradox, Coupon Collectors, Caching Algorithms and Self-Organizing Search , 1992, Discret. Appl. Math..

[3]  Herbert S. Wilf,et al.  Combinatorial Algorithms: An Update , 1987 .

[4]  Søren Christensen.  Decidability and decomposition in process algebras , 1993 .

[5]  Mike D. Atkinson,et al.  On computing the number of linear extensions of a tree , 1990 .

[6]  Mike Zabrocki,et al.  Analytic aspects of the shuffle product , 2008, STACS.

[7]  Charalambos A. Charalambides,et al.  Enumerative combinatorics , 2018, SIGA.

[8]  W. Wasow.  Asymptotic expansions for ordinary differential equations , 1965 .

[9]  Luca Aceto,et al.  On Relating Concurency and Nondeterminism , 1991, MFPS.

[10]  Luca Ace.  On Relating Concurrency and Nondeterminism * , .

[11]  L. Lipshitz,et al.  D-finite power series , 1989 .

[12]  Peter Winkler,et al.  Counting linear extensions is #P-complete , 1991, STOC '91.

[13]  Philippe Flajolet,et al.  Analytic Combinatorics , 2009 .

[14]  O. Bodini,et al.  Enumeration and Random Generation of Concurrent Computations , 2012 .

[15]  Robin Milner,et al.  A Calculus of Communicating Systems , 1980, Lecture Notes in Computer Science.

[16]  Alain Denise,et al.  Coverage-biased random exploration of large models and application to testing , 2011, International Journal on Software Tools for Technology Transfer.

[17]  L. Lipshitz,et al.  The diagonal of a D-finite power series is D-finite , 1988 .

[18]  Philippe Flajolet,et al.  A Calculus for the Random Generation of Labelled Combinatorial Structures , 1994, Theor. Comput. Sci..