1514 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 43, NO. 11, NOVEMBER 1998 Boundedness Analysis of Finitely Recursive Processes—Part I: Concurrent Processes Supratik Bose, Amit Patra, Member, IEEE, and Siddhartha Mukhopadhyay, Member, IEEE Abstract—In the finitely recursive process (FRP) model of dis- crete event systems (DES), concepts about processes and process operators have been introduced. An infinite set of process ex- pressions or functions can be built recursively through function composition using a few elementary operators. Given any process realization, it is important to find out whether the process is bounded, i.e., whether it has a finite state realization. In the FRP setting this translates to the problem of finding out whether the set of post-process expressions is finite or not. In Cieslak and Varaiya (1990) it has been shown that the boundedness problem is undecidable for general FRP’s. This paper investigates the decidability of the problem for subclasses of FRP. In Inan and Varaiya (1988), it was conjectured that the set of functions that can be recursively generated using the parallel composition operator ( ) and different change operators [i.e., without using the Sequential Composition Operator ( )] will be finite and FRP’s constructed over this set of functions will naturally be bounded. In the present work a counterexample has been provided to disprove the conjecture about the finiteness of the above set of functions. However, using a suitable post-process computation procedure, it has been shown here that the FRP’s, built recursively over this set of functions, are bounded. Index Terms—Boundedness, discrete-event systems, finitely re- cursive processes, process algebra models. NOMENCLATURE Fixed finite collection of event sym- bols. Set of finite length strings (also called traces) formed with elements of Null string. String containing a single event String containing Concatenation of the string Fol- lowed by string Language Prefix-closed Prefix closed if Family of prefix-closed languages over Length of the string repeated times. Manuscript received October 25, 1996; revised December 2, 1997. Recom- mended by Associate Editor, E. K. P. Chong. This work was supported in part by the University Grants Commission’s Young Teachers’ Career Development Award to S. Mukhopadhyay. S. Bose was with the Department of Electrical Engineering, Indian Institute of Technology, Kharagpur 721302, India. He is now with Siemens Information Systems Limited, Bangalore, India. A. Patra and S. Mukhopadhyay are with the Department of Electrical Engineering, Indian Institute of Technology, Kharagpur 721302, India. Publisher Item Identifier S 0018-9286(98)08494-3. Projection of on the process Traces of the process Alphabet function of the process Termination function of the process Deterministic projection operator. Deterministic partial order (subpro- cess). Deterministic marked process space. Deterministic marked nonterminat- ing process space. Constant nonterminating process Constant terminating process . Deterministic choice operator. Sequential composition operator. Parallel composition operator. Local change operator Local change operator Global Change operator Global change operator Set of constant symbols of Set of constant symbols of Signature over Syntax set over Semantics set over Realization of the FRP Algebraic process space over Set of post-process expressions of Length of function expression Subexpressions of and are syntactically equivalent. has a structure and are semantically equivalent. Minimum present alphabet function. Minimum absent alphabet function. where where 0018–9286/98$10.00  1998 IEEE