Read those three pages and after you read answer my question. you can answer it
ID: 3879491 • Letter: R
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Read those three pages and after you read answer my question. you can answer it by give a prove or a java program. Question: Show that there are 21 reachable states and that a state a state of type (4,4,......) is not among those 21 ? 1.7 Synchronization by Shared Variables h ea is to let them share a certain number of variables. Even if, from a theoreti point of view, it is possible to present the shared variables using a syncht nized product, their practical interest is such that we prefer to introdte Another way to have components of a system communicate wit 0- them explicitly. We saw earlier how variables could be "added" to auto It then is natural to allow one (or several) variable(s) to be shared by several automata if turn=A, printA turn := B Fig. 1.16. The user A Consider once again the case of the two users A et B who share a printer Their unhappy experience with the unfair printer manager from section 1.3 triggered their decision to share a variable turn keeping track of who has the risn seo a is thus modeled by the automaton on figure 1.16. The automaton describing the behavior of user B is of course symmetrical (see figure 1.1 if turn-B, printB turn := A Fig. 1.17. The user BExplanation / Answer
The state complexity of a regular language is the number of states of its minimal deterministic finite automaton (DFA). The nondeterministic state complexity of a regular language is the number of states of a minimal nondeterministic finite automaton (NFA) accepting the language. If we speak about the state complexity of an operation on regular languages, we ask how many states are sufficient and necessary in the worst case to accept the language resulting from the operation. Some early results concerning the state complexity of regular languages can be found in [19–21]. The state complexity of some operations on regular languages was investigated in [2,3,18]. Yu et al. [27] were the first to systematically study the complexity of regular language operations. Their paper was followed by several articles investigating the state complexity of finite languages operations and unary languages operations [5,22,23]. The nondeterministic state complexity of regular languages operations was studied by Holzer and Kutrib in [10–13]. Further results on this topic are presented in [6,7] and state-of-the-art surveys for DFAs can be found in [29,30]. In this paper, we investigate the state complexity of some operations on binary regular languages and provide answers to some problems which have been open for the binary case. In particular, we consider the concatenation of DFA languages, and the reversal and complementation of NFA languages. For the concatenation of DFA languages, the worst case m2n 2n1 was given by an m-state DFA language and an n-state DFA language over a three-letter alphabet in [27]. We show that the worst case can be reached by the concatenation of two binary DFA languages. The reversal of any n-state NFA language can be accepted by an (n + 1)-state NFA and this upper bound was shown to be tight for a three-letter alphabet by Holzer and Kutrib [13]. We give a binary n -state NFA language reaching the upper bound on the reversal. To accept the complement of any n-state NFA language 2n states suffice since we can simply convert a given NFA to an equivalent DFA and then exchange accepting and rejecting states. Birget [3] claimed that the upper bound is tight for a three-letter alphabet but later corrected this to a four-letter alphabet [4]. We prove that the upper bound is also tight for a binary alphabet by presenting a binary n-state NFA language such that any NFA accepting its complement needs at least 2n states. To prove the result for concatenation we show that a deterministic finite automaton is minimal. We obtain the lower bound on reversal using a counting argument. To obtain the lower bound on complementation we use a fooling-set lower-bound technique known from communication complexity theory [14], cf. also [2,3,9]. The paper consists of sixsections, including this introduction, and an appendix. The next section contains basic definitions and notations used throughout the paper. In Section 3 we present our result for concatenation. Section 4 deals with the reversal operation. In Section 5 we investigate the concatenation operation. The last section contains concluding remarks and open problems. In the appendix, we give some omitted proofs.
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