Contents
We have seen an example of use of finite automata in describing the operation of a simplified version of vending machine. Many other systems operating in practice can also be modeled by finite automata such as control circuits of computers, computer network communication protocols, lexical analysers for compilers etc. Many of those systems fall into the class of systems called reactive system. A reactive system is a system that changes its actions, outputs and conditions/status in response to stimuli from within or outside it. It is an event driven or control driven system continuously having to react to external and/or internal stimuli. The inputs for a reactive system are never ready unlike for example when two numbers are added together by an adder (Here we are considering an adder at a higher level of abstraction than physical devices level ignoring for example the transient states of the electronic circuit that realizes an adder). An adder does not respond unless the input i.e. two numbers to be added are ready. A system such as an adder is called a transformational system. In the case of vending machine or communication protocol, on the other hand, a system must respond to each stimulus, even to a fragment of input such as each coin tossed in for a can of soda or every message received.

It is generally agreed that finite automata are a natural medium to describe dynamic behaviors of reactive systems. Finite automata are formal and rigorous and computer programs can be easily written to simulate their behaviors. To model a reactive system with finite automaton, first the states the system goes in or the modes of its operation are identified. These become the states of the finite automaton that models it. Then the transitions between the states triggered by events and conditions, external or internal to the system, are identified and they become arcs in the transition diagram of the finite automaton. In addition actions that may take place in...

...Lesson 3 FiniteAutomata with Output
Three types of automata are studied in Formal Language Theory. *
Acceptor
The symbols of the sequence
s(1) s(2) … s(i) … s(t)
are presented sequentially to a machine M. M responds with a binary signal to each input. If the string scanned so far is accepted, then the light goes on, else the light is off.
A language acceptor
* Lesson 3 employs the treatment of this subject as found in Machines, Languages, and Computation by Denning, Dennis and Qualitz , Prentice-Hall.
Transducer
Abstract machines that operate as transducers are of interest in connection with the translation of languages. The following transducer produces a sentence
r(1) r(2) … r(n)
in response to the input sentence
s(1) s(2) … s(m)
If this machine is deterministic, then each sentence of an input language is translated into a specific sentence of an output language.
Generator
When M is started from its initial state, it emits a sequence of symbols
r(1) r(2) … r(i) … r(t)
from a set known as its output alphabet.
We will begin our study with the transducer model of abstract machine (or automaton). We often refer to such a device as a Finite State Machine (FSM) or as an automaton with output.
Finite State Machine (FSM)
The FSM model arises...

...C
H
A
P
T
E
R
Finite-State Machines and Pushdown Automata
The ﬁnite-state machine (FSM) and the pushdown automaton (PDA) enjoy a special place in computer science. The FSM has proven to be a very useful model for many practical tasks and deserves to be among the tools of every practicing computer scientist. Many simple tasks, such as interpreting the commands typed into a keyboard or running a calculator, can be modeled byﬁnite-state machines. The PDA is a model to which one appeals when writing compilers because it captures the essential architectural features needed to parse context-free languages, languages whose structure most closely resembles that of many programming languages. In this chapter we examine the language recognition capability of FSMs and PDAs. We show that FSMs recognize exactly the regular languages, languages deﬁned by regular expressions and generated by regular grammars. We also provide an algorithm to ﬁnd a FSM that is equivalent to a given FSM but has the fewest states. We examine language recognition by PDAs and show that PDAs recognize exactly the context-free languages, languages whose grammars satisfy less stringent requirements than regular grammars. Both regular and context-free grammar types are special cases of the phrasestructure grammars that are shown in Chapter 5 to be the languages accepted by Turing machines. It is desirable not only to classify languages by the...

...Implementations of:
Finiteautomata
Regular expression
Pushdown automata
Engineering applications of finiteautomata
The study of automata has been acquiring increasing importance for engineers in many fields. For some time, the capabilities of these automata have been of the greatest interest to logicians and mathematicians. However, the expanding literature on the use of finiteautomata as probabilistic models demonstrates the growing interest in the application of these mechanisms to engineering phenomena.
We, the authors, became interested in these probabilistic models in an effort to develop a general self-adaptive control scheme based on the prediction of the future of the process to be controlled. Conceivably, an adequate model of a particular process could be generated by simply observing the process parameters. With this goal in mind we began an investigation of several different modeling techniques. The ability to model stochastic data was our primary concern. We feel that the results of several modeling experiments presented here may be of interest to our readers, and we hope to encourage the use of these techniques, especially in control applications.
We have seen an example of use of finiteautomata in...

...(Union n in N and n >0)
=
Example : Let L = { a, ab }. Then we have,
=
= {e} {a, ab} {aa, aab, aba, abab} …
=
= {a, ab} {aa, aab, aba, abab} …
Note : e is in , for every language L, including .
The previously introduced definition of is an instance of Kleene star.
Automata and Grammars
Automata
An automata is an abstract computing device (or machine). There are different varities of such abstract machines (also called models of computation) which can be defined mathematically. Some of them are as powerful in principle as today's real computers, while the simpler ones are less powerful. ( Some models are considered even more powerful than any real computers as they have infinite memory and are not subject to physical constraints on memory unlike in real computers). Studying the simpler machines are still worth as it is easier to introduce some formalisms used in theory.
* Every automaton consists of some essential features as in real computers. It has a mechanism for reading input. The input is assumed to be a sequence of symbols over a given alphabet and is placed on an input tape(or written on an input file). The simpler automata can only read the input one symbol at a time from left to right but not change. Powerful versions can both read (from left to right or right to left) and change the input.
* The automaton can produce output of some form. If the output in...

...1
Equivalence of FiniteAutomata and Regular Expressions
FiniteAutomata Recognize Regular Languages Theorem 1. L is a regular language iﬀ there is a regular expression R such that L(R) = L iﬀ there is a DFA M such that L(M ) = L iﬀ there is a NFA N such that L(N ) = L. i.e., regular expressions, DFAs and NFAs have the same computational power. Proof. • Given regular expression R, will construct NFA N such that L(N ) = L(R)
• Given DFA M , will construct regular expression R such that L(M ) = L(R)
2
Regular Expressions to NFA
Regular Expressions to FiniteAutomata . . . to Non-determinstic FiniteAutomata Lemma 2. For any regex R, there is an NFA NR s.t. L(NR ) = L(R). Proof Idea We will build the NFA NR for R, inductively, based on the number of operators in R, #(R). • Base Case: #(R) = 0 means that R is ∅, , or a (from some a ∈ Σ). We will build NFAs for these cases. • Induction Hypothesis: Assume that for regular expressions R, with #(R) < n, there is an NFA NR s.t. L(NR ) = L(R). • Induction Step: Consider R with #(R) = n. Based on the form of R, the NFA NR will be built using the induction hypothesis.
Regular Expression to NFA Base Cases If R is an elementary regular expression, NFA NR is constructed as follows. R=∅ q0
R= q0
q0 a
R=a
q1
1
Induction Step: Union Case R = R1 ∪ R2 By induction hypothesis, there are N1 , N2...

...Practice Sheet 1
1. The language L = {w|w has exactly two 0’s and at least two 1’s } is the intersection
of two simpler languages. Construct DFA’s for the simpler languages and then combine them using the idea of a product automaton to obtain a DFA that accepts L.
Minimize this DFA, using the minimization algorithm, using the algorithm explained
in the class.
Soln: Similar to Problem 2.
2. The language L = {w|w has even length and an odd number of 0’s } is the intersection of two simpler languages. Construct DFA’s for the simpler languages and then
combine them using the idea of a product automaton to obtain a DFA that accepts L.
Minimize this DFA, using the minimization algorithm, using the algorithm explained
in the class.
Soln: Combine the two DFAs in Fig. 1 by constructing a product automaton.
0
q0
q1
1
0
1
0, 1
q0
q1
0, 1
Figure 1: Figure for Solution of Problem 2 in Practice Sheet 1
3. Draw the transition diagram of a DFA that accepts the following languages: (a) The
empty set (b) all strings over {0, 1} except the empty string (c) {w|w begins with a
0 and ends with a 1 }
Soln: See Fig. 2 attached.
4. For each of the following languages, give two strings that are in the language and two
that are not (assume Σ = {a, b}): (a) a(ba)∗ b (b) (a + ba + bb)Σ∗
Soln: (a) The strings
and a, for example. (b) The strings
and b, for example.
5. Construct an NFA that accepts the language described by the r.e. (01 + 001 + 010)∗
Son:...

...Lyceum of the Philippines University-Batangas
Capitol Site, Batangas City
School of Advanced Studies and Research
SUBJECT: Enrollment System
TOPIC: Problems met by the CITHM Students in the Sectioning and Encoding process during enrollment period
DISCUSSANTS: Patricia Isabel P. Maghirang
Joy S. Macatangay
PROFESSOR: Mrs. Jessica Cabaces
I. INTRODUCTION
This research activity focused on the slow processing of transactions in enrollment period, to know the problems, to recognize the reason, to give possible solutions.
II. Related Literature/Presentation
1. To know the problems met by the CITHM students in the Sectioning and Encoding process during enrollment period.
Processing of transactions was most difficult during the enrollment period. Some of the major problems encountered, particularly as concerns the enrollment are; Slow processing of transactions, tedious enrollment procedures, too much paper works, not enough rooms, settle financial obligations in the university, submit requested documents, comply with admission provisions, students have to go back and forth in different offices for evaluation and validation.
2. To recognize the reasons on the slow processing of transactions during enrollment period; Poor marketing/admissions, poor leadership, inaccessibility or unavailability of data/system that were needed immediately, and unorganized steps for enrollment, lack of enrollment personnel.
3. The possible solutions to the slow...

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