Introduction

In the previous chapters, an abstract model of a machine that accepts input values but doesn't produce any output values, was discussed. Some machines, however, produce output values after accepting input values. In the case of FSA (finite state automata), movements from state qi to qj depend on the input at qi and no output emerges. However, in the case of FSM, a move from a state qi to state qj results in an output. Consequently, a FSM possesses two special features: a finite set Γ of output symbols and an output function λ :Q × ∑ → Γ, where ∑ is the input alphabet. Thus, one major limitation of finite automata is that its output is limited to a binary signal: ‘accept’ or ‘don't accept’, to indicate the acceptance or rejection of an input string respectively. In order to make finite automata to have output capabilities, two classical machines are designed that transform input strings into output strings. They are

1. a. Moore machine and

2. b. Mealy machine.

A Moore machine is a FSM – M0, named after Edward Moore, who introduced it in 1956.

A Mealy machine is a FSM – Me, named after George H.Mealy, who introduced it in 1955.

These machines are basically DFAs, except that they associate an output symbol with each state or with each state transition. However, there are no final states, because there is no acceptance or rejection involved.

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This book is the culmination of our fascination for the subject of computation and its applications. It has been designed for students of computer science at the postgraduate level and is logically conceived, self-contained, well-organised and user-friendly.

The book contains an in-depth coverage of all the topics related to the theory of computation as mentioned in the syllabuses of B.E., M.C.A. and M.Sc. (Computer Science) of various universities. Sufficient amount of theoretical inputs supported by a number of illustrations are included for those who take deep interest in the subject.

The book includes 15 chapters, segregated into three sections. Part one ‘Fundamentals of Computation’, contains chapters 1 to 3. Chapter 1 is introductory in nature and gives a broad picture of the subject. Chapters 2 and 3 discuss Formal Language Theory and Regular Expressions and Languages. Part two ‘Models and Principles of Computation’, comprises chapters 4 to 7, and discusses topics like Finite State Machine, Equivalent Automata, Optimisation/Minimisation of DFA, Finite Automata and Regular Expressions, Transducers et cetera in a systematic way. Finally, part three ‘Advanced Models and Principles of Computation’ includes chapter 9 to 15, and addresses issues of Context- Free Grammar and Context-Free Languages, Simplification of Context-Free Grammar, Pushdown Automata, Pumping Lemma, Turing Machine, TM Extensions and Languages and Formal Languages/Grammar Hierarchy .

Introduction

In this chapter, we briefly sample a few additional topics in DFA, which are of interest.

Optimum DFA

It is possible to have more than one DFAs that accept the same language. Among these equivalent DFAs, it is often useful to find the smallest, i.e., the DFA with the minimum possible number of states. This is especially important, when DFAs are used for designing computer hardware circuits.

Definition

Minimisation/optimisation of a deterministic finite automaton refers to the detection of those states of a DFA, whose presence or absence in a DFA does not affect the language accepted by the automata.

The states that can be eliminated from automata, without affecting the language accepted

by automata, are:

1. ▪Unreachable or inaccessible states.

2. ▪Dead states.

3. ▪Non-distinguishable or indistinguishable state or equivalent states.

Unreachable States

These are the states that cannot possibly be reached from the initial state. Unreachable states of a DFA are not reachable from the initial state of DFA, by any possible input sequence.

EXAMPLE 6.1.1: (Unreachable states)

1. i. Here, state 5 is unreachable, from the initial state 0, with any input string (either b or a).

2. ii. Here, states q2 and q4 are unreachable, from the initial state q0, with any input string (0 or 1).

Dead State or Trap State

A state is dead, if it is not an accepting state and has no out-going transitions, except to itself. Alternatively, a dead state is a nonfinal state of a DFA, whose transitions on every input symbol terminates on itself.

Thermostat

Consider the problem of heating a room. Assume that the thermostat is used as a controller and we do not have the exact model of how the the thermostat functions. It is only known that the thermostat turns on the radiator when temperature is between 68 and 70 degrees and it turns off the radiator when temperature is between 80 and 82 degrees. This heating system can be modelled as an automaton shown in the figure below, where X denotes the temperature.

The automaton shown is a non deterministic automaton, in the sense that for a given initial condition it accepts the whole family of different executions (solutions).

Elevator Controller

Consider an elevator that serves two floors. Inputs to elevator are calls to a floor, either from inside the elevator or from the floor itself.

The following are the 3 possible inputs:

1. i) no calls (a)

2. ii) call to floor one (b)

3. iii) call to floor two. (c)

The following are the 6 possible states:

1. q1 – waiting on first floor

2. q2 – about to go up

3. q3 – going up

4. q4 – going down

5. q5 – waiting on second floor

6. q6 – about to go down.

The above transition diagram is without the accepting and rejecting states because the elevator design is simple and acceptance is not an issue. In designing a more complex elevator, with states like overloading, breakdown etc., accepting and rejecting states become significant.

Introduction

When we study computability, we study problems in an abstract sense. For example,

1. a. Addition is the problem of returning a third number that is the sum of two given numbers.

2. b. Travelling Salesman problem (TSP) is one in which a list of distances between some number of cities are given and the person is asked to find the shortest route so that he visits each city once and returns to the start.

3. c. Halting Problem (HP) is one in which a program is given some appropriate input and it needs to be decided whether the program, when run on that input, loops forever or halts.

Problems (b) and (c) are the two problems of computer interest.

In both these problems, the statement of the problem does not give the actual values needed to provide the result but just tells what kind of objects they are. A set of actual values for a problem is called an instance of the problem (by this terminology, all homework problems done by school students are considered as instances of problems).

In all these problems an instance is required that is the input and the relationship between the input and the output is to be understood. In order to solve these problems, there are certain things one should know.

1. Can it be solved algorithmically – is there any definite procedure that solves any instance of the problem in a finite amount of time? In other words, is it computable? Not all problems are computable.

Introduction

In any language such as English, Hindi or Sanskrit, words can be combined in several ways. Naturally, some combinations form valid sentences, while others do not. The validity of a sentence is determined by the grammar of a language, which comprises a set of rules. For instance, ‘The boy prepares tea quickly’, although meaningless, is a perfectly legal sentence. In other words, the sentences in a language may be nonsensical, but they must obey the rules of grammar. The discussion in this chapter deals with only the syntax of sentence (the way the words are combined) and not with the semantics of sentences (meaning).

In the previous chapters, two different (though equivalent) methods: finite automata and regular expressions were introduced for describing languages. These methods have their own limitations in the sense that some simple languages, such as {0n1n|n ≥ 0}, cannot be described by these methods. Formal languages and grammars are widely used in connection with programming languages. During programming, we proceed with an intuitive knowledge of the languages, which leads to errors. Therefore, a precise description of the language is needed, at almost every step, which helps to understand the syntax diagrams found in programming texts. Among the ways in which programming languages can be defined precisely, grammars or context-free grammars are most widely used. This method happens to be a very powerful method and such grammars can describe certain features which have a recursive structure. Context-free-grammars (CFG) were first used in the study of human languages.

Introduction

A language is a system of signs used to communicate information to others. However, the language of computation is a combination of both english and mathematics. Fundamentally, a computer is a symbol manipulator, in the sense, that it takes sequences of symbols as inputs and manipulates them as per the program specifications. These symbols are precise and unambiguous, unlike the language of humans. The first step in communicating a problem to a machine is the design of a proper language of computation and this is the fundamental object of computability.

While discussing about languages, it is important to note two cases:

1. a. During the evaluation of an input expression (e.g.: calculator), the language of arithmetic expression handles both the input and the output communication.

2. b. In the case of web form, the language simply describes all the legitimate inputs to the field and the output is simply a binary value-Yes or No.

Thus, the problems belonging to case (a), have both input and an output language, while, those belonging to case (b), have only an input language. It is of interest to note that, from the perspective of theory of computation, any type of problem can be expressed in terms of a language recognition.

Since a language is a medium of communication, it should be given some meaning (i.e. its semantics). But much of the manipulation of symbols, strings and language could be done effectively without understanding their semantics and this is the subject of this section. In other words, the mathematical study of the theory of computation begins with the understanding of the mathematics of symbols and strings.