construct a deterministic finite automaton

If it produces an output sequence in response to an input sequence, then it is called a transducer(or automaton with output). If the output in response to an input string is binary (say, accept or reject), then it is called an accepter. The automaton can produce output of some form.The simpler automata can only read the input one symbol at a time from left to right but not change.

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).

Every automaton consists of some essential features as in real computers.

Of such abstract machines (also called models of computation) which can be Here are some simple examples (though there is nothing new in it).Īn automata is an abstract computing device (or machine). Set operations on languages : Since languages are set of strings we can apply set operations to languages. with or without subscripts are normally used Ĭonvention : Capital letters A, B, C, L, etc. When more than one alphabets are considered for discussion, subscripts may be used (e.g. ∑ 1, ∑ 2 etc) or sometimes other symbol like G may also be introduced.Ī string or word over an alphabet ∑ is a finite sequence of concatenated symbols ∑.Įxample : 0110, 11, 001 are three strings over the binary alphabet. The alphabet of a language is denoted by ∑. Usually, characters from a typical keyboard are only used as symbols.Īn alphabet is a finite, nonempty set of symbols. A symbol is any single object such as ↑, a, 0, 1, #, begin, or do. That is, symbols are atoms of the world of languages. Symbols are indivisible objects or entity that cannot be defined. Programming languages belong to this category. That is, our focus here is on formal languages that need precise and formal definitions.

The languages we consider for our discussion is an abstraction of natural languages. (More formally, we might say: if integer x is a prime, then x is odd.)ĭISPROOF: The integer 2 is a prime, but 2 is even. As we mentioned, if S is any statement, then the statement “S is not a theorem” is itsel a statement without parameters, and thus can be regarded as an observation than aĪlleged Theorem : All primes are odd. It often is easier to prove that a statement is not a theorem than to prove it is a theorem. This form of proof is called proof by contradiction. Complete the proof by showing that something known to be false follows logically from H and not C. That is, start by assuming both the hypothesis H and the negation of the conclusion C. To see why “If H then C” and “I not C then not H” are logically equivalent, first observe that there are four cases to consider: The contrapositive o the statement “if H and C” is “if not C then not H.” A statement and its contrapositive are either both true or both false, so we can prove either to prove the other.įigure 1.6: Steps in the “only-if” part of Theorem 1.10 If a given relation is reflexive, symmentric and transitive then the relation is called equivalence relation.ĭeductive proof: Consists of sequence of statements whose truth lead us from some initialĮg: if a and b are odd numbers then product is also an odd number. Let a and b be two sets a relation R contains aXb. Logic relations: a b = > 7a U b 7(a ∩ b)= 7a U 7b (AUB)’ =A’ ∩ B’ (A ∩ B)’ = A’ U B’ Double compliment Uses of Automata: compiler design and parsing.Ībsorption law: AU(A ∩ B) = A, A ∩ (AUB) = A states (represented in the figure by circles),Īs the automaton sees a symbol of input, it makes a transition (or jump) to another state, according to its transition function (which takes the current state and the recent symbol as its inputs).These abstract machines are called automata. In theoretical computer science, automata theory is the study of abstract machines (or more appropriately, abstract 'mathematical' machines or systems) and the computational problems that can be solved using these machines. There are several models in use, but the most commonly examined is the Turing machine. In order to perform a rigorous study of computation, computer scientists work with a mathematical abstraction of computers called a model of computation. The field is divided into three major branches: automata theory, computability theory and computational complexity theory. L is exactly the set of strings accepted by M.In theoretical computer science, the theory of computation is the branch that deals with whether and how efficiently problems can be solved on a model of computation, using an algorithm. We don’t mean by this that L and maybe some other strings are accepted by M we mean L = L(M), i.e. Note that we sometimes use a slightly different phrasing and say that a language L is accepted by some machine M.

Construct a deterministic finite automaton