Turing machine for multiplication Last Updated : 11 Jul, 2025 Comments Improve Suggest changes Like Article Like Report Prerequisite - Turing Machine Problem: Draw a turing machine which multiply two numbers. Example: Steps: Step-1. First ignore 0's, C and go to right & then if B found convert it into C and go to left. Step-2. Then ignore 0's and go left & then convert C into C and go right. Step-3. Then convert all X into X and go right if 0 found convert it into X and go to left otherwise if C found convert it into B and go to right and stop the machine. Step-4. If then X found convert it into X and go left then C into C and left then Y into Y and left. Step-5. Then if B found convert it into B and right then if Y into 0 and right or if C into C and right and go to step 3 and repeat the process otherwise if 0 found after 4th step then convert it into Y and right then Y into Y and right then C into C and right then 0 into 0 or X into X and right then C into C and right then 0 into 0 and right then B into 0 and left then 0 into 0 and left then C into C and left then 0 into 0 or X into X and left then C into C and left. Step-6. Then repeat the 5th step. Here, q0 shows the initial state and q1, q2, ....., q10, q11are the transition states and q12shows the final state. And X, Y, 0, C are the variables used for multiplication and R, L shows right and left. Comment More infoAdvertise with us Next Article Construct a Turing Machine for language L = {wwr | w ∈ {0, 1}} U ujjwal57 Follow Improve Article Tags : Theory of Computation Similar Reads Introduction to Theory of Computation Automata theory, also known as the Theory of Computation, is a field within computer science and mathematics that focuses on studying abstract machines to understand the capabilities and limitations of computation by analyzing mathematical models of how machines can perform calculations.Why we study 7 min read TOC BasicsIntroduction to Theory of ComputationAutomata theory, also known as the Theory of Computation, is a field within computer science and mathematics that focuses on studying abstract machines to understand the capabilities and limitations of computation by analyzing mathematical models of how machines can perform calculations.Why we study 7 min read Chomsky Hierarchy in Theory of ComputationAccording to Chomsky hierarchy, grammar is divided into 4 types as follows: Type 0 is known as unrestricted grammar.Type 1 is known as context-sensitive grammar.Type 2 is known as a context-free grammar.Type 3 Regular Grammar.Type 0: Unrestricted Grammar: Type-0 grammars include all formal grammar. 2 min read Applications of various AutomataAutomata are used to design and analyze the behavior of computational systems. 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These concepts form the foundation of automata theory, compiler design, and text processing.Regular ExpressionsRegular expressions are symbolic notations used 7 min read Arden's Theorem in Theory of ComputationA Regular Expression (RE) is a way to describe patterns of strings using symbols and operators like union, concatenation, and star. A Deterministic Finite Automaton (DFA) is a machine that reads input strings and decides if they match the pattern by moving through a set of defined states without any 6 min read Conversion from NFA to DFAAn NFA can have zero, one or more than one move from a given state on a given input symbol. An NFA can also have NULL moves (moves without input symbol). On the other hand, DFA has one and only one move from a given state on a given input symbol. 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In this article we are going to discuss Moore Machines and Mealy Machines, the difference between these two machinesas well as Conversion from Moore to Mealy and Convers 3 min read CFG & PDASimplifying Context Free GrammarsA Context-Free Grammar (CFG) is a formal grammar that consists of a set of production rules used to generate strings in a language. However, many grammars contain redundant rules, unreachable symbols, or unnecessary complexities. Simplifying a CFG helps in reducing its size while preserving the gene 6 min read Converting Context Free Grammar to Chomsky Normal FormChomsky Normal Form (CNF) is a way to simplify context-free grammars (CFGs) so that all production rules follow specific patterns. In CNF, each rule either produces two non-terminal symbols, or a single terminal symbol, or, in some cases, the empty string. Converting a CFG to CNF is an important ste 5 min read Closure Properties of Context Free LanguagesContext-Free Languages (CFLs) are an essential class of languages in the field of automata theory and formal languages. They are generated by context-free grammars (CFGs) and are recognized by pushdown automata (PDAs). Understanding the closure properties of CFLs helps in determining which operation 11 min read Pumping Lemma in Theory of ComputationThere are two Pumping Lemmas, which are defined for 1. Regular Languages, and 2. Context - Free Languages Pumping Lemma for Regular Languages For any regular language L, there exists an integer n, such that for all x ? L with |x| ? n, there exists u, v, w ? ?*, such that x = uvw, and (1) |uv| ? n (2 4 min read Ambiguity in Context free Grammar and LanguagesContext-Free Grammars (CFGs) are essential in formal language theory and play a crucial role in programming language design, compiler construction, and automata theory. One key challenge in CFGs is ambiguity, which can lead to multiple derivations for the same string.Understanding Derivation in Cont 3 min read Context-sensitive Grammar (CSG) and Language (CSL)Context-Sensitive Grammar - A Context-sensitive grammar is an Unrestricted grammar in which all the productions are of form - Where α and β are strings of non-terminals and terminals. Context-sensitive grammars are more powerful than context-free grammars because there are some languages that can be 2 min read Introduction of Pushdown AutomataWe have already discussed finite automata. But finite automata can be used to accept only regular languages. Pushdown Automata is a finite automata with extra memory called stack which helps Pushdown automata to recognize Context Free Languages. This article describes pushdown automata in detail.Pus 5 min read Turing Machine & DecidabilityTuring Machine in TOCTuring Machines (TM) play a crucial role in the Theory of Computation (TOC). They are abstract computational devices used to explore the limits of what can be computed. Turing Machines help prove that certain languages and problems have no algorithmic solution. Their simplicity makes them an effecti 7 min read Recursive and Recursive Enumerable Languages in TOCRecursive and Recursive Enumerable Languages in TOC are two important classes of languages which are linked with Turing Machine. A recursive language is one where a Turing Machine always halts and decides whether a string belongs to the language or not. A recursively enumerable language is one where 6 min read Halting Problem in Theory of ComputationThe halting problem is a fundamental issue in theory and computation. The problem is to determine whether a computer program will halt or run forever.Definition: The Halting Problem asks whether a given program or algorithm will eventually halt (terminate) or continue running indefinitely for a part 4 min read Turing Machine as ComparatorPrerequisite – Turing MachineProblem : Draw a turing machine which compare two numbers. Using unary format to represent the number. For example, 4 is represented by 4 = 1 1 1 1 or 0 0 0 0 Lets use one's for representation. Example: Approach: Comparing two numbers by comparing number of '1's.Comparin 3 min read Decidable and Undecidable Problems in Theory of ComputationIn the Theory of Computation, problems can be classified into decidable and undecidable categories based on whether they can be solved using an algorithm. A decidable problem is one for which a solution can be found in a finite amount of time, meaning there exists an algorithm that can always provid 6 min read Undecidability and Reducibility in TOCDecidable Problems A problem is decidable if we can construct a Turing machine which will halt in finite amount of time for every input and give answer as ‘yes’ or ‘no’. A decidable problem has an algorithm to determine the answer for a given input. Examples Equivalence of two regular languages: Giv 5 min read Computable and non-computable problems in TOCIn the Theory of Computation, problems are classified as computable or non-computable based on whether they can be solved by an algorithm. Computable problems have a clear, step-by-step procedure that always lead to a correct solution while non-computable problems cannot be solved by any algorithm, 6 min read Problems on Finite AutomataDFA for Strings not ending with "THE"Problem - Accept Strings that not ending with substring "THE". Check if a given string is ending with "the" or not. The different forms of "the" which are avoided in the end of the string are: "THE", "ThE", "THe", "tHE", "thE", "The", "tHe" and "the" All those strings that are ending with any of the 12 min read DFA of a string with at least two 0’s and at least two 1’sProblem - Draw deterministic finite automata (DFA) of a string with at least two 0’s and at least two 1’s. The first thing that come to mind after reading this question us that we count the number of 1's and 0's. Thereafter if they both are at least 2 the string is accepted else not accepted. But we 3 min read DFA for accepting the language L = { anbm | n+m =even }ProblemDesign a deterministic finite automata(DFA) for accepting the language L = {an bm | n+m = even}Examples:Input: a a b b , n = 2, m = 2 2 + 2 = 4 (even)Output: ACCEPTEDInput: a a a b b b b ,n = 3, m = 43 + 4 = 7 (odd) Output: NOT ACCEPTEDInput: a a a b b b , n = 3, m = 33 + 3 = 6 (even)Output: 14 min read DFA machines accepting odd number of 0’s or/and even number of 1’sPrerequisite - Designing finite automata Problem - Construct a DFA machine over input alphabet \sum_= {0, 1}, that accepts: Odd number of 0’s or even number of 1’s Odd number of 0’s and even number of 1’s Either odd number of 0’s or even number of 1’s but not the both together Solution - Let first d 3 min read DFA of a string in which 2nd symbol from RHS is 'a'Draw deterministic finite automata (DFA) of the language containing the set of all strings over {a, b} in which 2nd symbol from RHS is 'a'. The strings in which 2nd last symbol is "a" are: aa, ab, aab, aaa, aabbaa, bbbab etc Input/Output INPUT : baba OUTPUT: NOT ACCEPTED INPUT: aaab OUTPUT: ACCEPTED 10 min read Problems on PDAConstruct Pushdown Automata for all length palindromeA Pushdown Automata (PDA) is like an epsilon Non deterministic Finite Automata (NFA) with infinite stack. PDA is a way to implement context free languages. Hence, it is important to learn, how to draw PDA. Here, take the example of odd length palindrome:Que-1: Construct a PDA for language L = {wcw' 6 min read Construct Pushdown automata for L = {0n1m2m3n | m,n ≥ 0}Prerequisite - Pushdown automata, Pushdown automata acceptance by final state Pushdown automata (PDA) plays a significant role in compiler design. Therefore there is a need to have a good hands on PDA. Our aim is to construct a PDA for L = {0n1m2m3n | m,n ≥ 0} Examples - Input : 00011112222333 Outpu 3 min read Construct Pushdown automata for L = {a2mc4ndnbm | m,n ≥ 0}Pushdown Automata plays a very important role in task of compiler designing. That is why there is a need to have a good practice on PDA. Our objective is to construct a PDA for L = {a2mc4ndn bm | m,n ≥ 0} Example:Input: aaccccdbOutput: AcceptedInput: aaaaccccccccddbbOutput: AcceptedInput: acccddbOut 3 min read NPDA for accepting the language L = {anbn | n>=1}Prerequisite: Basic knowledge of pushdown automata.Problem :Design a non deterministic PDA for accepting the language L = {an bn | n>=1}, i.e.,L = {ab, aabb, aaabbb, aaaabbbb, ......} In each of the string, the number of a's are followed by equal number of b's. ExplanationHere, we need to maintai 2 min read NPDA for accepting the language L = {ambncm+n | m,n ≥ 1}The problem below require basic knowledge of Pushdown Automata.Problem Design a non deterministic PDA for accepting the language L = {am bn cm+n | m,n ≥ 1} for eg. ,L = {abcc, aabccc, abbbcccc, aaabbccccc, ......} In each of the string, the total sum of the number of 'a’ and 'b' is equal to the numb 2 min read NPDA for accepting the language L = {aibjckdl | i==k or j==l,i>=1,j>=1}Prerequisite - Pushdown automata, Pushdown automata acceptance by final state Problem - Design a non deterministic PDA for accepting the language L = {a^i b^j c^k d^l : i==k or j==l, i>=1, j>=1}, i.e., L = {abcd, aabccd, aaabcccd, abbcdd, aabbccdd, aabbbccddd, ......} In each string, the numbe 3 min read NPDA for accepting the language L = {anb2n| n>=1} U {anbn| n>=1}To understand this question, you should first be familiar with pushdown automata and their final state acceptance mechanism.ProblemDesign a non deterministic PDA for accepting the language L = {an b2n : n>=1} U {an bn : n>=1}, i.e.,L = {abb, aabbbb, aaabbbbbb, aaaabbbbbbbb, ......} U {ab, aabb 2 min read Problems on Turing MachinesTuring Machine for additionPrerequisite - Turing Machine A number is represented in binary format in different finite automata. For example, 5 is represented as 101. However, in the case of addition using a Turing machine, unary format is followed. In unary format, a number is represented by either all ones or all zeroes. For 3 min read Turing machine for multiplicationPrerequisite - Turing Machine Problem: Draw a turing machine which multiply two numbers. Example: Steps: Step-1. First ignore 0's, C and go to right & then if B found convert it into C and go to left. Step-2. Then ignore 0's and go left & then convert C into C and go right. Step-3. Then conv 2 min read Construct a Turing Machine for language L = {wwr | w ∈ {0, 1}}The language L = {wwres | w ∈ {0, 1}} represents a kind of language where you use only 2 character, i.e., 0 and 1. The first part of language can be any string of 0 and 1. The second part is the reverse of the first part. Combining both these parts a string will be formed. Any such string that falls 5 min read Construct a Turing Machine for language L = {ww | w ∈ {0,1}}Prerequisite - Turing Machine The language L = {ww | w ∈ {0, 1}} tells that every string of 0's and 1's which is followed by itself falls under this language. The logic for solving this problem can be divided into 2 parts: Finding the mid point of the string After we have found the mid point we matc 7 min read Construct Turing machine for L = {an bm a(n+m) | n,m≥1}Problem : L = { anbma(n +m) | n , m ≥ 1} represents a kind of language where we use only 2 character, i.e., a and b. The first part of language can be any number of "a" (at least 1). The second part be any number of "b" (at least 1). The third part of language is a number of "a" whose count is sum o 3 min read Construct a Turing machine for L = {aibjck | i*j = k; i, j, k ≥ 1}Prerequisite – Turing Machine In a given language, L = {aibjck | i*j = k; i, j, k ≥ 1}, where every string of 'a', 'b' and 'c' has a certain number of a's, then a certain number of b's and then a certain number of c's. The condition is that each of these 3 symbols should occur at least once. 'a' and 2 min read Turing machine for 1's and 2’s complementProblem-1:Draw a Turing machine to find 1's complement of a binary number. 1’s complement of a binary number is another binary number obtained by toggling all bits in it, i.e., transforming the 0 bit to 1 and the 1 bit to 0. Example:1's ComplementApproach:Scanning input string from left to rightConv 3 min read PracticeLast Minute Notes - Theory of ComputationThe Theory of Computation (TOC) is a critical subject in the GATE Computer Science syllabus. It involves concepts like Finite Automata, Regular Expressions, Context-Free Grammars, and Turing Machines, which form the foundation of understanding computational problems and algorithms.This article provi 13 min read Topic wise multiple choice questions in computer scienceWe have covered multiple choice questions on several computer science topics like C programming, algorithms, data structures, computer networks, aptitude mock tests, etc. Practice for computer science topics by solving these practice mcq questions.This page specifically covers a lot of questions and 2 min read Theory of Computation - GATE CSE Previous Year QuestionsThe Theory of Computation(TOC) subject has high importance in GATE CSE exam because:large number of questions nearly 6-8% of the total papersignificant weightage (6-8 marks) across multiple years Below is the table for previous four year mark distribution of TOC in GATE CS:YearApprox. Marks from TOC 2 min read Like