all finite languages are decidable proof

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CS1010: Theory of Computation

Outline Decidable languages Regular languages are decidable Context-free languages are decidable Languages hierarchy From Sipser Chapter 4 1 Decidable Languages We start with problems that are decidable We first look at problems concerning regular languages and then those for context-free languages

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Decidability and Undecidability

Decidable Languages A language L is called decidable iff there is a decider M such that (ℒ M) = L Given a decider M you can learn whether or not a string w ∈ (ℒ M) Run M on w Although it might take a staggeringly long time M will eventually accept or reject w The set R is the set of all decidable languages L ∈ R iff L is decidable

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Decidability

Σ Infinite binary sequence is uncountable Theorem The set of Turing Machines is countable Proof For any given machine we can give a full description of the machine with a finite string M = (Q Σ Γ δ q0 qacc qrej) and each of the elements of the 7-tuple is finite The function δ for example is δ: Q × Γ → Q × Γ ×

Is a finite set decidable?
Every finite set is decidable since we can always "hard-code" a Turing machine to accept a given finite set: fixing a 1,..., a n, just write a program which on input k checks whether k = a 1, whether k = a 2 ,... , whether k = a n, and outputs "YES" if the answer to one of these questions is YES and outputs "NO" otherwise.
Can a finite language be accepted by a finite machine?
One-line proof: A finite language can be accepted by a finite machine. Detailed construction: Suppose the language L L consists of strings a1,a2, …,an a 1, a 2, …, a n. Consider the following NFA to accept L L: It has a start state S S and an accepting state A A. In between S S and A A there are n n different paths of states, one for each ai a i.
What are the closure properties of decidable languages?
Closure properties of Decidable languages. The class of decidable languages is closed under Union, Concatenation, Star, Intersection, and Complementation. Theorem. Closure properties of Recognizable languages. The class of recognizable languages is closed under Union, Concatenation, Star, and Intersection.
Are all semi-decidable+ languages undecidable?
semi-decidable+. Every TM for a semi-decidable+ language halts in the accept state for strings in the language but loops for some strings not in the language. Any language outside Dec is undecidable. All semi-decidable+ languages are undecidable, but we’ll see there are undecidable languages that aren’t semi-decidable+!

PDF	Solutions to Problem Set 4 Feb 23 2007 If. A is finite

PDF	CS 373: Theory of Computation decidable (or simply decidable) if there exists a TM M which decides L. • Every finite language is decidable: For example by a TM that has all the strings in ...

PDF	BBM401-Lecture 7: Decidable Languages and the Halting Problem Every finite language is decidable: For e.g. by a TM that has all the Proof. If Atm is recognizable

PDF	A note on algebras of languages study some immunity properties: for instance we prove that for every coinfinite decidable language L there exists a decidable language L′ such that L ⊆ L′ L′

PDF	CSE 135: Introduction to Theory of Computation Decidability and ▷ Every finite language is decidable. Page 10. Decidable and Recognizable Languages. Recall: Definition. A Turing machine M is said to recognize a language L

PDF	CSE 6321 - Solutions to Problem Set 1 Show that the collection of decidable languages is closed under the following operations. 1. complementation. Solution: Proof. Let L be a decidable language and

PDF	Adriana Palacio - University of California San Diego Instructor Jul 29 2004 Since all finite languages are regular

PDF	Practice Problems for Final Exam: Solutions CS 341: Foundations of In each part below if you need to prove that the given language L is decidable

PDF	Lecture 33: Reductions and Undecidability SD & Turing Enumerable • Can only happen if L is finite. • But all finite languages are decidable. • Fixes proof. • But not decidable whether L is finite!! Undecidable Problems. The

PDF	Decidability If any generate w accept

PDF	BA. B→b. A→a. Theorem. Every Context Free Language is decidable. Proof. Let L

PDF	CS 373: Theory of Computation Every finite language is decidable: For example by a TM that has all the Proposition 2. If L and L are recognizable

PDF	Solutions to Problem Set 4 23 févr. 2007 these machines only recognize regular languages). ... A is finite it is decidable because all finite languages are decidable (just hardwire ...

PDF	Practice Problems for Final Exam: Solutions CS 341: Foundations of In each part below if you need to prove that the given language L is decidable

PDF	BBM401-Lecture 7: Decidable Languages and the Halting Problem Every finite language is decidable: For e.g. by a TM that has Proposition. If L and L are recognizable

PDF	CMPE 350 - Spring 2018 26 avr. 2018 Prove or disprove: “The class of non-context-free languages is closed under com- ... decidable because all finite languages are decidable.

PDF	Languages and Finite Automata Every decidable language is Turing-Acceptable a finite language? ... If a language is decidable then its complement is decidable too. L. L. Proof:.

PDF	Undecidability - Lecture in INF4130 18 oct. 2018 Theorem. Any finite language is decidable. Proof. If A is a finite language a decider MD for A can be constructed by hard-coding all.

PDF	Co-finiteness of VASS coverability languages 24 juil. 2019 and ?fin(A) to denote the set of all finite subsets of A. ... for VASS coverability languages is decidable. Proof. Consider finite sets of ...

PDF	CSE 6321 - Solutions to Problem Set 1 Show that the collection of decidable languages is closed under the following operations. 1. complementation. Solution: Proof. Let L be a decidable language and

PDF	Equations over finite sets of words and equivalence problems in a given regular language. L i.e. whether cr(.x)=t(.~) holds for all x in L

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PDF	Note - CS 373: Theory of Computation Every finite language is decidable: For example, by a TM that has all the strings Proof If Atm is recognizable, since Atm is recognizable, the two languages will

PDF	Undecidability - UiO 18 oct 2018 · A Turing machine that halts on all inputs, it is called a decider Any finite language is decidable Proof If A is a finite language, a decider MD

PDF	BBM401-Lecture 7: Decidable Languages and the Halting Problem Every finite language is decidable: For e g , by a TM that has If L and L are recognizable, then L is decidable Proof Program P for deciding L, given programs

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Practice Problems for Final Exam: Solutions CS 341 - NJIT

Turing-decidable language Answer: A Answer: A set S is countable if it is finite or we can define a correspondence between S and the positive integers In other words, we can create a list of all the elements in S and each specific element will The typical approach to proving a language C is NP-Complete is as follows:

PDF	Homework Solutions - Santa Fe Institute Prove that such a machine can be simulated with a standard Since all finite languages are decidable (a hard-coded Turing machine can just look the input up)

PDF	Decidable Languages finite automaton accepts any strings at all E DFA = { A is a DFA and L(A) = ∅} Can you come up with an algorithm to perform this test? Proof: Determine

PDF	6045J Lecture 7: Decidability - MIT OpenCourseWare Claimed they are all equivalent, so the notion of Decidable and recognizable languages M 1 M 2 • Proof: ⇐ – M copies M's finite-state control keeps

PDF	Formal Languages, Automata and Computation - andrewcmued ENCODING FINITE AUTOMATA AS STRINGS Here is one With these conventions, all we need to encode is δ and F Each entry of δ, e g , δ(qi,aj) decidable language PROOF Convert NFA to DFA and use Theorem 4 1 N = “On input