Chapter4Section1_F11 - 91.304 Foundations of(Theoretical...

This preview shows page 1 - 9 out of 25 pages.

91.304 Foundations of (Thtil) CtS i(Theoretical) Computer ScienceChapter 4 Lecture Notes (Section 4.1: Decidable Languages)David Martin[email protected]With modifications by Prof. Karen Daniels, Fall 2011This work is licensed under the Creative Commons Attribution-ShareAlike License. To view a copy of this license, visit -1sa/2.0/ or send a letter to Creative Commons, 559 Nathan Abbott Way, Stanford, California 94305, USA.
Back to Σ1Thft tht Σi t ld dThe fact that Σ1is not closed under complement means that there exists some language L that is not some language L that is not recognizable by any TM.By Church Turing thesis this means By Church-Turing thesis this means that no imaginable finite computer, even with infinite memory could even with infinite memory, could recognize this language L!2
L ALL-Σ1A non-Σ1languageALLALLCFPPΣ1RPPCFLΣ0REGFINEach point is a language in this Venn diagram3
StrategyGoal: Explore limits of algorithmic Goal: Explore limits of algorithmic solvability.We’ll show (later in Section 4 2) that there We ll show (later in Section 4.2) that there are more (a lotmore) languages in ALL than there are in Σ1Namely, that Σ1is countable but ALL isn’t countableWhich implies that Σ1ALLWhich implies that Σ1ALLWhich implies that there exists some L that is not in Σ14
Overview of Section 4.1Decidable Languages (in Σ): to foster Decidable Languages (in Σ0): to foster later appreciation of undecidable languageslanguagesRegular LanguagesADFA: Acceptance problem for DFAsANFA: Acceptance problem for NFAsAREX: Acceptance problem for Regular ExpressionsEDFA: Emptiness testing for DFAsEDFA: Emptiness testing for DFAsEQDFA: 2 DFAs recognizing the same languageContext-Free Languages (see next slide)…5
Overview of Section 4.1 (cont.)DidblL(i Σ) tftDecidable Languages(in Σ0): to foster later appreciation of undecidable languageslanguagesContext-Free LanguagesA: Does a given CFG generate a given ACFG: Does a given CFG generate a given string?ECFG: Is the language of a given CFG empty?Every CFL is decidable by a Turing machine.6
Overview of Section 4.1DidblL(i Σ) tftDecidable Languages (in Σ0): to foster later appreciation of undecidable languageslanguagesRegular LanguagesA:Acceptance problem for DFAsADFA:Acceptance problem for DFAsAcceptance problem for NFAsAcceptance problem for Regular ExpressionsEmptiness testing for DFAs2 DFAs recognizing the same language7
Decidable Problems for Regular Languages: DFAsAcceptance problem for DFAsAcceptance problem for DFAsLanguage includes encodings of all DFAs and strings }stringgiven aacceptsDFA that ais|,{ADFAwBwB><=they accept.

  • Left Quote Icon

    Student Picture

  • Left Quote Icon

    Student Picture

  • Left Quote Icon

    Student Picture