Turing Decidable and Recognizable Languages

TMs and Infinite Loops

• Question: If a TM is given a string, what can happen to the computation?


• Answer: The machine can either
1. Accept the string (ie enter the Accept state and halt the computation)


2. Reject the string (ie enter the Reject state and halt the computation)


3. Enter an Infinite Loop (ie the computation never ends)

Looping on a String

• We say that a machine loops on a string if the string puts the machine into an infinite loop


• When a machine loops, it never enters the Accept state or the Reject state


• When a machine loops, it transitions forever among states that are neither the Accept state nor the Reject state

Looping and Defining Languages

• The possibility of looping affects how we define the relationship between a machine and the language it defines
• Notice that this is not an issue with DFA, NFA, PDA: they ALWAYS halt!


• TM M recognizes language L iff
  1. the strings L put M into the Accept state
  2. the strings NOT IN L
  • EITHER put M into the Reject state
  • OR cause M to loop
  
  
• TM M decides language L iff
  1. the strings L put M into the Accept state
  2. the strings NOT IN L put M into the Reject state

Recognizers and Deciders

• A recognizer of a language is a machine that recognizes that language
• A decider of a language is a machine that decides that language


• Both types of machine halt in the Accept state on strings that are in the language
• A Decider also halts if the string is not in the language
• A Recogizer MAY or MAY NOT halt on strings that are not in the language


• On all input:
• A Decider MUST halt (in Accept or Reject state)
• A Recogizer MAY or MAY NOT halt on some strings (Q: Which ones?)


• Is every Recognizer a Decider, or is every Decider a Recognizer?


• Finish this statement: TM M decides language L if M recognizes L and it also ...

Examples

• Think about the Turing machines for these languages:
• aⁱb^jc^k


• w#w


• Are they deciders or recognizers?

Example Recognizers

• Examples of recognizers? Hmmm, let's think about it ...


• If machine M ______ L, then M ______ L


• Can we find a machine M that recognizes L but that does not decide L?
• It's easy
• How can we modify the previous examples.


• More interesting question: Can we find a L that has a recognizer but not a decider?
• Let's think about language classes before answering this question.

Definition: Turing Decidable Language

• A language is Turing-decidable (or decidable) if some Turing machine decides it


• Aka Recursive Language

Review: Turing Recognizable Language

• A language is Turing-recognizable if some Turing machine recognizes it


• Aka Recursively Enumerable Language

Relationship among Language Classes

• How do these language classes fit togther
• Regular
• Context Free
• Turing Decidable
• Turing Recognizable


• (We haven't yet proved the relation between CF and TD langs)


• What are some examples in each class (and not in smaller class)?
• First three are easy: Regular, CF, decidable
• Are there languages that are recognizable but NOT decidable?
• Are the languages that are NOT recognizable?