Samenvatting Hoofdstuk 11: A Hierarchy of Formal Languages and Automata
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Course
Automaten en Berekenbaarheid
Institution
Vrije Universiteit Brussel (VUB)
Book
An Introduction to Formal Languages and Automata
Dit is de samenvatting van het elfde hoofdstuk van het vak Automaten en Berekenbaarheid. In deze samenvatting werd alle relevante informatie uit de slides alsook uit eigen notities opgenomen.
Eindresultaat: 16/20
Hoofdstuk 11: A hierarchy of formal
languages and automata
Inhoud:
• Relation between Turing machines and languages
• Is every language determined by some Turing machine?
• Hierarchy of languages and grammars
1 Recursive(ly enumerable) languages
Bij beslisbare taal → stopt in niet finale state indien woord niet geaccepteerd wordt
Semi-beslisbaar → loopt voor eeuwig indien woord niet geaccepteerd wordt.
2 Enumerating languages
• Can you enumerate all elements of recursive language?
• What about a recursively enumerable language?
2.1 Are all languages recursively enumerable?
Stelling:
For any non-empty alphabet, there exists a language that is not recursively enumerable.
Bewijs:
In hoofdstuk 10 aangetoond dat er eindig aantal TM zijn over een gegeven alfabet. Er zijn echter
oneindig veel talen over een gegeven niet leeg alfabet. Verder accepteert een TM maar 1 taal
waardoor er dus voor elk niet leeg alfabet minstens een taal zal bestaan die we niet met een TM
kunnen accepteren.
• This means that there exist languages for which we cannot even write a program that
accepts them.
• This means that there are computational problems we cannot solve with a computer (more
on that next chapter)
1
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