Chomsky hierarchy - Nobo Komagata

Power-Economy Tradeoff• Complex mechanisms are more powerfulthan simple ones (can represent morevariety of problems/languages).• Simple mechanisms are less expensivethan complex ones (can operate simplyand faster in general).• An appropriate level would be the leastpowerful one for the requirement.CSC460 C1 7Downgrading TMs• Imposing a specific tape organization– As a stack– As a queue• Limiting the tape size– Set some finite fixed bound– Limit the growth (e.g., linearly with the input)• Limiting the head mechanism– Read-only– One-way scanning onlyCSC460 C1 alternative models 8TM-Equivalent ModelsReviewIdentifying Alternative ModelsTM variationsURMComputers ?TM λ-calculusRecursive Functions• Problem ~ set– Most commonly as a language (set of strings)– Recognized by a TM• Whatever mechanism that can recognize(or generate) a string can be examined forthe equivalence with a TM (or somedowngraded forms).CSC460 C1 Other models? 9CSC460 C1 10Grammar• Grammar [unrestricted grammar, semi-Thue system]– Rewrite rules of the form: α → βwhere α, β are strings of symbols and |α| > 0• S → NP 1 V NP 2• NP 1 V NP 2 → NP 2 is Ved by NP 1• NP 1 → liisa | tiina• NP 2 → mikko | seppo• V → kisses | kicks• Ved → kissed | kickedCSC460 C1 11TM vs. grammar?Using a Grammar• Generating a string• Accepting a string (parsing)• S → NP 1 V NP 2• NP 1 V NP 2 → NP 2 is Ved by NP 1• NP 1 → liisa | tiina• NP 2 → mikko | seppo• V → kisses | kicks• Ved → kissed | kicked• Terminal symbols [lower case]: Cannot be rewritten• Nonterminal symbols [UPPER CASE]: Must be rewrittenCSC460 C1 12