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Chapter5Remainder

# Chapter5Remainder - 91.304 Foundations of(Theoretical C(Th...

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91.304 Foundations of (Theoretical) Computer Science Chapter 5 Lecture Notes (Remainder) David Martin (with modifications by Karen Daniels) [email protected] With thanks to Giam Pecelli This work is licensed under the Creative Commons Attribution-ShareAlike License. To view a copy of this license, visit http://creativecommons.org/licenses/by- 1 sa/2.0/ or send a letter to Creative Commons, 559 Nathan Abbott Way, Stanford, California 94305, USA.

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Computation Histories ecall the notion of onfiguration (state ± Recall the notion of configuration : (state, head_position, tape_contents). ± In a deterministic TM every transition takes us from one configuration to the next. ± The idea of " computation history " is based n sequences of configurations on sequences of configurations. 2
Computation Histories efinition 5 5 Let M be a TM and w a string: Definition 5.5 : Let M be a TM and w a string: - An accepting computation history for M on w is a sequence of configurations C , C , …, C , 1 2 k where C 1 is the start configuration of M on w, C k is an accepting configuration of M, and each C i ` C i+1 ccording to the rules of M according to the rules of M. - A rejecting computation history for M on w is defined similarly, except that C k is a rejecting configuration for M. 3

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Computation Histories efinition 5 6 A near bounded automaton Definition 5.6 : A linear bounded automaton is a standard TM whose tape head is not allowed to move beyond the tape squares containing the input. Note : If the tape alphabet is larger than the input lphabet the available memory can be increased alphabet, the available memory can be increased by a constant factor - but the amount of memory remains a linear function of the length of the input. 4
Linear Bounded Automata ± Example (using left and right end markers) : (board work) ± They are powerful: the following are LBAs: ² Decider for A DFA ² Decider for A CFG ² Decider for E DFA ² Decider for E CFG 5 ± Every CFL can be decided by an LBA.

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Computation Histories hy bother LBAs can be shown sufficient for Why bother ? LBAs can be shown sufficient for the recognition of almost all "realistic" classes of languages, in particular CFLs: - ughly, your C program will compile (actually, roughly, your C program will compile (actually, parse, but who's counting?) in a ("small") constant multiple of the number of bytes necessary to store it in memory. Definition : A LBA = { h M, w i | M is an LBA that accepts w} Question : is A LBA decidable? The "general" problem (A TM ) is not… 6
Computation Histories emma 5 8 Let M be an LBA with q states and g Lemma 5.8 : Let M be an LBA with q states and g symbols in the tape alphabet. Then there are exactly qng n distinct configurations of M for a tape of length n. Proof . A configuration is uniquely determined y the state M is in (q possibilities) 1. by the state M is in (q possibilities), 2. by the head position (n possibilities) and y the tape contents (g n ossibilities: g 3. by the tape contents (g possibilities: g possibilities for each of the n tape positions).

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Chapter5Remainder - 91.304 Foundations of(Theoretical C(Th...

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