218.55.22.50 at 08:28, 19 December 2013

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	The ~~writer~~ is ~~called Wilber Pegues~~. ~~What me~~ and ~~my family love psychics~~ ([http://~~black7~~.~~mireene~~.com/~~aqw~~/~~5741 click~~ the ~~next document]~~) is ~~performing ballet but I~~'~~ve been taking~~ on new ~~issues recently~~. ~~Office supervising~~ is ~~what she does~~ for a ~~living~~. ~~My spouse~~ and ~~I reside in Mississippi but now I~~'~~m contemplating other choices~~.
			In [[computer science]], a '''computation history''' is a sequence of steps taken by an [[abstract machine]] in the process of computing its result. Computation histories are frequently used in [[Mathematical proof\|proofs]] about the capabilities of certain machines, and particularly about the [[undecidable problem\|undecidability]] of various [[formal languages]].

			Formally, a computation history is a (normally [[Finite set\|finite]]) sequence of '''configurations''' of a formal [[automaton]]. Each configuration fully describes the status of the machine at a particular point. To be valid, certain conditions must hold:
			* the first configuration must be a valid initial configuration of the automaton and
			* each transition between adjacent configurations must be valid according to the transition rules of the automaton.
			In addition, to be '''complete''', a computation history must be finite and
			* the final configuration must be a valid terminal configuration of the automaton.
			The definitions of "valid initial configuration", "valid transition", and "valid terminal configuration" vary for different kinds of formal machines.

			A [[deterministic]] automaton has exactly one computation history for a given initial configuration, though the history may be infinite and therefore incomplete.

			== Finite State Machines ==
			For a [[finite state machine]] <math>M</math>, a configuration is simply
			the current state of the machine, together with the remaining input. The first configuration must be the initial state of <math>M</math> and the complete input. A transition from a configuration <math>(S,I)</math> to
			a configuration <math>(T,J)</math> is allowed if <math>I=aJ</math> for
			some input symbol <math>a</math> and if <math>M</math> has a transition from
			<math>S</math> to <math>T</math> on input <math>a</math>. The final
			configuration must have the empty string <math>\epsilon</math> as its remaining
			input; whether <math>M</math> has accepted or rejected the input depends
			on whether the final state is an accepting state. <ref name="MumfordJain2009">{{cite book\|author1=Christine L. Mumford\|author2=Lakhmi C. Jain\|title=Computational Intelligence: Collaboration, Fusion and Emergence\|url=http://books.google.com/books?id=eECU8WrC99wC&pg=PA337\|accessdate=25 March 2012\|year=2009\|publisher=Springer\|isbn=978-3-642-01798-8\|page=337}}</ref>

			== Turing Machines ==

			Computation histories are more commonly used in reference to [[Turing machines]]. The configuration of a single-tape Turing machine consists of the contents of the tape, the position of the read/write head on the tape, and the current state of the associated state machine; this is usually written

			<math>...0011010101q00110101010...</math>

			where <math>q</math> is the current state of the machine, represented in some
			way that's distinguishable from the tape language, and where <math>q</math> is
			positioned immediately before the position of the read/write head.

			Consider a Turing machine <math>M</math> on input <math>w</math>. The first
			configuration must be <math>q_0 w_0 w_1 ...</math>, where <math>q_0</math>
			is the initial state of the Turing machine. The machine's state in the final
			configuration must be either <math>q_a</math> (the accept state) or <math>q_r</math>
			(the reject state). A configuration <math>c_{i+1}</math> is a valid successor
			to configuration <math>c_i</math> if there's a transition from the state in
			<math>c_i</math> to the state in <math>c_{i+1}</math> which manipulates the
			tape and moves the read/write head in a way that produces the result in
			<math>c_{i+1}</math>.<ref name="Blass2010">{{cite book\|author=Andreas Blass\|title=Fields of Logic and Computation: Essays Dedicated to Yuri Gurevich on the Occasion of His 70th Birthday\|url=http://books.google.com/books?id=HqIRT51n74YC&pg=PA468\|accessdate=25 March 2012\|date=22 October 2010\|publisher=Springer\|isbn=978-3-642-15024-1\|page=468}}</ref>

			=== Decidability results ===

			Computation histories can be used to show that certain problems for
			[[pushdown automata]] are [[undecidable problem\|undecidable]]. This is because the language of
			non-accepting computation histories of a Turing machine <math>M</math>
			on input <math>w</math> is a [[context-free language]] recognizable by a
			non-deterministic pushdown automaton.

			We encode a Turing computation history <math>c_0,c_1,...,c_n</math> as the
			string <math>C_0 \# C^r_1 \# C_2 \# C^r_3 \# ... \# C_n</math>, where <math>C_i</math>
			is the encoding of configuration <math>c_i</math>, as discussed above, and where
			every other configuration is written in reverse. Before reading a particular
			configuration, the pushdown automaton makes a non-deterministic choice
			to either ignore the configuration or read it completely onto the stack.

			* If the pushdown automaton decides to ignore the configuration, it simply reads and discards it completely and makes the same choice for the next one.
			* If it decides to process the configuration, it pushes it completely onto the stack, then verifies that the next configuration is a valid successor according to the rules of <math>M</math>. Since successive configurations are always written in opposite orders, this can be done by, for each tape symbol in the new configuration, popping off a symbol from the stack and checking if they're the same. Where they disagree, it must be accountable for by a valid transition of <math>M</math>. If, at any point, the automaton decides that the transition is invalid, it immediately enters a special accept state which ignores the rest of the input and accepts at the end.

			In addition, the automaton verifies that the first configuration is the correct
			initial configuration (if not, it accepts) and that the state of the final
			configuration of the history is the accept state (if not, it accepts). Since
			a non-deterministic automaton accepts if there's any valid way for it to accept,
			the automaton described here will discover if the history is not a valid
			accepting history and will accept if so, and reject if not. <ref name="Blum1998">{{cite book\|author=Lenore Blum\|title=Complexity and real computation\|url=http://books.google.com/books?id=zxtrVqUP-AwC&pg=PA31\|accessdate=25 March 2012\|year=1998\|publisher=Springer\|isbn=978-0-387-98281-6\|page=31}}</ref>

			This same trick cannot be used to recognize ''accepting'' computation histories
			with an NPDA, since non-determinism could be used to skip past a test that would
			otherwise fail. A linear-bounded Turing machine is sufficient to recognize
			accepting computation histories.

			This result allows us to prove that <math>ALL_{PDA}</math>, the language
			of pushdown automata which accept all input, is undecidable. Suppose
			we have a decider for it, <math>D</math>. For any Turing machine
			<math>M</math> and input <math>w</math>, we can form the pushdown automaton
			<math>P</math> which accepts non-accepting computation histories for that
			machine. <math>D(P)</math> will accept if and only if there are no
			accepting computation histories for <math>M</math> on <math>w</math>; this
			would allow us to decide <math>A_{TM}</math>, which we know to be undecidable.
			==References==
			{{reflist}}

			[[Category:Theory of computation]]

en>قلی زادگان at 10:46, 24 March 2012

2012-03-24T10:46:57Z

New page

The writer is called Wilber Pegues. What me and my family love psychics ([http://black7.mireene.com/aqw/5741 click the next document]) is performing ballet but I've been taking on new issues recently. Office supervising is what she does for a living. My spouse and I reside in Mississippi but now I'm contemplating other choices.

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218.55.22.50 at 08:28, 19 December 2013

en>قلی زادگان at 10:46, 24 March 2012