Resting potential: Difference between revisions

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-In [[automata theory]], an '''alternating finite automaton''' (AFA) is a [[nondeterministic finite automaton]] whose transitions are divided into ''[[existential quantification|existential]]'' and ''[[universal quantification|universal]]'' transitions. For example, let ''A'' be an alternating [[automaton]].
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-* For an existential transition <math>(q, a, q_1 \vee q_2)</math>, ''A'' nondeterministically chooses to switch the state to either <math>q_1</math> or <math>q_2</math>, reading ''a''. Thus, behaving like a regular [[nondeterministic finite automaton]].
-* For a universal transition <math>(q, a, q_1 \wedge q_2)</math>, ''A'' moves to <math>q_1</math> '''and''' <math>q_2</math>, reading ''a'', simulating the behavior of a parallel machine.
-Note that due to the universal quantification a run is represented by a run ''tree''. ''A'' accepts a word ''w'', if there ''exists'' a run tree on ''w'' such that ''every'' path ends in an accepting state.
-A basic theorem tells that any AFA is equivalent to an [[non-deterministic finite automaton]] (NFA) by performing a similar kind of powerset construction as it is used for the transformation of an NFA to a [[deterministic finite automaton]] (DFA). This construction converts an AFA with ''k'' states to an NFA with up to <math>2^k</math> states.
-An alternative model which is frequently used is the one where Boolean combinations are represented as ''clauses''. For instance, one could assume the combinations to be in [[Disjunctive normal form|Disjunctive Normal Form]] so that <math>\{\{q_1\},\{q_2,q_3\}\}</math> would represent <math>q_1 \vee (q_2 \wedge q_3)</math>. The state '''tt''' (''true'') is represented by <math>\{\{\}\}</math> in this case and '''ff''' (''false'') by <math>\varnothing</math>.
-This clause representation is usually more efficient.
-==Formal Definition==
-An alternating finite automaton (AFA) is a [[n-tuple|6-tuple]],
-<math>(S(\exists), S(\forall), \Sigma, \delta, P_0, F)</math>, where
-*<math>S(\exists)</math> is a finite set of existential states. Also commonly represented as <math>S(\vee)</math>.
-*<math>S(\forall)</math> is a finite set of universal states. Also commonly represented as <math>S(\wedge)</math>.
-*<math>\ \Sigma</math> is a finite set of input symbols.
-*<math>\ \delta</math> is a set of transition [[function (mathematics)|functions]] to next state <math>(S(\exists) \cup S(\forall)) \times (\Sigma \cup \{ \varepsilon \} ) \to 2^{S(\exists) \cup S(\forall)}</math>.
-*<math>\ P_0</math> is the initial (start) state, such that <math>P_0 \in S(\exists) \cup S(\forall)</math>.
-*<math>\ F</math> is a set of accepting (final) states such that <math>F \subseteq S(\exists) \cup S(\forall)</math>.
-==References==
-* {{cite book | title=Theories of Computability | first=Nicholas | last=Pippenger | authorlink=Nick Pippenger | publisher=[[Cambridge University Press]] | year=1997 | isbn=0-521-55380-6 }}
-{{DEFAULTSORT:Alternating Finite Automaton}}
-[[Category:Automata theory]]
-{{comp-sci-theory-stub}}

Resting potential: Difference between revisions

Latest revision as of 20:45, 25 October 2014

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