lect08 - 6.841 Advanced Complexity Theory Lecture 08...

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6.841 Advanced Complexity Theory Mar 02, 2009 Lecture 08 Lecturer: Madhu Sudan Scribe: Vartika Singh 1 Alternation It is a notion which tries to capture languages in NP and CoNP in a unifying way. It defines the time-vs-space relation and hence Alternating Turing Ma- chines. It is a means for finding lower bounds for certain languages for which there is no obvious short certificate for membership and hence can not be characterized using non determinism alone. 1.1 Alternating Turing Machines, ATM Definition 1 Let M be an alternating TM. For a function T : N -→ N , we say that M is an T ( n ) -time ATM if for every input x (0 , 1) * and for every possible sequence of transition function choices, M will halt after at most T ( | x | ) steps. Alternating Turing Machines (ATM), are generalizations of TM with the addi- tional internal states of or . The ATM can be thought of as a tree, starting from the root node. If the state reached is a deterministic one, then there is only one move to make. If it is either of or , then either of (0 , 1) moves could be made. Each node can be thought of as a configuration of M on input x , and there is an edge from configuration C or C 1 , if latter can be obtained from former in one step. The nodes up a computation path are labeled by repeatedly applying the following rules, till they can not be applied anymore: The configuration C accept where the machine is in q accept is labeled “AC- CEPT”. If a configuration C is in a state labeled exists and one of the configu- rations C 1 reachable from it in one step is labeled “ACCEPT” then we label C “ACCEPT”. If a configuration C is in a state labeled and both the configurations C 1 and C 2 reachable from it one step is labeled “ACCEPT” then we label C “ACCEPT”. If a configuration C is in a deterministic state then just proceed up as in a unary computation. 08-1
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We say that M accepts x if at the end of this process the starting configuration C start is labeled ”ACCEPT”. The language accepted by M is the set of all x ’s such that M accepts x . Three resources are important in defining the complexity of a ATM. TIME : ATIME ( t ( n )) = { L | L is decided by some t(n)-time ATM } SPACE : ASPACE ( s ( n )) = { L | L is decided by some s(n)-space ATM } Alternations : Most important resource that will be used is maximum number of alternations done during the course of computation. (Note: Flip from to state is not a valid alternation. Similarly for .) 1.2 NDTMs and CoNDTMS NDTM can viewed as a TM with extra existential state. The machine accepts if one or more of its branches ends in an accept state.
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