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Index

accept 15adjacent 5alphabet 10AM 80and-function 44Arthur 79Arthur-Merlin game 52Arthur-Merlin hierarchy 79automorphism 7automorphism problem 7

binomial distribution 70Boolean circuit 92Boolean function 92bounded error 68bounded truth-table reduction 116BP-operator 74BPP 68

Cayley diagram 18certificate 1Chebyshev's inequality 38, 81, 127check 15Chernoff bound 70circuit 92circuit family 93code 1collapse 55, 84communication channel 60complete 23complexity class 13composition 9computation model 12computation problem 11conditional instruction 19configuration 95conjunctive reduction 116, 125construction problem 25Cook's hypothesis 55coset 9, 31

counting class 4counting problem 11, 12

decision problem 11degree 5, 20deterministic algorithm 15directed graph 17, 18disjunctive normal form 105disjunctive reduction 21,44,50, 110

edge 5efficient 14encoding 13

feasible 15fixpoint 10, 46

game 52i-complete 23, 88gate 92generating function 70GI-complete 19graph 5graph automorphism problem 3, 7graph isomorphism problem 1, 5Graph Reconstruction Conjecture 146graph union 8group isomorphism 18guess 15

halting problem 94hash function 127high 87high hierarchy 87highness 86

instance 12instantaneous description 95interactive proof system 58intermediate 2, 82, 86invariant 1

INDEX

IP 59isomorphic 1, 5isomorphic graphs 11isomorphism 5

isomorphism problem 5

label 7labeling algori thm 1language 10length 10lexicographical order 28, 107logarithmic cost 13low 84,87low hierarchy 87lowness 84LWPP 144

MA 83MA' 103majority reducible 75majority vote 69many-one reducibility 17Merlin 79multiple edges 18

node 5non-recursive 94non-uniform 91nondeterministic algorithm 15not-function 122NP 14NP-complete 23NP-equivalent 44

one-letter alphabet 85operator 53, 74or-function 44oracle 20, 56, 120orbit 9output gate 92

P 14pairwise independence 37parity function 36parity problem 119parsimonious transformation 121partial decision problem 39permutation 1, 5, 7

permutation group 8PH 55planar graph 2, 14player 51polynomial hierarchy 52, 54polynomial size circuits 4, 91, 93polynomial time 13prefix search 105primes 3, 24, 45, 69, 118, 123private coins 79probabilistic algorithm 66probabilistic construction 77probability amplification 60, 71probability distribution 66program checker 64promise problem 39protocol 60Prover 53, 60PSC 91PSPACE 56public coins 79publishable proof 82

quantifier 53quantifier simulation 78

RAM 95random access machine 12, 95random coin tosses 73random guess 59randomized reduction 35random move 52random number generator 35random permutation 60random reduction 35random variable 35, 59reducibility 16reflexive 23reject 15rigid 7round 59RP 69running time 13

Sat 24,25satisfiabili ty 26search circuit 99search problem 11

159

160

self-computable solutions 25self-inverse 43, 122self-reducible 25, 116semi-group isomorphism 19SemiGroupIso 19solution 11sparse 4, 105SPARSE 105sparse set 91stabilizer 10strong reducibility 87subgraph isomorphism 24subgroup 9subroutine 19

Taut 24tournament graph 3, 118, 122, 123transformation 17transitive 23transversal 9tree 6tree isomorphism 6truth table 44Turing machine 12,94Turing reducibility 17

undirected 5uniform 91uniform distribution 59, 74unique solution 33, 39universal hashing 36

valid computation 15variance 37Verifier 53, 60vertex 5

winning predicate 53winning strategy 51, 53witness 98, 99

XOR 36,40

zero-knowledge 65

INDEX