information that the verier need only inspect in a few places in order
to become convinced. The following denition makes this idea more
precise. A language A has a probabilistically checkable proof if there
exists an oracle BPP-machine M such that: For al
A, the interactive proof system (P,V) rejectsx with probability greater
than 1/2+. By substituting random choices for existential choices in the
proof that ATIME(t) DSPACE(t) (Theorem 5.11), it is straightforward to
show that IP PSPACE. It was originally
vetheseopenquestionsand replace conjecture with mathematical
wiak,Sung-ilPae,Leonard Pitt, Michael R
problem lies in NPco-NP. The complexity class IP comprises the
languages A for which there exists a verier V and a positive such that
There exists a prover P such that for all x in A, the interactive proof
system ( P,V) accepts x with probability greater
254;andTheComplexityofFiniteFunctions,by R.B. Boppana and M.
Sipser, pp. 757804, which covers circuit complexity. A collection of
articles edited by Hartmanis  includes an overview of complexity
MJ.Comput.,6(4):675695. Goldwasser, S., Micali, S., and Rackoff, C.
1989. The knowledge complexity of interactive proof systems. SIAM J.
Comput., 18(1):186208. Hartmanis, J., Ed. 1989. C
1977. On relating time and space to size and depth. SIAM J. Comput.,
6(4):733744. Bovet, D.P. and Crescenzi, P. 1994. Introduction to the
Theory of Complexity. Prentice Hall International Ltd; Hertfordshire, U.K.
understand that the prover P does not see the coins that V ips in
making its random choices; P sees only the graphs Gand H that V sends
as messages.) V accepts the interaction with P as proof that G and H
are non-isomorphic if P is able to pick the correc
ATuringmachineMisamodelofcomputationwithareadonlyinputtapeandmultiple worktapes.Ateachstep, M
its current state and the symbols in those cells, M changes state, writes
new symbols on the wor
[1990,1995],BovetandCrescenzi , Du and Ko ,
Hemaspaandra and Ogihara , and Papadimitriou . Wagner
and Wechsung  is an exhaustive survey of complexity theory that
FIGURE5.5 Probabilistic complexity classes.
ycomputer scientists consider BPP to be the class of practically feasible
computational problems. Next, we dene a cla
A, and for every language B, machine MB accepts x with probability
strictly less than 1/2.
Intuitively, the oracle language Bx represents a proof of membership of
x in A. Notice that Bx can be
also sponsored by the IEEE, is entirely devoted to complexity theory.
Research articles on complexity theory regularly appear in the following
journals, among others: Chicago Journal on Theoretical Computer Science, Computational Complexity, Information a
A. If L RP,thenforeverypolynomialq(n),thereexistsanRP-machine M
suchthat pM(x) > 11/2q(n) for every x in L.
It is important to note just how minuscule the probability of error is
(provided that the coin ips are
SIAM J. Comput., 31(5):15011526.
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Levin, L. 1973. Universal search problems. Pr
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closed under complementation. SIAM J. Comput., 17(5):935938.
Impagliazzo, R. and Wigderson, A. 1997. P = BPP if E requires
exponential circuits: Derandomizing the XOR lemma. Proc. 29th Annu
authors of this chapter. Many of these pages have downloadable
Volume A: Algorithms and Complexity. Elsevier Science, Amsterdam,
and M.I.T. Press, Cambridge, MA. Wagner, K. and Wechsung, G. 1986.
Computational Complexity. D. Reidel, Dordrecht, The Netherlands.
Wrathall, C. 1976. Complete sets and the polynomial-time
GOTO language with the additional property that it halts on all inputs.
Such programs will be called halting
6.1 Introduction 6.2 Computability and a Universal Algorithm Some
Computational Problems AUniversalAlgorithm 6.3 Undecidability
Diagonalization and Self-Reference ReductionsandMore
UndecidableProblems 6.4 Formal Languages and Grammars
there is a GOTO program that halts and outputs 1 on all
FIGURE6.2 An exampleoftiling.
thattheinputisastringof symbols from a nite input alphabet (which is a
subset of the tape alphabet), which is stored on the tape
problem 1 (if it exists) can be used to
More precisely, a Wang tile, after Hao Wang, who wrote the rst
research paper on it.
accomplishesthe simple task of doubling the number of 1s (Figure 6.1).
More precisely, on the input containing k 1s, the
knownexamplesofalgorithmicquestions.However,until the 1930s the
notion of algorithms was used informally (or rigorously but in a limited
context). It was a major triumph of logicians and mathematicians of this
century to offer a rigorous denition of this