Michael Sipser

A complexity theoretic approach to randomness

We study a time bounded variant of Kolmogorov complexity. This notion, together with universal hashing, can be used to show that problems solvable probabilistically in polynomial time are all within the second level of the polynomial time hierarchy. We also discuss applications to the theory of probabilistic constructions.

Private coins versus public coins in interactive proof systems

An interactive proof system is a method by which one party of unlimited resources, called the prover, can convince a party of limited resources, call the verifier, of the truth of a proposition. The verifier may toss coins, ask repeated questions of the prover, and run efficient tests upon the provers responses before deciding whether to be convinced. This extends the familiar proof system implicit in the notion of NP in that there the verifier may not toss coins or speak, but only listen and verify. Interactive proof systems may not yield proof in the strict mathematical sense: the proofs are probabilistic with an exponentially small, though non-zero chance of error. We consider two notions of interactive proof system. One, defined by Goldwasser, Micali, and Rackoff [GMR] permits the verifier a coin that can be tossed in private, i.e., a secret source of randomness. The Permission to copy without ice all or part of this material is granted provided that the copies are not made or distributed for direct commercial advantage, the ACM copyright notice and the title of the publication and its date appear, and notice is given that copying is by permission of the Association for Computing Machinery. To copy otherwise, or to republish, requires a fee and/or specific permission.

Nondeterminism and the size of two way finite automata

An important goal of the theory of computation is the classification of languages according to computational difficulty. Classes such as P, NP, and LOGSPACE provide a natural framework for this, though it is a fundamental open problem to demonstrate languages distinguishing them. The complete languages of Cook, Karp, and others [1-7] are candidates for such languages in the sense that, if the classes are in fact different, these languages witness the difference. We consider two questions on regular languages resembling these open problems. One of these questions concerns 2-way non-deterministic (2n) and 2-way deterministic (2d) finite automata:

GO Is Polynomial-Space Hard

It is shown that, given an arbitrary GO position on an n × n board, the problem of determining the winner is Pspace hard. New techniques are exploited to overcome the difficulties arising from the planar nature of board games. In particular, it is proved that GO is Pspace hard by reducing a Pspace-complete set, TQBF, to a game called generalized geography, then to a planar version of that game, and finally to GO.

Communication complexity

In this paper we prove several results concerning this complexity measure. First we establish (in a non-constructive manner) that there exist languages which cannot be recognized with less than <italic>n</italic> communication (obviously, communication <italic>n</italic> is always enough for recognizing any language). In fact, we show that for any function<italic>f(n)-&-lt; n,</italic> there are languages recognizable with communication<italic>f(n)</italic> but not with communication<italic>f (n)</italic>-&-minus;1. In other words, this complexity measure possesses a very dense hierarchy or complexity classes, as miniscule increments in communication add to the languages that can be recognized.

Halting space-bounded computations☆

The main result of this paper is that for any deterministic Turing machine which runs in space S (n) and which possibly rejects by looping, there is an equivalent Turing machine which runs in the same amount of space and always halts. Hopcroft and Ullman [2] have previously shown this for S (n) > log n but conjecture that it is false for small S. Their proof for S (n) > log n counts steps on a separate track of the tape and shuts the machine off if it has run for too long. Hartmanis and Berman [1] prove this for S (n) < log n on unary languages with a more complicated counting technique. In addition to space bounded Turing machines, our technique applies to other deterministic computation models. Using it, we can enforce termination on twoway finite automata without exponentially increasing the number of states and on two-way multihead finite automata without increasing the number of heads.

Interactive proof systems: Provers that never fail and random selection

An interactive proof system with Perfect Completeness (resp. Perfect Soundness) for a language L is an interactive proof (for L) in which for every x ∈ L (resp. x ∉ L) the verifier always accepts (resp. always rejects). Zachos and Fuerer showed that any language having a bounded interactive proof has one with perfect completeness. We extend their result and show that any language having a (possibly unbounded) interactive proof system has one with perfect completeness. On the other hand, only languages in NP have interactive proofs with perfect soundness. We present two proofs of the main result. One proof extends Lautemanns proof that BPP is in the polynomial-time hierarchy. The other proof, uses a new protocol for proving approximately lower bounds and random selection. The problem of random selection consists of a verifier selecting at random, with uniform probability distribution, an element from an arbitrary set held by the prover. Previous protocols known for approximate lower bound do not solve the random selection problem. Interestingly, random selection can be implemented by an unbounded Arthur-Merlin game but can not be implemented by a two-iteration game.

Lower bounds on the size of sweeping automata

Establishing good lower bounds on the complexity of languages is an important area of current research in the theory of computation. However, despite much effort, fundamental questions such as P &equil;? NP and L &equil;? NL remain open. To resolve these questions it may be necessary to develop a deep combinatorial understanding of polynomial time or log space computations, possibly a formidable task.n One avenue for approaching these problems is to study weaker models of computation for which the analogous problems may be easier to settle, perhaps yielding insight into the original problems. Sakoda and Sipser [3] raise the following question about finite automata:n Is there a polynomial p, such that every n-state 2nfa (two-way nondeterministic finite automaton) has an equivalent p(n)-state 2dfa? n They conjecture a negative answer to this. In this paper we take a step toward proving this conjecture by showing that 2nfa are exponentially more succinct than 2dfa of a certain restricted form.

GO is pspace hard

A great deal of effort has been spent in the search for optimal and computationally feasible game strategies. In some cases (e.g. Bridge-it, Nim), such strategies have been found, vhile in others the search has been unsuccessful. Recently, it has become possible to provide compelling evidence that such strategies may not always exist. Even and Tarjan [1] and Schaefer [2] have shown that determining which player has a winning strategy in certain combinatorial games is a polynomial space complete problem [3]. (See also [4;5].)