Showing papers by "Harry Buhrman published in 1995"

Optimal resiliency against mobile faults

Abstract: We consider a model where malicious agents can corrupt hosts and move around in a network of processors. We consider a family of mobile fault models MF(t/n-1,/spl rho/). In MF(t/n-1,/spl rho/) there are a total of n processors, the maximum number of mobile faults is t, and their roaming pace is /spl rho/ (for example, /spl rho/=3 means that it takes an agent at least 3 rounds to "hop" to the next host). We study in these models the classical testbed problem for fault tolerant distributed computing: Byzantine agreement. It has been shown that if /spl rho/=1, then agreement cannot be reached in the presence of even one fault, unless one of the processors remains uncorrupted for a certain amount of time. Subject to this proviso, we present a protocol for MF(/sup 1///sub 3/,1), which is optimal. The running time of the protocol is O(n) rounds, also optimal for these models. >

Long-lived renaming made fast

Abstract: In the long-lived renaming problem — a generalization of the classical one-time renaming problem — n processors with unique names ranging over a source name space {O, . . . . S – 1} repeatedly acquire and release unique names from a (smaller) destination name space {O, . . . . D – 1}. It is assumed that at most k out of n processors concurrently request or hold names. An efficient renaming protocol provides a useful front-end for protocols whose time complexity depends on the size of the name space containing the participating processes. We consider long-lived renaming in the context of asynchronous, shared-memory multiprocessing systems that provide only read and write operations. A renaming protocol is fast iff the time complexity of acquiring and releasing a name is polynomial in k and independent of n and S. We present a wait-free, read/write protocol for long-lived renaming that achieves a destination name space of size O(k2) with time complexity O(k3). If S is polynomial in k, we further improve the timecomplexity to O(k log k). This shows, for the first time, that fast, read/write protocols for long-lived renaming exist. Part of our wait-free solution uses mutual exclusion tournament trees, where we apply hashing based on polynomials over finite fields to avoid blocking. Thk technique may be of general int crest.

Using autoreducibility to separate complexity classes

Abstract: A language is autoreducible if it can be reduced to itself by a Turing machine that does not ask its own input to the oracle. We use autoreducibility to separate exponential space from doubly exponential space by showing that all Turing complete sets for exponential space are autoreducible but there exists some Turing complete set for doubly exponential space that is not. We immediately also get a separation of logarithmic space from polynomial space. Although we already know how to separate these classes using diagonalization, our proofs separate classes solely by showing they have different structural properties, thus applying Post's Program (E. Pos, 1944) to complexity theory. We feel such techniques may prove unknown separations in the future. In particular if we could settle the question as to whether all complete sets for doubly exponential time were autoreducible we would separate polynomial time from either logarithmic space or polynomial space. We also show several other theorems about autoreducibility.

An excursion to the Kolmogorov random strings

Abstract: We study the set of resource bounded Kolmogorov random strin,gs: Rt = {x 1 Kt(x) _> \XI} f or t a tame constructible function such that t(n) 2 2”’ and t(n) E 2”O‘l). We show that the class of sets that TUTing reduce to Rt has measure 0 in EXP with respect to the resoul-ce-bounded measure introduced by [I 71. From this we conclude that Rt is not Turing -complete for EXP. This contrasts the resource unbounded setting. There R is Turing -complete for CO-RE. we show that thti class of sets to which Rt bounded truthtable reduces; has p2 -measure 0 (therefore, measure 0 in EXP). Th,is answers an open question of Lutz, giving a natural example of a language that is not weaklycomplete for EXP and that reduces to a measure 0 class in EXP. It follows that the sets that are <:tt -hard for EXP have p2-measure 0.

SPARSE Reduces Conjunctively to TALLY

Abstract: Polynomials over finite fields are used to show that any sparse set can conjunctively reduce to a tally set. This leads to new results and to simple proofs of known results about various classes that lie between P and P/poly.

On the Sparse Set Conjecture for Sets with Low Density

Abstract: We study the sparse set conjecture for sets with low density. The sparse set conjecture states that P=NP if and only if there exists a sparse Turing hard set for NP. In this paper we study a weaker variant of the conjecture. We are interested in the consequences of NP having Turing hard sets of density f(n), for (unbounded) functions f(n), that are sub-polynomial, for example log(n). We establish a connection between Turing hard sets for NP with density f(n) and bounded nondeterminism: We prove that if NP has a Turing hard set of density f(n), then satisfiability is computable in polynomial time with O(log(n)*f(n c )) many nondeterministic bits for some constant c. As a consequence of the proof technique we obtain absolute results about the density of Turing hard sets for EXP. We show that no Turing hard set for EXP can have sub-polynomial density. On the other hand we show that these results are optimal w.r.t. relativizing computations. For unbounded functions f(n), there exists an oracle relative to which NP has a f(n) dense Turing hard tally set but still P≠NP.

Learnability of Kolmogorov-easy circuit expressions via queries

Abstract: Circuit expressions were introduced to provide a natural link between Computational Learning and certain aspects of Structural Complexity. Upper and lower bounds on the learnability of circuit expressions are known. We study here the case in which the circuit expressions are of low (time-bounded) Kolmogorov complexity. We show that these are polynomial-time learnable from membership queries in the presence of an NP oracle. We also exactly characterize the sets that have such circuit expressions, and precisely identify the subclass whose circuit expressions can be learned from membership queries alone. The extension of the results to various Kolmogorov complexity bounds is discussed.

Distinguishing Complexity and Symmetry of Information

Abstract: We describe how to use polynomial-time Kolmogorov distinguishing complexity to give an approximate measure of the size of sets. For any set S, we show that relative to S the polynomial time distinguishing complexity of every element of length n of S is bounded by 2 log jjS =n jj + O(log n). This lemma enables us to give a characterization of sparse sets using distinguishing complexity. We use this new lemma as a catalyst to study symmetry of information for polynomial-time distinguishing complexity. Longpr e and Mocas and Longpr e and Watanabe showed that if certain one-way functions exist then symmetry of information fails for the standard polynomial-time Kolmogorov complexity. We try to recover symmetry of information by studying Kolmogorov distinguishing complexity using our new approximating measure idea to avoid the problems with one-way functions and indexing of strings in (small) sets. We show problems with even formalizing the symmetry of information question for polynomial-time deterministic distinguishing complexity. Nondeterministic distinguishing complexity gives us more hope but we show that symmetry of information still seems unlikely due to the apparent inability of nondeterminism to approximately count.