We show that non-interactive statistically-secret bit commitment cannot be constructed from arbitrary black-box one-to-one trapdoor functions and thus from general public-key cryptosystems. Reducing the problems of non-interactive crypto-computing, rerandomizable encryption, and non-interactive statistically-sender-private oblivious transfer and low-communication private information retrieval to such commitment schemes, it follows that these primitives are neither constructible from one-to-one trapdoor functions and public-key encryption in general. Furthermore, our...

Given a set X of sequences over a nite alphabet, we investigate the following three quantities. (i) The feasible predictability of X is the highest success ratio that a polynomial-time randomized predictor can achieve on all sequences in X.

In the 1980's, Bennett introduced computational depth as a formal measure of the amount of computational history that is evident in an object's structure. In particular, Bennett identi ed the classes of weakly deep and strongly deep sequences, and showed that the halting problem is strongly deep. Juedes, Lathrop, and Lutz subsequently extended this result by de ning the class of weakly useful sequences, and proving that every weakly useful sequence is strongly deep. The present paper investigates re nements of Bennett's notions of weak and strong depth, called recursively weak depth (introduced by Fenner, Lutz and Mayordomo) and recursively strong depth (introduced here). It is argued that these re nements naturally capture Bennett's idea that deep objects are those which \contain internal evidence of a nontrivial causal history. " The fundamental properties of recursive computational depth are developed, and it is shown that the recursively weakly (respectively, strongly) deep sequences form a proper subclass of the class of weakly (respectively, strongly) deep sequences. The above-mentioned theorem of Juedes, Lathrop, and Lutz is then strengthened by proving that every weakly useful sequence is recursively strongly deep. It follows from these results that not every strongly deep sequence is weakly useful, thereby answering a question posed by Juedes.

The computational potential of artificial living systems can be studied without knowing the algorithms that govern their behavior. Modeling single organisms by means of socalled cognitive transducers, we will estimate the computational power of AL systems by viewing them as conglomerates of such organisms. We describe a scenario in which an artificial living (AL) system is involved in a potentially infinite, unpredictable interaction with an active or passive environment, to which it can react by learning and adjusting its behaviour. By making use of sequences of cognitive transducers one can also model the evolution of AL systems caused by `architectural' changes. Among the examples are `communities of agents', i.e. by communities of mobile, interactive cognitive transducers.

The word "martingale " has related, but different, meanings in probability theory and theoretical computer science. In computational complexity and algorithmic information theory, a martingale is typically a function d on strings such that E(d(wb)|w) = d(w) for all strings w, where the conditional expectation is computed over all possible values of the next symbol b. In modern probability theory a martingale is typically a sequence,0,,1,,2,... of random variables such that E(,n+1|,0,...,,n) =,n for all n.

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