Deterministic Computations Whose History Is Independent of the Order of Asynchronous Updating

Consider a network of processors (sites) in which each site x has a finite set N(x) of neighbors. There is a transition function f that for each site x computes the next state ξ(x) from the states in N(x). But these transitions (updates) are applied in arbitrary order, one or many at a time. If the state of site x at time t is η(x; t) then let us define the sequence ζ(x; 0); ζ(x; 1), ... by taking the sequence η(x; 0),η(x; 1), ... , and deleting each repetition, i.e. each element equal to the preceding one. The function f is said to have invariant histories if the sequence ζ(x; i), (while it lasts, in case it is finite) depends only on the initial configuration, not on the order of updates. This paper shows that though the invariant history property is typically undecidable, there is a useful simple sufficient condition, called commutativity: For any configuration, for any pair x; y of neighbors, if the updating would change both ξ(x) and ξ(y) then the result of updating first x and then y is the same as the result of doing this in the reverse order. This fact is derivable from known results on the confluence of term-rewriting systems but the self-contained proof given here may be justifiable.National Science Foundation (CCR-920484

Clairvoyant embedding in one dimension

Let v, w be infinite 0-1 sequences, and m a positive integer. We say that w is m-embeddable in v, if there exists an increasing sequence n_{i} of integers with n_{0}=0, such that 0 0. Let X and Y be independent coin-tossing sequences. We will show that there is an m with the property that Y is m-embeddable into X with positive probability. This answers a question that was open for a while. The proof generalizes somewhat the hierarchical method of an earlier paper of the author on dependent percolation.Comment: 49 pages. Some errors corrected. arXiv admin note: substantial text overlap with arXiv:math/010915

Deterministic computations whose history is independent of the order of asynchronous updating

Consider a network of processors (sites) in which each site x has a finite set N(x) of neighbors. There is a transition function f that for each site x computes the next state \xi(x) from the states in N(x). But these transitions (updates) are applied in arbitrary order, one or many at a time. If the state of site x at time t is \eta(x,t) then let us define the sequence \zeta(x,0), \zeta(x,1), ... by taking the sequence \eta(x,0), \eta(x,1), ..., and deleting repetitions. The function f is said to have invariant histories if the sequence \zeta(x,i), (while it lasts, in case it is finite) depends only on the initial configuration, not on the order of updates. This paper shows that though the invariant history property is typically undecidable, there is a useful simple sufficient condition, called commutativity: For any configuration, for any pair x,y of neighbors, if the updating would change both \xi(x) and \xi(y) then the result of updating first x and then y is the same as the result of doing this in the reverse order

Compatible sequences and a slow Winkler percolation

Two infinite 0-1 sequences are called compatible when it is possible to cast out 0's from both in such a way that they become complementary to each other. Answering a question of Peter Winkler, we show that if the two 0-1-sequences are random i.i.d. and independent from each other, with probability p of 1's, then if p is sufficiently small they are compatible with positive probability. The question is equivalent to a certain dependent percolation with a power-law behavior: the probability that the origin is blocked at distance n but not closer decreases only polynomially fast and not, as usual, exponentially.Comment: 33 pages, 8 figures. Submitted to Combinatorics, Probability and Computing. Some errors correcte

The angel wins

The angel-devil game is played on an infinite two-dimensional ``chessboard''. The squares of the board are all white at the beginning. The players called angel and devil take turns in their steps. When it is the devil's turn, he can turn a square black. The angel always stays on a white square, and when it is her turn she can fly at a distance of at most J steps (each of which can be horizontal, vertical or diagonal) to a new white square. Here J is a constant. The devil wins if the angel does not find any more white squares to land on. The result of the paper is that if J is sufficiently large then the angel has a strategy such that the devil will never capture her. This deceptively easy-sounding result has been a conjecture, surprisingly, for about thirty years. Several other independent solutions have appeared simultaneously, some of them prove that J=2 is sufficient (see the Wikipedia on the angel problem). Still, it is hoped that the hierarchical solution presented here may prove useful for some generalizations.Comment: 28 pages, 8 figure

Gac two-component system in Pseudomonas syringae pv. tabaci is required for virulence but not for hypersensitive reaction

Pseudomonas syringae pv. tabaci 6605 causes wildfire disease on host tobacco plants. To investigate the regulatory mechanism of the expression of virulence, Gac two-Component system-defective mutants, Delta gacA and Delta gacS, and a double mutant, Delta gacA Delta gacS, were generated. These mutants produced smaller amounts of N-acyl homoserine lactones required for quorum sensing, had lost swarming motility, and had reduced expression of virulence-related hrp genes and the algT gene required for exopolysaccharide production. The ability of the mutants to cause disease symptoms in their host tobacco plant was remarkably reduced, while they retained the ability to induce hypersensitive reaction (HR) in the nonhost plants. These results indicated that the Gac two-component system of P. syringae pv. tabaci 6605 is indispensable for virulence on the host plant, but not for HR induction in the nonhost plants.</p

Uniform test of algorithmic randomness over a general space

The algorithmic theory of randomness is well developed when the underlying space is the set of finite or infinite sequences and the underlying probability distribution is the uniform distribution or a computable distribution. These restrictions seem artificial. Some progress has been made to extend the theory to arbitrary Bernoulli distributions (by Martin-Loef), and to arbitrary distributions (by Levin). We recall the main ideas and problems of Levin's theory, and report further progress in the same framework. - We allow non-compact spaces (like the space of continuous functions, underlying the Brownian motion). - The uniform test (deficiency of randomness) d_P(x) (depending both on the outcome x and the measure P should be defined in a general and natural way. - We see which of the old results survive: existence of universal tests, conservation of randomness, expression of tests in terms of description complexity, existence of a universal measure, expression of mutual information as "deficiency of independence. - The negative of the new randomness test is shown to be a generalization of complexity in continuous spaces; we show that the addition theorem survives. The paper's main contribution is introducing an appropriate framework for studying these questions and related ones (like statistics for a general family of distributions).Comment: 40 pages. Journal reference and a slight correction in the proof of Theorem 7 adde

Parallel compensatory evolution stabilizes plasmids across the parasitism-mutualism continuum

Plasmids drive genomic diversity in bacteria via horizontal gene transfer [1 and 2]; nevertheless, explaining their survival in bacterial populations is challenging [3]. Theory predicts that irrespective of their net fitness effects, plasmids should be lost: when parasitic (costs outweigh benefits), plasmids should decline due to purifying selection [4, 5 and 6], yet under mutualism (benefits outweigh costs), selection favors the capture of beneficial accessory genes by the chromosome and loss of the costly plasmid backbone [4]. While compensatory evolution can enhance plasmid stability within populations [7, 8, 9, 10, 11, 12, 13, 14 and 15], the propensity for this to occur across the parasitism-mutualism continuum is unknown. We experimentally evolved Pseudomonas fluorescens and its mercury resistance mega-plasmid, pQBR103 [ 16], across an environment-mediated parasitism-mutualism continuum. Compensatory evolution stabilized plasmids by rapidly ameliorating the cost of plasmid carriage in all environments. Genomic analysis revealed that, in both parasitic and mutualistic treatments, evolution repeatedly targeted the gacA/gacS bacterial two-component global regulatory system while leaving the plasmid sequence intact. Deletion of either gacA or gacS was sufficient to completely ameliorate the cost of plasmid carriage. Mutation of gacA/gacS downregulated the expression of ∼17% of chromosomal and plasmid genes and appears to have relieved the translational demand imposed by the plasmid. Chromosomal capture of mercury resistance accompanied by plasmid loss occurred throughout the experiment but very rarely invaded to high frequency, suggesting that rapid compensatory evolution can limit this process. Compensatory evolution can explain the widespread occurrence of plasmids and allows bacteria to retain horizontally acquired plasmids even in environments where their accessory genes are not immediately useful

The clairvoyant demon has a hard task

Consider the integer lattice L = ℤ2. For some m [ges ] 4, let us colour each column of this lattice independently and uniformly with one of m colours. We do the same for the rows, independently of the columns. A point of L will be called blocked if its row and column have the same colour. We say that this random configuration percolates if there is a path in L starting at the origin, consisting of rightward and upward unit steps, avoiding the blocked points. As a problem arising in distributed computing, it has been conjectured that for m [ges ] 4 the configuration percolates with positive probability. This question remains open, but we prove that the probability that there is percolation to distance n but not to infinity is not exponentially small in n. This narrows the range of methods available for proving the conjecture