A generalized asynchronous computability theorem

We consider the models of distributed computation defined as subsets of the runs of the iterated immediate snapshot model. Given a task $T$ and a model $M$, we provide topological conditions for $T$ to be solvable in $M$. When applied to the wait-free model, our conditions result in the celebrated Asynchronous Computability Theorem (ACT) of Herlihy and Shavit. To demonstrate the utility of our characterization, we consider a task that has been shown earlier to admit only a very complex $t$-resilient solution. In contrast, our generalized computability theorem confirms its $t$-resilient solvability in a straightforward manner.Comment: 16 pages, 5 figure

Strong Equivalence Relations for Iterated Models

The Iterated Immediate Snapshot model (IIS), due to its elegant geometrical representation, has become standard for applying topological reasoning to distributed computing. Its modular structure makes it easier to analyze than the more realistic (non-iterated) read-write Atomic-Snapshot memory model (AS). It is known that AS and IIS are equivalent with respect to \emph{wait-free task} computability: a distributed task is solvable in AS if and only if it solvable in IIS. We observe, however, that this equivalence is not sufficient in order to explore solvability of tasks in \emph{sub-models} of AS (i.e. proper subsets of its runs) or computability of \emph{long-lived} objects, and a stronger equivalence relation is needed. In this paper, we consider \emph{adversarial} sub-models of AS and IIS specified by the sets of processes that can be \emph{correct} in a model run. We show that AS and IIS are equivalent in a strong way: a (possibly long-lived) object is implementable in AS under a given adversary if and only if it is implementable in IIS under the same adversary. %This holds whether the object is one-shot or long-lived. Therefore, the computability of any object in shared memory under an adversarial AS scheduler can be equivalently investigated in IIS

Tight Bounds for Connectivity and Set Agreement in Byzantine Synchronous Systems

In this paper, we show that the protocol complex of a Byzantine synchronous system can remain $(k - 1)$-connected for up to $\lceil t/k \rceil$ rounds, where $t$ is the maximum number of Byzantine processes, and $t \ge k \ge 1$. This topological property implies that $\lceil t/k \rceil + 1$ rounds are necessary to solve $k$-set agreement in Byzantine synchronous systems, compared to $\lfloor t/k \rfloor + 1$ rounds in synchronous crash-failure systems. We also show that our connectivity bound is tight as we indicate solutions to Byzantine $k$-set agreement in exactly $\lceil t/k \rceil + 1$ synchronous rounds, at least when $n$ is suitably large compared to $t$. In conclusion, we see how Byzantine failures can potentially require one extra round to solve $k$-set agreement, and, for $n$ suitably large compared to $t$, at most that

Approximate Consensus in Highly Dynamic Networks: The Role of Averaging Algorithms

In this paper, we investigate the approximate consensus problem in highly dynamic networks in which topology may change continually and unpredictably. We prove that in both synchronous and partially synchronous systems, approximate consensus is solvable if and only if the communication graph in each round has a rooted spanning tree, i.e., there is a coordinator at each time. The striking point in this result is that the coordinator is not required to be unique and can change arbitrarily from round to round. Interestingly, the class of averaging algorithms, which are memoryless and require no process identifiers, entirely captures the solvability issue of approximate consensus in that the problem is solvable if and only if it can be solved using any averaging algorithm. Concerning the time complexity of averaging algorithms, we show that approximate consensus can be achieved with precision of $\varepsilon$ in a coordinated network model in $O(n^{n+1} \log\frac{1}{\varepsilon})$ synchronous rounds, and in $O(\Delta n^{n\Delta+1} \log\frac{1}{\varepsilon})$ rounds when the maximum round delay for a message to be delivered is $\Delta$. While in general, an upper bound on the time complexity of averaging algorithms has to be exponential, we investigate various network models in which this exponential bound in the number of nodes reduces to a polynomial bound. We apply our results to networked systems with a fixed topology and classical benign fault models, and deduce both known and new results for approximate consensus in these systems. In particular, we show that for solving approximate consensus, a complete network can tolerate up to 2n-3 arbitrarily located link faults at every round, in contrast with the impossibility result established by Santoro and Widmayer (STACS '89) showing that exact consensus is not solvable with n-1 link faults per round originating from the same node

Continuous Tasks and the Asynchronous Computability Theorem

The celebrated 1999 Asynchronous Computability Theorem (ACT) of Herlihy and Shavit characterized distributed tasks that are wait-free solvable and uncovered deep connections with combinatorial topology. We provide an alternative characterization of those tasks by means of the novel concept of continuous tasks, which have an input/output specification that is a continuous function between the geometric realizations of the input and output complex: We state and prove a precise characterization theorem (CACT) for wait-free solvable tasks in terms of continuous tasks. Its proof utilizes a novel chromatic version of a foundational result in algebraic topology, the simplicial approximation theorem, which is also proved in this paper. Apart from the alternative proof of the ACT implied by our CACT, we also demonstrate that continuous tasks have an expressive power that goes beyond classic task specifications, and hence open up a promising venue for future research: For the well-known approximate agreement task, we show that one can easily encode the desired proportion of the occurrence of specific outputs, namely, exact agreement, in the continuous task specification

A logical basis for constructive systems

The work is devoted to Computability Logic (CoL) -- the philosophical/mathematical platform and long-term project for redeveloping classical logic after replacing truth} by computability in its underlying semantics (see http://www.cis.upenn.edu/~giorgi/cl.html). This article elaborates some basic complexity theory for the CoL framework. Then it proves soundness and completeness for the deductive system CL12 with respect to the semantics of CoL, including the version of the latter based on polynomial time computability instead of computability-in-principle. CL12 is a sequent calculus system, where the meaning of a sequent intuitively can be characterized as "the succedent is algorithmically reducible to the antecedent", and where formulas are built from predicate letters, function letters, variables, constants, identity, negation, parallel and choice connectives, and blind and choice quantifiers. A case is made that CL12 is an adequate logical basis for constructive applied theories, including complexity-oriented ones