The computational power of networks of small resource-limited mobile agents is explored. Two new models of computation based on pairwise interactions of finite-state agents in populations of finite but unbounded size are defined. With a fairness condition on interactions, the concept of stable computation of a function or predicate is defined. Protocols are given that stably compute any predicate in the class definable by formulas of Presburger arithmetic, which includes Boolean combinations of threshold-k, majority, and equivalence modulo m. All stably computable predicates are shown to be in NL. Assuming uniform random sampling of interacting pairs yields the model of conjugating automata. Any counter machine with O (1) counters of capacity O (n) can be simulated with high probability by a conjugating automaton in a population of size n. All predicates computable with high probability in this model are shown to be in P; they can also be computed by a randomized logspace machine in exponential time. Several open problems and promising future directions are discussed.

Computation in networks of passively mobile finite-state sensors

Fault tolerance in distributed computing is a wide area with a significant body of literature that is vastly diverse in methodology and terminology. This paper aims at structuring the area and thus guiding readers into this interesting field. We use a formal approach to define important terms like fault, fault tolerance, and redundancy. This leads to four distinct forms of fault tolerance and to two main phases in achieving them: detection and correction. We show that this can help to reveal inherently fundamental structures that contribute to understanding and unifying methods and terminology. By doing this, we survey many existing methodologies and discuss their relations. The underlying system model is the close-to-reality asynchronous message-passing model of distributed computing.

Fundamentals of fault-tolerant distributed computing in asynchronous environments

We explore the computational power of networks of small resource-limited mobile agents. We define two new models of computation based on pairwise interactions of finite-state agents in populations of finite but unbounded size. With a fairness condition on interactions, we define the concept of stable computation of a function or predicate, and give protocols that stably compute functions in a class including Boolean combinations of threshold-k, parity, majority, and simple arithmetic. We prove that all stably computable predicates are in NL. With uniform random sampling of pairs to interact, we define the model of conjugating automata and show that any counter machine with O(1) counters of capacity O(n) can be simulated with high probability by a protocol in a population of size n. We prove that all predicates computable with high probability in this model are in P ∩ RL. Several open problems and promising future directions are discussed.

/pdf/computation-in-networks-of-passively-mobile-finite-state-3svdcrrvsf.pdf

In this paper we introduce the concept of a Petri net component and show how systems can be composed from components. A component communicates with its environment via distinguished input and output places, which formalizes communication by message passing. Then, we present a compositional semantics for components. The semantics is an extension of processes for place/transition systems (partial order semantics). We show that the semantics is fully abstract with respect to the behaviour of closed components (essentially, processes of place/transition systems). A main feature of the compositional semantics is that composition of components corresponds to conjunction. This feature makes the compositional semantics applicable in combination with a temporal logic, which then allows to reason about systems in a compositional way. This is demonstrated by help of a simple temporal logic.

A Compositional Partial Order Semantics for Petri Net Components

This paper studies how to adjoin probability to event structures, leading to the model of probabilistic event structures. In their simplest form, probabilistic choice is localised to cells, where conflict arises; in which case probabilistic independence coincides with causal independence. An event structure is associated with a domain--that of its configurations ordered by inclusion. In domain theory, probabilistic processes are denoted by continuous valuations on a domain. A key result of this paper is a representation theorem showing how continuous valuations on the domain of a confusion-free event structure correspond to the probabilistic event structures it supports. We explore how to extend probability to event structures which are not confusion-free via two notions of probabilistic runs of a general event structure. Finally, we show how probabilistic correlation and probabilistic event structures with confusion can arise from event structures which are originally confusion-free by using morphisms to rename and hide events.

/pdf/probabilistic-event-structures-and-domains-5dobnua233.pdf

Probabilistic event structures and domains

This paper studies how to adjoin probability to event structures, leading to the model of probabilistic event structures. In their simplest form probabilistic choice is localised to cells, where conflict arises; in which case probabilistic independence coincides with causal independence. An application to the semantics of a probabilistic CCS is sketched. An event structure is associated with a domain-that of its configurations ordered by inclusion. In domain theory probabilistic processes are denoted by continuous valuations on a domain. A key result of this paper is a representation theorem showing how continuous valuations on the domain of a confusion-free event structure correspond to the probabilistic event structures it supports. We explore how to extend probability to event structures which are not confusion-free via two notions of probabilistic runs of a general event structure. Finally, we show how probabilistic correlation and probabilistic event structures with confusion can arise from event structures which are originally confusion-free by using morphisms to rename and hide events.

A technique to model and to verify distributed algorithms is suggested. This technique (based on Petri nets) reduces the modelling and analysis effort to a reasonable level. The paper outlines the technique using the example of a typical network algorithm, theecho algorithm.

Petri net based verification of distributed algorithms: An example

Algebraic Petri nets as defined by Reisig [17] lack a feature for modelling distributed network algorithms, viz. flexible arcs. In this paper we equip algebraic Petri nets with flexible arcs and we call the resulting extension algebraic system nets. We demonstrate that algebraic system nets are better suited for modelling distributed algorithms.

/pdf/flexibility-in-algebraic-nets-2x425ovqcp.pdf

Flexibility in Algebraic Nets

A non-sequential, i.e. "true concurrency", semantics for randomized distributed algorithms is suggested. It is based on Petri nets and their branching processes. We introduce randomized Petri nets and their semantics, probabilistic branching processes. As a main result, we show that each probabilistic branching process defines a unique canonical probability space. Finally, we show that the non-sequential semantics differs from the classical sequential semantics, modelling a new adversary, called the distributed adversary.

Hagen Völzer

Papers

Probabilistic event structures and domains

Probabilistic event structures and domains

Petri net based verification of distributed algorithms: An example

Flexibility in Algebraic Nets

Randomized Non-sequential Processes