144 research outputs found
Population Protocols with Unordered Data
Population protocols form a well-established model of computation of passively mobile anonymous agents with constant-size memory. It is well known that population protocols compute Presburger-definable predicates, such as absolute majority and counting predicates. In this work, we initiate the study of population protocols operating over arbitrarily large data domains. More precisely, we introduce population protocols with unordered data as a formalism to reason about anonymous crowd computing over unordered sequences of data. We first show that it is possible to determine whether an unordered sequence from an infinite data domain has a datum with absolute majority. We then establish the expressive power of the "immediate observation" restriction of our model, namely where, in each interaction, an agent observes another agent who is unaware of the interaction
Towards Efficient Verification of Population Protocols
Population protocols are a well established model of computation by
anonymous, identical finite state agents. A protocol is well-specified if from
every initial configuration, all fair executions reach a common consensus. The
central verification question for population protocols is the
well-specification problem: deciding if a given protocol is well-specified.
Esparza et al. have recently shown that this problem is decidable, but with
very high complexity: it is at least as hard as the Petri net reachability
problem, which is EXPSPACE-hard, and for which only algorithms of non-primitive
recursive complexity are currently known.
In this paper we introduce the class WS3 of well-specified strongly-silent
protocols and we prove that it is suitable for automatic verification. More
precisely, we show that WS3 has the same computational power as general
well-specified protocols, and captures standard protocols from the literature.
Moreover, we show that the membership problem for WS3 reduces to solving
boolean combinations of linear constraints over N. This allowed us to develop
the first software able to automatically prove well-specification for all of
the infinitely many possible inputs.Comment: 29 pages, 1 figur
Succinct Population Protocols for Presburger Arithmetic
International audienceIn [5], Angluin et al. proved that population protocols compute exactly the predicates definable in Presburger arithmetic (PA), the first-order theory of addition. As part of this result, they presented a procedure that translates any formula of quantifier-free PA with remainder predicates (which has the same expressive power as full PA) into a population protocol with states that computes . More precisely, the number of states of the protocol is exponential in both the bit length of the largest coefficient in the formula, and the number of nodes of its syntax tree. In this paper, we prove that every formula of quantifier-free PA with remainder predicates is computable by a leaderless population protocol with states. Our proof is based on several new constructions, which may be of independent interest. Given a formula of quantifier-free PA with remainder predicates, a first construction produces a succinct protocol (with leaders) that computes Ï•; this completes the work initiated in [8], where we constructed such protocols for a fragment of PA. For large enough inputs, we can get rid of these leaders. If the input is not large enough, then it is small, and we design another construction producing a succinct protocol with one leader that computes . Our last construction gets rid of this leader for small inputs
Automatic Analysis of Expected Termination Time for Population Protocols
Population protocols are a formal model of sensor networks consisting of identical mobile devices. Two devices can interact and thereby change their states. Computations are infinite sequences of interactions in which the interacting devices are chosen uniformly at random.
In well designed population protocols, for every initial configuration of devices, and for every computation starting at this configuration, all devices eventually agree on a consensus value. We address the problem of automatically computing a parametric bound on the expected time the protocol needs to reach this consensus. We present the first algorithm that, when successful, outputs a function f(n) such that the expected time to consensus is bound by O(f(n)), where n is the number of devices executing the protocol. We experimentally show that our algorithm terminates and provides good bounds for many of the protocols found in the literature
Separators in Continuous Petri Nets
Leroux has proved that unreachability in Petri nets can be witnessed by a
Presburger separator, i.e. if a marking cannot reach a
marking , then there is a formula of Presburger
arithmetic such that: holds; is forward
invariant, i.e., and imply
); and holds. While these
separators could be used as explanations and as formal certificates of
unreachability, this has not yet been the case due to their
(super-)Ackermannian worst-case size and the (super-)exponential complexity of
checking that a formula is a separator.
We show that, in continuous Petri nets, these two problems can be overcome.
We introduce locally closed separators, and prove that: (a) unreachability can
be witnessed by a locally closed separator computable in polynomial time; (b)
checking whether a formula is a locally closed separator is in NC (so, simpler
than unreachablity, which is P-complete).
We further consider the more general problem of (existential) set-to-set
reachability, where two sets of markings are given as convex polytopes. We show
that, while our approach does not extend directly, we can still efficiently
certify unreachability via an altered Petri.Comment: Submitted to LMCS as an extension of the FoSSaCS'22 conference
versio
Population Protocols with Unordered Data
Population protocols form a well-established model of computation of
passively mobile anonymous agents with constant-size memory. It is well known
that population protocols compute Presburger-definable predicates, such as
absolute majority and counting predicates. In this work, we initiate the study
of population protocols operating over arbitrarily large data domains. More
precisely, we introduce population protocols with unordered data as a formalism
to reason about anonymous crowd computing over unordered sequences of data. We
first show that it is possible to determine whether an unordered sequence from
an infinite data domain has a datum with absolute majority. We then establish
the expressive power of the immediate observation restriction of our model,
namely where, in each interaction, an agent observes another agent who is
unaware of the interaction.Comment: accepted at ICALP 202
Large Flocks of Small Birds: on the Minimal Size of Population Protocols
Population protocols are a well established model of distributed computation by mobile finite-state agents with very limited storage. A classical result establishes that population protocols compute exactly predicates definable in Presburger arithmetic. We initiate the study of the minimal amount of memory required to compute a given predicate as a function of its size. We present results on the predicates x >= n for n in N, and more generally on the predicates corresponding to systems of linear inequalities. We show that they can be computed by protocols with O(log n) states (or, more generally, logarithmic in the coefficients of the predicate), and that, surprisingly, some families of predicates can be computed by protocols with O(log log n) states. We give essentially matching lower bounds for the class of 1-aware protocols
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