scholarly journals The complexity of verifying population protocols

2021 ◽  
Vol 34 (2) ◽  
pp. 133-177
Author(s):  
Javier Esparza ◽  
Stefan Jaax ◽  
Mikhail Raskin ◽  
Chana Weil-Kennedy
Keyword(s):  
Expressive Power ◽  
Property A ◽  
Reachability Problem ◽  
Seminal Paper ◽  
Population Protocols ◽  
Finite State ◽  
State Agents ◽  
Exhaustive Study ◽  
Main Model ◽  
Main Population

AbstractPopulation protocols (Angluin et al. in PODC, 2004) are a model of distributed computation in which indistinguishable, finite-state agents interact in pairs to decide if their initial configuration, i.e., the initial number of agents in each state, satisfies a given property. In a seminal paper Angluin et al. classified population protocols according to their communication mechanism, and conducted an exhaustive study of the expressive power of each class, that is, of the properties they can decide (Angluin et al. in Distrib Comput 20(4):279–304, 2007). In this paper we study the correctness problem for population protocols, i.e., whether a given protocol decides a given property. A previous paper (Esparza et al. in Acta Inform 54(2):191–215, 2017) has shown that the problem is decidable for the main population protocol model, but at least as hard as the reachability problem for Petri nets, which has recently been proved to have non-elementary complexity. Motivated by this result, we study the computational complexity of the correctness problem for all other classes introduced by Angluin et al., some of which are less powerful than the main model. Our main results show that for the class of observation models the complexity of the problem is much lower, ranging from $$\varPi _2^p$$ Π 2 p to .

scholarly journals Towards efficient verification of population protocols

2021 ◽  
Author(s):  
Michael Blondin ◽  
Javier Esparza ◽  
Stefan Jaax ◽  
Philipp J. Meyer
Keyword(s):  
Linear Constraints ◽  
Computational Power ◽  
High Complexity ◽  
Reachability Problem ◽  
Population Protocols ◽  
Finite State ◽  
State Agents ◽  
Efficient Verification ◽  
Primitive Recursive ◽  
Very High

AbstractPopulation 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 of the protocol 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 -hard, and for which only algorithms of non-primitive recursive complexity are currently known. In this paper we introduce the class $${ WS}^3$$ WS 3 of well-specified strongly-silent protocols and we prove that it is suitable for automatic verification. More precisely, we show that $${ WS}^3$$ WS 3 has the same computational power as general well-specified protocols, and captures standard protocols from the literature. Moreover, we show that the membership and correctness problems for $${ WS}^3$$ WS 3 reduce to solving boolean combinations of linear constraints over $${\mathbb {N}}$$ N . This allowed us to develop the first software able to automatically prove correctness for all of the infinitely many possible inputs.

scholarly journals Finding Cut-Offs in Leaderless Rendez-Vous Protocols is Easy

2021 ◽  
pp. 42-61
Author(s):  
A. R. Balasubramanian ◽  
Javier Esparza ◽  
Mikhail Raskin
Keyword(s):  
Lower Bounds ◽  
Petri Net ◽  
Special Class ◽  
Upper Bounds ◽  
Initial State ◽  
Reachability Problem ◽  
Final State ◽  
Final Configuration ◽  
Finite State ◽  
State Agents

AbstractIn rendez-vous protocols an arbitrarily large number of indistinguishable finite-state agents interact in pairs. The cut-off problem asks if there exists a number B such that all initial configurations of the protocol with at least B agents in a given initial state can reach a final configuration with all agents in a given final state. In a recent paper [17], Horn and Sangnier prove that the cut-off problem is equivalent to the Petri net reachability problem for protocols with a leader, and in "Image missing" for leaderless protocols. Further, for the special class of symmetric protocols they reduce these bounds to "Image missing" and "Image missing" , respectively. The problem of lowering these upper bounds or finding matching lower bounds is left open. We show that the cut-off problem is "Image missing" -complete for leaderless protocols, "Image missing" -complete for symmetric protocols with a leader, and in "Image missing" for leaderless symmetric protocols, thereby solving all the problems left open in [17].

scholarly journals On the Expressive Power of Extended Process Rewrite Systems

2004 ◽  
Vol 11 (7) ◽  
Author(s):  
Mojmír Kretínský ◽  
Vojtech Rehák ◽  
Jan Strejcek
Keyword(s):  
Petri Nets ◽  
Process Algebra ◽  
Control Unit ◽  
Expressive Power ◽  
State Control ◽  
Reachability Problem ◽  
Rewrite Systems ◽  
Finite State ◽  
Unified View ◽  
Finite State Control

We provide a unified view on three extensions of Process rewrite systems (PRS) and compare their and PRS's expressive power. We show that the class of Petri Nets is less expressible up to bisimulation than the class of Process Algebra extended with finite state control unit. Further we show our main result that the reachability problem for PRS extended with a so called weak finite state unit is decidable.

scholarly journals Running Time Analysis of Broadcast Consensus Protocols

2021 ◽  
pp. 164-183
Author(s):  
Philipp Czerner ◽  
Stefan Jaax
Keyword(s):  
Turing Machine ◽  
Polynomial Time ◽  
Execution Time ◽  
Time Analysis ◽  
Asymptotically Optimal ◽  
Population Protocols ◽  
Running Time ◽  
Running Time Analysis ◽  
Finite State ◽  
State Agents

AbstractBroadcast consensus protocols (BCPs) are a model of computation, in which anonymous, identical, finite-state agents compute by sending/receiving global broadcasts. BCPs are known to compute all number predicates in $$\mathsf {NL}=\mathsf {NSPACE}(\log n)$$ NL = NSPACE ( log n ) where n is the number of agents. They can be considered an extension of the well-established model of population protocols. This paper investigates execution time characteristics of BCPs. We show that every predicate computable by population protocols is computable by a BCP with expected $$\mathcal {O}(n \log n)$$ O ( n log n ) interactions, which is asymptotically optimal. We further show that every log-space, randomized Turing machine can be simulated by a BCP with $$\mathcal {O}(n \log n \cdot T)$$ O ( n log n · T ) interactions in expectation, where T is the expected runtime of the Turing machine. This allows us to characterise polynomial-time BCPs as computing exactly the number predicates in $$\mathsf {ZPL}$$ ZPL , i.e. predicates decidable by log-space, randomised Turing machine with zero-error in expected polynomial time where the input is encoded as unary.

scholarly journals Copyful Streaming String Transducers

2021 ◽  
Vol 178 (1-2) ◽  
pp. 59-76
Author(s):  
Emmanuel Filiot ◽  
Pierre-Alain Reynier
Keyword(s):  
Polynomial Time ◽  
Equivalence Problem ◽  
Expressive Power ◽  
The Other ◽  
Finite State Automata ◽  
Finite State ◽  
Variable Content ◽  
Deterministic Automata ◽  
Intermediate Output ◽  
Linear Manner

Copyless streaming string transducers (copyless SST) have been introduced by R. Alur and P. Černý in 2010 as a one-way deterministic automata model to define transductions of finite strings. Copyless SST extend deterministic finite state automata with a set of variables in which to store intermediate output strings, and those variables can be combined and updated all along the run, in a linear manner, i.e., no variable content can be copied on transitions. It is known that copyless SST capture exactly the class of MSO-definable string-to-string transductions, and are as expressive as deterministic two-way transducers. They enjoy good algorithmic properties. Most notably, they have decidable equivalence problem (in PSpace). On the other hand, HDT0L systems have been introduced for a while, the most prominent result being the decidability of the equivalence problem. In this paper, we propose a semantics of HDT0L systems in terms of transductions, and use it to study the class of deterministic copyful SST. Our contributions are as follows: (i)HDT0L systems and total deterministic copyful SST have the same expressive power, (ii)the equivalence problem for deterministic copyful SST and the equivalence problem for HDT0L systems are inter-reducible, in quadratic time. As a consequence, equivalence of deterministic SST is decidable, (iii)the functionality of non-deterministic copyful SST is decidable, (iv)determining whether a non-deterministic copyful SST can be transformed into an equivalent non-deterministic copyless SST is decidable in polynomial time.

Mirror, mirror in my hand: a duality between specifications and models of process behaviour

1996 ◽  
Vol 6 (4) ◽  
pp. 353-373 ◽  
Author(s):  
J. L. Fiadeiro ◽  
J. F. Costa
Keyword(s):  
Temporal Logic ◽  
Structural Property ◽  
Reactive Systems ◽  
Expressive Power ◽  
Seminal Paper ◽  
Process Specification ◽  
Temporal Specifications ◽  
Mirror Mirror

SummarySince Pnueli’s seminal paper in 1977, Temporal Logic has been used as a formalism for specifying and verifying the correctness of reactive systems. In this paper, we show that, besides its expressive power, Temporal Logic enjoys a very strong structural property: it is categorical on processes. That is, we show how temporal specifications (as theories) can be embedded in categories of process behaviour, and out of this adjunction we build an institution that is categorical in the sense of Meseguer. This characterisation means that temporal logic is, in a sense, ‘sound and complete’ with respect to process specification and interconnection techniques.

SOME COMPLEXITY RESULTS FOR RINGS OF PETRI NETS

1994 ◽  
Vol 05 (03n04) ◽  
pp. 281-292
Author(s):  
HSU-CHUN YEN ◽  
BOW-YAW WANG ◽  
MING-SHANG YANG
Keyword(s):  
Petri Nets ◽  
State Machines ◽  
Slight Improvement ◽  
Reachability Problem ◽  
Encoding Scheme ◽  
N Cycle ◽  
Vector Addition ◽  
Finite State ◽  
Boundedness Problem ◽  
Compare And Contrast

We define a subclass of Petri nets called m–state n–cycle Petri nets, each of which can be thought of as a ring of n bounded (by m states) Petri nets using n potentially unbounded places as joins. Let Ring(n, l, m) be the class of m–state n–cycle Petri nets in which the largest integer mentioned can be represented in l bits (when the standard binary encoding scheme is used). As it turns out, both the reachability problem and the boundedness problem can be decided in O(n(l+log m)) nondeterministic space. Our results provide a slight improvement over previous results for the so-called cyclic communicating finite state machines. We also compare and contrast our results with that of VASS(n, l, s), which represents the class of n-dimensional s-state vector addition systems with states where the largest integer mentioned can be described in l bits.

scholarly journals First order quantifiers in monadic second order logic

2004 ◽  
Vol 69 (1) ◽  
pp. 118-136 ◽  
Author(s):  
H. Jerome Keisler ◽  
Wafik Boulos Lotfallah
Keyword(s):  
Expressive Power ◽  
Second Order ◽  
Order Logic ◽  
Existential Quantifier ◽  
Reachability Problem ◽  
Open Problems ◽  
First Order ◽  
Second Order Logic ◽  
Monadic Second Order Logic ◽  
Quantifier Alternation

AbstractThis paper studies the expressive power that an extra first order quantifier adds to a fragment of monadic second order logic, extending the toolkit of Janin and Marcinkowski [JM01].We introduce an operation existsn (S) on properties S that says “there are n components having S”. We use this operation to show that under natural strictness conditions, adding a first order quantifier word u to the beginning of a prefix class V increases the expressive power monotonically in u. As a corollary, if the first order quantifiers are not already absorbed in V, then both the quantifier alternation hierarchy and the existential quantifier hierarchy in the positive first order closure of V are strict.We generalize and simplify methods from Marcinkowski [Mar99] to uncover limitations of the expressive power of an additional first order quantifier, and show that for a wide class of properties S, S cannot belong to the positive first order closure of a monadic prefix class W unless it already belongs to W.We introduce another operation alt(S) on properties which has the same relationship with the Circuit Value Problem as reach(S) (defined in [JM01]) has with the Directed Reachability Problem. We use alt(S) to show that Πn ⊈ FO(Σn), Σn ⊈ FO(∆n). and ∆n+1 ⊈ FOB(Σn), solving some open problems raised in [Mat98].

scholarly journals The Decidability of Verification under PS 2.0

2021 ◽  
pp. 1-29
Author(s):  
Parosh Aziz Abdulla ◽  
Mohamed Faouzi Atig ◽  
Adwait Godbole ◽  
S. Krishna ◽  
Viktor Vafeiadis
Keyword(s):  
Global Memory ◽  
Program Transformations ◽  
Reachability Problem ◽  
Context Switching ◽  
Input Program ◽  
Finite State ◽  
Memory Accesses ◽  
Primitive Recursive

AbstractWe consider the reachability problem for finite-state multi-threaded programs under thepromising semantics() of Lee et al., which captures most common program transformations. Since reachability is already known to be undecidable in the fragment of with only release-acquire accesses (-), we consider the fragment with only relaxed accesses and promises (). We show that reachability under is undecidable in general and that it becomes decidable, albeit non-primitive recursive, if we bound the number of promises.Given these results, we consider a bounded version of the reachability problem. To this end, we bound both the number of promises and of “view-switches”, i.e., the number of times the processes may switch their local views of the global memory. We provide a code-to-code translation from an input program under (with relaxed and release-acquire memory accesses along with promises) to a program under SC, thereby reducing the bounded reachability problem under to the bounded context-switching problem under SC. We have implemented a tool and tested it on a set of benchmarks, demonstrating that typical bugs in programs can be found with a small bound.

FUZZY ATTRIBUTE IMPLICATIONS AND THEIR EXPRESSIVE POWER

2013 ◽  
Vol 21 (04) ◽  
pp. 483-496 ◽  
Author(s):  
VILEM VYCHODIL
Keyword(s):  
Fuzzy Sets ◽  
Additional Condition ◽  
Expressive Power ◽  
Closure System ◽  
Property A ◽  
Concept Lattices ◽  
Practical Consequence ◽  
Alternative Description ◽  
Linguistic Hedges ◽  
Truth Degrees

We deal with the expressive power of if-then rules called fuzzy attribute implications (FAIs) which can be seen as formulas A ⇒ B where both A and B are conjunctions of subformulas containing propositional variables and constants for truth degrees and whose interpretation is parameterized by linguistic hedges. The formulas admit the following model-theoretical property: a system of fuzzy sets [Formula: see text] is a fuzzy closure system satisfying an additional condition of being closed under a*-shifts (so-called L*-closure system) if and only if [Formula: see text] is a system of models of a set of FAIs. In this paper, we point out the importance of constants of truth degrees in the antecedents of formulas by showing that simpler formulas are not sufficient to describe all L*-closure systems. As a practical consequence, the simpler formulas cannot be used as an alternative description of concept lattices with linguistic hedges.

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