On the Complexity of Adding Convergence

Abstract

Abstract. This paper investigates the complexity of designing SelfStabilizing (SS) distributed programs, where an SS program meets two properties, namely closure and convergence. Convergence requires that, from any state, the computations of an SS program reach a set of legitimate states (a.k.a. invariant). Upon reaching a legitimate state, the computations of an SS program remain in the set of legitimate states as long as no faults occur; i.e., Closure. We illustrate that, in general, the problem of augmenting a distributed program with convergence, i.e., adding convergence, is NP-complete (in the size of its state space). An implication of our NP-completeness result is the hardness of adding nonmasking fault tolerance to distributed programs, which has been an open problem for the past decade.

Specification-Based Synthesis of Distributed Self-Stabilizing Protocols

“…Synthesis of Self-Stabilizing Systems. In [21], the authors show that adding strong convergence is NP-complete in the size of the state space, which itself is exponential in the number of variables of the protocol. Ebnenasir and Farahat [10] also proposed an automated method to synthesize self-stabilizing algorithms.…”

Logic-Based Program Synthesis and Transformation

“…Previous work [16,19] shows that weak convergence can be added in polynomial time (in the size of the state space), whereas adding strong convergence is known to be an NP-complete problem [26]. Farahat and Ebnenasir [13,16] present a sound and complete algorithm for the addition of weak convergence and a set of heuristics for efficient addition of strong convergence.…”

On the Applications of Interactive Theorem Proving in Computational Sciences and Engineering

“…For example, the design of parameterized self-stabilizing protocols algorithmically is known to be an open problem. Moreover, the problem of adding convergence to finite state automata is NP-hard (in the size of state space) [12,13,14]. Most existing techniques for the design of self-stabilization rely on a 'manual design and after-the-fact verification' methods which are limited to specific heuristics.…”

“…The proposed method includes a synthesis step and a theorem proving step. In [26,25,14,40] the authors enable the synthesis step where they take a non-stabilizing protocol and generate a selfstabilizing version thereof that is correct by construction up to a certain number of processes. This paper investigates the second step where we use the theorem prover PVS to prove (or disprove) the correctness of the synthesized protocol for an arbitrary number of processes; i.e., generalize the synthesized protocol.…”

“…This paper investigates the second step where we use the theorem prover PVS to prove (or disprove) the correctness of the synthesized protocol for an arbitrary number of processes; i.e., generalize the synthesized protocol. The synthesis algorithms in [26,25,14,40] incorporate weak and strong convergence in existing network protocols; i.e., adding convergence. Weak (respectively, Strong) convergence requires that from every state there exists an execution that (respectively, every execution) reaches an invariant state in finite number of steps.…”