Complexity of fixed point counting problems in Boolean networks

Bridoux, Florian; Durbec, Amélia; Perrot, Kévin; Richard, Adrien

doi:10.1016/j.jcss.2022.01.004

Cited by 8 publications

(3 citation statements)

References 36 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Suppose that f u (x) i = 1, the other case being symmetric. By (8) there is a word v over J such that f uv (x) J = f uv (y) J = 0 and |v| ≤ 2|J|. We deduce from (5) that f uv (x) K = f uv (y) K .…”

Section: F (T + 2)mentioning

confidence: 89%

“…By Lemma 5, there is a word u over V such that f u (x) i = 1 and |u| < F (n + 2). So by (8) there is a word v over J such that f uv (x) J = f uv (y) J = 0 with |v| ≤ 2|J|. We deduce from (5) that f uv (x) K = f uv (y) K .…”

Section: F (T + 2)mentioning

confidence: 89%

“…Our reductions from 3-SAT are based on the simple signed digraph H ψ defined as follows; see Figure 1 for an illustration: • For all s ∈ [m], the arc (µ s , c s ) is negative, and all the other arcs are positive. This signed digraph H ψ has been recently introduced in [8]. Actually, in this paper, the authors consider the signed digraph H ′ ψ obtained from H ψ by adding, for all r ∈ [n], a new vertex i r with a positive arc (i r , λ + r ) and a negative arc (i r , λ − r ), and then prove the following: ψ is satisfiable if and only if there is a BN on H ′ ψ with at least two fixed points.…”

Section: Strong-synchronizing-problemmentioning

confidence: 99%

See 2 more Smart Citations

Synchronizing Boolean networks asynchronously

Aracena¹,

Richard²,

Salinas³

2022

Preprint

Self Cite

View full text Add to dashboard Cite

The asynchronous automaton associated with a Boolean network f : {0, 1} n → {0, 1} n , considered in many applications, is the finite deterministic automaton where the set of states is {0, 1} n , the alphabet is [n], and the action of letter i on a state x consists in either switching the ith component if f i (x) = x i or doing nothing otherwise. These actions are extended to words in the natural way. A word is then synchronizing if the result of its action is the same for every state. In this paper, we ask for the existence of synchronizing words, and their minimal length, for a basic class of Boolean networks called and-or-nets: given an arc-signed digraph G on [n], we say that f is an and-or-net on G if, for every i ∈ [n], there is a such that, for all state x, f i (x) = a if and only if x j = a (x j = a) for every positive (negative) arc from j to i; so if a = 1 (a = 0) then f i is a conjunction (disjunction) of positive or negative literals. Our main result is that if G is strongly connected and has no positive cycles, then either every and-or-net on G has a synchronizing word of length at most 10( √ 5 + 1) n , much smaller than the bound (2 n − 1) 2 given by the well known Černý's conjecture, or G is a cycle and no and-or-net on G has a synchronizing word. This contrasts with the following complexity result: it is coNP-hard to decide if every and-or-net on G has a synchronizing word, even if G is strongly connected or has no positive cycles.

show abstract

Section: F (T + 2)mentioning

confidence: 89%

Section: F (T + 2)mentioning

confidence: 89%

Section: Strong-synchronizing-problemmentioning

confidence: 99%

See 1 more Smart Citation