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Computational Complexity of Minimal Trap Spaces in Boolean Networks

Kyungduk Moon, Kangbok Lee, Loïc Paulevé

TL;DR

In the general case, it is demonstrated that the trap space property is coNP-complete, and the minimality and the membership properties are $\Pi_2^{\text P}$-complete.

Abstract

A Boolean network (BN) is a discrete dynamical system defined by a Boolean function that maps to the domain itself. A trap space of a BN is a generalization of a fixed point, which is defined as the sub-hypercubes closed by the function of the BN. A trap space is minimal if it does not contain any smaller trap space. Minimal trap spaces have applications for the analysis of attractors of BNs with various update modes. This paper establishes the computational complexity results of three decision problems related to minimal trap spaces: the decision of the trap space property of a sub-hypercube, the decision of its minimality, and the decision of the membership of a given configuration to a minimal trap space. Under several cases on Boolean function representations, we investigate the computational complexity of each problem. In the general case, we demonstrate that the trap space property is coNP-complete, and the minimality and the membership properties are $Π_2^{\text P}$-complete. The complexities drop by one level in the polynomial hierarchy whenever the local functions of the BN are either unate, or are represented using truth-tables, binary decision diagrams, or double DNFs (Petri net encoding): the trap space property can be decided in a polynomial time, whereas deciding the minimality and the membership are coNP- complete. When the BN is given as its functional graph, all these problems are in P.

Computational Complexity of Minimal Trap Spaces in Boolean Networks

TL;DR

In the general case, it is demonstrated that the trap space property is coNP-complete, and the minimality and the membership properties are -complete.

Abstract

A Boolean network (BN) is a discrete dynamical system defined by a Boolean function that maps to the domain itself. A trap space of a BN is a generalization of a fixed point, which is defined as the sub-hypercubes closed by the function of the BN. A trap space is minimal if it does not contain any smaller trap space. Minimal trap spaces have applications for the analysis of attractors of BNs with various update modes. This paper establishes the computational complexity results of three decision problems related to minimal trap spaces: the decision of the trap space property of a sub-hypercube, the decision of its minimality, and the decision of the membership of a given configuration to a minimal trap space. Under several cases on Boolean function representations, we investigate the computational complexity of each problem. In the general case, we demonstrate that the trap space property is coNP-complete, and the minimality and the membership properties are -complete. The complexities drop by one level in the polynomial hierarchy whenever the local functions of the BN are either unate, or are represented using truth-tables, binary decision diagrams, or double DNFs (Petri net encoding): the trap space property can be decided in a polynomial time, whereas deciding the minimality and the membership are coNP- complete. When the BN is given as its functional graph, all these problems are in P.
Paper Structure (18 sections, 11 theorems, 10 equations, 2 figures, 1 table)

This paper contains 18 sections, 11 theorems, 10 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Representations and the unate property of local functions in Boolean networks
  4. As propositional formula
  5. As disjunctive normal form (DNF)
  6. Unate (monotone) case
  7. As truth table (TT)
  8. As binary decision diagram (BDD)
  9. As double DNF (DNF01)
  10. Boolean networks (BNs)
  11. Sub-hypercubes and minimal trap spaces of BNs
  12. Upper bounds results to the computational complexity
  13. Results
  14. Local functions given as propositional formulas \newlabelsec:hardness-prop-general0
  15. Locally-monotone BNs with local functions given as propositional formulas
  16. ...and 3 more sections

Key Result

Theorem 2.4

\newlabelthm:trapspace-tt0 Given hypercube $\mathbf{h}$ and BN $f$ with its local functions represented as truth tables, BDDs, or double DNFs, TRAPSPACE($f,\mathbf{h}$) can be solved in a polynomial time.

Figures (2)

  • Figure 1: The first two levels of the polynomial hierarchy
  • Figure 3: The functional graph and minimal trap spaces of the Boolean network $f$ in Example \ref{['example:locally-monotone']}

Theorems & Definitions (30)

  • Example 2.1
  • Example 2.2
  • Example 2.3
  • Theorem 2.4
  • Theorem 3.1
  • Proof 1
  • Remark 3.2
  • Proof 2
  • Remark 3.3
  • Proof 3
  • ...and 20 more