Computational complexity of ecological and evolutionary spatial dynamics

There are deep, yet largely unexplored, connections between computer science and biology. Both disciplines examine how information proliferates in time and space. Central results in computer science describe the complexity of algorithms that solve certain classes of problems. An algorithm is deemed efficient if it can solve a problem in polynomial time, which means the running time of the algorithm is a polynomial function of the length of the input. There are classes of harder problems for which the fastest possible algorithm requires exponential time. Another criterion is the space requirement of the algorithm. There is a crucial distinction between algorithms that can find a solution, verify a solution, or list several distinct solutions in given time and space. The complexity hierarchy that is generated in this way is the foundation of theoretical computer science. Precise complexity results can be notoriously difficult. The famous question whether polynomial time equals nondeterministic polynomial time (i.e., P = NP) is one of the hardest open problems in computer science and all of mathematics. Here, we consider simple processes of ecological and evolutionary spatial dynamics. The basic question is: What is the probability that a new invader (or a new mutant) will take over a resident population? We derive precise complexity results for a variety of scenarios. We therefore show that some fundamental questions in this area cannot be answered by simple equations (assuming that P is not equal to NP).

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Introduction

Combinatorics Probability Computing ◽

10.1017/s0963548318000317 ◽

2018 ◽

Vol 27 (4) ◽

pp. 441-441

Author(s):

PAUL BALISTER ◽

BÉLA BOLLOBÁS ◽

IMRE LEADER ◽

ROB MORRIS ◽

OLIVER RIORDAN

Keyword(s):

Computer Science ◽

Theoretical Computer Science ◽

Special Issue ◽

Open Problems ◽

Theoretical Computer ◽

Discrete Probability ◽

The Common

This special issue is devoted to papers from the meeting on Combinatorics and Probability, held at the Mathematisches Forschungsinstitut in Oberwolfach from the 17th to the 23rd April 2016. The lectures at this meeting focused on the common themes of Combinatorics and Discrete Probability, with many of the problems studied originating in Theoretical Computer Science. The lectures, many of which were given by young participants, stimulated fruitful discussions. The fact that the participants work in different and yet related topics, and the open problems session held during the meeting, encouraged interesting discussions and collaborations.

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ON UNIFORMLY RECURRENT MORPHIC SEQUENCES

International Journal of Foundations of Computer Science ◽

10.1142/s0129054109006966 ◽

2009 ◽

Vol 20 (05) ◽

pp. 919-940 ◽

Cited By ~ 6

Author(s):

FRANCOIS NICOLAS ◽

YURI PRITYKIN

Keyword(s):

Positive Integer ◽

Computer Science ◽

Polynomial Time ◽

Linear Growth ◽

Fibonacci Sequence ◽

Theoretical Computer Science ◽

Symbolic Sequence ◽

Theoretical Computer ◽

Automatic Sequences ◽

The Status

A pure morphic sequence is a right-infinite, symbolic sequence obtained by iterating a letter-to-word substitution. For instance, the Fibonacci sequence and the Thue–Morse sequence, which play an important role in theoretical computer science, are pure morphic. Define a coding as a letter-to-letter substitution. The image of a pure morphic sequence under a coding is called a morphic sequence.A sequence x is called uniformly recurrent if for each finite subword u of x there exists an integer l such that u occurs in every l-length subword of x.The paper mainly focuses on the problem of deciding whether a given morphic sequence is uniformly recurrent. Although the status of the problem remains open, we show some evidence for its decidability: in particular, we prove that it can be solved in polynomial time on pure morphic sequences and on automatic sequences.In addition, we prove that the complexity of every uniformly recurrent, morphic sequence has at most linear growth: here, complexity is understood as the function that maps each positive integer n to the number of distinct n-length subwords occurring in the sequence.

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Introduction

Combinatorics Probability Computing ◽

10.1017/s0963548310000362 ◽

2010 ◽

Vol 19 (5-6) ◽

pp. 641-641

Author(s):

Noga Alon ◽

Béla Bollobás

Keyword(s):

Computer Science ◽

Theoretical Computer Science ◽

Special Issue ◽

Open Problems ◽

Theoretical Computer ◽

Discrete Probability ◽

The Common

This special issue is devoted to papers from the meeting on Combinatorics and Probability, held at the Mathematisches Forschungsinstitut in Oberwolfach from 26 April to 2 May. This meeting focused on the common themes of Combinatorics, Discrete Probability and Theoretical Computer Science, and the lectures, many of which were given by young participants, stimulated fruitful discussions. The open problems session held during the meeting, and the fact that the participants work in different and related topics, encouraged interesting discussions and collaborations.

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P versus NP

10.20944/preprints201908.0037.v5 ◽

2020 ◽

Author(s):

Frank Vega

Keyword(s):

Computer Science ◽

Polynomial Time ◽

Polynomial Time Algorithm ◽

Time Algorithm ◽

Complexity Class ◽

Decision Problems ◽

Open Problems ◽

Precise Statement ◽

Np Problem

P versus NP is considered as one of the most important open problems in computer science. This consists in knowing the answer of the following question: Is P equal to NP? The precise statement of the P versus NP problem was introduced independently by Stephen Cook and Leonid Levin. Since that date, all efforts to find a proof for this problem have failed. Another major complexity class is P-Sel. P-Sel is the class of decision problems for which there is a polynomial time algorithm (called a selector) with the following property: Whenever it’s given two instances, a “yes” and a “no” instance, the algorithm can always decide which is the “yes” instance. It is known that if NP is contained in P-Sel, then P = NP. We claim a possible selector for 3SAT and thus, P = NP.

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Introduction: Where Statistical Physics Meets Computation

Computational Complexity and Statistical Physics ◽

10.1093/oso/9780195177374.003.0007 ◽

2005 ◽

Author(s):

Allon G. Percus ◽

Gabriel Istrate

Keyword(s):

Computational Complexity ◽

Computer Science ◽

Polynomial Time ◽

Statistical Physics ◽

Computation Time ◽

Combinatorial Problem ◽

Theoretical Computer Science ◽

Complexity Class ◽

Input Size ◽

Digital Computers

Computer science and physics have been closely linked since the birth of modern computing. This book is about that link. John von Neumann’s original design for digital computing in the 1940s was motivated by applications in ballistics and hydrodynamics, and his model still underlies today’s hardware architectures. Within several years of the invention of the first digital computers, the Monte Carlo method was developed, putting these devices to work simulating natural processes using the principles of statistical physics. It is difficult to imagine how computing might have evolved without the physical insights that nurtured it. It is impossible to imagine how physics would have evolved without computation. While digital computers quickly became indispensable, a true theoretical understanding of the efficiency of the computation process did not occur until twenty years later. In 1965, Hartmanis and Stearns [227] as well as Edmonds [139, 140] articulated the notion of computational complexity, categorizing algorithms according to how rapidly their time and space requirements grow with input size. The qualitative distinctions that computational complexity draws between algorithms form the foundation of theoretical computer science. Chief among these distinctions is that of polynomial versus exponential time. A combinatorial problem belongs in the complexity class P (polynomial time) if there exists an algorithm guaranteeing a solution in a computation time, or number of elementary steps of the algorithm, that grows at most polynomially with input size. Loosely speaking, such problems are considered computationally feasible. An example might be sorting a list of n numbers: even a particularly naive and inefficient algorithm for this will run in a number of steps that grows as O(n2), and so sorting is in the class P. A problem belongs in the complexity class NP (non-deterministic polynomial time) if it is merely possible to test, in polynomial time, whether a specific presumed solution is correct. Of course, P ⊆ NP: for any problem whose solution can be found in polynomial time, one can surely verify the validity of a presumed solution in polynomial time.

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Introduction

Combinatorics Probability Computing ◽

10.1017/s0963548315000085 ◽

2015 ◽

Vol 24 (4) ◽

pp. 584-584

Author(s):

PAUL BALISTER

Keyword(s):

Computer Science ◽

Theoretical Computer Science ◽

Special Issue ◽

Open Problems ◽

Theoretical Computer ◽

Discrete Probability ◽

The Common

This special issue is devoted to papers from the meeting on Combinatorics and Probability, held at the Mathematisches Forschungsinstitut in Oberwolfach from the 14th to 20th April 2013. The lectures at this meeting focused on the common themes of Combinatorics and Discrete Probability, with many of the problems studied originating in Theoretical Computer Science. The lectures, many of which were given by young participants, stimulated fruitful discussions. The fact that the participants work in different and yet related topics, and the open problems session held during the meeting, encouraged interesting discussions and collaborations.

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Complexity Results for 1-safe Nets

DAIMI Report Series ◽

10.7146/dpb.v22i455.6773 ◽

1993 ◽

Vol 22 (455) ◽

Cited By ~ 2

Author(s):

Allan Cheng ◽

Javier Esparza ◽

Jens Palsberg

Keyword(s):

Petri Nets ◽

Computer Science ◽

Free Choice ◽

Theoretical Computer Science ◽

Theoretical Computer ◽

Software Technology ◽

Complexity Results ◽

Np Complete

We study the complexity of several standard problems for 1-safe Petri nets and some of its subclasses. We prove that reachability, liveness, and deadlock are all PSPACE-complete for 1-safe nets. We also prove that deadlock is NP-complete for free-choice nets and for 1-safe free-choice nets. Finally, we prove that for arbitrary Petri nets, deadlock is equivalent to reachability and liveness. This paper is to be presented at FST & TCS, Foundations of Software Technology & Theoretical Computer Science, to be held 15-17 December 1993, in Bombay, India.A version of the paper with most proofs omitted is to appear in the proceedings.

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