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).evolutionary games | fixation probability | complexity classes E volution occurs in populations of reproducing individuals. Mutation generates distinct types. Selection favors some types over others. The mathematical formalism of evolution describes how populations change in their genetic (or phenotypic) composition over time. Deterministic models of evolution are based on differential equations. They assume infinitely large population size and ignore demographic and other stochasticity. The more precise descriptions of evolutionary dynamics, however, use stochastic processes, which take into account the intrinsic randomness of when and where individuals reproduce and how many of their offspring survive. They also describe populations of finite size.A well-known stochastic process of evolution was formulated by Moran in 1958 (1). In any one-time step, a random individual is chosen proportional to fitness for reproduction and a random individual is chosen for death. The offspring of the first individual is added to the population. The total population size remains constant and is given by N. The original process was formulated for constant fitness, which means the fitness value of individuals does not depend on the relative abundance of various types in the population; it is a fixed number. The crucial question is: What is the probability that a newly introduced mutant will generate a lineage that takes over the entire population? This quantity is called the fixation probability. For the original Moran process, the...
In two-player games on graphs, the players move a token through a graph to produce an infinite path, which determines the winner or payoff of the game. Such games are central in formal verification since they model the interaction between a non-terminating system and its environment. We study bidding games in which the players bid for the right to move the token. Bidding games with variants of firstprice auctions were previously studied: in each round, the players simultaneously submit bids, the higher bidder moves the token, and, in Richman bidding, pays his bid to the other player whereas in poorman bidding, pays his bid to the "bank". While reachability poorman games have been studied before, we present, for the first time, results on infinite-duration poorman games. A central quantity in these games is the ratio between the two players' initial budgets. We show that the favorable properties of reachability poorman games extend to complex qualitative objectives such as parity, similarly to the Richman case: each vertex has a threshold value, which is a necessary and sufficient ratio with which a player can achieve a goal. Our most interesting results concern quantitative poorman games, namely mean-payoff poorman games, where we construct optimal strategies depending on the initial ratio. The crux of the proof shows that strongly-connected mean-payoff poorman games are equivalent to biased random-turn games. The equivalence in itself is interesting, because it does not hold for reachability poorman games and it is richer than the equivalence with uniform random-turn games that Richman bidding exhibit. We also solve the complexity problems that arise in poorman games. * This paper is a full version of [7]. it visits t. The simplest mode of moving is turn based: the vertices are partitioned between the two players and whenever the token reaches a vertex that is controlled by a player, he decides how to move the token.We study a new mode of moving in infinite-duration games, which is called bidding, and in which the players bid for the right to move the token. The bidding mode of moving was introduced in [31, 32] for reachability games, where two variants of first-price auctions where studied: Each player has a budget, and before each move, the players submit sealed bids simultaneously, where a bid is legal if it does not exceed the available budget, and the higher bidder moves the token. The bidding rules differ in where the higher bidder pays his bid. In Richman bidding (named after David Richman), the higher bidder pays the lower bidder. In poorman bidding, which is the bidding rule that we focus on in this paper, the higher bidder pays the "bank". Thus, the bid is deducted from his budget and the money is lost. Note that while the sum of budgets is constant in Richman bidding, in poorman bidding, the sum of budgets shrinks as the game proceeds. One needs to devise a mechanism that resolves ties in biddings, and our results are not affected by the tie-breaking mechanism that is used.Bidding games naturally mode...
Interprocedural analysis is at the heart of numerous applications in programming languages, such as alias analysis, constant propagation, etc. Recursive state machines (RSMs) are standard models for interprocedural analysis. We consider a general framework with RSMs where the transitions are labeled from a semiring, and path properties are algebraic with semiring operations. RSMs with algebraic path properties can model interprocedural dataflow analysis problems, the shortest path problem, the most probable path problem, etc. The traditional algorithms for interprocedural analysis focus on path properties where the starting point is fixed as the entry point of a specific method. In this work, we consider possible multiple queries as required in many applications such as in alias analysis. The study of multiple queries allows us to bring in a very important algorithmic distinction between the resource usage of the one-time preprocessing vs for each individual query. The second aspect that we consider is that the control flow graphs for most programs have constant treewidth.Our main contributions are simple and implementable algorithms that support multiple queries for algebraic path properties for RSMs that have constant treewidth. Our theoretical results show that our algorithms have small additional one-time preprocessing, but can answer subsequent queries significantly faster as compared to the current best-known solutions for several important problems, such as interprocedural reachability and shortest path. We provide a prototype implementation for interprocedural reachability and intraprocedural shortest path that gives a significant speed-up on several benchmarks.
Two standard algorithms for approximately solving two-player zerosum concurrent reachability games are value iteration and strategy iteration. We prove upper and lower bounds of 2 m Θ(N ) on the worst case number of iterations needed by both of these algorithms for providing non-trivial approximations to the value of a game with N non-terminal positions and m actions for each player in each position. In particular, both algorithms have doubly-exponential complexity. Even when the game given as input has only one non-terminal position, we prove an exponential lower bound on the worst case number of iterations needed to provide non-trivial approximations.
Interprocedural data-flow analyses form an expressive and useful paradigm of numerous static analysis applications, such as live variables analysis, alias analysis and null pointers analysis. The most widely-used framework for interprocedural data-flow analysis is IFDS, which encompasses distributive data-flow functions over a finite domain. On-demand data-flow analyses restrict the focus of the analysis on specific program locations and data facts. This setting provides a natural split between (i) an offline (or preprocessing) phase, where the program is partially analyzed and analysis summaries are created, and (ii) an online (or query) phase, where analysis queries arrive on demand and the summaries are used to speed up answering queries. In this work, we consider on-demand IFDS analyses where the queries concern program locations of the same procedure (aka same-context queries). We exploit the fact that flow graphs of programs have low treewidth to develop faster algorithms that are space and time optimal for many common data-flow analyses, in both the preprocessing and the query phase. We also use treewidth to develop query solutions that are embarrassingly parallelizable, i.e. the total work for answering each query is split to a number of threads such that each thread performs only a constant amount of work. Finally, we implement a static analyzer based on our algorithms, and perform a series of on-demand analysis experiments on standard benchmarks. Our experimental results show a drastic speed-up of the queries after only a lightweight preprocessing phase, which significantly outperforms existing techniques.
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