TY - GEN AB - We study the problem of learning controllers for discrete-time non-linear stochastic dynamical systems with formal reach-avoid guarantees. This work presents the first method for providing formal reach-avoid guarantees, which combine and generalize stability and safety guarantees, with a tolerable probability threshold $p\in[0,1]$ over the infinite time horizon. Our method leverages advances in machine learning literature and it represents formal certificates as neural networks. In particular, we learn a certificate in the form of a reach-avoid supermartingale (RASM), a novel notion that we introduce in this work. Our RASMs provide reachability and avoidance guarantees by imposing constraints on what can be viewed as a stochastic extension of level sets of Lyapunov functions for deterministic systems. Our approach solves several important problems -- it can be used to learn a control policy from scratch, to verify a reach-avoid specification for a fixed control policy, or to fine-tune a pre-trained policy if it does not satisfy the reach-avoid specification. We validate our approach on $3$ stochastic non-linear reinforcement learning tasks. AU - Zikelic, Dorde AU - Lechner, Mathias AU - Henzinger, Thomas A AU - Chatterjee, Krishnendu ID - 14600 T2 - arXiv TI - Learning control policies for stochastic systems with reach-avoid guarantees ER - TY - CONF AB - A deterministic finite automaton (DFA) 𝒜 is composite if its language L(𝒜) can be decomposed into an intersection ⋂_{i = 1}^k L(𝒜_i) of languages of smaller DFAs. Otherwise, 𝒜 is prime. This notion of primality was introduced by Kupferman and Mosheiff in 2013, and while they proved that we can decide whether a DFA is composite, the precise complexity of this problem is still open, with a doubly-exponential gap between the upper and lower bounds. In this work, we focus on permutation DFAs, i.e., those for which the transition monoid is a group. We provide an NP algorithm to decide whether a permutation DFA is composite, and show that the difficulty of this problem comes from the number of non-accepting states of the instance: we give a fixed-parameter tractable algorithm with the number of rejecting states as the parameter. Moreover, we investigate the class of commutative permutation DFAs. Their structural properties allow us to decide compositionality in NL, and even in LOGSPACE if the alphabet size is fixed. Despite this low complexity, we show that complex behaviors still arise in this class: we provide a family of composite DFAs each requiring polynomially many factors with respect to its size. We also consider the variant of the problem that asks whether a DFA is k-factor composite, that is, decomposable into k smaller DFAs, for some given integer k ∈ ℕ. We show that, for commutative permutation DFAs, restricting the number of factors makes the decision computationally harder, and yields a problem with tight bounds: it is NP-complete. Finally, we show that in general, this problem is in PSPACE, and it is in LOGSPACE for DFAs with a singleton alphabet. AU - Jecker, Ismael R AU - Mazzocchi, Nicolas AU - Wolf, Petra ID - 10052 SN - 1868-8969 T2 - 32nd International Conference on Concurrency Theory TI - Decomposing permutation automata VL - 203 ER - TY - CONF AB - Graphs and games on graphs are fundamental models for the analysis of reactive systems, in particular, for model-checking and the synthesis of reactive systems. The class of ω-regular languages provides a robust specification formalism for the desired properties of reactive systems. In the classical infinitary formulation of the liveness part of an ω-regular specification, a "good" event must happen eventually without any bound between the good events. A stronger notion of liveness is bounded liveness, which requires that good events happen within d transitions. Given a graph or a game graph with n vertices, m edges, and a bounded liveness objective, the previous best-known algorithmic bounds are as follows: (i) O(dm) for graphs, which in the worst-case is O(nÂł); and (ii) O(nÂČ dÂČ) for games on graphs. Our main contributions improve these long-standing algorithmic bounds. For graphs we present: (i) a randomized algorithm with one-sided error with running time O(n^{2.5} log n) for the bounded liveness objectives; and (ii) a deterministic linear-time algorithm for the complement of bounded liveness objectives. For games on graphs, we present an O(nÂČ d) time algorithm for the bounded liveness objectives. AU - Chatterjee, Krishnendu AU - Henzinger, Monika H AU - Kale, Sagar Sudhir AU - Svozil, Alexander ID - 10054 SN - 1868-8969 T2 - 48th International Colloquium on Automata, Languages, and Programming TI - Faster algorithms for bounded liveness in graphs and game graphs VL - 198 ER - TY - CONF AB - We study the expressiveness and succinctness of good-for-games pushdown automata (GFG-PDA) over finite words, that is, pushdown automata whose nondeterminism can be resolved based on the run constructed so far, but independently of the remainder of the input word. We prove that GFG-PDA recognise more languages than deterministic PDA (DPDA) but not all context-free languages (CFL). This class is orthogonal to unambiguous CFL. We further show that GFG-PDA can be exponentially more succinct than DPDA, while PDA can be double-exponentially more succinct than GFG-PDA. We also study GFGness in visibly pushdown automata (VPA), which enjoy better closure properties than PDA, and for which we show GFGness to be ExpTime-complete. GFG-VPA can be exponentially more succinct than deterministic VPA, while VPA can be exponentially more succinct than GFG-VPA. Both of these lower bounds are tight. Finally, we study the complexity of resolving nondeterminism in GFG-PDA. Every GFG-PDA has a positional resolver, a function that resolves nondeterminism and that is only dependant on the current configuration. Pushdown transducers are sufficient to implement the resolvers of GFG-VPA, but not those of GFG-PDA. GFG-PDA with finite-state resolvers are determinisable. AU - Guha, Shibashis AU - Jecker, Ismael R AU - Lehtinen, Karoliina AU - Zimmermann, Martin ID - 10075 SN - 1868-8969 T2 - 46th International Symposium on Mathematical Foundations of Computer Science TI - A bit of nondeterminism makes pushdown automata expressive and succinct VL - 202 ER - TY - CONF AB - In the Intersection Non-emptiness problem, we are given a list of finite automata A_1, A_2,
 , A_m over a common alphabet ÎŁ as input, and the goal is to determine whether some string w ∈ ÎŁ^* lies in the intersection of the languages accepted by the automata in the list. We analyze the complexity of the Intersection Non-emptiness problem under the promise that all input automata accept a language in some level of the dot-depth hierarchy, or some level of the Straubing-ThĂ©rien hierarchy. Automata accepting languages from the lowest levels of these hierarchies arise naturally in the context of model checking. We identify a dichotomy in the dot-depth hierarchy by showing that the problem is already NP-complete when all input automata accept languages of the levels B_0 or B_{1/2} and already PSPACE-hard when all automata accept a language from the level B_1. Conversely, we identify a tetrachotomy in the Straubing-ThĂ©rien hierarchy. More precisely, we show that the problem is in AC^0 when restricted to level L_0; complete for L or NL, depending on the input representation, when restricted to languages in the level L_{1/2}; NP-complete when the input is given as DFAs accepting a language in L_1 or L_{3/2}; and finally, PSPACE-complete when the input automata accept languages in level L_2 or higher. Moreover, we show that the proof technique used to show containment in NP for DFAs accepting languages in L_1 or L_{3/2} does not generalize to the context of NFAs. To prove this, we identify a family of languages that provide an exponential separation between the state complexity of general NFAs and that of partially ordered NFAs. To the best of our knowledge, this is the first superpolynomial separation between these two models of computation. AU - Arrighi, Emmanuel AU - Fernau, Henning AU - Hoffmann, Stefan AU - Holzer, Markus AU - Jecker, Ismael R AU - De Oliveira Oliveira, Mateus AU - Wolf, Petra ID - 10630 SN - 1868-8969 T2 - 41st IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science TI - On the complexity of intersection non-emptiness for star-free language classes VL - 213 ER - TY - CONF AB - Product graphs arise naturally in formal verification and program analysis. For example, the analysis of two concurrent threads requires the product of two component control-flow graphs, and for language inclusion of deterministic automata the product of two automata is constructed. In many cases, the component graphs have constant treewidth, e.g., when the input contains control-flow graphs of programs. We consider the algorithmic analysis of products of two constant-treewidth graphs with respect to three classic specification languages, namely, (a) algebraic properties, (b) mean-payoff properties, and (c) initial credit for energy properties. Our main contributions are as follows. Consider a graph G that is the product of two constant-treewidth graphs of size n each. First, given an idempotent semiring, we present an algorithm that computes the semiring transitive closure of G in time Õ(n⁎). Since the output has size Θ(n⁎), our algorithm is optimal (up to polylog factors). Second, given a mean-payoff objective, we present an O(nÂł)-time algorithm for deciding whether the value of a starting state is non-negative, improving the previously known O(n⁎) bound. Third, given an initial credit for energy objective, we present an O(n⁔)-time algorithm for computing the minimum initial credit for all nodes of G, improving the previously known O(n⁞) bound. At the heart of our approach lies an algorithm for the efficient construction of strongly-balanced tree decompositions of constant-treewidth graphs. Given a constant-treewidth graph G' of n nodes and a positive integer λ, our algorithm constructs a binary tree decomposition of G' of width O(λ) with the property that the size of each subtree decreases geometrically with rate (1/2 + 2^{-λ}). AU - Chatterjee, Krishnendu AU - Ibsen-Jensen, Rasmus AU - Pavlogiannis, Andreas ID - 10629 SN - 1868-8969 T2 - 41st IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science TI - Quantitative verification on product graphs of small treewidth VL - 213 ER - TY - CONF AB - In a two-player zero-sum graph game the players move a token throughout a graph to produce an infinite path, which determines the winner or payoff of the game. Traditionally, the players alternate turns in moving the token. In bidding games, however, the players have budgets, and in each turn, we hold an “auction” (bidding) to determine which player moves the token: both players simultaneously submit bids and the higher bidder moves the token. The bidding mechanisms differ in their payment schemes. Bidding games were largely studied with variants of first-price bidding in which only the higher bidder pays his bid. We focus on all-pay bidding, where both players pay their bids. Finite-duration all-pay bidding games were studied and shown to be technically more challenging than their first-price counterparts. We study for the first time, infinite-duration all-pay bidding games. Our most interesting results are for mean-payoff objectives: we portray a complete picture for games played on strongly-connected graphs. We study both pure (deterministic) and mixed (probabilistic) strategies and completely characterize the optimal and almost-sure (with probability 1) payoffs the players can respectively guarantee. We show that mean-payoff games under all-pay bidding exhibit the intriguing mathematical properties of their first-price counterparts; namely, an equivalence with random-turn games in which in each turn, the player who moves is selected according to a (biased) coin toss. The equivalences for all-pay bidding are more intricate and unexpected than for first-price bidding. AU - Avni, Guy AU - Jecker, Ismael R AU - Zikelic, Dorde ED - Marx, DĂĄniel ID - 10694 SN - 978-1-61197-646-5 T2 - Proceedings of the 2021 ACM-SIAM Symposium on Discrete Algorithms TI - Infinite-duration all-pay bidding games ER - TY - CONF AB - We study the two-player zero-sum extension of the partially observable stochastic shortest-path problem where one agent has only partial information about the environment. We formulate this problem as a partially observable stochastic game (POSG): given a set of target states and negative rewards for each transition, the player with imperfect information maximizes the expected undiscounted total reward until a target state is reached. The second player with the perfect information aims for the opposite. We base our formalism on POSGs with one-sided observability (OS-POSGs) and give the following contributions: (1) we introduce a novel heuristic search value iteration algorithm that iteratively solves depth-limited variants of the game, (2) we derive the bound on the depth guaranteeing an arbitrary precision, (3) we propose a novel upper-bound estimation that allows early terminations, and (4) we experimentally evaluate the algorithm on a pursuit-evasion game. AU - TomĂĄĆĄek, Petr AU - HorĂĄk, Karel AU - Aradhye, Aditya AU - BoĆĄanskĂœ, Branislav AU - Chatterjee, Krishnendu ID - 10847 SN - 1045-0823 T2 - 30th International Joint Conference on Artificial Intelligence TI - Solving partially observable stochastic shortest-path games ER - TY - CONF AB - matching is compatible to two or more labeled point sets of size n with labels {1,
,n} if its straight-line drawing on each of these point sets is crossing-free. We study the maximum number of edges in a matching compatible to two or more labeled point sets in general position in the plane. We show that for any two labeled convex sets of n points there exists a compatible matching with ⌊2n−−√⌋ edges. More generally, for any ℓ labeled point sets we construct compatible matchings of size Ω(n1/ℓ) . As a corresponding upper bound, we use probabilistic arguments to show that for any ℓ given sets of n points there exists a labeling of each set such that the largest compatible matching has O(n2/(ℓ+1)) edges. Finally, we show that Θ(logn) copies of any set of n points are necessary and sufficient for the existence of a labeling such that any compatible matching consists only of a single edge. AU - Aichholzer, Oswin AU - Arroyo Guevara, Alan M AU - MasĂĄrovĂĄ, Zuzana AU - Parada, Irene AU - Perz, Daniel AU - Pilz, Alexander AU - Tkadlec, Josef AU - Vogtenhuber, Birgit ID - 9296 SN - 03029743 T2 - 15th International Conference on Algorithms and Computation TI - On compatible matchings VL - 12635 ER - TY - CHAP AB - Optimal decision making requires individuals to know their available options and to anticipate correctly what consequences these options have. In many social interactions, however, we refrain from gathering all relevant information, even if this information would help us make better decisions and is costless to obtain. This chapter examines several examples of “deliberate ignorance.” Two simple models are proposed to illustrate how ignorance can evolve among self-interested and payoff - maximizing individuals, and open problems are highlighted that lie ahead for future research to explore. AU - Schmid, Laura AU - Hilbe, Christian ED - Hertwig, Ralph ED - Engel, Christoph ID - 9403 SN - 978-0-262-04559-9 T2 - Deliberate Ignorance: Choosing Not To Know TI - The evolution of strategic ignorance in strategic interaction VL - 29 ER - TY - CONF AB - Several problems in planning and reactive synthesis can be reduced to the analysis of two-player quantitative graph games. Optimization is one form of analysis. We argue that in many cases it may be better to replace the optimization problem with the satisficing problem, where instead of searching for optimal solutions, the goal is to search for solutions that adhere to a given threshold bound. This work defines and investigates the satisficing problem on a two-player graph game with the discounted-sum cost model. We show that while the satisficing problem can be solved using numerical methods just like the optimization problem, this approach does not render compelling benefits over optimization. When the discount factor is, however, an integer, we present another approach to satisficing, which is purely based on automata methods. We show that this approach is algorithmically more performant – both theoretically and empirically – and demonstrates the broader applicability of satisficing over optimization. AU - Bansal, Suguman AU - Chatterjee, Krishnendu AU - Vardi, Moshe Y. ID - 12767 SN - 0302-9743 T2 - 27th International Conference on Tools and Algorithms for the Construction and Analysis of Systems TI - On satisficing in quantitative games VL - 12651 ER - TY - CONF AB - Bayesian neural networks (BNNs) place distributions over the weights of a neural network to model uncertainty in the data and the network's prediction. We consider the problem of verifying safety when running a Bayesian neural network policy in a feedback loop with infinite time horizon systems. Compared to the existing sampling-based approaches, which are inapplicable to the infinite time horizon setting, we train a separate deterministic neural network that serves as an infinite time horizon safety certificate. In particular, we show that the certificate network guarantees the safety of the system over a subset of the BNN weight posterior's support. Our method first computes a safe weight set and then alters the BNN's weight posterior to reject samples outside this set. Moreover, we show how to extend our approach to a safe-exploration reinforcement learning setting, in order to avoid unsafe trajectories during the training of the policy. We evaluate our approach on a series of reinforcement learning benchmarks, including non-Lyapunovian safety specifications. AU - Lechner, Mathias AU - Ćœikelić, Ðorđe AU - Chatterjee, Krishnendu AU - Henzinger, Thomas A ID - 10667 T2 - 35th Conference on Neural Information Processing Systems TI - Infinite time horizon safety of Bayesian neural networks ER - TY - JOUR AB - We study optimal election sequences for repeatedly selecting a (very) small group of leaders among a set of participants (players) with publicly known unique ids. In every time slot, every player has to select exactly one player that it considers to be the current leader, oblivious to the selection of the other players, but with the overarching goal of maximizing a given parameterized global (“social”) payoff function in the limit. We consider a quite generic model, where the local payoff achieved by a given player depends, weighted by some arbitrary but fixed real parameter, on the number of different leaders chosen in a round, the number of players that choose the given player as the leader, and whether the chosen leader has changed w.r.t. the previous round or not. The social payoff can be the maximum, average or minimum local payoff of the players. Possible applications include quite diverse examples such as rotating coordinator-based distributed algorithms and long-haul formation flying of social birds. Depending on the weights and the particular social payoff, optimal sequences can be very different, from simple round-robin where all players chose the same leader alternatingly every time slot to very exotic patterns, where a small group of leaders (at most 2) is elected in every time slot. Moreover, we study the question if and when a single player would not benefit w.r.t. its local payoff when deviating from the given optimal sequence, i.e., when our optimal sequences are Nash equilibria in the restricted strategy space of oblivious strategies. As this is the case for many parameterizations of our model, our results reveal that no punishment is needed to make it rational for the players to optimize the social payoff. AU - Zeiner, Martin AU - Schmid, Ulrich AU - Chatterjee, Krishnendu ID - 8793 IS - 1 JF - Discrete Applied Mathematics SN - 0166218X TI - Optimal strategies for selecting coordinators VL - 289 ER - TY - JOUR AB - A game of rock-paper-scissors is an interesting example of an interaction where none of the pure strategies strictly dominates all others, leading to a cyclic pattern. In this work, we consider an unstable version of rock-paper-scissors dynamics and allow individuals to make behavioural mistakes during the strategy execution. We show that such an assumption can break a cyclic relationship leading to a stable equilibrium emerging with only one strategy surviving. We consider two cases: completely random mistakes when individuals have no bias towards any strategy and a general form of mistakes. Then, we determine conditions for a strategy to dominate all other strategies. However, given that individuals who adopt a dominating strategy are still prone to behavioural mistakes in the observed behaviour, we may still observe extinct strategies. That is, behavioural mistakes in strategy execution stabilise evolutionary dynamics leading to an evolutionary stable and, potentially, mixed co-existence equilibrium. AU - Kleshnina, Maria AU - Streipert, Sabrina S. AU - Filar, Jerzy A. AU - Chatterjee, Krishnendu ID - 9381 IS - 4 JF - PLoS Computational Biology SN - 1553734X TI - Mistakes can stabilise the dynamics of rock-paper-scissors games VL - 17 ER - TY - JOUR AB - Selection and random drift determine the probability that novel mutations fixate in a population. Population structure is known to affect the dynamics of the evolutionary process. Amplifiers of selection are population structures that increase the fixation probability of beneficial mutants compared to well-mixed populations. Over the past 15 years, extensive research has produced remarkable structures called strong amplifiers which guarantee that every beneficial mutation fixates with high probability. But strong amplification has come at the cost of considerably delaying the fixation event, which can slow down the overall rate of evolution. However, the precise relationship between fixation probability and time has remained elusive. Here we characterize the slowdown effect of strong amplification. First, we prove that all strong amplifiers must delay the fixation event at least to some extent. Second, we construct strong amplifiers that delay the fixation event only marginally as compared to the well-mixed populations. Our results thus establish a tight relationship between fixation probability and time: Strong amplification always comes at a cost of a slowdown, but more than a marginal slowdown is not needed. AU - Tkadlec, Josef AU - Pavlogiannis, Andreas AU - Chatterjee, Krishnendu AU - Nowak, Martin A. ID - 9640 IS - 1 JF - Nature Communications TI - Fast and strong amplifiers of natural selection VL - 12 ER - TY - CONF AB - We consider the fundamental problem of deriving quantitative bounds on the probability that a given assertion is violated in a probabilistic program. We provide automated algorithms that obtain both lower and upper bounds on the assertion violation probability. The main novelty of our approach is that we prove new and dedicated fixed-point theorems which serve as the theoretical basis of our algorithms and enable us to reason about assertion violation bounds in terms of pre and post fixed-point functions. To synthesize such fixed-points, we devise algorithms that utilize a wide range of mathematical tools, including repulsing ranking supermartingales, Hoeffding's lemma, Minkowski decompositions, Jensen's inequality, and convex optimization. On the theoretical side, we provide (i) the first automated algorithm for lower-bounds on assertion violation probabilities, (ii) the first complete algorithm for upper-bounds of exponential form in affine programs, and (iii) provably and significantly tighter upper-bounds than the previous approaches. On the practical side, we show our algorithms can handle a wide variety of programs from the literature and synthesize bounds that are remarkably tighter than previous results, in some cases by thousands of orders of magnitude. AU - Wang, Jinyi AU - Sun, Yican AU - Fu, Hongfei AU - Chatterjee, Krishnendu AU - Goharshady, Amir Kafshdar ID - 9646 SN - 9781450383912 T2 - Proceedings of the 42nd ACM SIGPLAN International Conference on Programming Language Design and Implementation TI - Quantitative analysis of assertion violations in probabilistic programs ER - TY - CONF AB - We consider the fundamental problem of reachability analysis over imperative programs with real variables. Previous works that tackle reachability are either unable to handle programs consisting of general loops (e.g. symbolic execution), or lack completeness guarantees (e.g. abstract interpretation), or are not automated (e.g. incorrectness logic). In contrast, we propose a novel approach for reachability analysis that can handle general and complex loops, is complete, and can be entirely automated for a wide family of programs. Through the notion of Inductive Reachability Witnesses (IRWs), our approach extends ideas from both invariant generation and termination to reachability analysis. We first show that our IRW-based approach is sound and complete for reachability analysis of imperative programs. Then, we focus on linear and polynomial programs and develop automated methods for synthesizing linear and polynomial IRWs. In the linear case, we follow the well-known approaches using Farkas' Lemma. Our main contribution is in the polynomial case, where we present a push-button semi-complete algorithm. We achieve this using a novel combination of classical theorems in real algebraic geometry, such as Putinar's Positivstellensatz and Hilbert's Strong Nullstellensatz. Finally, our experimental results show we can prove complex reachability objectives over various benchmarks that were beyond the reach of previous methods. AU - Asadi, Ali AU - Chatterjee, Krishnendu AU - Fu, Hongfei AU - Goharshady, Amir Kafshdar AU - Mahdavi, Mohammad ID - 9645 SN - 9781450383912 T2 - Proceedings of the 42nd ACM SIGPLAN International Conference on Programming Language Design and Implementation TI - Polynomial reachability witnesses via StellensĂ€tze ER - TY - CONF AB - We present a faster symbolic algorithm for the following central problem in probabilistic verification: Compute the maximal end-component (MEC) decomposition of Markov decision processes (MDPs). This problem generalizes the SCC decomposition problem of graphs and closed recurrent sets of Markov chains. The model of symbolic algorithms is widely used in formal verification and model-checking, where access to the input model is restricted to only symbolic operations (e.g., basic set operations and computation of one-step neighborhood). For an input MDP with n vertices and m edges, the classical symbolic algorithm from the 1990s for the MEC decomposition requires O(n2) symbolic operations and O(1) symbolic space. The only other symbolic algorithm for the MEC decomposition requires O(nm−−√) symbolic operations and O(m−−√) symbolic space. A main open question is whether the worst-case O(n2) bound for symbolic operations can be beaten. We present a symbolic algorithm that requires O˜(n1.5) symbolic operations and O˜(n−−√) symbolic space. Moreover, the parametrization of our algorithm provides a trade-off between symbolic operations and symbolic space: for all 0<ϔ≀1/2 the symbolic algorithm requires O˜(n2−ϔ) symbolic operations and O˜(nÏ”) symbolic space ( O˜ hides poly-logarithmic factors). Using our techniques we present faster algorithms for computing the almost-sure winning regions of ω -regular objectives for MDPs. We consider the canonical parity objectives for ω -regular objectives, and for parity objectives with d -priorities we present an algorithm that computes the almost-sure winning region with O˜(n2−ϔ) symbolic operations and O˜(nÏ”) symbolic space, for all 0<ϔ≀1/2 . AU - Chatterjee, Krishnendu AU - Dvorak, Wolfgang AU - Henzinger, Monika H AU - Svozil, Alexander ID - 10002 KW - Computer science KW - Computational modeling KW - Markov processes KW - Probabilistic logic KW - Formal verification KW - Game Theory SN - 1043-6871 T2 - Proceedings of the 36th Annual ACM/IEEE Symposium on Logic in Computer Science TI - Symbolic time and space tradeoffs for probabilistic verification ER - TY - CONF AB - Markov chains are the de facto finite-state model for stochastic dynamical systems, and Markov decision processes (MDPs) extend Markov chains by incorporating non-deterministic behaviors. Given an MDP and rewards on states, a classical optimization criterion is the maximal expected total reward where the MDP stops after T steps, which can be computed by a simple dynamic programming algorithm. We consider a natural generalization of the problem where the stopping times can be chosen according to a probability distribution, such that the expected stopping time is T, to optimize the expected total reward. Quite surprisingly we establish inter-reducibility of the expected stopping-time problem for Markov chains with the Positivity problem (which is related to the well-known Skolem problem), for which establishing either decidability or undecidability would be a major breakthrough. Given the hardness of the exact problem, we consider the approximate version of the problem: we show that it can be solved in exponential time for Markov chains and in exponential space for MDPs. AU - Chatterjee, Krishnendu AU - Doyen, Laurent ID - 10004 KW - Computer science KW - Heuristic algorithms KW - Memory management KW - Automata KW - Markov processes KW - Probability distribution KW - Complexity theory SN - 1043-6871 T2 - Proceedings of the 36th Annual ACM/IEEE Symposium on Logic in Computer Science TI - Stochastic processes with expected stopping time ER - TY - CONF AB - Repeated idempotent elements are commonly used to characterise iterable behaviours in abstract models of computation. Therefore, given a monoid M, it is natural to ask how long a sequence of elements of M needs to be to ensure the presence of consecutive idempotent factors. This question is formalised through the notion of the Ramsey function R_M associated to M, obtained by mapping every k ∈ ℕ to the minimal integer R_M(k) such that every word u ∈ M^* of length R_M(k) contains k consecutive non-empty factors that correspond to the same idempotent element of M. In this work, we study the behaviour of the Ramsey function R_M by investigating the regular 𝒟-length of M, defined as the largest size L(M) of a submonoid of M isomorphic to the set of natural numbers {1,2, 
, L(M)} equipped with the max operation. We show that the regular 𝒟-length of M determines the degree of R_M, by proving that k^L(M) ≀ R_M(k) ≀ (k|M|⁎)^L(M). To allow applications of this result, we provide the value of the regular 𝒟-length of diverse monoids. In particular, we prove that the full monoid of n × n Boolean matrices, which is used to express transition monoids of non-deterministic automata, has a regular 𝒟-length of (nÂČ+n+2)/2. AU - Jecker, Ismael R ID - 10055 SN - 1868-8969 T2 - 38th International Symposium on Theoretical Aspects of Computer Science TI - A Ramsey theorem for finite monoids VL - 187 ER - TY - CONF AB - Stateless model checking (SMC) is one of the standard approaches to the verification of concurrent programs. As scheduling non-determinism creates exponentially large spaces of thread interleavings, SMC attempts to partition this space into equivalence classes and explore only a few representatives from each class. The efficiency of this approach depends on two factors: (a) the coarseness of the partitioning, and (b) the time to generate representatives in each class. For this reason, the search for coarse partitionings that are efficiently explorable is an active research challenge. In this work we present RVF-SMC , a new SMC algorithm that uses a novel reads-value-from (RVF) partitioning. Intuitively, two interleavings are deemed equivalent if they agree on the value obtained in each read event, and read events induce consistent causal orderings between them. The RVF partitioning is provably coarser than recent approaches based on Mazurkiewicz and “reads-from” partitionings. Our experimental evaluation reveals that RVF is quite often a very effective equivalence, as the underlying partitioning is exponentially coarser than other approaches. Moreover, RVF-SMC generates representatives very efficiently, as the reduction in the partitioning is often met with significant speed-ups in the model checking task. AU - Agarwal, Pratyush AU - Chatterjee, Krishnendu AU - Pathak, Shreya AU - Pavlogiannis, Andreas AU - Toman, Viktor ID - 9987 SN - 0302-9743 T2 - 33rd International Conference on Computer-Aided Verification TI - Stateless model checking under a reads-value-from equivalence VL - 12759 ER - TY - JOUR AB - In this work we solve the algorithmic problem of consistency verification for the TSO and PSO memory models given a reads-from map, denoted VTSO-rf and VPSO-rf, respectively. For an execution of n events over k threads and d variables, we establish novel bounds that scale as nk+1 for TSO and as nk+1· min(nk2, 2k· d) for PSO. Moreover, based on our solution to these problems, we develop an SMC algorithm under TSO and PSO that uses the RF equivalence. The algorithm is exploration-optimal, in the sense that it is guaranteed to explore each class of the RF partitioning exactly once, and spends polynomial time per class when k is bounded. Finally, we implement all our algorithms in the SMC tool Nidhugg, and perform a large number of experiments over benchmarks from existing literature. Our experimental results show that our algorithms for VTSO-rf and VPSO-rf provide significant scalability improvements over standard alternatives. Moreover, when used for SMC, the RF partitioning is often much coarser than the standard Shasha-Snir partitioning for TSO/PSO, which yields a significant speedup in the model checking task. AU - Bui, Truc Lam AU - Chatterjee, Krishnendu AU - Gautam, Tushar AU - Pavlogiannis, Andreas AU - Toman, Viktor ID - 10191 IS - OOPSLA JF - Proceedings of the ACM on Programming Languages KW - safety KW - risk KW - reliability and quality KW - software TI - The reads-from equivalence for the TSO and PSO memory models VL - 5 ER - TY - THES AB - The design and verification of concurrent systems remains an open challenge due to the non-determinism that arises from the inter-process communication. In particular, concurrent programs are notoriously difficult both to be written correctly and to be analyzed formally, as complex thread interaction has to be accounted for. The difficulties are further exacerbated when concurrent programs get executed on modern-day hardware, which contains various buffering and caching mechanisms for efficiency reasons. This causes further subtle non-determinism, which can often produce very unintuitive behavior of the concurrent programs. Model checking is at the forefront of tackling the verification problem, where the task is to decide, given as input a concurrent system and a desired property, whether the system satisfies the property. The inherent state-space explosion problem in model checking of concurrent systems causes naĂŻve explicit methods not to scale, thus more inventive methods are required. One such method is stateless model checking (SMC), which explores in memory-efficient manner the program executions rather than the states of the program. State-of-the-art SMC is typically coupled with partial order reduction (POR) techniques, which argue that certain executions provably produce identical system behavior, thus limiting the amount of executions one needs to explore in order to cover all possible behaviors. Another method to tackle the state-space explosion is symbolic model checking, where the considered techniques operate on a succinct implicit representation of the input system rather than explicitly accessing the system. In this thesis we present new techniques for verification of concurrent systems. We present several novel POR methods for SMC of concurrent programs under various models of semantics, some of which account for write-buffering mechanisms. Additionally, we present novel algorithms for symbolic model checking of finite-state concurrent systems, where the desired property of the systems is to ensure a formally defined notion of fairness. AU - Toman, Viktor ID - 10199 KW - concurrency KW - verification KW - model checking SN - 2663-337X TI - Improved verification techniques for concurrent systems ER - TY - JOUR AB - We consider planning problems for graphs, Markov Decision Processes (MDPs), and games on graphs in an explicit state space. While graphs represent the most basic planning model, MDPs represent interaction with nature and games on graphs represent interaction with an adversarial environment. We consider two planning problems with k different target sets: (a) the coverage problem asks whether there is a plan for each individual target set; and (b) the sequential target reachability problem asks whether the targets can be reached in a given sequence. For the coverage problem, we present a linear-time algorithm for graphs, and quadratic conditional lower bound for MDPs and games on graphs. For the sequential target problem, we present a linear-time algorithm for graphs, a sub-quadratic algorithm for MDPs, and a quadratic conditional lower bound for games on graphs. Our results with conditional lower bounds, based on the boolean matrix multiplication (BMM) conjecture and strong exponential time hypothesis (SETH), establish (i) model-separation results showing that for the coverage problem MDPs and games on graphs are harder than graphs, and for the sequential reachability problem games on graphs are harder than MDPs and graphs; and (ii) problem-separation results showing that for MDPs the coverage problem is harder than the sequential target problem. AU - Chatterjee, Krishnendu AU - Dvoƙák, Wolfgang AU - Henzinger, Monika H AU - Svozil, Alexander ID - 9293 IS - 8 JF - Artificial Intelligence SN - 0004-3702 TI - Algorithms and conditional lower bounds for planning problems VL - 297 ER - TY - JOUR AB - We consider the core algorithmic problems related to verification of systems with respect to three classical quantitative properties, namely, the mean-payoff, the ratio, and the minimum initial credit for energy property. The algorithmic problem given a graph and a quantitative property asks to compute the optimal value (the infimum value over all traces) from every node of the graph. We consider graphs with bounded treewidth—a class that contains the control flow graphs of most programs. Let n denote the number of nodes of a graph, m the number of edges (for bounded treewidth 𝑚=𝑂(𝑛)) and W the largest absolute value of the weights. Our main theoretical results are as follows. First, for the minimum initial credit problem we show that (1) for general graphs the problem can be solved in 𝑂(𝑛2⋅𝑚) time and the associated decision problem in 𝑂(𝑛⋅𝑚) time, improving the previous known 𝑂(𝑛3⋅𝑚⋅log(𝑛⋅𝑊)) and 𝑂(𝑛2⋅𝑚) bounds, respectively; and (2) for bounded treewidth graphs we present an algorithm that requires 𝑂(𝑛⋅log𝑛) time. Second, for bounded treewidth graphs we present an algorithm that approximates the mean-payoff value within a factor of 1+𝜖 in time 𝑂(𝑛⋅log(𝑛/𝜖)) as compared to the classical exact algorithms on general graphs that require quadratic time. Third, for the ratio property we present an algorithm that for bounded treewidth graphs works in time 𝑂(𝑛⋅log(|𝑎⋅𝑏|))=𝑂(𝑛⋅log(𝑛⋅𝑊)), when the output is 𝑎𝑏, as compared to the previously best known algorithm on general graphs with running time 𝑂(𝑛2⋅log(𝑛⋅𝑊)). We have implemented some of our algorithms and show that they present a significant speedup on standard benchmarks. AU - Chatterjee, Krishnendu AU - Ibsen-Jensen, Rasmus AU - Pavlogiannis, Andreas ID - 9393 JF - Formal Methods in System Design SN - 0925-9856 TI - Faster algorithms for quantitative verification in bounded treewidth graphs VL - 57 ER - TY - CONF AB - We present a new approach to proving non-termination of non-deterministic integer programs. Our technique is rather simple but efficient. It relies on a purely syntactic reversal of the program's transition system followed by a constraint-based invariant synthesis with constraints coming from both the original and the reversed transition system. The latter task is performed by a simple call to an off-the-shelf SMT-solver, which allows us to leverage the latest advances in SMT-solving. Moreover, our method offers a combination of features not present (as a whole) in previous approaches: it handles programs with non-determinism, provides relative completeness guarantees and supports programs with polynomial arithmetic. The experiments performed with our prototype tool RevTerm show that our approach, despite its simplicity and stronger theoretical guarantees, is at least on par with the state-of-the-art tools, often achieving a non-trivial improvement under a proper configuration of its parameters. AU - Chatterjee, Krishnendu AU - Goharshady, Ehsan Kafshdar AU - NovotnĂœ, Petr AU - Zikelic, Dorde ID - 9644 SN - 9781450383912 T2 - Proceedings of the 42nd ACM SIGPLAN International Conference on Programming Language Design and Implementation TI - Proving non-termination by program reversal ER - TY - CONF AB - We consider the almost-sure (a.s.) termination problem for probabilistic programs, which are a stochastic extension of classical imperative programs. Lexicographic ranking functions provide a sound and practical approach for termination of non-probabilistic programs, and their extension to probabilistic programs is achieved via lexicographic ranking supermartingales (LexRSMs). However, LexRSMs introduced in the previous work have a limitation that impedes their automation: all of their components have to be non-negative in all reachable states. This might result in LexRSM not existing even for simple terminating programs. Our contributions are twofold: First, we introduce a generalization of LexRSMs which allows for some components to be negative. This standard feature of non-probabilistic termination proofs was hitherto not known to be sound in the probabilistic setting, as the soundness proof requires a careful analysis of the underlying stochastic process. Second, we present polynomial-time algorithms using our generalized LexRSMs for proving a.s. termination in broad classes of linear-arithmetic programs. AU - Chatterjee, Krishnendu AU - Goharshady, Ehsan Kafshdar AU - NovotnĂœ, Petr AU - ZĂĄrevĂșcky, Jiƙí AU - Zikelic, Dorde ID - 10414 SN - 0302-9743 T2 - 24th International Symposium on Formal Methods TI - On lexicographic proof rules for probabilistic termination VL - 13047 ER - TY - THES AB - In this thesis, we consider several of the most classical and fundamental problems in static analysis and formal verification, including invariant generation, reachability analysis, termination analysis of probabilistic programs, data-flow analysis, quantitative analysis of Markov chains and Markov decision processes, and the problem of data packing in cache management. We use techniques from parameterized complexity theory, polyhedral geometry, and real algebraic geometry to significantly improve the state-of-the-art, in terms of both scalability and completeness guarantees, for the mentioned problems. In some cases, our results are the first theoretical improvements for the respective problems in two or three decades. AU - Goharshady, Amir Kafshdar ID - 8934 SN - 2663-337X TI - Parameterized and algebro-geometric advances in static program analysis ER - TY - THES AB - Indirect reciprocity in evolutionary game theory is a prominent mechanism for explaining the evolution of cooperation among unrelated individuals. In contrast to direct reciprocity, which is based on individuals meeting repeatedly, and conditionally cooperating by using their own experiences, indirect reciprocity is based on individuals’ reputations. If a player helps another, this increases the helper’s public standing, benefitting them in the future. This lets cooperation in the population emerge without individuals having to meet more than once. While the two modes of reciprocity are intertwined, they are difficult to compare. Thus, they are usually studied in isolation. Direct reciprocity can maintain cooperation with simple strategies, and is robust against noise even when players do not remember more than their partner’s last action. Meanwhile, indirect reciprocity requires its successful strategies, or social norms, to be more complex. Exhaustive search previously identified eight such norms, called the “leading eight”, which excel at maintaining cooperation. However, as the first result of this thesis, we show that the leading eight break down once we remove the fundamental assumption that information is synchronized and public, such that everyone agrees on reputations. Once we consider a more realistic scenario of imperfect information, where reputations are private, and individuals occasionally misinterpret or miss observations, the leading eight do not promote cooperation anymore. Instead, minor initial disagreements can proliferate, fragmenting populations into subgroups. In a next step, we consider ways to mitigate this issue. We first explore whether introducing “generosity” can stabilize cooperation when players use the leading eight strategies in noisy environments. This approach of modifying strategies to include probabilistic elements for coping with errors is known to work well in direct reciprocity. However, as we show here, it fails for the more complex norms of indirect reciprocity. Imperfect information still prevents cooperation from evolving. On the other hand, we succeeded to show in this thesis that modifying the leading eight to use “quantitative assessment”, i.e. tracking reputation scores on a scale beyond good and bad, and making overall judgments of others based on a threshold, is highly successful, even when noise increases in the environment. Cooperation can flourish when reputations are more nuanced, and players have a broader understanding what it means to be “good.” Finally, we present a single theoretical framework that unites the two modes of reciprocity despite their differences. Within this framework, we identify a novel simple and successful strategy for indirect reciprocity, which can cope with noisy environments and has an analogue in direct reciprocity. We can also analyze decision making when different sources of information are available. Our results help highlight that for sustaining cooperation, already the most simple rules of reciprocity can be sufficient. AU - Schmid, Laura ID - 10293 SN - 2663-337X TI - Evolution of cooperation via (in)direct reciprocity under imperfect information ER - TY - JOUR AB - Indirect reciprocity is a mechanism for the evolution of cooperation based on social norms. This mechanism requires that individuals in a population observe and judge each other’s behaviors. Individuals with a good reputation are more likely to receive help from others. Previous work suggests that indirect reciprocity is only effective when all relevant information is reliable and publicly available. Otherwise, individuals may disagree on how to assess others, even if they all apply the same social norm. Such disagreements can lead to a breakdown of cooperation. Here we explore whether the predominantly studied ‘leading eight’ social norms of indirect reciprocity can be made more robust by equipping them with an element of generosity. To this end, we distinguish between two kinds of generosity. According to assessment generosity, individuals occasionally assign a good reputation to group members who would usually be regarded as bad. According to action generosity, individuals occasionally cooperate with group members with whom they would usually defect. Using individual-based simulations, we show that the two kinds of generosity have a very different effect on the resulting reputation dynamics. Assessment generosity tends to add to the overall noise and allows defectors to invade. In contrast, a limited amount of action generosity can be beneficial in a few cases. However, even when action generosity is beneficial, the respective simulations do not result in full cooperation. Our results suggest that while generosity can favor cooperation when individuals use the most simple strategies of reciprocity, it is disadvantageous when individuals use more complex social norms. AU - Schmid, Laura AU - Shati, Pouya AU - Hilbe, Christian AU - Chatterjee, Krishnendu ID - 9997 IS - 1 JF - Scientific Reports KW - Multidisciplinary TI - The evolution of indirect reciprocity under action and assessment generosity VL - 11 ER - TY - JOUR AB - Direct and indirect reciprocity are key mechanisms for the evolution of cooperation. Direct reciprocity means that individuals use their own experience to decide whether to cooperate with another person. Indirect reciprocity means that they also consider the experiences of others. Although these two mechanisms are intertwined, they are typically studied in isolation. Here, we introduce a mathematical framework that allows us to explore both kinds of reciprocity simultaneously. We show that the well-known ‘generous tit-for-tat’ strategy of direct reciprocity has a natural analogue in indirect reciprocity, which we call ‘generous scoring’. Using an equilibrium analysis, we characterize under which conditions either of the two strategies can maintain cooperation. With simulations, we additionally explore which kind of reciprocity evolves when members of a population engage in social learning to adapt to their environment. Our results draw unexpected connections between direct and indirect reciprocity while highlighting important differences regarding their evolvability. AU - Schmid, Laura AU - Chatterjee, Krishnendu AU - Hilbe, Christian AU - Nowak, Martin A. ID - 9402 IS - 10 JF - Nature Human Behaviour TI - A unified framework of direct and indirect reciprocity VL - 5 ER - TY - CONF AB - The Price of Anarchy (PoA) is a well-established game-theoretic concept to shed light on coordination issues arising in open distributed systems. Leaving agents to selfishly optimize comes with the risk of ending up in sub-optimal states (in terms of performance and/or costs), compared to a centralized system design. However, the PoA relies on strong assumptions about agents' rationality (e.g., resources and information) and interactions, whereas in many distributed systems agents interact locally with bounded resources. They do so repeatedly over time (in contrast to "one-shot games"), and their strategies may evolve. Using a more realistic evolutionary game model, this paper introduces a realized evolutionary Price of Anarchy (ePoA). The ePoA allows an exploration of equilibrium selection in dynamic distributed systems with multiple equilibria, based on local interactions of simple memoryless agents. Considering a fundamental game related to virus propagation on networks, we present analytical bounds on the ePoA in basic network topologies and for different strategy update dynamics. In particular, deriving stationary distributions of the stochastic evolutionary process, we find that the Nash equilibria are not always the most abundant states, and that different processes can feature significant off-equilibrium behavior, leading to a significantly higher ePoA compared to the PoA studied traditionally in the literature. AU - Schmid, Laura AU - Chatterjee, Krishnendu AU - Schmid, Stefan ID - 7346 T2 - Proceedings of the 23rd International Conference on Principles of Distributed Systems TI - The evolutionary price of anarchy: Locally bounded agents in a dynamic virus game VL - 153 ER - TY - CONF AB - A vector addition system with states (VASS) consists of a finite set of states and counters. A transition changes the current state to the next state, and every counter is either incremented, or decremented, or left unchanged. A state and value for each counter is a configuration; and a computation is an infinite sequence of configurations with transitions between successive configurations. A probabilistic VASS consists of a VASS along with a probability distribution over the transitions for each state. Qualitative properties such as state and configuration reachability have been widely studied for VASS. In this work we consider multi-dimensional long-run average objectives for VASS and probabilistic VASS. For a counter, the cost of a configuration is the value of the counter; and the long-run average value of a computation for the counter is the long-run average of the costs of the configurations in the computation. The multi-dimensional long-run average problem given a VASS and a threshold value for each counter, asks whether there is a computation such that for each counter the long-run average value for the counter does not exceed the respective threshold. For probabilistic VASS, instead of the existence of a computation, we consider whether the expected long-run average value for each counter does not exceed the respective threshold. Our main results are as follows: we show that the multi-dimensional long-run average problem (a) is NP-complete for integer-valued VASS; (b) is undecidable for natural-valued VASS (i.e., nonnegative counters); and (c) can be solved in polynomial time for probabilistic integer-valued VASS, and probabilistic natural-valued VASS when all computations are non-terminating. AU - Chatterjee, Krishnendu AU - Henzinger, Thomas A AU - Otop, Jan ID - 8600 SN - 18688969 T2 - 31st International Conference on Concurrency Theory TI - Multi-dimensional long-run average problems for vector addition systems with states VL - 171 ER - TY - CONF AB - Game of Life is a simple and elegant model to study dynamical system over networks. The model consists of a graph where every vertex has one of two types, namely, dead or alive. A configuration is a mapping of the vertices to the types. An update rule describes how the type of a vertex is updated given the types of its neighbors. In every round, all vertices are updated synchronously, which leads to a configuration update. While in general, Game of Life allows a broad range of update rules, we focus on two simple families of update rules, namely, underpopulation and overpopulation, that model several interesting dynamics studied in the literature. In both settings, a dead vertex requires at least a desired number of live neighbors to become alive. For underpopulation (resp., overpopulation), a live vertex requires at least (resp. at most) a desired number of live neighbors to remain alive. We study the basic computation problems, e.g., configuration reachability, for these two families of rules. For underpopulation rules, we show that these problems can be solved in polynomial time, whereas for overpopulation rules they are PSPACE-complete. AU - Chatterjee, Krishnendu AU - Ibsen-Jensen, Rasmus AU - Jecker, Ismael R AU - Svoboda, Jakub ID - 8533 SN - 18688969 T2 - 45th International Symposium on Mathematical Foundations of Computer Science TI - Simplified game of life: Algorithms and complexity VL - 170 ER - TY - CONF AB - A regular language L of finite words is composite if there are regular languages L₁,L₂,
,L_t such that L = ⋂_{i = 1}^t L_i and the index (number of states in a minimal DFA) of every language L_i is strictly smaller than the index of L. Otherwise, L is prime. Primality of regular languages was introduced and studied in [O. Kupferman and J. Mosheiff, 2015], where the complexity of deciding the primality of the language of a given DFA was left open, with a doubly-exponential gap between the upper and lower bounds. We study primality for unary regular languages, namely regular languages with a singleton alphabet. A unary language corresponds to a subset of ℕ, making the study of unary prime languages closer to that of primality in number theory. We show that the setting of languages is richer. In particular, while every composite number is the product of two smaller numbers, the number t of languages necessary to decompose a composite unary language induces a strict hierarchy. In addition, a primality witness for a unary language L, namely a word that is not in L but is in all products of languages that contain L and have an index smaller than L’s, may be of exponential length. Still, we are able to characterize compositionality by structural properties of a DFA for L, leading to a LogSpace algorithm for primality checking of unary DFAs. AU - Jecker, Ismael R AU - Kupferman, Orna AU - Mazzocchi, Nicolas ID - 8534 SN - 18688969 T2 - 45th International Symposium on Mathematical Foundations of Computer Science TI - Unary prime languages VL - 170 ER - TY - CONF AB - Simple stochastic games are turn-based 2Âœ-player games with a reachability objective. The basic question asks whether one player can ensure reaching a given target with at least a given probability. A natural extension is games with a conjunction of such conditions as objective. Despite a plethora of recent results on the analysis of systems with multiple objectives, the decidability of this basic problem remains open. In this paper, we present an algorithm approximating the Pareto frontier of the achievable values to a given precision. Moreover, it is an anytime algorithm, meaning it can be stopped at any time returning the current approximation and its error bound. AU - Ashok, Pranav AU - Chatterjee, Krishnendu AU - Kretinsky, Jan AU - Weininger, Maximilian AU - Winkler, Tobias ID - 7955 SN - 9781450371049 T2 - Proceedings of the 35th Annual ACM/IEEE Symposium on Logic in Computer Science TI - Approximating values of generalized-reachability stochastic games ER - TY - JOUR AB - Resources are rarely distributed uniformly within a population. Heterogeneity in the concentration of a drug, the quality of breeding sites, or wealth can all affect evolutionary dynamics. In this study, we represent a collection of properties affecting the fitness at a given location using a color. A green node is rich in resources while a red node is poorer. More colors can represent a broader spectrum of resource qualities. For a population evolving according to the birth-death Moran model, the first question we address is which structures, identified by graph connectivity and graph coloring, are evolutionarily equivalent. We prove that all properly two-colored, undirected, regular graphs are evolutionarily equivalent (where “properly colored” means that no two neighbors have the same color). We then compare the effects of background heterogeneity on properly two-colored graphs to those with alternative schemes in which the colors are permuted. Finally, we discuss dynamic coloring as a model for spatiotemporal resource fluctuations, and we illustrate that random dynamic colorings often diminish the effects of background heterogeneity relative to a proper two-coloring. AU - Kaveh, Kamran AU - McAvoy, Alex AU - Chatterjee, Krishnendu AU - Nowak, Martin A. ID - 8767 IS - 11 JF - PLOS Computational Biology KW - Ecology KW - Modelling and Simulation KW - Computational Theory and Mathematics KW - Genetics KW - Ecology KW - Evolution KW - Behavior and Systematics KW - Molecular Biology KW - Cellular and Molecular Neuroscience SN - 1553-734X TI - The Moran process on 2-chromatic graphs VL - 16 ER - TY - JOUR AB - Cooperation is a ubiquitous and beneficial behavioural trait despite being prone to exploitation by free-riders. Hence, cooperative populations are prone to invasions by selfish individuals. However, a population consisting of only free-riders typically does not survive. Thus, cooperators and free-riders often coexist in some proportion. An evolutionary version of a Snowdrift Game proved its efficiency in analysing this phenomenon. However, what if the system has already reached its stable state but was perturbed due to a change in environmental conditions? Then, individuals may have to re-learn their effective strategies. To address this, we consider behavioural mistakes in strategic choice execution, which we refer to as incompetence. Parametrising the propensity to make such mistakes allows for a mathematical description of learning. We compare strategies based on their relative strategic advantage relying on both fitness and learning factors. When strategies are learned at distinct rates, allowing learning according to a prescribed order is optimal. Interestingly, the strategy with the lowest strategic advantage should be learnt first if we are to optimise fitness over the learning path. Then, the differences between strategies are balanced out in order to minimise the effect of behavioural uncertainty. AU - Kleshnina, Maria AU - Streipert, Sabrina AU - Filar, Jerzy AU - Chatterjee, Krishnendu ID - 8789 IS - 11 JF - Mathematics TI - Prioritised learning in snowdrift-type games VL - 8 ER - TY - JOUR AB - We consider a real-time setting where an environment releases sequences of firm-deadline tasks, and an online scheduler chooses on-the-fly the ones to execute on a single processor so as to maximize cumulated utility. The competitive ratio is a well-known performance measure for the scheduler: it gives the worst-case ratio, among all possible choices for the environment, of the cumulated utility of the online scheduler versus an offline scheduler that knows these choices in advance. Traditionally, competitive analysis is performed by hand, while automated techniques are rare and only handle static environments with independent tasks. We present a quantitative-verification framework for precedence-aware competitive analysis, where task releases may depend on preceding scheduling choices, i.e., the environment can respond to scheduling decisions dynamically . We consider two general classes of precedences: 1) follower precedences force the release of a dependent task upon the completion of a set of precursor tasks, while and 2) pairing precedences modify the characteristics of a dependent task provided the completion of a set of precursor tasks. Precedences make competitive analysis challenging, as the online and offline schedulers operate on diverging sequences. We make a formal presentation of our framework, and use a GPU-based implementation to analyze ten well-known schedulers on precedence-based application examples taken from the existing literature: 1) a handshake protocol (HP); 2) network packet-switching; 3) query scheduling (QS); and 4) a sporadic-interrupt setting. Our experimental results show that precedences and task parameters can vary drastically the best scheduler. Our framework thus supports application designers in choosing the best scheduler among a given set automatically. AU - Pavlogiannis, Andreas AU - Schaumberger, Nico AU - Schmid, Ulrich AU - Chatterjee, Krishnendu ID - 8788 IS - 11 JF - IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems SN - 02780070 TI - Precedence-aware automated competitive analysis of real-time scheduling VL - 39 ER - TY - JOUR AB - In this paper we introduce and study all-pay bidding games, a class of two player, zero-sum games on graphs. The game proceeds as follows. We place a token on some vertex in the graph and assign budgets to the two players. Each turn, each player submits a sealed legal bid (non-negative and below their remaining budget), which is deducted from their budget and the highest bidder moves the token onto an adjacent vertex. The game ends once a sink is reached, and Player 1 pays Player 2 the outcome that is associated with the sink. The players attempt to maximize their expected outcome. Our games model settings where effort (of no inherent value) needs to be invested in an ongoing and stateful manner. On the negative side, we show that even in simple games on DAGs, optimal strategies may require a distribution over bids with infinite support. A central quantity in bidding games is the ratio of the players budgets. On the positive side, we show a simple FPTAS for DAGs, that, for each budget ratio, outputs an approximation for the optimal strategy for that ratio. We also implement it, show that it performs well, and suggests interesting properties of these games. Then, given an outcome c, we show an algorithm for finding the necessary and sufficient initial ratio for guaranteeing outcome c with probability 1 and a strategy ensuring such. Finally, while the general case has not previously been studied, solving the specific game in which Player 1 wins iff he wins the first two auctions, has been long stated as an open question, which we solve. AU - Avni, Guy AU - Ibsen-Jensen, Rasmus AU - Tkadlec, Josef ID - 9197 IS - 02 JF - Proceedings of the AAAI Conference on Artificial Intelligence SN - 2159-5399 TI - All-pay bidding games on graphs VL - 34 ER - TY - JOUR AB - Coinfections with multiple pathogens can result in complex within‐host dynamics affecting virulence and transmission. While multiple infections are intensively studied in solitary hosts, it is so far unresolved how social host interactions interfere with pathogen competition, and if this depends on coinfection diversity. We studied how the collective disease defences of ants – their social immunity – influence pathogen competition in coinfections of same or different fungal pathogen species. Social immunity reduced virulence for all pathogen combinations, but interfered with spore production only in different‐species coinfections. Here, it decreased overall pathogen sporulation success while increasing co‐sporulation on individual cadavers and maintaining a higher pathogen diversity at the community level. Mathematical modelling revealed that host sanitary care alone can modulate competitive outcomes between pathogens, giving advantage to fast‐germinating, thus less grooming‐sensitive ones. Host social interactions can hence modulate infection dynamics in coinfected group members, thereby altering pathogen communities at the host level and population level. AU - Milutinovic, Barbara AU - Stock, Miriam AU - Grasse, Anna V AU - Naderlinger, Elisabeth AU - Hilbe, Christian AU - Cremer, Sylvia ID - 7343 IS - 3 JF - Ecology Letters SN - 1461-023X TI - Social immunity modulates competition between coinfecting pathogens VL - 23 ER - TY - GEN AB - Coinfections with multiple pathogens can result in complex within-host dynamics affecting virulence and transmission. Whilst multiple infections are intensively studied in solitary hosts, it is so far unresolved how social host interactions interfere with pathogen competition, and if this depends on coinfection diversity. We studied how the collective disease defenses of ants – their social immunity ­– influence pathogen competition in coinfections of same or different fungal pathogen species. Social immunity reduced virulence for all pathogen combinations, but interfered with spore production only in different-species coinfections. Here, it decreased overall pathogen sporulation success, whilst simultaneously increasing co-sporulation on individual cadavers and maintaining a higher pathogen diversity at the community-level. Mathematical modeling revealed that host sanitary care alone can modulate competitive outcomes between pathogens, giving advantage to fast-germinating, thus less grooming-sensitive ones. Host social interactions can hence modulate infection dynamics in coinfected group members, thereby altering pathogen communities at the host- and population-level. AU - Milutinovic, Barbara AU - Stock, Miriam AU - Grasse, Anna V AU - Naderlinger, Elisabeth AU - Hilbe, Christian AU - Cremer, Sylvia ID - 13060 TI - Social immunity modulates competition between coinfecting pathogens ER - TY - CONF AB - Multiple-environment Markov decision processes (MEMDPs) are MDPs equipped with not one, but multiple probabilistic transition functions, which represent the various possible unknown environments. While the previous research on MEMDPs focused on theoretical properties for long-run average payoff, we study them with discounted-sum payoff and focus on their practical advantages and applications. MEMDPs can be viewed as a special case of Partially observable and Mixed observability MDPs: the state of the system is perfectly observable, but not the environment. We show that the specific structure of MEMDPs allows for more efficient algorithmic analysis, in particular for faster belief updates. We demonstrate the applicability of MEMDPs in several domains. In particular, we formalize the sequential decision-making approach to contextual recommendation systems as MEMDPs and substantially improve over the previous MDP approach. AU - Chatterjee, Krishnendu AU - Chmelik, Martin AU - Karkhanis, Deep AU - NovotnĂœ, Petr AU - Royer, AmĂ©lie ID - 8193 SN - 23340835 T2 - Proceedings of the 30th International Conference on Automated Planning and Scheduling TI - Multiple-environment Markov decision processes: Efficient analysis and applications VL - 30 ER - TY - CONF AB - We study turn-based stochastic zero-sum games with lexicographic preferences over reachability and safety objectives. Stochastic games are standard models in control, verification, and synthesis of stochastic reactive systems that exhibit both randomness as well as angelic and demonic non-determinism. Lexicographic order allows to consider multiple objectives with a strict preference order over the satisfaction of the objectives. To the best of our knowledge, stochastic games with lexicographic objectives have not been studied before. We establish determinacy of such games and present strategy and computational complexity results. For strategy complexity, we show that lexicographically optimal strategies exist that are deterministic and memory is only required to remember the already satisfied and violated objectives. For a constant number of objectives, we show that the relevant decision problem is in NP∩coNP , matching the current known bound for single objectives; and in general the decision problem is PSPACE -hard and can be solved in NEXPTIME∩coNEXPTIME . We present an algorithm that computes the lexicographically optimal strategies via a reduction to computation of optimal strategies in a sequence of single-objectives games. We have implemented our algorithm and report experimental results on various case studies. AU - Chatterjee, Krishnendu AU - Katoen, Joost P AU - Weininger, Maximilian AU - Winkler, Tobias ID - 8272 SN - 03029743 T2 - International Conference on Computer Aided Verification TI - Stochastic games with lexicographic reachability-safety objectives VL - 12225 ER - TY - JOUR AB - We study relations between evidence theory and S-approximation spaces. Both theories have their roots in the analysis of Dempsterchr('39')s multivalued mappings and lower and upper probabilities, and have close relations to rough sets. We show that an S-approximation space, satisfying a monotonicity condition, can induce a natural belief structure which is a fundamental block in evidence theory. We also demonstrate that one can induce a natural belief structure on one set, given a belief structure on another set, if the two sets are related by a partial monotone S-approximation space. AU - Shakiba, A. AU - Goharshady, Amir Kafshdar AU - Hooshmandasl, M.R. AU - Alambardar Meybodi, M. ID - 8671 IS - 2 JF - Iranian Journal of Mathematical Sciences and Informatics SN - 1735-4463 TI - A note on belief structures and s-approximation spaces VL - 15 ER - TY - JOUR AB - The fixation probability of a single mutant invading a population of residents is among the most widely-studied quantities in evolutionary dynamics. Amplifiers of natural selection are population structures that increase the fixation probability of advantageous mutants, compared to well-mixed populations. Extensive studies have shown that many amplifiers exist for the Birth-death Moran process, some of them substantially increasing the fixation probability or even guaranteeing fixation in the limit of large population size. On the other hand, no amplifiers are known for the death-Birth Moran process, and computer-assisted exhaustive searches have failed to discover amplification. In this work we resolve this disparity, by showing that any amplification under death-Birth updating is necessarily bounded and transient. Our boundedness result states that even if a population structure does amplify selection, the resulting fixation probability is close to that of the well-mixed population. Our transience result states that for any population structure there exists a threshold r⋆ such that the population structure ceases to amplify selection if the mutant fitness advantage r is larger than r⋆. Finally, we also extend the above results to ÎŽ-death-Birth updating, which is a combination of Birth-death and death-Birth updating. On the positive side, we identify population structures that maintain amplification for a wide range of values r and ÎŽ. These results demonstrate that amplification of natural selection depends on the specific mechanisms of the evolutionary process. AU - Tkadlec, Josef AU - Pavlogiannis, Andreas AU - Chatterjee, Krishnendu AU - Nowak, Martin A. ID - 7212 JF - PLoS computational biology TI - Limits on amplifiers of natural selection under death-Birth updating VL - 16 ER - TY - THES AB - In this thesis we study certain mathematical aspects of evolution. The two primary forces that drive an evolutionary process are mutation and selection. Mutation generates new variants in a population. Selection chooses among the variants depending on the reproductive rates of individuals. Evolutionary processes are intrinsically random – a new mutation that is initially present in the population at low frequency can go extinct, even if it confers a reproductive advantage. The overall rate of evolution is largely determined by two quantities: the probability that an invading advantageous mutation spreads through the population (called fixation probability) and the time until it does so (called fixation time). Both those quantities crucially depend not only on the strength of the invading mutation but also on the population structure. In this thesis, we aim to understand how the underlying population structure affects the overall rate of evolution. Specifically, we study population structures that increase the fixation probability of advantageous mutants (called amplifiers of selection). Broadly speaking, our results are of three different types: We present various strong amplifiers, we identify regimes under which only limited amplification is feasible, and we propose population structures that provide different tradeoffs between high fixation probability and short fixation time. AU - Tkadlec, Josef ID - 7196 TI - A role of graphs in evolutionary processes ER - TY - GEN AB - Data and mathematica notebooks for plotting figures from Language learning with communication between learners AU - Ibsen-Jensen, Rasmus AU - Tkadlec, Josef AU - Chatterjee, Krishnendu AU - Nowak, Martin ID - 9814 TI - Data and mathematica notebooks for plotting figures from language learning with communication between learners from language acquisition with communication between learners ER - TY - CONF AB - The notion of program sensitivity (aka Lipschitz continuity) specifies that changes in the program input result in proportional changes to the program output. For probabilistic programs the notion is naturally extended to expected sensitivity. A previous approach develops a relational program logic framework for proving expected sensitivity of probabilistic while loops, where the number of iterations is fixed and bounded. In this work, we consider probabilistic while loops where the number of iterations is not fixed, but randomized and depends on the initial input values. We present a sound approach for proving expected sensitivity of such programs. Our sound approach is martingale-based and can be automated through existing martingale-synthesis algorithms. Furthermore, our approach is compositional for sequential composition of while loops under a mild side condition. We demonstrate the effectiveness of our approach on several classical examples from Gambler's Ruin, stochastic hybrid systems and stochastic gradient descent. We also present experimental results showing that our automated approach can handle various probabilistic programs in the literature. AU - Wang, Peixin AU - Fu, Hongfei AU - Chatterjee, Krishnendu AU - Deng, Yuxin AU - Xu, Ming ID - 8324 IS - POPL T2 - Proceedings of the ACM on Programming Languages TI - Proving expected sensitivity of probabilistic programs with randomized variable-dependent termination time VL - 4 ER - TY - JOUR AB - Markov decision processes (MDPs) are the defacto framework for sequential decision making in the presence of stochastic uncertainty. A classical optimization criterion for MDPs is to maximize the expected discounted-sum payoff, which ignores low probability catastrophic events with highly negative impact on the system. On the other hand, risk-averse policies require the probability of undesirable events to be below a given threshold, but they do not account for optimization of the expected payoff. We consider MDPs with discounted-sum payoff with failure states which represent catastrophic outcomes. The objective of risk-constrained planning is to maximize the expected discounted-sum payoff among risk-averse policies that ensure the probability to encounter a failure state is below a desired threshold. Our main contribution is an efficient risk-constrained planning algorithm that combines UCT-like search with a predictor learned through interaction with the MDP (in the style of AlphaZero) and with a risk-constrained action selection via linear programming. We demonstrate the effectiveness of our approach with experiments on classical MDPs from the literature, including benchmarks with an order of 106 states. AU - BrĂĄzdil, TomĂĄĆĄ AU - Chatterjee, Krishnendu AU - NovotnĂœ, Petr AU - Vahala, Jiƙí ID - 15055 IS - 06 JF - Proceedings of the 34th AAAI Conference on Artificial Intelligence KW - General Medicine SN - 2374-3468 TI - Reinforcement learning of risk-constrained policies in Markov decision processes VL - 34 ER -