arXiv subjects

Krishnendu Chatterjee

Publications and source records attributed to Krishnendu Chatterjee.

At least 91 records · Page 5Linked to original sources

arXiv2020

Inductive Reachability Witnesses

In this work, we consider the fundamental problem of reachability analysis over imperative programs with real variables. The reachability property requires that a program can reach certain target states during its execution. Previous works that tackle reachability analysis 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/reverse Hoare logic). In contrast, we propose a novel approach for reachability analysis that can handle general programs, is (semi-)complete, and can be entirely automated for a wide family of programs. Our approach extends techniques from both invariant generation and ranking-function synthesis to reachability analysis through the notion of (Universal) Inductive Reachability Witnesses (IRWs/UIRWs). While traditional invariant generation uses over-approximations of reachable states, we consider the natural dual problem of under-approximating the set of program states that can reach a target state. We then apply an argument similar to ranking functions to ensure that all states in our under-approximation can indeed reach the target set in finitely many steps.

cs.PL↗

arXiv2020

Multi-dimensional Long-Run Average Problems for Vector Addition Systems with States

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.

cs.FL↗

arXiv2020

Simplified Game of Life: Algorithms and Complexity

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.

nlin.CG↗

arXiv2020

Stochastic Games with Lexicographic Reachability-Safety Objectives

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 $\cap$ coNP, matching the current known bound for single objectives; and in general the decision problem is PSPACE-hard and can be solved in NEXPTIME $\cap$ 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.

cs.GT↗

arXiv2020

Approximating Values of Generalized-Reachability Stochastic Games

Simple stochastic games are turn-based 2.5-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.

cs.GT↗

arXiv2020

Faster Algorithms for Quantitative Analysis of Markov Chains and Markov Decision Processes with Small Treewidth

Discrete-time Markov Chains (MCs) and Markov Decision Processes (MDPs) are two standard formalisms in system analysis. Their main associated quantitative objectives are hitting probabilities, discounted sum, and mean payoff. Although there are many techniques for computing these objectives in general MCs/MDPs, they have not been thoroughly studied in terms of parameterized algorithms, particularly when treewidth is used as the parameter. This is in sharp contrast to qualitative objectives for MCs, MDPs and graph games, for which treewidth-based algorithms yield significant complexity improvements. In this work, we show that treewidth can also be used to obtain faster algorithms for the quantitative problems. For an MC with $n$ states and $m$ transitions, we show that each of the classical quantitative objectives can be computed in $O((n+m)\cdot t^2)$ time, given a tree decomposition of the MC that has width $t$. Our results also imply a bound of $O(κ\cdot (n+m)\cdot t^2)$ for each objective on MDPs, where $κ$ is the number of strategy-iteration refinements required for the given input and objective. Finally, we make an experimental evaluation of our new algorithms on low-treewidth MCs and MDPs obtained from the DaCapo benchmark suite. Our experimental results show that on MCs and MDPs with small treewidth, our algorithms outperform existing well-established methods by one or more orders of magnitude.

cs.DS↗

arXiv2020

Optimal and Perfectly Parallel Algorithms for On-demand Data-flow Analysis

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.

cs.DS↗

arXiv2020

Polynomial Invariant Generation for Non-deterministic Recursive Programs

We consider the classical problem of invariant generation for programs with polynomial assignments and focus on synthesizing invariants that are a conjunction of strict polynomial inequalities. We present a sound and semi-complete method based on positivstellensaetze, i.e. theorems in semi-algebraic geometry that characterize positive polynomials over a semi-algebraic set. To the best of our knowledge, this is the first invariant generation method to provide completeness guarantees for invariants consisting of polynomial inequalities. Moreover, on the theoretical side, the worst-case complexity of our approach is subexponential, whereas the worst-case complexity of the previously-known complete method (Colon et al, CAV 2003), which could only handle linear invariants, is exponential. On the practical side, we reduce the invariant generation problem to quadratic programming (QCLP), which is a classical optimization problem with many industrial solvers. Finally, we demonstrate the applicability of our approach by providing experimental results on several academic benchmarks.

cs.LO↗

arXiv2020

Reinforcement Learning of Risk-Constrained Policies in Markov Decision Processes

Markov decision processes (MDPs) are the defacto frame-work for sequential decision making in the presence ofstochastic uncertainty. A classical optimization criterion forMDPs is to maximize the expected discounted-sum pay-off, which ignores low probability catastrophic events withhighly negative impact on the system. On the other hand,risk-averse policies require the probability of undesirableevents to be below a given threshold, but they do not accountfor optimization of the expected payoff. We consider MDPswith discounted-sum payoff with failure states which repre-sent catastrophic outcomes. The objective of risk-constrainedplanning is to maximize the expected discounted-sum payoffamong risk-averse policies that ensure the probability to en-counter a failure state is below a desired threshold. Our maincontribution is an efficient risk-constrained planning algo-rithm that combines UCT-like search with a predictor learnedthrough interaction with the MDP (in the style of AlphaZero)and with a risk-constrained action selection via linear pro-gramming. We demonstrate the effectiveness of our approachwith experiments on classical MDPs from the literature, in-cluding benchmarks with an order of 10^6 states.

cs.AI↗

arXiv2019

Proving Expected Sensitivity of Probabilistic Programs with Randomized Variable-Dependent Termination Time

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.

cs.PL↗

arXiv2019

Optimal Dyck Reachability for Data-Dependence and Alias Analysis

A fundamental algorithmic problem at the heart of static analysis is Dyck reachability. The input is a graph where the edges are labeled with different types of opening and closing parentheses, and the reachability information is computed via paths whose parentheses are properly matched. We present new results for Dyck reachability problems with applications to alias analysis and data-dependence analysis. Our main contributions, that include improved upper bounds as well as lower bounds that establish optimality guarantees, are as follows. First, we consider Dyck reachability on bidirected graphs, which is the standard way of performing field-sensitive points-to analysis. Given a bidirected graph with $n$ nodes and $m$ edges, we present: (i)~an algorithm with worst-case running time $O(m + n \cdot α(n))$, where $α(n)$ is the inverse Ackermann function, improving the previously known $O(n^2)$ time bound; (ii)~a matching lower bound that shows that our algorithm is optimal wrt to worst-case complexity; and (iii)~an optimal average-case upper bound of $O(m)$ time, improving the previously known $O(m \cdot \log n)$ bound. Second, we consider the problem of context-sensitive data-dependence analysis, where the task is to obtain analysis summaries of library code in the presence of callbacks. Our algorithm preprocesses libraries in almost linear time, after which the contribution of the library in the complexity of the client analysis is only linear, and only wrt the number of call sites. Third, we prove that combinatorial algorithms for Dyck reachability on general graphs with truly sub-cubic bounds cannot be obtained without obtaining sub-cubic combinatorial algorithms for Boolean Matrix Multiplication, which is a long-standing open problem. We also show that the same hardness holds for graphs of constant treewidth.

cs.PL↗

arXiv2019

Near-Linear Time Algorithms for Streett Objectives in Graphs and MDPs

The fundamental model-checking problem, given as input a model and a specification, asks for the algorithmic verification of whether the model satisfies the specification. Two classical models for reactive systems are graphs and Markov decision processes (MDPs). A basic specification formalism in the verification of reactive systems is the strong fairness (aka Streett) objective, where given different types of requests and corresponding grants, the requirement is that for each type, if the request event happens infinitely often, then the corresponding grant event must also happen infinitely often. All omega-regular objectives can be expressed as Streett objectives and hence they are canonical in verification. Consider graphs/MDPs with n vertices, m edges, and a Streett objectives with k pairs, and let b denote the size of the description of the Streett objective for the sets of requests and grants. The current best-known algorithm for the problem requires time $O(min(n^2, m \sqrt{m \log n}) + b \log n)$. In this work, we present randomized near-linear time algorithms, with expected running time $\widetilde{O}(m + b)$, where the $\widetilde{O}$ notation hides poly-log factors. Our randomized algorithms are near-linear in the size of the input, and hence optimal up to poly-log factors.

cs.GT↗

arXiv2019

Quasipolynomial Set-Based Symbolic Algorithms for Parity Games

Solving parity games, which are equivalent to modal $μ$-calculus model checking, is a central algorithmic problem in formal methods. Besides the standard computation model with the explicit representation of games, another important theoretical model of computation is that of set-based symbolic algorithms. Set-based symbolic algorithms use basic set operations and one-step predecessor operations on the implicit description of games, rather than the explicit representation. The significance of symbolic algorithms is that they provide scalable algorithms for large finite-state systems, as well as for infinite-state systems with finite quotient. Consider parity games on graphs with $n$ vertices and parity conditions with $d$ priorities. While there is a rich literature of explicit algorithms for parity games, the main results for set-based symbolic algorithms are as follows: (a) an algorithm that requires $O(n^d)$ symbolic operations and $O(d)$ symbolic space; and (b) an improved algorithm that requires $O(n^{d/3+1})$ symbolic operations and $O(n)$ symbolic space. Our contributions are as follows: (1) We present a black-box set-based symbolic algorithm based on the explicit progress measure algorithm. Two important consequences of our algorithm are as follows: (a) a set-based symbolic algorithm for parity games that requires quasi-polynomially many symbolic operations and $O(n)$ symbolic space; and (b) any future improvement in progress measure based explicit algorithms imply an efficiency improvement in our set-based symbolic algorithm for parity games. (2) We present a set-based symbolic algorithm that requires quasi-polynomially many symbolic operations and $O(d \cdot \log n)$ symbolic space. Moreover, for the important special case of $d \leq \log n$, our algorithm requires only polynomially many symbolic operations and poly-logarithmic symbolic space.

cs.GT↗

arXiv2019

Value-centric Dynamic Partial Order Reduction

The verification of concurrent programs remains an open challenge, as thread interaction has to be accounted for, which leads to state-space explosion. Stateless model checking battles this problem by exploring traces rather than states of the program. As there are exponentially many traces, dynamic partial-order reduction (DPOR) techniques are used to partition the trace space into equivalence classes, and explore a few representatives from each class. The standard equivalence that underlies most DPOR techniques is the happens-before equivalence, however recent works have spawned a vivid interest towards coarser equivalences. The efficiency of such approaches is a product of two parameters: (i) the size of the partitioning induced by the equivalence, and (ii) the time spent by the exploration algorithm in each class of the partitioning. In this work, we present a new equivalence, called value-happens-before and show that it has two appealing features. First, value-happens-before is always at least as coarse as the happens-before equivalence, and can be even exponentially coarser. Second, the value-happens-before partitioning is efficiently explorable when the number of threads is bounded. We present an algorithm called value-centric DPOR (VCDPOR), which explores the underlying partitioning using polynomial time per class. Finally, we perform an experimental evaluation of VCDPOR on various benchmarks, and compare it against other state-of-the-art approaches. Our results show that value-happens-before typically induces a significant reduction in the size of the underlying partitioning, which leads to a considerable reduction in the running time for exploring the whole partitioning.

cs.PL↗

arXiv2019

Modular Verification for Almost-Sure Termination of Probabilistic Programs

In this work, we consider the almost-sure termination problem for probabilistic programs that asks whether a given probabilistic program terminates with probability 1. Scalable approaches for program analysis often rely on modularity as their theoretical basis. In non-probabilistic programs, the classical variant rule (V-rule) of Floyd-Hoare logic provides the foundation for modular analysis. Extension of this rule to almost-sure termination of probabilistic programs is quite tricky, and a probabilistic variant was proposed in [Fioriti and Hermanns 2015]. While the proposed probabilistic variant cautiously addresses the key issue of integrability, we show that the proposed modular rule is still not sound for almost-sure termination of probabilistic programs. Besides establishing unsoundness of the previous rule, our contributions are as follows: First, we present a sound modular rule for almost-sure termination of probabilistic programs. Our approach is based on a novel notion of descent supermartingales. Second, for algorithmic approaches, we consider descent supermartingales that are linear and show that they can be synthesized in polynomial time. Finally, we present experimental results on a variety of benchmarks and several natural examples that model various types of nested while loops in probabilistic programs and demonstrate that our approach is able to efficiently prove their almost-sure termination property.

cs.LO↗

arXiv2019

Quantitative Automata under Probabilistic Semantics

Automata with monitor counters, where the transitions do not depend on counter values, and nested weighted automata are two expressive automata-theoretic frameworks for quantitative properties. For a well-studied and wide class of quantitative functions, we establish that automata with monitor counters and nested weighted automata are equivalent. We study for the first time such quantitative automata under probabilistic semantics. We show that several problems that are undecidable for the classical questions of emptiness and universality become decidable under the probabilistic semantics. We present a complete picture of decidability for such automata, and even an almost-complete picture of computational complexity, for the probabilistic questions we consider.

cs.FL↗

arXiv2019

Deciding Fast Termination for Probabilistic VASS with Nondeterminism

A probabilistic vector addition system with states (pVASS) is a finite state Markov process augmented with non-negative integer counters that can be incremented or decremented during each state transition, blocking any behaviour that would cause a counter to decrease below zero. The pVASS can be used as abstractions of probabilistic programs with many decidable properties. The use of pVASS as abstractions requires the presence of nondeterminism in the model. In this paper, we develop techniques for checking fast termination of pVASS with nondeterminism. That is, for every initial configuration of size n, we consider the worst expected number of transitions needed to reach a configuration with some counter negative (the expected termination time). We show that the problem whether the asymptotic expected termination time is linear is decidable in polynomial time for a certain natural class of pVASS with nondeterminism. Furthermore, we show the following dichotomy: if the asymptotic expected termination time is not linear, then it is at least quadratic, i.e., in $Ω(n^2)$.

cs.FL↗

arXiv2019

Strategy Representation by Decision Trees with Linear Classifiers

Graph games and Markov decision processes (MDPs) are standard models in reactive synthesis and verification of probabilistic systems with nondeterminism. The class of $ω$-regular winning conditions; e.g., safety, reachability, liveness, parity conditions; provides a robust and expressive specification formalism for properties that arise in analysis of reactive systems. The resolutions of nondeterminism in games and MDPs are represented as strategies, and we consider succinct representation of such strategies. The decision-tree data structure from machine learning retains the flavor of decisions of strategies and allows entropy-based minimization to obtain succinct trees. However, in contrast to traditional machine-learning problems where small errors are allowed, for winning strategies in graph games and MDPs no error is allowed, and the decision tree must represent the entire strategy. In this work we propose decision trees with linear classifiers for representation of strategies in graph games and MDPs. We have implemented strategy representation using this data structure and we present experimental results for problems on graph games and MDPs, which show that this new data structure presents a much more efficient strategy representation as compared to standard decision trees.

cs.LO↗