Abstract We present a parallel abstract interpretation technique for polynomial programs with assertions presented as unions of range bound constraints. We use the powerset domain of hyper-rectangles to over-approximate sets of reachable states. Our key technical contributions include novel abstract transformers and refinement operators that account for the semantics of polynomial assignments and guards more precisely than earlier work, while remaining amenable to parallelization and efficient implementation. This is achieved by appealing to Farkas’ Lemma and Handelman’s Theorem, and by exploiting geometric properties of unions of hyper-rectangles. Our abstract interpretation technique proves safety properties of many polynomial programs that state-of-the-art abstract interpretation tools fail to prove. We have implemented our approach in a tool called PolyAbs , and experimentally evaluated it on a suite of benchmarks. Our experiments demonstrate the improved precision and broader coverage of PolyAbs vis-a-vis state-of-the-art abstract interpretation tools, including a commercial-grade tool.

Differential privacy (DP) has established itself as one of the standards for ensuring privacy of individual data. However, reasoning about DP is a challenging and error-prone task, hence methods for formal verification and refutation of DP properties have received significant interest in recent years. In this work, we present a novel method for automated formal refutation of є-DP. Our method refutes є-DP by searching for a pair of inputs together with a non-negative function over outputs whose expected value on these two inputs differs by a significant amount. The two inputs and the non-negative function over outputs are computed simultaneously, by utilizing upper expectation supermartingales and lower expectation submartingales from probabilistic program analysis, which we leverage to introduce a sound and complete proof rule for є-DP refutation. To the best of our knowledge, our method is the first method for є-DP refutation to offer the following four desirable features: (1) it is fully automated, (2) it is applicable to stochastic mechanisms with sampling instructions from both discrete and continuous distributions, (3) it provides soundness guarantees, and (4) it provides semi-completeness guarantees. Our experiments show that our prototype tool SuperDP achieves superior performance compared to the state of the art and manages to refute є-DP for a number of challenging examples collected from the literature, including ones that were out of the reach of prior methods.

The problem of checking satisfiability of linear real arithmetic (LRA) and non-linear real arithmetic (NRA) formulas has broad applications, in particular, they are at the heart of logic-related applications such as logic for artificial intelligence, program analysis, etc. While there has been much work on checking satisfiability of unquantified LRA and NRA formulas, the problem of checking satisfiability of quantified LRA and NRA formulas remains a significant challenge. The main bottleneck in the existing methods is a computationally expensive quantifier elimination step. In this work, we propose a novel method for efficient quantifier elimination in quantified LRA and NRA formulas. We propose a template-based Skolemization approach, where we automatically synthesize linear/polynomial Skolem functions in order to eliminate quantifiers in the formula. The key technical ingredient in our approach are Positivstellensätze theorems from algebraic geometry, which allow for an efficient manipulation of polynomial inequalities. Our method offers a range of appealing theoretical properties combined with a strong practical performance. On the theory side, our method is sound, semi-complete, and runs in subexponential time and polynomial space, as opposed to existing sound and complete quantifier elimination methods that run in doubly-exponential time and at least exponential space. On the practical side, our experiments show superior performance compared to state of the art SMT solvers in terms of the number of solved instances and runtime, both on LRA and on NRA benchmarks.

Learning-based methods provide a promising approach to solving highly non-linear control tasks that are often challenging for classical control methods. To ensure the satisfaction of a safety property, learning-based methods jointly learn a control policy together with a certificate function for the property. Popular examples include barrier functions for safety and Lyapunov functions for asymptotic stability. While there has been significant progress on learning-based control with certificate functions in the white-box setting, where the correctness of the certificate function can be formally verified, there has been little work on ensuring their reliability in the black-box setting where the system dynamics are unknown. In this work, we consider the problems of certifying and repairing neural network control policies and certificate functions in the black-box setting. We propose a novel framework that utilizes runtime monitoring to detect system behaviors that violate the property of interest under some initially trained neural network policy and certificate. These violating behaviors are used to extract new training data, that is used to re-train the neural network policy and the certificate function and to ultimately repair them. We demonstrate the effectiveness of our approach empirically by using it to repair and to boost the safety rate of neural network policies learned by a state-of-the-art method for learning-based control on two autonomous system control tasks.

Abstract We present the first supermartingale certificate for quantitative $$\omega $$ ω -regular properties of discrete-time infinite-state stochastic systems. Our certificate is defined on the product of the stochastic system and a limit-deterministic Büchi automaton that specifies the property of interest; hence we call it a limit-deterministic Büchi supermartingale (LDBSM). Previously known supermartingale certificates applied only to quantitative reachability, safety, or reach-avoid properties, and to qualitative (i.e., probability 1) $$\omega $$ ω -regular properties.We also present fully automated algorithms for the template-based synthesis of LDBSMs, for the case when the stochastic system dynamics and the controller can be represented in terms of polynomial inequalities. Our experiments demonstrate the ability of our method to solve verification and control tasks for stochastic systems that were beyond the reach of previous supermartingale-based approaches.

Weighted model integration (WMI) is a relatively recent formalism that has received significant interest as a technique for solving probabilistic inference tasks with complicated weight functions. Existing methods and tools are mostly focused on linear and polynomial functions and provide limited support for WMI of rational or radical functions, which naturally arise in several applications. In this work, we present a novel method for approximate WMI, which provides more effective support for the wide class of semi-algebraic functions that includes rational and radical functions, with literals defined over non-linear real arithmetic. Our algorithm leverages Farkas’ lemma and Handelman's theorem from real algebraic geometry to reduce WMI to solving a number of linear programming (LP) instances. The algorithm provides formal guarantees on the error bound of the obtained approximation and can reduce it to any user-defined value epsilon. Furthermore, our approach is perfectly parallelizable. Finally, we present extensive experimental results, demonstrating the superior performance of our method on a range of WMI tasks for rational and radical functions when compared to state-of-the-art tools for WMI, in terms of both applicability and tightness.

Abstract We consider the problem of refuting equivalence of probabilistic programs, i.e., the problem of proving that two probabilistic programs induce different output distributions. We study this problem in the context of programs with conditioning (i.e., with observe and score statements), where the output distribution is conditioned by the event that all the observe statements along a run evaluate to true, and where the probability densities of different runs may be updated via the score statements. Building on a recent work on programs without conditioning, we present a new equivalence refutation method for programs with conditioning. Our method is based on weighted restarting, a novel transformation of probabilistic programs with conditioning to the output equivalent probabilistic programs without conditioning that we introduce in this work. Our method is the first to be both a) fully automated, and b) providing provably correct answers. We demonstrate the applicability of our method on a set of programs from the probabilistic inference literature.

Abstract We study the classical problem of verifying programs with respect to formal specifications given in the linear temporal logic (LTL). We first present novel sound and complete witnesses for LTL verification over imperative programs. Our witnesses are applicable to both verification (proving) and refutation (finding bugs) settings. We then consider LTL formulas in which atomic propositions can be polynomial constraints and turn our focus to polynomial arithmetic programs, i.e. programs in which every assignment and guard consists only of polynomial expressions. For this setting, we provide an efficient algorithm to automatically synthesize such LTL witnesses. Our synthesis procedure is both sound and semi-complete. Finally, we present experimental results demonstrating the effectiveness of our approach and that it can handle programs which were beyond the reach of previous state-of-the-art tools.

We study selfish mining attacks in longest-chain blockchains like Bitcoin, but where the proof of work is replaced with efficient proof systems - like proofs of stake or proofs of space - and consider the problem of computing an optimal selfish mining attack which maximizes expected relative revenue of the adversary, thus minimizing the chain quality. To this end, we propose a novel selfish mining attack that aims to maximize this objective and formally model the attack as a Markov decision process (MDP). We then present a formal analysis procedure which computes an ϵ-tight lower bound on the optimal expected relative revenue in the MDP and a strategy that achieves this ϵ-tight lower bound, where ϵ > 0 may be any specified precision. Our analysis is fully automated and provides formal guarantees on the correctness. We evaluate our selfish mining attack and observe that it achieves superior expected relative revenue compared to two considered baselines.

Cost analysis, also known as resource usage analysis, is the task of finding bounds on the total cost of a program and is a well-studied problem in static analysis. In this work, we consider two classical quantitative problems in cost analysis for probabilistic programs. The first problem is to find a bound on the expected total cost of the program. This is a natural measure for the resource usage of the program and can also be directly applied to average-case runtime analysis. The second problem asks for a tail bound, i.e. ‍given a threshold t the goal is to find a probability bound p such that ℙ[total cost ≥ t ] ≤ p . Intuitively, given a threshold t on the resource, the problem is to find the likelihood that the total cost exceeds this threshold. First, for expectation bounds, a major obstacle in previous works on cost analysis is that they can handle only non-negative costs or bounded variable updates. In contrast, we provide a new variant of the standard notion of cost martingales, that allows us to find expectation bounds for a class of programs with general positive or negative costs and no restriction on the variable updates. More specifically, our approach is applicable as long as there is a lower bound on the total cost incurred along every path. Second, for tail bounds, all previous methods are limited to programs in which the expected total cost is finite. In contrast, we present a novel approach, based on a combination of our martingale-based method for expectation bounds with a quantitative safety analysis, to obtain a solution to the tail bound problem that is applicable even to programs with infinite expected cost. Specifically, this allows us to obtain runtime tail bounds for programs that do not terminate almost-surely. In summary, we provide a novel combination of martingale-based cost analysis and quantitative safety analysis that is able to find expectation and tail cost bounds for probabilistic programs, without the restrictions of non-negative costs, bounded updates, or finiteness of the expected total cost. Finally, we provide experimental results showcasing that our approach can solve instances that were beyond the reach of previous methods.

We consider the problems of statically refuting equivalence and similarity of output distributions defined by a pair of probabilistic programs. Equivalence and similarity are two fundamental relational properties of probabilistic programs that are essential for their correctness both in implementation and in compilation. In this work, we present a new method for static equivalence and similarity refutation. Our method refutes equivalence and similarity by computing a function over program outputs whose expected value with respect to the output distributions of two programs is different. The function is computed simultaneously with an upper expectation supermartingale and a lower expectation submartingale for the two programs, which we show to together provide a formal certificate for refuting equivalence and similarity. To the best of our knowledge, our method is the first approach to relational program analysis to offer the combination of the following desirable features: (1) it is fully automated, (2) it is applicable to infinite-state probabilistic programs, and (3) it provides formal guarantees on the correctness of its results. We implement a prototype of our method and our experiments demonstrate the effectiveness of our method to refute equivalence and similarity for a number of examples collected from the literature. CCS Concepts: • Theory of computation → Program verification; Program analysis; • Software and its engineering → Formal software verification; • Mathematics of computing → Probability and statistics .

Two-player zero-sum "graph games" are central in logic, verification, and multi-agent systems. The game proceeds by placing a token on a vertex of a graph, and allowing the players to move it 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, an auction (bidding) determines which player moves the token. So far, bidding games have only been studied as full-information games. In this work we initiate the study of partial-information bidding games: we study bidding games in which a player's initial budget is drawn from a known probability distribution. We show that while for some bidding mechanisms and objectives, it is straightforward to adapt the results from the full-information setting to the partial-information setting, for others, the analysis is significantly more challenging, requires new techniques, and gives rise to interesting results. Specifically, we study games with "mean-payoff" objectives in combination with "poorman" bidding. We construct optimal strategies for a partially-informed player who plays against a fully-informed adversary. We show that, somewhat surprisingly, the "value" under pure strategies does not necessarily exist in such games.

We study the problem of training and certifying adversarially robust quantized neural networks (QNNs). Quantization is a technique for making neural networks more efficient by running them using low-bit integer arithmetic and is therefore commonly adopted in industry. Recent work has shown that floating-point neural networks that have been verified to be robust can become vulnerable to adversarial attacks after quantization, and certification of the quantized representation is necessary to guarantee robustness. In this work, we present quantization-aware interval bound propagation (QA-IBP), a novel method for training robust QNNs. Inspired by advances in robust learning of non-quantized networks, our training algorithm computes the gradient of an abstract representation of the actual network. Unlike existing approaches, our method can handle the discrete semantics of QNNs. Based on QA-IBP, we also develop a complete verification procedure for verifying the adversarial robustness of QNNs, which is guaranteed to terminate and produce a correct answer. Compared to existing approaches, the key advantage of our verification procedure is that it runs entirely on GPU or other accelerator devices. We demonstrate experimentally that our approach significantly outperforms existing methods and establish the new state-of-the-art for training and certifying the robustness of QNNs.

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.

Abstract Markov decision processes can be viewed as transformers of probability distributions. While this view is useful from a practical standpoint to reason about trajectories of distributions, basic reachability and safety problems are known to be computationally intractable (i.e., Skolem-hard) to solve in such models. Further, we show that even for simple examples of MDPs, strategies for safety objectives over distributions can require infinite memory and randomization. In light of this, we present a novel overapproximation approach to synthesize strategies in an MDP, such that a safety objective over the distributions is met. More precisely, we develop a new framework for template-based synthesis of certificates as affine distributional and inductive invariants for safety objectives in MDPs. We provide two algorithms within this framework. One can only synthesize memoryless strategies, but has relative completeness guarantees, while the other can synthesize general strategies. The runtime complexity of both algorithms is in PSPACE. We implement these algorithms and show that they can solve several non-trivial examples.

We consider bidding games, a class of two-player zero-sum graph games. The game proceeds as follows. Both players have bounded budgets. A token is placed on a vertex of a graph, in each turn the players simultaneously submit bids, and the higher bidder moves the token, where we break bidding ties in favor of Player 1. Player 1 wins the game iff the token visits a designated target vertex. We consider, for the first time, poorman discrete-bidding in which the granularity of the bids is restricted and the higher bid is paid to the bank. Previous work either did not impose granularity restrictions or considered Richman bidding (bids are paid to the opponent). While the latter mechanisms are technically more accessible, the former is more appealing from a practical standpoint. Our study focuses on threshold budgets, which is the necessary and sufficient initial budget required for Player 1 to ensure winning against a given Player 2 budget. We first show existence of thresholds. In DAGs, we show that threshold budgets can be approximated with error bounds by thresholds under continuous-bidding and that they exhibit a periodic behavior. We identify closed-form solutions in special cases. We implement and experiment with an algorithm to find threshold budgets.

Reinforcement learning has shown promising results in learning neural network policies for complicated control tasks. However, the lack of formal guarantees about the behavior of such policies remains an impediment to their deployment. We propose a novel method for learning a composition of neural network policies in stochastic environments, along with a formal certificate which guarantees that a specification over the policy's behavior is satisfied with the desired probability. Unlike prior work on verifiable RL, our approach leverages the compositional nature of logical specifications provided in SpectRL, to learn over graphs of probabilistic reach-avoid specifications. The formal guarantees are provided by learning neural network policies together with reach-avoid supermartingales (RASM) for the graph’s sub-tasks and then composing them into a global policy. We also derive a tighter lower bound compared to previous work on the probability of reach-avoidance implied by a RASM, which is required to find a compositional policy with an acceptable probabilistic threshold for complex tasks with multiple edge policies. We implement a prototype of our approach and evaluate it on a Stochastic Nine Rooms environment.

Abstract Reinforcement learning has received much attention for learning controllers of deterministic systems. We consider a learner-verifier framework for stochastic control systems and survey recent methods that formally guarantee a conjunction of reachability and safety properties. Given a property and a lower bound on the probability of the property being satisfied, our framework jointly learns a control policy and a formal certificate to ensure the satisfaction of the property with a desired probability threshold. Both the control policy and the formal certificate are continuous functions from states to reals, which are learned as parameterized neural networks. While in the deterministic case, the certificates are invariant and barrier functions for safety, or Lyapunov and ranking functions for liveness, in the stochastic case the certificates are supermartingales. For certificate verification, we use interval arithmetic abstract interpretation to bound the expected values of neural network functions.

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 a LexRSM not existing even for simple terminating programs. Our contributions are twofold. First, we introduce a generalization of LexRSMs that 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.

We consider the problem of formally verifying almost-sure (a.s.) asymptotic stability in discrete-time nonlinear stochastic control systems. While verifying stability in deterministic control systems is extensively studied in the literature, verifying stability in stochastic control systems is an open problem. The few existing works on this topic either consider only specialized forms of stochasticity or make restrictive assumptions on the system, rendering them inapplicable to learning algorithms with neural network policies. In this work, we present an approach for general nonlinear stochastic control problems with two novel aspects: (a) instead of classical stochastic extensions of Lyapunov functions, we use ranking supermartingales (RSMs) to certify a.s. asymptotic stability, and (b) we present a method for learning neural network RSMs. We prove that our approach guarantees a.s. asymptotic stability of the system and provides the first method to obtain bounds on the stabilization time, which stochastic Lyapunov functions do not. Finally, we validate our approach experimentally on a set of nonlinear stochastic reinforcement learning environments with neural network policies.

Abstract We consider the quantitative problem of obtaining lower-bounds on the probability of termination of a given non-deterministic probabilistic program. Specifically, given a non-termination threshold $$p \in [0, 1],$$ p ∈ [ 0 , 1 ] , we aim for certificates proving that the program terminates with probability at least $$1-p$$ 1 - p . The basic idea of our approach is to find a terminating stochastic invariant, i.e. a subset $$ SI $$ SI of program states such that (i) the probability of the program ever leaving $$ SI $$ SI is no more than p, and (ii) almost-surely, the program either leaves $$ SI $$ SI or terminates. While stochastic invariants are already well-known, we provide the first proof that the idea above is not only sound, but also complete for quantitative termination analysis. We then introduce a novel sound and complete characterization of stochastic invariants that enables template-based approaches for easy synthesis of quantitative termination certificates, especially in affine or polynomial forms. Finally, by combining this idea with the existing martingale-based methods that are relatively complete for qualitative termination analysis, we obtain the first automated, sound, and relatively complete algorithm for quantitative termination analysis. Notably, our completeness guarantees for quantitative termination analysis are as strong as the best-known methods for the qualitative variant. Our prototype implementation demonstrates the effectiveness of our approach on various probabilistic programs. We also demonstrate that our algorithm certifies lower bounds on termination probability for probabilistic programs that are beyond the reach of previous methods.

Given a Markov chain M = (V, v_0, δ), with state space V and a starting state v_0, and a probability threshold ε, an ε-core is a subset C of states that is left with probability at most ε. More formally, C ⊆ V is an ε-core, iff ℙ[reach (V\C)] ≤ ε. Cores have been applied in a wide variety of verification problems over Markov chains, Markov decision processes, and probabilistic programs, as a means of discarding uninteresting and low-probability parts of a probabilistic system and instead being able to focus on the states that are likely to be encountered in a real-world run. In this work, we focus on the problem of computing a minimal ε-core in a Markov chain. Our contributions include both negative and positive results: (i) We show that the decision problem on the existence of an ε-core of a given size is NP-complete. This solves an open problem posed in [Jan Kretínský and Tobias Meggendorfer, 2020]. We additionally show that the problem remains NP-complete even when limited to acyclic Markov chains with bounded maximal vertex degree; (ii) We provide a polynomial time algorithm for computing a minimal ε-core on Markov chains over control-flow graphs of structured programs. A straightforward combination of our algorithm with standard branch prediction techniques allows one to apply the idea of cores to find a subset of program lines that are left with low probability and then focus any desired static analysis on this core subset.

We present a novel approach to differential cost analysis that, given a program revision, attempts to statically bound the difference in resource usage, or cost, between the two program versions. Differential cost analysis is particularly interesting because of the many compelling applications for it, such as detecting resource-use regressions at code-review time or proving the absence of certain side-channel vulnerabilities. One prior approach to differential cost analysis is to apply relational reasoning that conceptually constructs a product program on which one can over-approximate the difference in costs between the two program versions. However, a significant challenge in any relational approach is effectively aligning the program versions to get precise results. In this paper, our key insight is that we can avoid the need for and the limitations of program alignment if, instead, we bound the difference of two cost-bound summaries rather than directly bounding the concrete cost difference. In particular, our method computes a threshold value for the maximal difference in cost between two program versions simultaneously using two kinds of cost-bound summaries---a potential function that evaluates to an upper bound for the cost incurred in the first program and an anti-potential function that evaluates to a lower bound for the cost incurred in the second. Our method has a number of desirable properties: it can be fully automated, it allows optimizing the threshold value on relative cost, it is suitable for programs that are not syntactically similar, and it supports non-determinism. We have evaluated an implementation of our approach on a number of program pairs collected from the literature, and we find that our method computes tight threshold values on relative cost in most examples.

Formal verification of neural networks is an active topic of research, and recent advances have significantly increased the size of the networks that verification tools can handle. However, most methods are designed for verification of an idealized model of the actual network which works over real arithmetic and ignores rounding imprecisions. This idealization is in stark contrast to network quantization, which is a technique that trades numerical precision for computational efficiency and is, therefore, often applied in practice. Neglecting rounding errors of such low-bit quantized neural networks has been shown to lead to wrong conclusions about the network's correctness. Thus, the desired approach for verifying quantized neural networks would be one that takes these rounding errors into account. In this paper, we show that verifying the bit-exact implementation of quantized neural networks with bit-vector specifications is PSPACE-hard, even though verifying idealized real-valued networks and satisfiability of bit-vector specifications alone are each in NP. Furthermore, we explore several practical heuristics toward closing the complexity gap between idealized and bit-exact verification. In particular, we propose three techniques for making SMT-based verification of quantized neural networks more scalable. Our experiments demonstrate that our proposed methods allow a speedup of up to three orders of magnitude over existing approaches.

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.

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.

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.

A graph game proceeds as follows: two players move a token through a graph to produce a finite or infinite path, which determines the payoff of the game. We study bidding games in which in each turn, an auction determines which player moves the token. Bidding games were largely studied in combination with two variants of first-price auctions called “Richman” and “poorman” bidding. We study taxman bidding, which span the spectrum between the two. The game is parameterized by a constant τ ∈ [ 0 , 1 ] : portion τ of the winning bid is paid to the other player, and portion 1 − τ to the bank. While finite-duration (reachability) taxman games have been studied before, we present, for the first time, results on infinite-duration taxman games: we unify, generalize, and simplify previous equivalences between bidding games and a class of stochastic games called random-turn games .

A standard objective in partially-observable Markov decision processes (POMDPs) is to find a policy that maximizes the expected discounted-sum payoff. However, such policies may still permit unlikely but highly undesirable outcomes, which is problematic especially in safety-critical applications. Recently, there has been a surge of interest in POMDPs where the goal is to maximize the probability to ensure that the payoff is at least a given threshold, but these approaches do not consider any optimization beyond satisfying this threshold constraint. In this work we go beyond both the “expectation” and “threshold” approaches and consider a “guaranteed payoff optimization (GPO)” problem for POMDPs, where we are given a threshold t and the objective is to find a policy σ such that a) each possible outcome of σ yields a discounted-sum payoff of at least t, and b) the expected discounted-sum payoff of σ is optimal (or near-optimal) among all policies satisfying a). We present a practical approach to tackle the GPO problem and evaluate it on standard POMDP benchmarks.

Termination is one of the basic liveness properties, and we study the termination problem for probabilistic programs with real-valued variables. Previous works focused on the qualitative problem that asks whether an input program terminates with probability 1 (almost-sure termination). A powerful approach for this qualitative problem is the notion of ranking supermartingales with respect to a given set of invariants. The quantitative problem (probabilistic termination) asks for bounds on the termination probability, and this problem has not been addressed yet. A fundamental and conceptual drawback of the existing approaches to address probabilistic termination is that even though the supermartingales consider the probabilistic behaviour of the programs, the invariants are obtained completely ignoring the probabilistic aspect (i.e., the invariants are obtained considering all behaviours with no information about the probability).