Memory models are formal specifications of concurrent-program executions, accounting for weak behaviors introduced by compiler and architectural optimizations. The increase of their number and complexity has spawned efforts for uniform verification across whole classes of models, by axiomatizing the models in an adequate metatheory that admits a uniform treatment. In this work, we formally study Monadic Second-Order logic (MSO) as a metatheory for weak memory, by proving results on the treewidth and MSO-expressibility of various popular weak-memory models, as this combination allows us to uniformly tackle several verification problems. In summary, our results are as follows. First, we prove that executions under Sequential Consistency (SC) have bounded treewidth, while already those under Total Store Order (TSO) do not. Second, we prove that a broad range of models, including Release/Acquire and the full RC20, are MSO-axiomatizable, while others, such as Strong Release/Acquire and TSO, are not, unless the Orthogonal Vectors problem - which requires quadratic time under SETH - can be solved in linear time. Finally, we introduce the notion of reads-from robustness, as an extension to recent work on coarse robustness criteria. We show that our treewidth bounds (both upper and lower) have far-reaching algorithmic implications for any of our MSO-axiomatizable models MM: there is an algorithm that, for every program 𝖯, either verifies 𝖯 under MM or reports that 𝖯 is not reads-from robust against MM. Overall, our results establish a rich and versatile theoretical framework for weak-memory verification and robustness.

A key computational question underpinning the automated testing and verification of concurrent programs is the consistency question — given a partial execution history, can it be completed in a consistent manner? Due to its importance, consistency testing has been studied extensively for memory models, as well as for database isolation levels. A common theme in all these settings is the use of shared-memory as the primal mode of interthread communication. On the other hand, modern programming languages, such as Go, Rust and Kotlin, advocate a paradigm shift towards channel-based (i.e., message-passing) communication. However, the consistency question for channel-based concurrency is currently poorly understood. In this paper we lift the study of fundamental consistency problems to channels, taking into account various input parameters, such as the number of threads executing, the number of channels, and the channel capacities. We draw a rich complexity landscape, including upper bounds that become polynomial when certain input parameters are fixed, as well as hardness lower bounds. Our upper bounds are based on algorithms that can drive the verification of channel consistency in automated verification tools. Our lower bounds characterize minimal input parameters that are sufficient for hardness to arise, and thus shed light on the intricacies of testing channel-based concurrency. In combination, our upper and lower bounds characterize the boundary of tractability/intractability of verifying channel consistency, and imply that our algorithms are often (nearly) optimal. We have also implemented our main consistency checking algorithm and designed optimizations to enhance its performance. We evaluated the performance of our implementation over a set of 103 instances obtained from open source Go projects, and compared it against a constraint-solving based algorithm. Our experimental results demonstrate the power of our consistency-checking algorithm; it scales to around 1M events, and is significantly faster in running-time performance, compared to a constraint-solving approach.

Atomicity is a fundamental abstraction in concurrency, specifying that program behavior can be understood by considering specific code blocks executing atomically. However, atomicity invariants are tricky to maintain while also optimizing for code efficiency, and atomicity violations are a common root cause of many concurrency bugs. To address this problem, several dynamic techniques have been developed for testing whether a program execution adheres to an atomicity specification, most often instantiated as conflict serializability . The efficiency of the analysis has been targeted in various papers, with the state-of-the-art algorithms RegionTrack and Aerodrome achieving a time complexity O ( nk 3 ) and O ( nk ( k + v + ℓ)), respectively, for a trace σ of n events, k threads, v locations, and ℓ locks. In this paper we introduce AtomSanitizer, a new algorithm for testing conflict serializability, with time complexity O ( nk 2 ). AtomSanitizer operates in an efficient streaming style, is theoretically faster than all existing algorithms, and also has a smaller memory footprint. Moreover, AtomSanitizer is the first algorithm designed to incur minimal locking when deployed in a concurrent monitoring setting. Experiments on standard benchmarks indicate that AtomSanitizer is always faster in practice than all existing conflict-serializability testers. Finally, we also implement AtomSanitizer inside the TSAN framework, for monitoring atomicity in real time. Our experiments reveal that AtomSanitizer incurs minimal time and space overhead compared to the data-race detection engine of TSAN, and thus is the first algorithm for conflict serializability demonstrated to be suitable for a runtime monitoring setting.

Abstract GPU computing is embracing weak memory concurrency for performance improvement. However, compared to CPUs, modern GPUs provide more fine-grained concurrency features such as scopes, have additional properties like divergence, and thereby follow different weak memory consistency models. These features and properties make concurrent programming on GPUs more complex and error-prone. To this end, we present $$\textsf{GPUMC}$$ GPUMC , a stateless model checker to check the correctness of GPU shared-memory concurrent programs under scoped-RC11 weak memory concurrency model. $$\textsf{GPUMC}$$ GPUMC explores all possible executions in GPU programs to reveal various errors - races, barrier divergence, and assertion violations. In addition, $$\textsf{GPUMC}$$ GPUMC also automatically repairs these errors in the appropriate cases. We evaluate $$\textsf{GPUMC}$$ GPUMC on benchmarks and real-life GPU programs. $$\textsf{GPUMC}$$ GPUMC is efficient both in time and memory in verifying large GPU programs where state-of-the-art tools are timed out. In addition, $$\textsf{GPUMC}$$ GPUMC identifies all known errors in these benchmarks compared to the state-of-the-art tools.

Context-free language (CFL) reachability is a standard approach in static analyses, where the analysis question (e.g., is there a dataflow from x to y ?) is phrased as a language reachability problem on a graph G wrt a CFL L . However, CFLs lack the expressiveness needed for high analysis precision. On the other hand, common formalisms for context-sensitive languages are too expressive, in the sense that the corresponding reachability problem becomes undecidable. Are there useful context-sensitive language-reachability models for static analysis? In this paper, we introduce Multiple Context-Free Language (MCFL) reachability as an expressive yet tractable model for static program analysis. MCFLs form an infinite hierarchy of mildly context sensitive languages parameterized by a dimension d and a rank r . Larger d and r yield progressively more expressive MCFLs, offering tunable analysis precision. We showcase the utility of MCFL reachability by developing a family of MCFLs that approximate interleaved Dyck reachability, a common but undecidable static analysis problem. Given the increased expressiveness of MCFLs, one natural question pertains to their algorithmic complexity, i.e., how fast can MCFL reachability be computed ? We show that the problem takes O n 2 d + 1 time on a graph of n nodes when r = 1 , and O n d r + 1 time when r > 1 . Moreover, we show that when r = 1 , even the simpler membership problem has a lower bound of n 2 d based on the Strong Exponential Time Hypothesis, while reachability for d = 1 has a lower bound of n 3 based on the combinatorial Boolean Matrix Multiplication Hypothesis. Thus, for

Database isolation is a formal contract concerning the level of data consistency that a database provides to its clients. In order to achieve low latency, high throughput, and partition tolerance, modern databases forgo strong transaction isolation for weak isolation guarantees. However, several production databases have been found to suffer from isolation bugs , breaking their data-consistency contract. Black-box testing is a prominent technique for detecting isolation bugs, by checking whether histories of database transactions adhere to a prescribed isolation level. In order to test databases on realistic workloads of large size, isolation testers must be as efficient as possible, a requirement that has initiated a study of the complexity of isolation testing. Although testing strong isolation has been known to be NP-complete, weak isolation levels were recently shown to be testable in polynomial time, which has propelled the scalability of testing tools. However, existing testers have a large polynomial complexity, restricting testing to workloads of only moderate size, which is not typical of large-scale databases. How efficiently can we provably test weak database isolation? In this work, we develop AWDIT, a highly-efficient and provably optimal tester for weak database isolation . Given a history H of size n and k sessions, AWDIT tests whether H satisfies the most common weak isolation levels of Read Committed (RC), Read Atomic (RA), and Causal Consistency (CC) in time O ( n 3/2 ), O ( n 3/2 ), and O ( n · k ), respectively, improving significantly over the state of the art. Moreover, we prove that AWDIT is essentially optimal , in the sense that there is a lower bound of n 3/2 , based on the combinatorial BMM hypothesis, for any weak isolation level between RC and CC. Our experiments show that AWDIT is significantly faster than existing, highly optimized testers; e.g., for the ∼20﹪ largest histories, AWDIT obtains an average speedup of 245×, 193×, and 62× for RC, RA, and CC, respectively, over the best baseline.

Abstract Interleaved Dyck Reachability is a standard, graph-based formulation of a plethora of static analyses that seek to be context- and field-sensitive, where each type of sensitivity is expressed via a CFL/Dyck language. Unfortunately, the problem is well known to be undecidable in general, and thus existing approaches resort to clever overapproximations. Recently, a mutual-refinement algorithm, that iteratively considers each of the two sensitivities in isolation until a fixpoint is reached, was shown to achieve high precision. In this work we present a more precise approximation of Interleaved Dyck Reachability, by extending the mutual-refinement algorithm in three general directions. First, we develop refined CFLs to express each type of sensitivity precisely, while simultaneously also lightly overapproximating the opposite type. Second, we modify the graph information in a way to highlight unreachable paths not previously captured by the algorithm. Third, we apply the resulting algorithm on an on-demand basis, which effectively masks out imprecision incurred by parts of the graph that are irrelevant for the query at hand. Our experiments show that the new approach offers significantly higher precision than the vanilla mutual-refinement algorithm and other common baselines; for a particularly challenging benchmark, we report, on average, $51\%$ 51 % of the reachable pairs compared to the most recent alternative.

Dynamic analyses are a standard approach to analyzing and testing concurrent programs. Such techniques observe program traces σ and analyze them to infer the presence or absence of bugs. At its core, each analysis maintains a partial order P that represents order dependencies between the events of σ. Naturally, the scalability of the analysis largely depends on maintaining P efficiently. The standard data structure for this task has thus far been Vector Clocks. These, however, are slow for analyses that follow a non-streaming style, costing O(n) time for inserting (and propagating) each new ordering in P, where n is the size of σ, while they cannot handle the deletion of existing orderings.

Dynamic data race detection has emerged as a key technique for ensuring reliability of concurrent software in practice. However, dynamic approaches can often miss data races owing to non-determinism in the thread scheduler. Predictive race detection techniques cater to this shortcoming by inferring alternate executions that may expose data races without re-executing the underlying program. More formally, the dynamic data race prediction problem asks, given a trace σ of an execution of a concurrent program, can σ be correctly reordered to expose a data race? Existing state-of-the art techniques for data race prediction either do not scale to executions arising from real world concurrent software, or only expose a limited class of data races, such as those that can be exposed without reversing the order of synchronization operations.

Abstract The computation of bottom strongly connected components (BSCCs) is a fundamental task in model checking, as well as in characterizing the attractors of dynamical systems. As such, symbolic algorithms for BSCCs have received special attention, and are based on the idea that the computation of an SCC can be stopped early, as soon as it is deemed to be non-bottom. In this paper we introduce $$\textsc {Pendant}$$ P E N D A N T , a new symbolic algorithm for computing BSCCs which runs in linear symbolic time. In contrast to the standard approach of escaping non-bottom SCCs, $$\textsc {Pendant}$$ P E N D A N T aims to start the computation from nodes that are likely to belong to BSCCs, and thus is more effective in sidestepping SCCs that are non-bottom. Moreover, we employ a simple yet powerful deadlock-detection technique, that quickly identifies singleton BSCCs before the main algorithm is run. Our experimental evaluation on three diverse datasets of 553 models demonstrates the efficacy of our two methods: $$\textsc {Pendant}$$ P E N D A N T is decisively faster than the standard existing algorithm for BSCC computation, while deadlock detection improves the performance of each algorithm significantly.

Weak-memory models are standard formal specifications of concurrency across hardware, programming languages, and distributed systems. A fundamental computational problem is consistency testing : is the observed execution of a concurrent program in alignment with the specification of the underlying system? The problem has been studied extensively across Sequential Consistency (SC) and weak memory, and proven to be N P -complete when some aspect of the input (e.g., number of threads/memory locations) is unbounded. This unboundedness has left a natural question open: are there efficient parameterized algorithms for testing? The main contribution of this paper is a deep hardness result for consistency testing under many popular weak-memory models: the problem remains N P -complete even in its bounded setting, where candidate executions contain a bounded number of threads, memory locations, and values. This hardness spreads across several Release-Acquire variants of C11, a popular variant of its Relaxed fragment, popular Causal Consistency models, and the POWER architecture. To our knowledge, this is the first result that fully exposes the hardness of weak-memory testing and proves that the problem admits no parameterization under standard input parameters. It also yields a computational separation of these models from SC, x86-TSO, PSO, and Relaxed, for which bounded consistency testing is either known (for SC), or shown here (for the rest), to be in polynomial time.

Dyck reachability is a principled, graph-based formulation of a plethora of static analyses. Bidirected graphs are used for capturing dataflow through mutable heap data, and are usual formalisms of demand-driven points-to and alias analyses. The best (offline) algorithm runs in O ( m + n · α ( n ) ) time, where n is the number of nodes and m is the number of edges in the flow graph, which becomes O ( n 2 ) in the worst case. In the everyday practice of program analysis, the analyzed code is subject to continuous change, with source code being added and removed. On-the-fly static analysis under such continuous updates gives rise to dynamic Dyck reachability , where reachability queries run on a dynamically changing graph, following program updates. Naturally, executing the offline algorithm in this online setting is inadequate, as the time required to process a single update is prohibitively large. In this work we develop a novel dynamic algorithm for bidirected Dyck reachability that has O ( n · α ( n ) ) worst-case performance per update, thus beating the O ( n 2 ) bound, and is also optimal in certain settings. We also implement our algorithm and evaluate its performance on on-the-fly data-dependence and alias analyses, and compare it with two best known alternatives, namely (i) the optimal offline algorithm, and (ii) a fully dynamic Datalog solver. Our experiments show that our dynamic algorithm is consistently, and by far, the top performing algorithm, exhibiting speedups in the order of 1000X. The running time of each update is almost always unnoticeable to the human eye, making it ideal for the on-the-fly analysis setting.

The Voter model is a well-studied stochastic process that models the invasion of a novel trait A (e.g., a new opinion, social meme, genetic mutation, magnetic spin) in a network of individuals (agents, people, genes, particles) carrying an existing resident trait B. Individuals change traits by occasionally sampling the trait of a neighbor, while an invasion bias δ ≥ 0 expresses the stochastic preference to adopt the novel trait A over the resident trait B. The strength of an invasion is measured by the probability that eventually the whole population adopts trait A, i.e., the fixation probability. In more realistic settings, however, the invasion bias is not ubiquitous, but rather manifested only in parts of the network. For instance, when modeling the spread of a social trait, the invasion bias represents localized incentives. In this paper, we generalize the standard biased Voter model to the positional Voter model, in which the invasion bias is effectuated only on an arbitrary subset of the network nodes, called biased nodes. We study the ensuing optimization problem, which is, given a budget k, to choose k biased nodes so as to maximize the fixation probability of a randomly occurring invasion. We show that the problem is NP-hard both for finite δ and when δ → ∞ (strong bias), while the objective function is not submodular in either setting, indicating strong computational hardness. On the other hand, we show that, when δ → 0 (weak bias), we can obtain a tight approximation in O(n^2ω ) time, where ω is the matrix-multiplication exponent. We complement our theoretical results with an experimental evaluation of some proposed heuristics.

Abstract Decomposing a directed graph to its strongly connected components (SCCs) is a fundamental task in model checking. To deal with the state-space explosion problem, graphs are often represented symbolically using binary decision diagrams (BDDs), which have exponential compression capabilities. The theoretically-best symbolic algorithm for SCC decomposition is Gentilini et al’s $$\textsc {Skeleton}$$ algorithm, that uses O(n) symbolic steps on a graph of n nodes. However, $$\textsc {Skeleton}$$ uses $$\Theta (n)$$ symbolic objects, as opposed to (poly-)logarithmically many, which is the norm for symbolic algorithms, thereby relinquishing its symbolic nature. Here we present $$\textsc {Chain}$$ , a new symbolic algorithm for SCC decomposition that also makes O(n) symbolic steps, but further uses logarithmic space, and is thus truly symbolic. We then extend $$\textsc {Chain}$$ to $$\textsc {ColoredChain}$$ , an algorithm for SCC decomposition on edge-colored graphs, which arise naturally in model-checking a family of systems. Finally, we perform an experimental evaluation of $$\textsc {Chain}$$ among other standard symbolic SCC algorithms in the literature. The results show that $$\textsc {Chain}$$ is competitive on almost all benchmarks, and often faster, while it clearly outperforms all other algorithms on challenging inputs.

Deadlocks are one of the most notorious concurrency bugs, and significant research has focused on detecting them efficiently. Dynamic predictive analyses work by observing concurrent executions, and reason about alternative interleavings that can witness concurrency bugs. Such techniques offer scalability and sound bug reports, and have emerged as an effective approach for concurrency bug detection, such as data races. Effective dynamic deadlock prediction, however, has proven a challenging task, as no deadlock predictor currently meets the requirements of soundness, high-precision, and efficiency. In this paper, we first formally establish that this tradeoff is unavoidable, by showing that (a) sound and complete deadlock prediction is intractable, in general, and (b) even the seemingly simpler task of determining the presence of potential deadlocks, which often serve as unsound witnesses for actual predictable deadlocks, is intractable. The main contribution of this work is a new class of predictable deadlocks, called sync(hronization)-preserving deadlocks. Informally, these are deadlocks that can be predicted by reordering the observed execution while preserving the relative order of conflicting critical sections. We present two algorithms for sound deadlock prediction based on this notion. Our first algorithm SPDOffline detects all sync-preserving deadlocks, with running time that is linear per abstract deadlock pattern, a novel notion also introduced in this work. Our second algorithm SPDOnline predicts all sync-preserving deadlocks that involve two threads in a strictly online fashion, runs in overall linear time, and is better suited for a runtime monitoring setting. We implemented both our algorithms and evaluated their ability to perform offline and online deadlock-prediction on a large dataset of standard benchmarks. Our results indicate that our new notion of sync-preserving deadlocks is highly effective, as (i) it can characterize the vast majority of deadlocks and (ii) it can be detected using an online, sound, complete and highly efficient algorithm.

Over the years, several memory models have been proposed to capture the subtle concurrency semantics of C/C++. One of the most fundamental problems associated with a memory model M is consistency checking: given an execution X , is X consistent with M ? This problem lies at the heart of numerous applications, including specification testing and litmus tests, stateless model checking, and dynamic analyses. As such, it has been explored extensively and its complexity is well-understood for traditional models like SC and TSO. However, less is known for the numerous model variants of C/C++, for which the problem becomes challenging due to the intricacies of their concurrency primitives. In this work we study the problem of consistency checking for popular variants of the C11 memory model, in particular, the RC 20 model, its release-acquire ( RA ) fragment, the strong and weak variants of RA ( SRA and WRA ), as well as the Relaxed fragment of RC 20. Motivated by applications in testing and model checking, we focus on reads-from consistency checking. The input is an execution X specifying a set of events, their program order and their reads-from relation, and the task is to decide the existence of a modification order on the writes of X that makes X consistent in a memory model. We draw a rich complexity landscape for this problem; our results include (i) nearly-linear-time algorithms for certain variants, which improve over prior results, (ii) fine-grained optimality results, as well as (iii) matching upper and lower bounds (NP-hardness) for other variants. To our knowledge, this is the first work to characterize the complexity of consistency checking for C11 memory models. We have implemented our algorithms inside the TruSt model checker and the C11Tester testing tool. Experiments on standard benchmarks show that our new algorithms improve consistency checking, often by a significant margin.

The Moran process is a classic stochastic process that models invasion dynamics on graphs. A single mutant (e.g., a new opinion, strain, social trait etc.) invades a population of residents spread over the nodes of a graph. The mutant fitness advantage δ>=0 determines how aggressively mutants propagate to their neighbors. The quantity of interest is the fixation probability, i.e., the probability that the initial mutant eventually takes over the whole population. However, in realistic settings, the invading mutant has an advantage only in certain locations. E.g., the ability to metabolize a certain sugar is an advantageous trait to bacteria only when the sugar is actually present in their surroundings. In this paper we introduce the positional Moran process, a natural generalization in which the mutant fitness advantage is only realized on specific nodes called active nodes, and study the problem of fixation maximization: given a budget k, choose a set of k active nodes that maximize the fixation probability of the invading mutant. We show that the problem is NP-hard, while the optimization function is not submodular, thus indicating strong computational hardness. We focus on two natural limits. In the limit of δ to infinity (strong selection), although the problem remains NP-hard, the optimization function becomes submodular and thus admits a constant-factor approximation using a simple greedy algorithm. In the limit of δ to 0 (weak selection), we show that we can obtain a tight approximation in O(n^{2×ω}) time, where ω is the matrix-multiplication exponent. An experimental evaluation of the new algorithms along with some proposed heuristics corroborates our results.

Dynamic techniques are a scalable and effective way to analyze concurrent programs. Instead of analyzing all behaviors of a program, these techniques detect errors by focusing on a single program execution. Often a crucial step in these techniques is to define a causal ordering between events in the execution, which is then computed using vector clocks, a simple data structure that stores logical times of threads. The two basic operations of vector clocks, namely join and copy, require Θ(k) time, where k is the number of threads. Thus they are a computational bottleneck when k is large.

A pushdown vector addition system with states (PVASS) extends the model of vector addition systems with a pushdown store. A PVASS is said to be bidirected if every transition (pushing/popping a symbol or modifying a counter) has an accompanying opposite transition that reverses the effect. Bidirectedness arises naturally in many models; it can also be seen as a overapproximation of reachability. We show that the reachability problem for bidirected PVASS is decidable in Ackermann time and primitive recursive for any fixed dimension. For the special case of one-dimensional bidirected PVASS, we show reachability is in PSPACE, and in fact in polynomial time if the stack is polynomially bounded. Our results are in contrast to the directed setting, where decidability of reachability is a long-standing open problem already for one dimensional PVASS, and there is a PSPACE-lower bound already for one-dimensional PVASS with bounded stack. The reachability relation in the bidirected (stateless) case is a congruence over ℕ^d. Our upper bounds exploit saturation techniques over congruences. In particular, we show novel elementary-time constructions of semilinear representations of congruences generated by finitely many vector pairs. In the case of one-dimensional PVASS, we employ a saturation procedure over bounded-size counters. We complement our upper bound with a TOWER-hardness result for arbitrary dimension and k-EXPSPACE hardness in dimension 2k+6 using a technique by Lazić and Totzke to implement iterative exponentiations.

The voter process is a classic stochastic process that models the invasion of a mutant trait A (e.g., a new opinion, belief, legend, genetic mutation, magnetic spin) in a population of agents (e.g., people, genes, particles) who share a resident trait B, spread over the nodes of a graph. An agent may adopt the trait of one of its neighbors at any time, while the invasion bias r quantifies the stochastic preference towards (r>1) or against (r<1) adopting A over B. Success is measured in terms of the fixation probability, i.e., the probability that eventually all agents have adopted the mutant trait A. In this paper we study the problem of fixation probability maximization under this model: given a budget k, find a set of k agents to initiate the invasion that maximizes the fixation probability. We show that the problem is NP-hard for both regimes r>1 and r<1, while the latter case is also inapproximable within any multiplicative factor that is independent of r. On the positive side, we show that when r>1, the optimization function is submodular and thus can be greedily approximated within a factor 1-1/e. An experimental evaluation of some proposed heuristics corroborates our results.

There is a huge and growing gap between the speed of accesses to data stored in main memory vs cache. Thus, cache misses account for a significant portion of runtime overhead in virtually every program and minimizing them has been an active research topic for decades. The primary and most classical formal model for this problem is that of Cache-conscious Data Placement (CDP): given a commutative cache with constant capacity k and a sequence Σ of accesses to data elements, the goal is to map each data element to a cache line such that the total number of cache misses over Σ is minimized. Note that we are considering an offline single-threaded setting in which Σ is known a priori. CDP has been widely studied since the 1990s. In POPL 2002, Petrank and Rawitz proved a notoriously strong hardness result: They showed that for every k ≥ 3, CDP is not only NP-hard but also hard-to-approximate within any non-trivial factor unless P=NP. As such, all subsequent works gave up on theoretical improvements and instead focused on heuristic algorithms with no theoretical guarantees.

Data races are the most common concurrency bugs and considerable efforts are put in ensuring data-race-free (DRF) programs. The most popular approach is via dynamic analyses, which soundly report DRF violations by analyzing program executions. Recently, there has been a prevalent shift to predictive analysis techniques. Such techniques attempt to predict DRF violations even in unobserved program executions, while making sure that the analysis is sound (does not raise false positives).

Dyck reachability is the standard formulation of a large domain of static analyses, as it achieves the sweet spot between precision and efficiency, and has thus been studied extensively. Interleaved Dyck reachability (denoted D k ⊙ D k ) uses two Dyck languages for increased precision (e.g., context and field sensitivity) but is well-known to be undecidable. As many static analyses yield a certain type of bidirected graphs, they give rise to interleaved bidirected Dyck reachability problems. Although these problems have seen numerous applications, their decidability and complexity has largely remained open. In a recent work, Li et al. made the first steps in this direction, showing that (i) D 1 ⊙ D 1 reachability (i.e., when both Dyck languages are over a single parenthesis and act as counters) is computable in O ( n 7 ) time, while (ii) D k ⊙ D k reachability is NP-hard. However, despite this recent progress, most natural questions about this intricate problem are open. In this work we address the decidability and complexity of all variants of interleaved bidirected Dyck reachability. First, we show that D 1 ⊙ D 1 reachability can be computed in O ( n 3 · α( n )) time, significantly improving over the existing O ( n 7 ) bound. Second, we show that D k ⊙ D 1 reachability (i.e., when one language acts as a counter) is decidable, in contrast to the non-bidirected case where decidability is open. We further consider D k ⊙ D 1 reachability where the counter remains linearly bounded. Our third result shows that this bounded variant can be solved in O ( n 2 · α( n )) time, while our fourth result shows that the problem has a (conditional) quadratic lower bound, and thus our upper bound is essentially optimal. Fifth, we show that full D k ⊙ D k reachability is undecidable. This improves the recent NP-hardness lower-bound, and shows that the problem is equivalent to the non-bidirected case. Our experiments on standard benchmarks show that the new algorithms are very fast in practice, offering many orders-of-magnitude speedups over previous methods.

Abstract 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 $${\text {RVF-SMC}}$$ RVF-SMC , a new SMC algorithm that uses a novelreads-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, $${\text {RVF-SMC}}$$ RVF-SMC generates representatives very efficiently, as the reduction in the partitioning is often met with significant speed-ups in the model checking task.

Data races are among the most common bugs in concurrency. The standard approach to data-race detection is via dynamic analyses, which work over executions of concurrent programs, instead of the program source code. The rich literature on the topic has created various notions of dynamic data races, which are known to be detected efficiently when certain parameters (e.g., number of threads) are small. However, the fine-grained complexity of all these notions of races has remained elusive, making it impossible to characterize their trade-offs between precision and efficiency. In this work we establish several fine-grained separations between many popular notions of dynamic data races. The input is an execution trace σ with 𝒩 events, 𝒯 threads and ℒ locks. Our main results are as follows. First, we show that happens-before HB races can be detected in O(𝒩⋅ min(𝒯, ℒ)) time, improving over the standard O(𝒩⋅ 𝒯) bound when ℒ = o(𝒯). Moreover, we show that even reporting an HB race that involves a read access is hard for 2-orthogonal vectors (2-OV). This is the first rigorous proof of the conjectured quadratic lower-bound in detecting HB races. Second, we show that the recently introduced synchronization-preserving races are hard to detect for 3-OV and thus have a cubic lower bound, when 𝒯 = Ω(𝒩). This establishes a complexity separation from HB races which are known to be strictly less expressive. Third, we show that lock-cover races are hard for 2-OV, and thus have a quadratic lower-bound, even when 𝒯 = 2 and ℒ = ω(log 𝒩). The similar notion of lock-set races is known to be detectable in O(𝒩⋅ ℒ) time, and thus we achieve a complexity separation between the two. Moreover, we show that lock-set races become hitting-set (HS)-hard when ℒ = Θ(𝒩), and thus also have a quadratic lower bound, when the input is sufficiently complex. To our knowledge, this is the first work that characterizes the complexity of well-established dynamic race-detection techniques, allowing for a rigorous comparison between them.

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(n⁴). Since the output has size Θ(n⁴), our algorithm is optimal (up to polylog factors). Second, given a mean-payoff objective, we present an O(n³)-time algorithm for deciding whether the value of a starting state is non-negative, improving the previously known O(n⁴) bound. Third, given an initial credit for energy objective, we present an O(n⁵)-time algorithm for computing the minimum initial credit for all nodes of G, improving the previously known O(n⁸) 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}).

We consider the problem of reachability on One-Counter Systems (OCSs) and Pushdown Systems (PDSs). The problem has a well-known O(n3) bound on both models, while the bound is believed to be tight for PDSs. Here we establish new upper and lower bounds for reachability on OCSs and restricted PDSs. We show that the problem can be solved (i) in O(nω⋅log2⁡n) time on OCSs, and (ii) in O(n2) time on sparse PDSs when restricted to witness executions of stack height bounded by log⁡n. Moreover, we prove similar lower bounds.

Concurrent programs are notoriously hard to write correctly, as scheduling nondeterminism introduces subtle errors that are both hard to detect and to reproduce. The most common concurrency errors are (data) races, which occur when memory-conflicting actions are executed concurrently. Consequently, considerable effort has been made towards developing efficient techniques for race detection. The most common approach is dynamic race prediction: given an observed, race-free trace σ of a concurrent program, the task is to decide whether events of σ can be correctly reordered to a trace σ * that witnesses a race hidden in σ. In this work we introduce the notion of sync(hronization)-preserving races. A sync-preserving race occurs in σ when there is a witness σ * in which synchronization operations (e.g., acquisition and release of locks) appear in the same order as in σ. This is a broad definition that strictly subsumes the famous notion of happens-before races. Our main results are as follows. First, we develop a sound and complete algorithm for predicting sync-preserving races. For moderate values of parameters like the number of threads, the algorithm runs in Õ( N ) time and space, where N is the length of the trace σ. Second, we show that the problem has a Ω( N /log 2 N ) space lower bound, and thus our algorithm is essentially time and space optimal. Third, we show that predicting races with even just a single reversal of two sync operations is NP-complete and even W1-hard when parameterized by the number of threads. Thus, sync-preservation characterizes exactly the tractability boundary of race prediction, and our algorithm is nearly optimal for the tractable side. Our experiments show that our algorithm is fast in practice, while sync-preservation characterizes races often missed by state-of-the-art methods.

The verification of concurrent programs remains an open challenge due to the non-determinism in inter-process communication. One recurring algorithmic problem in this challenge is the consistency verification of concurrent executions. In particular, consistency verification under a reads-from map allows to compute the reads-from (RF) equivalence between concurrent traces, with direct applications to areas such as Stateless Model Checking (SMC). Importantly, the RF equivalence was recently shown to be coarser than the standard Mazurkiewicz equivalence, leading to impressive scalability improvements for SMC under SC (sequential consistency). However, for the relaxed memory models of TSO and PSO (total/partial store order), the algorithmic problem of deciding the RF equivalence, as well as its impact on SMC, has been elusive. 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 n k +1 for TSO and as n k +1 · min( n k 2 , 2 k · 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.

Pointer analysis is one of the fundamental problems in static program analysis. Given a set of pointers, the task is to produce a useful over-approximation of the memory locations that each pointer may point-to at runtime. The most common formulation is Andersen’s Pointer Analysis (APA), defined as an inclusion-based set of m pointer constraints over a set of n pointers. Scalability is extremely important, as points-to information is a prerequisite to many other components in the static-analysis pipeline. Existing algorithms solve APA in O ( n 2 · m ) time, while it has been conjectured that the problem has no truly sub-cubic algorithm, with a proof so far having remained elusive. It is also well-known that APA can be solved in O ( n 2 ) time under certain sparsity conditions that hold naturally in some settings. Besides these simple bounds, the complexity of the problem has remained poorly understood. In this work we draw a rich fine-grained and parallel complexity landscape of APA, and present upper and lower bounds. First, we establish an O ( n 3 ) upper-bound for general APA, improving over O ( n 2 · m ) as n = O ( m ). Second, we show that even on-demand APA (“may a specific pointer a point to a specific location b ?”) has an Ω( n 3 ) (combinatorial) lower bound under standard complexity-theoretic hypotheses. This formally establishes the long-conjectured “cubic bottleneck” of APA, and shows that our O ( n 3 )-time algorithm is optimal. Third, we show that under mild restrictions, APA is solvable in Õ( n ω ) time, where ω<2.373 is the matrix-multiplication exponent. It is believed that ω=2+ o (1), in which case this bound becomes quadratic. Fourth, we show that even under such restrictions, even the on-demand problem has an Ω( n 2 ) lower bound under standard complexity-theoretic hypotheses, and hence our algorithm is optimal when ω=2+ o (1). Fifth, we study the parallelizability of APA and establish lower and upper bounds: (i) in general, the problem is P-complete and hence unlikely parallelizable, whereas (ii) under mild restrictions, the problem is parallelizable. Our theoretical treatment formalizes several insights that can lead to practical improvements in the future.

Abstract 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.

Writing concurrent programs is notoriously hard due to scheduling non-determinism. The most common concurrency bugs are data races, which are accesses to a shared resource that can be executed concurrently. Dynamic data-race prediction is the most standard technique for detecting data races: given an observed, data-race-free trace t, the task is to determine whether t can be reordered to a trace t* that exposes a data-race. Although the problem has received significant practical attention for over three decades, its complexity has remained elusive. In this work, we address this lacuna, identifying sources of intractability and conditions under which the problem is efficiently solvable. Given a trace t of size n over k threads, our main results are as follows.

Writing concurrent programs is highly error-prone due to the nondeterminism in interprocess communication. The most reliable indicators of errors in concurrency are data races , which are accesses to a shared resource that can be executed concurrently. We study the problem of predicting data races in lock-based concurrent programs. The input consists of a concurrent trace t , and the task is to determine all pairs of events of t that constitute a data race. The problem lies at the heart of concurrent verification and has been extensively studied for over three decades. However, existing polynomial-time sound techniques are highly incomplete and can miss simple races. In this work we develop M 2: a new polynomial-time algorithm for this problem, which has no false positives. In addition, our algorithm is complete for input traces that consist of two processes, i.e., it provably detects all races in the trace. We also develop sufficient criteria for detecting completeness dynamically in cases of more than two processes. We make an experimental evaluation of our algorithm on a challenging set of benchmarks taken from recent literature on the topic. Our algorithm soundly reports hundreds of real races, many of which are missed by existing methods. In addition, using our dynamic completeness criteria, M 2 concludes that it has detected all races in the benchmark set, hence the reports are both sound and complete. Finally, its running times are comparable, and often smaller than the theoretically fastest, yet highly incomplete, existing methods. To our knowledge, M 2 is the first sound algorithm that achieves such a level of performance on both running time and completeness of the reported races.

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 δ-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 δ. These results demonstrate that amplification of natural selection depends on the specific mechanisms of the evolutionary process.

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.

There is a huge gap between the speeds of modern caches and main memories, and therefore cache misses account for a considerable loss of efficiency in programs. The predominant technique to address this issue has been Data Packing: data elements that are frequently accessed within time proximity are packed into the same cache block, thereby minimizing accesses to the main memory. We consider the algorithmic problem of Data Packing on a two-level memory system. Given a reference sequence R of accesses to data elements, the task is to partition the elements into cache blocks such that the number of cache misses on R is minimized. The problem is notoriously difficult: it is NP-hard even when the cache has size 1, and is hard to approximate for any cache size larger than 4. Therefore, all existing techniques for Data Packing are based on heuristics and lack theoretical guarantees. In this work, we present the first positive theoretical results for Data Packing, along with new and stronger negative results. We consider the problem under the lens of the underlying access hypergraphs, which are hypergraphs of affinities between the data elements, where the order of an access hypergraph corresponds to the size of the affinity group. We study the problem parameterized by the treewidth of access hypergraphs, which is a standard notion in graph theory to measure the closeness of a graph to a tree. Our main results are as follows: we show that there is a number q * depending on the cache parameters such that (a) if the access hypergraph of order q * has constant treewidth, then there is a linear-time algorithm for Data Packing; (b) the Data Packing problem remains NP-hard even if the access hypergraph of order q * −1 has constant treewidth. Thus, we establish a fine-grained dichotomy depending on a single parameter, namely, the highest order among access hypegraphs that have constant treewidth; and establish the optimal value q * of this parameter. Finally, we present an experimental evaluation of a prototype implementation of our algorithm. Our results demonstrate that, in practice, access hypergraphs of many commonly-used algorithms have small treewidth. We compare our approach with several state-of-the-art heuristic-based algorithms and show that our algorithm leads to significantly fewer cache-misses.

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.

Interprocedural analysis is at the heart of numerous applications in programming languages, such as alias analysis, constant propagation, and so on. Recursive state machines (RSMs) are standard models for interprocedural analysis. We consider a general framework with RSMs where the transitions are labeled from a semiring and path properties are algebraic with semiring operations. RSMs with algebraic path properties can model interprocedural dataflow analysis problems, the shortest path problem, the most probable path problem, and so on. The traditional algorithms for interprocedural analysis focus on path properties where the starting point is fixed as the entry point of a specific method. In this work, we consider possible multiple queries as required in many applications such as in alias analysis. The study of multiple queries allows us to bring in an important algorithmic distinction between the resource usage of the one-time preprocessing vs for each individual query. The second aspect we consider is that the control flow graphs for most programs have constant treewidth. Our main contributions are simple and implementable algorithms that support multiple queries for algebraic path properties for RSMs that have constant treewidth. Our theoretical results show that our algorithms have small additional one-time preprocessing but can answer subsequent queries significantly faster as compared to the current algorithmic solutions for interprocedural dataflow analysis. We have also implemented our algorithms and evaluated their performance for performing on-demand interprocedural dataflow analysis on various domains, such as for live variable analysis and reaching definitions, on a standard benchmark set. Our experimental results align with our theoretical statements and show that after a lightweight preprocessing, on-demand queries are answered much faster than the standard existing algorithmic approaches.

We present a new dynamic partial-order reduction method for stateless model checking of concurrent programs. A common approach for exploring program behaviors relies on enumerating the traces of the program, without storing the visited states (aka stateless exploration). As the number of distinct traces grows exponentially, dynamic partial-order reduction (DPOR) techniques have been successfully used to partition the space of traces into equivalence classes ( Mazurkiewicz partitioning), with the goal of exploring only few representative traces from each class. We introduce a new equivalence on traces under sequential consistency semantics, which we call the observation equivalence. Two traces are observationally equivalent if every read event observes the same write event in both traces. While the traditional Mazurkiewicz equivalence is control-centric, our new definition is data-centric. We show that our observation equivalence is coarser than the Mazurkiewicz equivalence, and in many cases even exponentially coarser. We devise a DPOR exploration of the trace space, called data-centric DPOR, based on the observation equivalence. For acyclic architectures, our algorithm is guaranteed to explore exactly one representative trace from each observation class, while spending polynomial time per class. Hence, our algorithm is optimal wrt the observation equivalence, and in several cases explores exponentially fewer traces than any enumerative method based on the Mazurkiewicz equivalence. For cyclic architectures, we consider an equivalence between traces which is finer than the observation equivalence; but coarser than the Mazurkiewicz equivalence, and in some cases is exponentially coarser. Our data-centric DPOR algorithm remains optimal under this trace equivalence. Finally, we perform a basic experimental comparison between the existing Mazurkiewicz-based DPOR and our data-centric DPOR on a set of academic benchmarks. Our results show a significant reduction in both running time and the number of explored equivalence classes.

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 · 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 · 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. Thus we establish that the existing combinatorial algorithms for Dyck reachability are (conditionally) optimal for general graphs. We also show that the same hardness holds for graphs of constant treewidth. Finally, we provide a prototype implementation of our algorithms for both alias analysis and data-dependence analysis. Our experimental evaluation demonstrates that the new algorithms significantly outperform all existing methods on the two problems, over real-world benchmarks.

This paper is devoted to automatic competitive analysis of real-time scheduling algorithms for firm-deadline tasksets, where only completed tasks contribute some utility to the system. Given such a taskset $${\mathcal {T}}$$ , the competitive ratio of an on-line scheduling algorithm $${\mathcal {A}}$$ for $${\mathcal {T}}$$ is the worst-case utility ratio of $${\mathcal {A}}$$ over the utility achieved by a clairvoyant algorithm. We leverage the theory of quantitative graph games to address the competitive analysis and competitive synthesis problems. For the competitive analysis case, given any taskset $${\mathcal {T}}$$ and any finite-memory on-line scheduling algorithm $${\mathcal {A}}$$ , we show that the competitive ratio of $${\mathcal {A}}$$ in $${\mathcal {T}}$$ can be computed in polynomial time in the size of the state space of $${\mathcal {A}}$$ . Our approach is flexible as it also provides ways to model meaningful constraints on the released task sequences that determine the competitive ratio. We provide an experimental study of many well-known on-line scheduling algorithms, which demonstrates the feasibility of our competitive analysis approach that effectively replaces human ingenuity (required for finding worst-case scenarios) by computing power. For the competitive synthesis case, we are just given a taskset $${\mathcal {T}}$$ , and the goal is to automatically synthesize an optimal on-line scheduling algorithm $${\mathcal {A}}$$ , i.e., one that guarantees the largest competitive ratio possible for $${\mathcal {T}}$$ . We show how the competitive synthesis problem can be reduced to a two-player graph game with partial information, and establish that the computational complexity of solving this game is Np-complete. The competitive synthesis problem is hence in Np in the size of the state space of the non-deterministic labeled transition system encoding the taskset. Overall, the proposed framework assists in the selection of suitable scheduling algorithms for a given taskset, which is in fact the most common situation in real-time systems design.

We study algorithmic questions wrt algebraic path properties in concurrent systems, where the transitions of the system are labeled from a complete, closed semiring. The algebraic path properties can model dataflow analysis problems, the shortest path problem, and many other natural problems that arise in program analysis. We consider that each component of the concurrent system is a graph with constant treewidth, a property satisfied by the controlflow graphs of most programs. We allow for multiple possible queries, which arise naturally in demand driven dataflow analysis. The study of multiple queries allows us to consider the tradeoff between the resource usage of the one-time preprocessing and for each individual query. The traditional approach constructs the product graph of all components and applies the best-known graph algorithm on the product. In this approach, even the answer to a single query requires the transitive closure (i.e., the results of all possible queries), which provides no room for tradeoff between preprocessing and query time. Our main contributions are algorithms that significantly improve the worst-case running time of the traditional approach, and provide various tradeoffs depending on the number of queries. For example, in a concurrent system of two components, the traditional approach requires hexic time in the worst case for answering one query as well as computing the transitive closure, whereas we show that with one-time preprocessing in almost cubic time, each subsequent query can be answered in at most linear time, and even the transitive closure can be computed in almost quartic time. Furthermore, we establish conditional optimality results showing that the worst-case running time of our algorithms cannot be improved without achieving major breakthroughs in graph algorithms (i.e., improving the worst-case bound for the shortest path problem in general graphs). Preliminary experimental results show that our algorithms perform favorably on several benchmarks.

We consider data-structures for answering reachability and distance queries on constant-treewidth graphs with n nodes, on the standard RAM computational model with wordsize W=Theta(log n). Our first contribution is a data-structure that after O(n) preprocessing time, allows (1) pair reachability queries in O(1) time; and (2) single-source reachability queries in O(n/log n) time. This is (asymptotically) optimal and is faster than DFS/BFS when answering more than a constant number of single-source queries. The data-structure uses at all times O(n) space. Our second contribution is a space-time tradeoff data-structure for distance queries. For any epsilon in [1/2,1], we provide a data-structure with polynomial preprocessing time that allows pair queries in O(n^{1-\epsilon} alpha(n)) time, where alpha is the inverse of the Ackermann function, and at all times uses O(n^epsilon) space. The input graph G is not considered in the space complexity.

We study algorithmic questions for concurrent systems where the transitions are labeled from a complete, closed semiring, and path properties are algebraic with semiring operations. The algebraic path properties can model dataflow analysis problems, the shortest path problem, and many other natural problems that arise in program analysis. We consider that each component of the concurrent system is a graph with constant treewidth, a property satisfied by the controlflow graphs of most programs. We allow for multiple possible queries, which arise naturally in demand driven dataflow analysis. The study of multiple queries allows us to consider the tradeoff between the resource usage of the one-time preprocessing and for each individual query. The traditional approach constructs the product graph of all components and applies the best-known graph algorithm on the product. In this approach, even the answer to a single query requires the transitive closure (i.e., the results of all possible queries), which provides no room for tradeoff between preprocessing and query time. Our main contributions are algorithms that significantly improve the worst-case running time of the traditional approach, and provide various tradeoffs depending on the number of queries. For example, in a concurrent system of two components, the traditional approach requires hexic time in the worst case for answering one query as well as computing the transitive closure, whereas we show that with one-time preprocessing in almost cubic time, each subsequent query can be answered in at most linear time, and even the transitive closure can be computed in almost quartic time. Furthermore, we establish conditional optimality results showing that the worst-case running time of our algorithms cannot be improved without achieving major breakthroughs in graph algorithms (i.e., improving the worst-case bound for the shortest path problem in general graphs). Preliminary experimental results show that our algorithms perform favorably on several benchmarks.

Interprocedural analysis is at the heart of numerous applications in programming languages, such as alias analysis, constant propagation, etc. Recursive state machines (RSMs) are standard models for interprocedural analysis. We consider a general framework with RSMs where the transitions are labeled from a semiring, and path properties are algebraic with semiring operations. RSMs with algebraic path properties can model interprocedural dataflow analysis problems, the shortest path problem, the most probable path problem, etc. The traditional algorithms for interprocedural analysis focus on path properties where the starting point is fixed as the entry point of a specific method. In this work, we consider possible multiple queries as required in many applications such as in alias analysis. The study of multiple queries allows us to bring in a very important algorithmic distinction between the resource usage of the one-time preprocessing vs for each individual query. The second aspect that we consider is that the control flow graphs for most programs have constant treewidth. Our main contributions are simple and implementable algorithms that support multiple queries for algebraic path properties for RSMs that have constant treewidth. Our theoretical results show that our algorithms have small additional one-time preprocessing, but can answer subsequent queries significantly faster as compared to the current best-known solutions for several important problems, such as interprocedural reachability and shortest path. We provide a prototype implementation for interprocedural reachability and intraprocedural shortest path that gives a significant speed-up on several benchmarks.

We consider the quantitative analysis problem for interprocedural control-flow graphs (ICFGs). The input consists of an ICFG, a positive weight function that assigns every transition a positive integer-valued number, and a labelling of the transitions (events) as good, bad, and neutral events. The weight function assigns to each transition a numerical value that represents a measure of how good or bad an event is. The quantitative analysis problem asks whether there is a run of the ICFG where the ratio of the sum of the numerical weights of good events versus the sum of weights of bad events in the long-run is at least a given threshold (or equivalently, to compute the maximal ratio among all valid paths in the ICFG). The quantitative analysis problem for ICFGs can be solved in polynomial time, and we present an efficient and practical algorithm for the problem.

We present a flexible framework for the automated competitive analysis of on-line scheduling algorithms for firm-deadline real-time tasks based on multi-objective graphs: Given a task set and an on-line scheduling algorithm specified as a labeled transition system, along with some optional safety, liveness, and/or limit-average constraints for the adversary, we automatically compute the competitive ratio of the algorithm w.r.t. A clairvoyant scheduler. We demonstrate the flexibility and power of our approach by comparing the competitive ratio of several on-line algorithms, including Dover, that have been proposed in the past, for various task sets. Our experimental results reveal that none of these algorithms is universally optimal, in the sense that there are task sets where other schedulers provide better performance. Our framework is hence a very useful design tool for selecting optimal algorithms for a given application.

A fundamental question in biology is the following: what is the time scale that is needed for evolutionary innovations? There are many results that characterize single steps in terms of the fixation time of new mutants arising in populations of certain size and structure. But here we ask a different question, which is concerned with the much longer time scale of evolutionary trajectories: how long does it take for a population exploring a fitness landscape to find target sequences that encode new biological functions? Our key variable is the length, L, of the genetic sequence that undergoes adaptation. In computer science there is a crucial distinction between problems that require algorithms which take polynomial or exponential time. The latter are considered to be intractable. Here we develop a theoretical approach that allows us to estimate the time of evolution as function of L. We show that adaptation on many fitness landscapes takes time that is exponential in L, even if there are broad selection gradients and many targets uniformly distributed in sequence space. These negative results lead us to search for specific mechanisms that allow evolution to work on polynomial time scales. We study a regeneration process and show that it enables evolution to work in polynomial time.

BACKGROUND: Biological networks tend to have high interconnectivity, complex topologies and multiple types of interactions. This renders difficult the identification of sub-networks that are involved in condition- specific responses. In addition, we generally lack scalable methods that can reveal the information flow in gene regulatory and biochemical pathways. Doing so will help us to identify key participants and paths under specific environmental and cellular context. RESULTS: This paper introduces the theory of network flooding, which aims to address the problem of network minimization and regulatory information flow in gene regulatory networks. Given a regulatory biological network, a set of source (input) nodes and optionally a set of sink (output) nodes, our task is to find (a) the minimal sub-network that encodes the regulatory program involving all input and output nodes and (b) the information flow from the source to the sink nodes of the network. Here, we describe a novel, scalable, network traversal algorithm and we assess its potential to achieve significant network size reduction in both synthetic and E. coli networks. Scalability and sensitivity analysis show that the proposed method scales well with the size of the network, and is robust to noise and missing data. CONCLUSIONS: The method of network flooding proves to be a useful, practical approach towards information flow analysis in gene regulatory networks. Further extension of the proposed theory has the potential to lead in a unifying framework for the simultaneous network minimization and information flow analysis across various "omics" levels.

We propose a new theoretical model for passively mobile wireless sensor networks, called P M , standing for passively mobile machines . The main modification w.r.t. the population protocol model (Angluin et al., 2006) [30] is that agents now, instead of being automata, are Turing Machines. We provide general definitions for unbounded memories, but we are mainly interested in computations upper-bounded by plausible space limitations. However, we prove that our results hold for more general cases. We focus on complete interaction graphs and define the complexity classes PMSPACE ( f ( n ) ) parametrically, consisting of all predicates that are stably computable by some PM protocol that uses O ( f ( n ) ) memory in each agent. We provide a protocol that generates unique identifiers from scratch only by using O ( log n ) memory, and use it to provide an exact characterization of the classes PMSPACE ( f ( n ) ) when f ( n ) = Ω ( log n ) : they are precisely the classes of all symmetric predicates in NSPACE ( n f ( n ) ) . As a consequence, we obtain a space hierarchy of the PM model when the memory bounds are Ω ( log n ) . We next explore the computability of the PM model when the protocols use o ( log log n ) space per machine and prove that SEM = PMSPACE ( f ( n ) ) when f ( n ) = o ( log log n ) , where SEM denotes the class of the semilinear predicates. Finally, we establish that the minimal space requirement for the computation of non-semilinear predicates is O ( log log n ) .