Users of program analyses expect that results change predictably in response to changes in their programs, but many analyses do not ensure such robustness. This paper introduces a theoretical framework that provides a unified language to articulate robustness properties. We adopt a categorical view in which programs and their properties form a category, and robust analyses are characterized as structure-preserving functors. A diverse range of robustness properties---e.g., invariance under variable renaming and monotonicity---arise from instantiating the category's arrows accordingly. Beyond formulating the meaning of robustness, this paper provides methods for achieving it. The first is a general recipe for designing robust analyses, by lifting a sound and robust analysis from a restricted (sub-Turing) model of computation to a sound and robust analysis for general programs. This recipe demystifies the design of several existing loop summarization and termination analyses by showing they are instantiations of this general recipe, and furthermore elucidates their robustness properties. The second is a characterization of a sense in which an algebraic program analysis is robust, provided that it is comprised of robust operators. In particular, we show that such analyses behave predictably under common refactoring patterns, such as variable renaming and loop unrolling.

We extend the scope of context-free-language (CFL) reachability to a new class of infinite-state systems. Parikh's Theorem is a useful tool for solving CFL-reachability problems for transition systems that consist of commuting transition relations. It implies that the image of a context-free language under a homomorphism into a commutative monoid is semi-linear, and that there is a linear-time algorithm for constructing a Presburger arithmetic formula that represents it. However, for many transition systems of interest, transitions do not commute. In this paper, we introduce almost-commuting transition systems , which pair finite-state control with commutative components, but which are in general not commutative. We extend Parikh's theorem to show that the image of a context-free language under a homomorphism into an almost-commuting monoid is semi-linear and that there is a polynomial-time algorithm for constructing a Presburger arithmetic formula that represents it. This result yields a general framework for solving CFL-reachability problems over almost commuting transition systems . We describe several examples of systems within this class. Finally, we examine closure properties of almost-commuting monoids that can be used to modularly compose almost-commuting transition systems while remaining in the class.

Relational NetKAT (RN) is a new specification language for network change validation. Engineers use RN to specify intended changes by providing a trace relation R , which maps existing packet traces in the pre-change network to intended packet traces in the post-change network. The intended set of traces may then be checked against the actual post-change traces to uncover errors in implementation. Trace relations are constructed compositionally from a language of combinators that include trace insertion, trace deletion, and packet transformation, as well as regular operators for concatenation, union, and iteration of relations. We provide algorithms for converting trace relations into a new form of NetKAT transducer and also for constructing an automaton that recognizes the image of a NetKAT automaton under a NetKAT transducer. These algorithms, together with existing decision procedures for NetKAT automaton equivalence, suffice for validating network changes. We provide a denotational semantics for our specification language, prove our compilation algorithms correct, implement a tool for network change validation, and evaluate it on a set of benchmarks drawn from a production network and Amazon’s Batfish toolkit.

In this work, we describe a new software model-checking algorithm called GPS. GPS treats the task of model checking a program as a directed search of the program states, guided by a compositional, summary-based static analysis. The summaries produced by static analysis are used both to prune away infeasible paths and to drive test generation to reach new, unexplored program states. GPS can find both proofs of safety and counter-examples to safety (i.e., inputs that trigger bugs), and features a novel two-layered search strategy that renders it particularly efficient at finding bugs in programs featuring long, input-dependent error paths. To make GPS refutationally complete (in the sense that it will find an error if one exists, if it is allotted enough time), we introduce an instrumentation technique and show that it helps GPS achieve refutation-completeness without sacrificing overall performance. We benchmarked GPS on a diverse suite of benchmarks including programs from the Software Verification Competition (SV-COMP), from prior literature, as well as synthetic programs based on examples in this paper. We found that our implementation of GPS outperforms state-of-the-art software model checkers (including the top performers in SV-COMP ReachSafety-Loops category), both in terms of the number of benchmarks solved and in terms of running time.

Abstract Checking satisfiability of formulae in the theory of linear arithmetic has far reaching applications, including program verification and synthesis. Many satisfiability solvers excel at proving and disproving satisfiability of quantifier-free linear arithmetic formulas and have recently begun to support quantified formulas. Beyond simply checking satisfiability of formulas, fine-grained strategies for satisfiability games enables solving additional program verification and synthesis tasks. Quantified satisfiability games are played between two players—SAT and UNSAT—who take turns instantiating quantifiers and choosing branches of boolean connectives to evaluate the given formula. A winning strategy for SAT (resp. UNSAT) determines the choices of SAT (resp. UNSAT) as a function of UNSAT ’s (resp. SAT ’s) choices such that the given formula evaluates to true (resp. false) no matter what choices UNSAT (resp. SAT) may make. As we are interested in both checking satisfiability and synthesizing winning strategies, we must avoid conversion to normal-forms that alter the game semantics of the formula (e.g. prenex normal form). We present fine-grained strategy improvement and strategy synthesis, the first technique capable of synthesizing winning fine-grained strategies for linear arithmetic satisfiability games, which may be used in higher-level applications. We experimentally evaluate our technique and find it performs favorably compared with state-of-the-art solvers.

Abstract This paper studies the problem of synthesizing (lexicographic) polynomial ranking functions for loops that can be described in polynomial arithmetic over integers and reals. While the analogous ranking function synthesis problem for linear arithmetic is decidable, even checking whether a given function ranks an integer loop is undecidable in the nonlinear setting. We side-step the decidability barrier by working within the theory of linear integer/real rings (LIRR) rather than the standard model of arithmetic. We develop a termination analysis that is guaranteed to succeed if a loop (expressed as a formula) admits a (lexicographic) polynomial ranking function. In contrast to template-based ranking function synthesis in real arithmetic, our completeness result holds for lexicographic ranking functions of unbounded dimension and degree, and effectively subsumes linear lexicographic ranking function synthesis for linear integer loops.

Relational network verification is a new approach for validating network changes. In contrast to traditional network verification, which analyzes specifications for a single network snapshot, it analyzes specifications that capture similarities and differences between two network snapshots (e.g., pre- and post-change snapshots). Relational specifications are compact and precise because they focus on the flows and paths that change between snapshots and then simply mandate that all other network behaviors "stay the same", without enumerating them. To achieve similar guarantees, single-snapshot specifications would need to enumerate all flow and path behaviors that are not expected to change in order to enable checking that nothing has accidentally changed. Such specifications are proportional to network size, which makes them impractical to generate for many real-world networks.

This paper presents a program analysis method that generates program summaries involving polynomial arithmetic. Our approach builds on prior techniques that use solvable polynomial maps for summarizing loops. These techniques are able to generate all polynomial invariants for a restricted class of programs, but cannot be applied to programs outside of this class—for instance, programs with nested loops, conditional branching, unstructured control flow, etc. There currently lacks approaches to apply these prior methods to the case of general programs. This paper bridges that gap. Instead of restricting the kinds of programs we can handle, our method abstracts every loop into a model that can be solved with prior techniques, bringing to bear prior work on solvable polynomial maps to general programs. While no method can generate all polynomial invariants for arbitrary programs, our method establishes its merit through a monotonicty result. We have implemented our techniques, and tested them on a suite of benchmarks from the literature. Our experiments indicate our techniques show promise on challenging verification tasks requiring non-linear reasoning.

This paper presents a technique for summarizing recursive procedures operating on integer variables. The motivation of our work is to create more predictable program analyzers, and in particular to formally guarantee compositionality and monotonicity of procedure summarization. To summarize a procedure, we compute its best abstraction as a vector addition system with resets (VASR) and exactly summarize the executions of this VASR over the context-free language of syntactic paths through the procedure. We improve upon this technique by refining the language of syntactic paths using (automatically synthesized) linear potential functions that bound the number of recursive calls within valid executions of the input program. We implemented our summarization technique in an automated program verification tool; our experimental evaluation demonstrates that our technique computes more precise summaries than existing abstract interpreters and that our tool’s verification capabilities are comparable with state-of-the-art software model checkers.

This paper presents a theory of non-linear integer/real arithmetic and algorithms for reasoning about this theory. The theory can be conceived of as an extension of linear integer/real arithmetic with a weakly-axiomatized multiplication symbol, which retains many of the desirable algorithmic properties of linear arithmetic. In particular, we show that the conjunctive fragment of the theory can be effectively manipulated (analogously to the usual operations on convex polyhedra, the conjunctive fragment of linear arithmetic). As a result, we can solve the following consequence-finding problem: given a ground formula F , find the strongest conjunctive formula that is entailed by F . As an application of consequence-finding, we give a loop invariant generation algorithm that is monotone with respect to the theory and (in a sense) complete. Experiments show that the invariants generated from the consequences are effective for proving safety properties of programs that require non-linear reasoning.

Abstract This paper is a tutorial on algebraic program analysis. It explains the foundations of algebraic program analysis, its strengths and limitations, and gives examples of algebraic program analyses for numerical invariant generation and termination analysis.

Abstract This paper shows how techniques for linear dynamical systems can be used to reason about the behavior of general loops. We present two main results. First, we show that every loop that can be expressed as a transition formula in linear integer arithmetic has a best model as a deterministic affine transition system. Second, we show that for any linear dynamical system f with integer eigenvalues and any integer arithmetic formula G, there is a linear integer arithmetic formula that holds exactly for the states of f for which G is eventually invariant. Combining the two, we develop a monotone conditional termination analysis for general loops.

Determining whether a given program terminates is the quintessential undecidable problem. Algorithms for termination analysis may be classified into two groups: (1) algorithms with strong behavioral guarantees that work in limited circumstances (e.g., complete synthesis of linear ranking functions for polyhedral loops), and (2) algorithms that are widely applicable, but have weak behavioral guarantees (e.g., Terminator). This paper investigates the space in between: how can we design practical termination analyzers with useful behavioral guarantees?

This paper is the confluence of two streams of ideas in the literature on generating numerical invariants, namely: (1) template-based methods, and (2) recurrence-based methods.

This paper presents a technique for computing numerical loop summaries. The method synthesizes a rational vector addition system with resets ( $$\mathbb {Q}$$ -VASR) that simulates the action of an input loop, and then uses the reachability relation of that $$\mathbb {Q}$$ -VASR to over-approximate the behavior of the loop. The key technical problem solved in this paper is to automatically synthesize a $$\mathbb {Q}$$ -VASR that is a best abstraction of a given loop in the sense that (1) it simulates the loop and (2) it is simulated by any other $$\mathbb {Q}$$ -VASR that simulates the loop. Since our loop summarization scheme is based on computing the exact reachability relation of a best abstraction of a loop, we can make theoretical guarantees about its behavior. Moreover, we show experimentally that the technique is precise and performant in practice.

Algebraic program analyses compute information about a program’s behavior by first (a) computing a valid path expression —i.e., a regular expression that recognizes all feasible execution paths (and usually more)—and then (b) interpreting the path expression in a semantic algebra that defines the analysis. There are an infinite number of different regular expressions that qualify as valid path expressions, which raises the question “ Which one should we choose? ” While any choice yields a sound result, for many analyses the choice can have a drastic effect on the precision of the results obtained. This paper investigates the following two questions: (1) What does it mean for one valid path expression to be “better” than another ? (2) Can we compute a valid path expression that is “better,” and if so, how ? We show that it is not satisfactory to compare two path expressions E 1 and E 2 solely by means of the languages that they generate . Counter to one’s intuition, it is possible for L ( E 2 ) ⊊ L ( E 1 ), yet for E 2 to produce a less-precise analysis result than E 1 —and thus we would not want to perform the transformation E 1 → E 2 . However, the exclusion of paths so as to analyze a smaller language of paths is exactly the refinement criterion used by some prior methods. In this paper, we develop an algorithm that takes as input a valid path expression E , and returns a valid path expression E ′ that is guaranteed to yield analysis results that are at least as good as those obtained using E . While the algorithm sometimes returns E itself, it typically does not: (i) we prove a no-degradation result for the algorithm’s base case—for transforming a leaf loop (i.e., a most-deeply-nested loop); (ii) at a non-leaf loop L , the algorithm treats each loop L ′ in the body of L as an indivisible atom, and applies the leaf-loop algorithm to L ; the no-degradation result carries over to (ii), as well. Our experiments show that the technique has a substantial impact: the loop-refinement algorithm allows the implementation of Compositional Recurrence Analysis to prove over 25% more assertions for a collection of challenging loop micro-benchmarks.

This paper investigates the problem of reasoning about non-linear behavior of simple numerical loops. Our approach builds on classical techniques for analyzing the behavior of linear dynamical systems. It is well-known that a closed-form representation of the behavior of a linear dynamical system can always be expressed using algebraic numbers, but this approach can create formulas that present an obstacle for automated-reasoning tools. This paper characterizes when linear loops have closed forms in simpler theories that are more amenable to automated reasoning. The algorithms for computing closed forms described in the paper avoid the use of algebraic numbers, and produce closed forms expressed using polynomials and exponentials over rational numbers. We show that the logic for expressing closed forms is decidable, yielding decision procedures for verifying safety and termination of a class of numerical loops over rational numbers. We also show that the procedure for computing closed forms for this class of numerical loops can be used to over-approximate the behavior of arbitrary numerical programs (with unrestricted control flow, non-deterministic assignments, and recursive procedures).

Many problems in formal methods can be formalized as two-player games. For several applications—program synthesis, for example—in addition to determining which player wins the game, we are interested in computing a winning strategy for that player. This paper studies the strategy synthesis problem for games defined within the theory of linear rational arithmetic. Two types of games are considered. A satisfiability game , described by a quantified formula, is played by two players that take turns instantiating quantifiers. The objective of each player is to prove (or disprove) satisfiability of the formula. A reachability game , described by a pair of formulas defining the legal moves of each player, is played by two players that take turns choosing positions—rational vectors of some fixed dimension. The objective of each player is to reach a position where the opposing player has no legal moves (or to play the game forever). We give a complete algorithm for synthesizing winning strategies for satisfiability games and a sound (but necessarily incomplete) algorithm for synthesizing winning strategies for reachability games.

Automatic generation of non-linear loop invariants is a long-standing challenge in program analysis, with many applications. For instance, reasoning about exponentials provides a way to find invariants of digital-filter programs, and reasoning about polynomials and/or logarithms is needed for establishing invariants that describe the time or memory usage of many well-known algorithms. An appealing approach to this challenge is to exploit the powerful recurrence-solving techniques that have been developed in the field of computer algebra, which can compute exact characterizations of non-linear repetitive behavior. However, there is a gap between the capabilities of recurrence solvers and the needs of program analysis: (1) loop bodies are not merely systems of recurrence relations---they may contain conditional branches, nested loops, non-deterministic assignments, etc., and (2) a client program analyzer must be able to reason about the closed-form solutions produced by a recurrence solver (e.g., to prove assertions). This paper presents a method for generating non-linear invariants of general loops based on analyzing recurrence relations. The key components are an abstract domain for reasoning about non-linear arithmetic, a semantics-based method for extracting recurrence relations from loop bodies, and a recurrence solver that avoids closed forms that involve complex or irrational numbers. Our technique has been implemented in a program analyzer that can analyze general loops and mutually recursive procedures. Our experiments show that our technique shows promise for non-linear assertion-checking and resource-bound generation.

This paper presents a high level overview of Proof Spaces [Farzan, Kincaid, and Podelski, 2015] as an instance of a new approach to compositional verification of concurrent programs and discusses potential future work extending the approach beyond its current scope of applicability.

Compositional recurrence analysis (CRA) is a static-analysis method based on a combination of symbolic analysis and abstract interpretation. This paper addresses the problem of creating a context-sensitive interprocedural version of CRA that handles recursive procedures. The problem is non-trivial because there is an "impedance mismatch" between CRA, which relies on analysis techniques based on regular languages (i.e., Tarjan's path-expression method), and the context-free-language underpinnings of context-sensitive analysis.

Correctness of multi-threaded programs typically requires that they satisfy liveness properties. For example, a program may require that no thread is starved of a shared resource, or that all threads eventually agree on a single value. This paper presents a method for proving that such liveness properties hold. Two particular challenges which are addressed in this work are that (1) the correctness argument may rely on global behaviour of the system (e.g., the correctness argument may require that all threads collectively progress towards "the good thing" rather than one thread progressing while the others do not interfere), and (2) such programs are often designed to be executed by any number of threads, and the desired liveness properties must hold no matter how many threads are active in the system.

This paper presents a new method for automatically generating numerical invariants for imperative programs. The procedure computes a transition formula which overapproximates the behaviour of a given input program. It is compositional in the sense that it operates by decomposing the program into parts, computing a transition formula for each part, and then composing them. Transition formulas for loops are computed by extracting recurrence relations from a transition formula for the loop body and then computing their closed forms. Experimentation demonstrates that this method is competitive with leading verification techniques based on abstraction refinement.

In this paper, we present a new approach to automatically verify multi-threaded programs which are executed by an unbounded number of threads running in parallel.

Applications in many areas of computing make discrete decisions under uncertainty, for reasons such as limited numerical precision in calculations and errors in sensor-derived inputs. As a result, individual decisions made by such programs may be nondeterministic, and lead to contradictory decisions at different points of an execution. This means that an otherwise correct program may execute along paths, that it would not follow under its ideal semantics, violating essential program invariants on the way. A program is said to be consistent if it does not suffer from this problem despite uncertainty in decisions.

Counting arguments are among the most basic proof methods in mathematics. Within the field of formal verification, they are useful for reasoning about programs with infinite control, such as programs with an unbounded number of threads, or (concurrent) programs with recursive procedures. While counting arguments are common in informal, hand-written proofs of such programs, there are no fully automated techniques to construct counting arguments. The key questions involved in automating counting arguments are: how to decide what should be counted?, and how to decide when a counting argument is valid? In this paper, we present a technique for automatically constructing and checking counting arguments, which includes novel solutions to these questions.

The rise in efficiency of Satisfiability Modulo Theories (SMT) solvers has created numerous uses for them in software verification, program synthesis, functional programming, refinement types, etc. In all of these applications, SMT solvers are used for generating satisfying assignments (e.g., a witness for a bug) or proving unsatisfiability/validity(e.g., proving that a subtyping relation holds). We are often interested in finding not just an arbitrary satisfying assignment, but one that optimizes (minimizes/maximizes) certain criteria. For example, we might be interested in detecting program executions that maximize energy usage (performance bugs), or synthesizing short programs that do not make expensive API calls. Unfortunately, none of the available SMT solvers offer such optimization capabilities.

The correctness of a sequential program can be shown by the annotation of its control flow graph with inductive assertions. We propose inductive data flow graphs, data flow graphs with incorporated inductive assertions, as the basis of an approach to verifying concurrent programs. An inductive data flow graph accounts for a set of dependencies between program actions in interleaved thread executions, and therefore stands as a representation for the set of concurrent program traces which give rise to these dependencies. The approach first constructs an inductive data flow graph and then checks whether all program traces are represented. The size of the inductive data flow graph is polynomial in the number of data dependencies (in a sense that can be made formal); it does not grow exponentially in the number of threads unless the data dependencies do. The approach shifts the burden of the exponential explosion towards the check whether all program traces are represented, i.e., to a combinatorial problem (over finite graphs).

In this paper, we consider the problem of verifying thread-state properties of multithreaded programs in which the number of active threads cannot be statically bounded. Our approach is based on decomposing the task into two modules, where one reasons about data and the other reasons about control. The data module computes thread-state invariants (e.g., linear constraints over global variables and local variables of one thread) using the thread interference information computed by the control module. The control module computes a representation of thread interference, as an incrementally constructed data flow graph, using the data invariants provided by the data module. These invariants are used to rule out patterns of thread interference that can not occur in a real program execution. The two modules are incorporated into a feedback loop, so that the abstractions of data and interference are iteratively coarsened as the algorithm progresses (that is, they become weaker) until a fixed point is reached. Our approach is sound and terminating, and applicable to programs with infinite state (e.g., unbounded integers) and unboundedly many threads. The verification method presented in this paper has been implemented into a tool, called Duet. We demonstrate the effectiveness of our technique by verifying properties of a selection of Linux device drivers using Duet, and also compare Duet with previous work on verification of parameterized Boolean program using the Boolean abstractions of these drivers.