Invariant Inference Research Articles

Invalidator: Automated Patch Correctness Assessment via Semantic and Syntactic Reasoning

Automated program repair (APR) faces the challenge of test overfitting, where generated patches pass validation tests but fail to generalize. Existing methods for patch assessment involve generating new tests or manual inspection, which can be time-consuming or biased. In this paper, we propose a novel technique, INVALIDATOR, to automatically assess the correctness of APR-generated patches via semantic and syntactic reasoning. INVALIDATOR leverages program invariants to reason about program semantics while also capturing program syntax through language semantics learned from a large code corpus using a pre-trained language model. Given a buggy program and the developer-patched program, INVALIDATOR infers likely invariants on both programs. Then, INVALIDATOR determines that an APR-generated patch overfits if: (1) it violates correct specifications or (2) maintains erroneous behaviors from the original buggy program. In case our approach fails to determine an overfitting patch based on invariants, INVALIDATOR utilizes a trained model from labeled patches to assess patch correctness based on program syntax. The benefit of INVALIDATOR is threefold. First, INVALIDATOR leverages both semantic and syntactic reasoning to enhance its discriminative capability. Second, INVALIDATOR does not require new test cases to be generated, but instead only relies on the current test suite and uses invariant inference to generalize program behaviors. Third, INVALIDATOR is fully automated. Experimental results demonstrate that INVALIDATOR outperforms existing methods in terms of Accuracy and F-measure, correctly identifying 79% of overfitting patches and detecting 23% more overfitting patches than the best baseline.

Using Symbolic States to Infer Numerical Invariants

Automatically inferring invariant specifications has proven valuable in enabling a wide range of software verification and validation approaches over the past two decades. Recent approaches have shifted from using observation of concrete program states to exploiting symbolic encodings of sets of concrete program states in order to improve the quality of inferred invariants. In this paper, we demonstrate that working directly with symbolic states generated by symbolic execution approaches can improve invariant inference further. Our technique uses a counterexample-based algorithm that iteratively creates concrete states from symbolic states, infers candidate invariants from both concrete and symbolic states, and then validates or refutes candidate invariants using symbolic states. The refutation process serves both to eliminate spurious invariants and to drive the inference process to produce more precise invariants. This framework can be employed to infer complex invariants that capture nonlinear polynomial relations among program variables. The open-source SymInfer tool implements these ideas to automatically generate invariants at arbitrary locations in Java or C programs. Our preliminary results show that across a collection of four benchmarks SymInfer improves on the state-of-the-art by efficiently inferring more informative invariants than prior work.

Property-directed reachability as abstract interpretation in the monotone theory

Inferring inductive invariants is one of the main challenges of formal verification. The theory of abstract interpretation provides a rich framework to devise invariant inference algorithms. One of the latest breakthroughs in invariant inference is property-directed reachability (PDR), but the research community views PDR and abstract interpretation as mostly unrelated techniques. This paper shows that, surprisingly, propositional PDR can be formulated as an abstract interpretation algorithm in a logical domain. More precisely, we define a version of PDR, called Λ-PDR, in which all generalizations of counterexamples are used to strengthen a frame. In this way, there is no need to refine frames after their creation, because all the possible supporting facts are included in advance. We analyze this algorithm using notions from Bshouty’s monotone theory, originally developed in the context of exact learning. We show that there is an inherent overapproximation between the algorithm’s frames that is related to the monotone theory. We then define a new abstract domain in which the best abstract transformer performs this overapproximation, and show that it captures the invariant inference process, i.e., Λ-PDR corresponds to Kleene iterations with the best transformer in this abstract domain. We provide some sufficient conditions for when this process converges in a small number of iterations, with sometimes an exponential gap from the number of iterations required for naive exact forward reachability. These results provide a firm theoretical foundation for the benefits of how PDR tackles forward reachability.

Induction duality: primal-dual search for invariants

Many invariant inference techniques reason simultaneously about states and predicates, and it is well-known that these two kinds of reasoning are in some sense dual to each other. We present a new formal duality between states and predicates, and use it to derive a new primal-dual invariant inference algorithm. The new induction duality is based on a notion of provability by incremental induction that is formally dual to reachability, and the duality is surprisingly symmetric. The symmetry allows us to derive the dual of the well-known Houdini algorithm, and by combining Houdini with its dual image we obtain primal-dual Houdini , the first truly primal-dual invariant inference algorithm. An early prototype of primal-dual Houdini for the domain of distributed protocol verification can handle difficult benchmarks from the literature.

Learning the boundary of inductive invariants

We study the complexity of invariant inference and its connections to exact concept learning. We define a condition on invariants and their geometry, called the fence condition, which permits applying theoretical results from exact concept learning to answer open problems in invariant inference theory. The condition requires the invariant's boundary---the states whose Hamming distance from the invariant is one---to be backwards reachable from the bad states in a small number of steps. Using this condition, we obtain the first polynomial complexity result for an interpolation-based invariant inference algorithm, efficiently inferring monotone DNF invariants with access to a SAT solver as an oracle. We further harness Bshouty's seminal result in concept learning to efficiently infer invariants of a larger syntactic class of invariants beyond monotone DNF. Lastly, we consider the robustness of inference under program transformations. We show that some simple transformations preserve the fence condition, and that it is sensitive to more complex transformations.

Verification and refutation of C programs based on k-induction and invariant inference

DepthK is a source-to-source transformation tool that employs bounded model checking (BMC) to verify and falsify safety properties in single- and multi-threaded C programs, without manual annotation of loop invariants. Here, we describe and evaluate a proof-by-induction algorithm that combines k-induction with invariant inference to prove and refute safety properties. We apply two invariant generators to produce program invariants and feed these into a k-induction-based verification algorithm implemented in DepthK, which uses the efficient SMT-based context-bounded model checker (ESBMC) as sequential verification back-end. A set of C benchmarks from the International Competition on Software Verification (SV-COMP) and embedded-system applications extracted from the available literature are used to evaluate the effectiveness of the proposed approach. Experimental results show that k-induction with invariants can handle a wide variety of safety properties, in typical programs with loops and embedded software applications from the telecommunications, control systems, and medical domains. The results of our comparative evaluation extend the knowledge about approaches that rely on both BMC and k-induction for software verification, in the following ways. (1) The proposed method outperforms the existing implementations that use k-induction with an interval-invariant generator (e.g., 2LS and ESBMC), in the category ConcurrencySafety, and overcame, in others categories, such as SoftwareSystems, other software verifiers that use plain BMC (e.g., CBMC). Also, (2) it is more precise than other verifiers based on the property-directed reachability (PDR) algorithm (i.e., SeaHorn, Vvt and CPAchecker-CTIGAR). This way, our methodology demonstrated improvement over existing BMC and k-induction-based approaches.

Complexity and information in invariant inference

This paper addresses the complexity of SAT-based invariant inference, a prominent approach to safety verification. We consider the problem of inferring an inductive invariant of polynomial length given a transition system and a safety property. We analyze the complexity of this problem in a black-box model, called the Hoare-query model, which is general enough to capture algorithms such as IC3/PDR and its variants. An algorithm in this model learns about the system's reachable states by querying the validity of Hoare triples. We show that in general an algorithm in the Hoare-query model requires an exponential number of queries. Our lower bound is information-theoretic and applies even to computationally unrestricted algorithms, showing that no choice of generalization from the partial information obtained in a polynomial number of Hoare queries can lead to an efficient invariant inference procedure in this class. We then show, for the first time, that by utilizing rich Hoare queries, as done in PDR, inference can be exponentially more efficient than approaches such as ICE learning, which only utilize inductiveness checks of candidates. We do so by constructing a class of transition systems for which a simple version of PDR with a single frame infers invariants in a polynomial number of queries, whereas every algorithm using only inductiveness checks and counterexamples requires an exponential number of queries. Our results also shed light on connections and differences with the classical theory of exact concept learning with queries, and imply that learning from counterexamples to induction is harder than classical exact learning from labeled examples. This demonstrates that the convergence rate of Counterexample-Guided Inductive Synthesis depends on the form of counterexamples.

Precise reasoning with structured time, structured heaps, and collective operations

Despite decades of progress, static analysis tools still have great difficulty dealing with programs that combine arithmetic, loops, dynamic memory allocation, and linked data structures. In this paper we draw attention to two fundamental reasons for this difficulty: First, typical underlying program abstractions are low-level and inherently scalar, characterizing compound entities like data structures or results computed through iteration only indirectly. Second, to ensure termination, analyses typically project away the dimension of time, and merge information per program point, which incurs a loss in precision. As a remedy, we propose to make collective operations first-class in program analysis – inspired by Σ-notation in mathematics, and also by the success of high-level intermediate languages based on @map/reduce@ operations in program generators and aggressive optimizing compilers for domain-specific languages (DSLs). We further propose a novel structured heap abstraction that preserves a symbolic dimension of time, reflecting the program’s loop structure and thus unambiguously correlating multiple temporal points in the dynamic execution with a single point in the program text. This paper presents a formal model, based on a high-level intermediate analysis language, a practical realization in a prototype tool that analyzes C code, and an experimental evaluation that demonstrates competitive results on a series of benchmarks. Remarkably, our implementation achieves these results in a fully semantics-preserving strongest-postcondition model, which is a worst-case for analysis/verification. The underlying ideas, however, are not tied to this model and would equally apply in other settings, e.g., demand-driven invariant inference in a weakest-precondition model. Given its semantics-preserving nature, our implementation is not limited to analysis for verification, but can also check program equivalence, and translate legacy C code to high-performance DSLs.

Cyber-Physical Specification Mismatches

Embedded systems use increasingly complex software and are evolving into cyber-physical systems (CPS) with sophisticated interaction and coupling between physical and computational processes. Many CPS operate in safety-critical environments and have stringent certification, reliability, and correctness requirements. These systems undergo changes throughout their lifetimes, where either the software or physical hardware is updated in subsequent design iterations. One source of failure in safety-critical CPS is when there are unstated assumptions in either the physical or cyber parts of the system, and new components do not match those assumptions. In this work, we present an automated method toward identifying unstated assumptions in CPS. Dynamic specifications in the form of candidate invariants of both the software and physical components are identified using dynamic analysis (executing and/or simulating the system implementation or model thereof). A prototype tool called Hynger (for HYbrid iNvariant GEneratoR) was developed that instruments Simulink/Stateflow (SLSF) model diagrams to generate traces in the input format compatible with the Daikon invariant inference tool, which has been extensively applied to software systems. Hynger, in conjunction with Daikon, is able to detect candidate invariants of several CPS case studies. We use the running example of a DC-to-DC power converter and demonstrate that Hynger can detect a specification mismatch where a tolerance assumed by the software is violated due to a plant change. Another case study of an automotive control system is also introduced to illustrate the power of Hynger and Daikon in automatically identifying cyber-physical specification mismatches.

Verifying Relative Safety, Accuracy, and Termination for Program Approximations

Approximate computing is an emerging area for trading off the accuracy of an application for improved performance, lower energy costs, and tolerance to unreliable hardware. However, developers must ensure that the leveraged approximations do not introduce significant, intolerable divergence from the reference implementation, as specified by several established robustness criteria. In this work, we show the application of automated differential verification towards verifying relative safety, accuracy, and termination criteria for a class of program approximations. We use mutual summaries to express relative specifications for approximations, and SMT-based invariant inference to automate the verification of such specifications. We perform a detailed feasibility study showing promise of applying automated verification to the domain of approximate computing in a cost-effective manner.

Out-of-sample equity premium predictability and sample split–invariant inference

For a comprehensive set of 21 equity premium predictors we find extreme variation in out-of-sample predictability results depending on the choice of the sample split date. To resolve this issue we propose reporting in graphical form the out-of-sample predictability criteria for every possible sample split, and two out-of-sample tests that are invariant to the sample split choice. We provide Monte Carlo evidence that our bootstrap-based inference is valid. The in-sample, and the sample split invariant out-of-sample mean and maximum tests that we propose, are in broad agreement. Finally we demonstrate how one can construct sample split invariant out-of-sample predictability tests that simultaneously control for data mining across many variables.

Data-driven precondition inference with learned features

We extend the data-driven approach to inferring preconditions for code from a set of test executions. Prior work requires a fixed set of features, atomic predicates that define the search space of possible preconditions, to be specified in advance. In contrast, we introduce a technique for on-demand feature learning, which automatically expands the search space of candidate preconditions in a targeted manner as necessary. We have instantiated our approach in a tool called PIE. In addition to making precondition inference more expressive, we show how to apply our feature-learning technique to the setting of data-driven loop invariant inference. We evaluate our approach by using PIE to infer rich preconditions for black-box OCaml library functions and using our loop-invariant inference algorithm as part of an automatic program verifier for C++ programs.

Less is More

Model-based reliability estimation of systems can provide useful insights early in the development process. However, computational complexity of estimating metrics such as mean time to first failure (MTTFF), turnaround time (TAT), or other domain-based quantitative measures can be prohibitive both in time, space, and precision. In this article, we present an alternative to exhaustive model exploration, as in probabilistic model checking, and partial random exploration, as in statistical model checking. Our hypothesis is that a (carefully crafted) partial systematic exploration of a system model can provide better bounds for these quantitative model metrics at lower computation cost. We present a novel automated technique for metric estimation that combines simulation, invariant inference, and probabilistic model checking. Simulation produces a probabilistically relevant set of traces from which a state invariant is inferred. The invariant characterises a partial model, which is then exhaustively explored using probabilistic model checking. We report on experiments that suggest that metric estimation using this technique (for both fully probabilistic models and those exhibiting nondeterminism) can be more effective than (full-model) probabilistic and statistical model checking, especially for system models for which the events of interest are rare.

From invariant checking to invariant inference using randomized search

We describe a general framework c2i for generating an invariant inference procedure from an invariant checking procedure. Given a checker and a language of possible invariants, c2i generates an inference procedure that iteratively invokes two phases. The search phase uses randomized search to discover candidate invariants and the validate phase uses the checker to either prove or refute that the candidate is an actual invariant. To demonstrate the applicability of c2i , we use it to generate inference procedures that prove safety properties of numerical programs, prove non-termination of numerical programs, prove functional specifications of array manipulating programs, prove safety properties of string manipulating programs, and prove functional specifications of heap manipulating programs that use linked list data structures.

Decidability of inferring inductive invariants

Induction is a successful approach for verification of hardware and software systems. A common practice is to model a system using logical formulas, and then use a decision procedure to verify that some logical formula is an inductive safety invariant for the system. A key ingredient in this approach is coming up with the inductive invariant, which is known as invariant inference. This is a major difficulty, and it is often left for humans or addressed by sound but incomplete abstract interpretation. This paper is motivated by the problem of inductive invariants in shape analysis and in distributed protocols. This paper approaches the general problem of inferring first-order inductive invariants by restricting the language L of candidate invariants. Notice that the problem of invariant inference in a restricted language L differs from the safety problem, since a system may be safe and still not have any inductive invariant in L that proves safety. Clearly, if L is finite (and if testing an inductive invariant is decidable), then inferring invariants in L is decidable. This paper presents some interesting cases when inferring inductive invariants in L is decidable even when L is an infinite language of universal formulas. Decidability is obtained by restricting L and defining a suitable well-quasi-order on the state space. We also present some undecidability results that show that our restrictions are necessary. We further present a framework for systematically constructing infinite languages while keeping the invariant inference problem decidable. We illustrate our approach by showing the decidability of inferring invariants for programs manipulating linked-lists, and for distributed protocols.

SPEEDY: An Eclipse-based IDE for invariant inference

SPEEDY is an Eclipse-based IDE for exploring techniques that assist users in generating correct specifications, particularly including invariant inference algorithms and tools. It integrates with several back-end tools that propose invariants and will incorporate published algorithms for inferring object and loop invariants. Though the architecture is language-neutral, current SPEEDY targets C programs. Building and using SPEEDY has confirmed earlier experience demonstrating the importance of showing and editing specifications in the IDEs that developers customarily use, automating as much of the production and checking of specifications as possible, and showing counterexample information directly in the source code editing environment. As in previous work, automation of specification checking is provided by back-end SMT solvers. However, reducing the effort demanded of software developers using formal methods also requires a GUI design that guides users in writing, reviewing, and correcting specifications and automates specification inference.

Loop invariants

Software verification has emerged as a key concern for ensuring the continued progress of information technology. Full verification generally requires, as a crucial step, equipping each loop with a “loop invariant.” Beyond their role in verification, loop invariants help program understanding by providing fundamental insights into the nature of algorithms. In practice, finding sound and useful invariants remains a challenge. Fortunately, many invariants seem intuitively to exhibit a common flavor. Understanding these fundamental invariant patterns could therefore provide help for understanding and verifying a large variety of programs. We performed a systematic identification, validation, and classification of loop invariants over a range of fundamental algorithms from diverse areas of computer science. This article analyzes the patterns, as uncovered in this study, governing how invariants are derived from postconditions; it proposes a taxonomy of invariants according to these patterns; and it presents its application to the algorithms reviewed. The discussion also shows the need for high-level specifications based on “domain theory.” It describes how the invariants and the corresponding algorithms have been mechanically verified using an automated program prover; the proof source files are available. The contributions also include suggestions for invariant inference and for model-based specification.

Probability Functionals for Self-Consistent and Invariant Inference: Entropy and Fisher Information

Two existing methods of probabilistic inference are based on variational principles: maximum entropy and minimum Fisher information. In each case, a probability density function is inferred by setting the first variation of a functional to zero, subject to information constraints. This study considers whether other functionals could be used for this purpose, and by starting with requirements for self-consistency and invariance, it is shown that the most general admissible functional is just a linear combination of entropy and Fisher information, with the proviso that the normal definition of Fisher information is modified by the inclusion of a prior. This amounts to an axiomatic derivation of entropy and Fisher information. The concern is with continuous random variables and both the single- and multivariable cases are considered. A number of examples are considered to compare inference based on entropy with that based on Fisher information, and to highlight the role of boundary conditions for inference based on Fisher information.

A survey of new trends in symbolic execution for software testing and analysis

Symbolic execution is a well-known program analysis technique which represents program inputs with symbolic values instead of concrete, initialized, data and executes the program by manipulating program expressions involving the symbolic values. Symbolic execution has been proposed over three decades ago but recently it has found renewed interest in the research community, due in part to the progress in decision procedures, availability of powerful computers and new algorithmic developments. We provide here a survey of some of the new research trends in symbolic execution, with particular emphasis on applications to test generation and program analysis. We first describe an approach that handles complex programming constructs such as input recursive data structures, arrays, as well as multithreading. Furthermore, we describe recent hybrid techniques that combine concrete and symbolic execution to overcome some of the inherent limitations of symbolic execution, such as handling native code or availability of decision procedures for the application domain. We follow with a discussion of techniques that can be used to limit the (possibly infinite) number of symbolic configurations that need to be analyzed for the symbolic execution of looping programs. Finally, we give a short survey of interesting new applications, such as predictive testing, invariant inference, program repair, analysis of parallel numerical programs and differential symbolic execution.

OPTIMAL INVARIANT INFERENCE WHEN THE NUMBER OF INSTRUMENTS IS LARGE

This paper studies the asymptotic behavior of a Gaussian linear instrumental variables model in which the number of instruments diverges with the sample size. Asymptotic efficiency bounds are obtained for rotation invariant inference procedures and are shown to be attainable by procedures based on the limited information maximum likelihood estimator. The bounds are obtained by characterizing the limiting experiment associated with the model induced by the rotation invariance restriction.