Modular Construction of Shape-Numeric Analyzers

The aim of static analysis is to infer invariants about programs that are precise enough to establish semantic properties, such as the absence of run-time errors. Broadly speaking, there are two major branches of static analysis for imperative programs. Pointer and shape analyses focus on inferring properties of pointers, dynamically-allocated memory, and recursive data structures, while numeric analyses seek to derive invariants on numeric values. Although simultaneous inference of shape-numeric invariants is often needed, this case is especially challenging and is not particularly well explored. Notably, simultaneous shape-numeric inference raises complex issues in the design of the static analyzer itself. In this paper, we study the construction of such shape-numeric, static analyzers. We set up an abstract interpretation framework that allows us to reason about simultaneous shape-numeric properties by combining shape and numeric abstractions into a modular, expressive abstract domain. Such a modular structure is highly desirable to make its formalization and implementation easier to do and get correct. To achieve this, we choose a concrete semantics that can be abstracted step-by-step, while preserving a high level of expressiveness. The structure of abstract operations (i.e., transfer, join, and comparison) follows the structure of this semantics. The advantage of this construction is to divide the analyzer in modules and functors that implement abstractions of distinct features.Comment: In Proceedings Festschrift for Dave Schmidt, arXiv:1309.455

Combining Forward and Backward Abstract Interpretation of Horn Clauses

Alternation of forward and backward analyses is a standard technique in abstract interpretation of programs, which is in particular useful when we wish to prove unreachability of some undesired program states. The current state-of-the-art technique for combining forward (bottom-up, in logic programming terms) and backward (top-down) abstract interpretation of Horn clauses is query-answer transformation. It transforms a system of Horn clauses, such that standard forward analysis can propagate constraints both forward, and backward from a goal. Query-answer transformation is effective, but has issues that we wish to address. For that, we introduce a new backward collecting semantics, which is suitable for alternating forward and backward abstract interpretation of Horn clauses. We show how the alternation can be used to prove unreachability of the goal and how every subsequent run of an analysis yields a refined model of the system. Experimentally, we observe that combining forward and backward analyses is important for analysing systems that encode questions about reachability in C programs. In particular, the combination that follows our new semantics improves the precision of our own abstract interpreter, including when compared to a forward analysis of a query-answer-transformed system.Comment: Francesco Ranzato. 24th International Static Analysis Symposium (SAS), Aug 2017, New York City, United States. Springer, Static Analysi

On Verifying Complex Properties using Symbolic Shape Analysis

One of the main challenges in the verification of software systems is the analysis of unbounded data structures with dynamic memory allocation, such as linked data structures and arrays. We describe Bohne, a new analysis for verifying data structures. Bohne verifies data structure operations and shows that 1) the operations preserve data structure invariants and 2) the operations satisfy their specifications expressed in terms of changes to the set of objects stored in the data structure. During the analysis, Bohne infers loop invariants in the form of disjunctions of universally quantified Boolean combinations of formulas. To synthesize loop invariants of this form, Bohne uses a combination of decision procedures for Monadic Second-Order Logic over trees, SMT-LIB decision procedures (currently CVC Lite), and an automated reasoner within the Isabelle interactive theorem prover. This architecture shows that synthesized loop invariants can serve as a useful communication mechanism between different decision procedures. Using Bohne, we have verified operations on data structures such as linked lists with iterators and back pointers, trees with and without parent pointers, two-level skip lists, array data structures, and sorted lists. We have deployed Bohne in the Hob and Jahob data structure analysis systems, enabling us to combine Bohne with analyses of data structure clients and apply it in the context of larger programs. This report describes the Bohne algorithm as well as techniques that Bohne uses to reduce the ammount of annotations and the running time of the analysis

A Static Analyzer for Large Safety-Critical Software

We show that abstract interpretation-based static program analysis can be made efficient and precise enough to formally verify a class of properties for a family of large programs with few or no false alarms. This is achieved by refinement of a general purpose static analyzer and later adaptation to particular programs of the family by the end-user through parametrization. This is applied to the proof of soundness of data manipulation operations at the machine level for periodic synchronous safety critical embedded software. The main novelties are the design principle of static analyzers by refinement and adaptation through parametrization, the symbolic manipulation of expressions to improve the precision of abstract transfer functions, the octagon, ellipsoid, and decision tree abstract domains, all with sound handling of rounding errors in floating point computations, widening strategies (with thresholds, delayed) and the automatic determination of the parameters (parametrized packing)

A simple abstraction of arrays and maps by program translation

We present an approach for the static analysis of programs handling arrays, with a Galois connection between the semantics of the array program and semantics of purely scalar operations. The simplest way to implement it is by automatic, syntactic transformation of the array program into a scalar program followed analysis of the scalar program with any static analysis technique (abstract interpretation, acceleration, predicate abstraction,.. .). The scalars invariants thus obtained are translated back onto the original program as universally quantified array invariants. We illustrate our approach on a variety of examples, leading to the " Dutch flag " algorithm

Program Analysis in A Combined Abstract Domain

Automated verification of heap-manipulating programs is a challenging task due to the complexity of aliasing and mutability of data structures used in these programs. The properties of a number of important data structures do not only relate to one domain, but to combined multiple domains, such as sorted list, priority queues, height-balanced trees and so on. The safety and sometimes efficiency of programs do rely on the properties of those data structures. This thesis focuses on developing a verification system for both functional correctness and memory safety of such programs which involve heap-based data structures. Two automated inference mechanisms are presented for heap-manipulating programs in this thesis. Firstly, an abstract interpretation based approach is proposed to synthesise program invariants in a combined pure and shape domain. Newly designed abstraction, join and widening operators have been defined for the combined domain. Furthermore, a compositional analysis approach is described to discover both pre-/post-conditions of programs with a bi-abduction technique in the combined domain. As results of my thesis, both inference approaches have been implemented and the obtained results validate the feasibility and precision of proposed approaches. The outcomes of the thesis confirm that it is possible and practical to analyse heap-manipulating programs automatically and precisely by using abstract interpretation in a sophisticated combined domain

Non-polynomial Worst-Case Analysis of Recursive Programs

We study the problem of developing efficient approaches for proving worst-case bounds of non-deterministic recursive programs. Ranking functions are sound and complete for proving termination and worst-case bounds of nonrecursive programs. First, we apply ranking functions to recursion, resulting in measure functions. We show that measure functions provide a sound and complete approach to prove worst-case bounds of non-deterministic recursive programs. Our second contribution is the synthesis of measure functions in nonpolynomial forms. We show that non-polynomial measure functions with logarithm and exponentiation can be synthesized through abstraction of logarithmic or exponentiation terms, Farkas' Lemma, and Handelman's Theorem using linear programming. While previous methods obtain worst-case polynomial bounds, our approach can synthesize bounds of the form $\mathcal{O}(n\log n)$ as well as $\mathcal{O}(n^r)$ where $r$ is not an integer. We present experimental results to demonstrate that our approach can obtain efficiently worst-case bounds of classical recursive algorithms such as (i) Merge-Sort, the divide-and-conquer algorithm for the Closest-Pair problem, where we obtain $\mathcal{O}(n \log n)$ worst-case bound, and (ii) Karatsuba's algorithm for polynomial multiplication and Strassen's algorithm for matrix multiplication, where we obtain $\mathcal{O}(n^r)$ bound such that $r$ is not an integer and close to the best-known bounds for the respective algorithms.Comment: 54 Pages, Full Version to CAV 201

A Simple and Scalable Static Analysis for Bound Analysis and Amortized Complexity Analysis

We present the first scalable bound analysis that achieves amortized complexity analysis. In contrast to earlier work, our bound analysis is not based on general purpose reasoners such as abstract interpreters, software model checkers or computer algebra tools. Rather, we derive bounds directly from abstract program models, which we obtain from programs by comparatively simple invariant generation and symbolic execution techniques. As a result, we obtain an analysis that is more predictable and more scalable than earlier approaches. Our experiments demonstrate that our analysis is fast and at the same time able to compute bounds for challenging loops in a large real-world benchmark. Technically, our approach is based on lossy vector addition systems (VASS). Our bound analysis first computes a lexicographic ranking function that proves the termination of a VASS, and then derives a bound from this ranking function. Our methodology achieves amortized analysis based on a new insight how lexicographic ranking functions can be used for bound analysis