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Abstract. We present a program logic for verifying the heap consumption of low-level programs. The proof rules employ a uniform assertion format and have been derived from a general purpose program logic [1]. In a proof-carrying code scenario, the inference of invariants is delegated to the code provider, who employs a certifying compiler that generates a certificate from program annotations and analysis. The granularity of the proof rules matches that of the linear type system presented in [6], which enables us to perform verification by replaying typing derivations in a theorem prover, given the specifications of individual methods. The resulting verification conditions are of limited complexity, and are automatically discharged. We also outline a proof system that relaxes the linearity restrictions and relates to the type system of usage aspects presented in [2].

A b str a c t. We present a size-aware type system for first-order shapely function definitions. Here, a function definition is called shapely when the size of the result is determined exactly by a polynomial in the sizes of the arguments. Examples of shapely function definitions may be matrix multiplication and the Cartesian product of two lists. The type checking problem for the type system is shown to be undecidable in general. We define a natural syntactic restriction such that the type checking becomes decidable, even though size polynomials are not necessarily linear or monotonic. Furthermore, a method that infers poly nomial size dependencies for a non-trivial class of function definitions is suggested.1

This paper presents an interpolation-based method of inferring arbitrary degree loop-bound functions for Java programs. Given a loop, by its "loop-bound function" we mean a function with the numeric program variables as its parameters, that is used to bound the number of loop-iterations. Using our analysis, loopbound functions that are polynomials with natural, rational or real coefficients can be found.Analysis of loop bounds is important in several different areas, including worst-case execution time (WCET) and heap consumption analysis, optimising compilers and termination-analysis. While several other methods exist to infer numerical loop bounds, we know of no other research on the inference of non-linear loopbound functions. Additionally, the inferred bounds are provable using external tools, e.g. KeY.To infer a loop-bound function for a given loop it is instrumented with a counter and executed on a well-chosen set of values of the numerical program variables. By well-chosen we mean that using these test values and the corresponding values of the counter, one can construct a unique interpolating polynomial. The uniqueness and the existence of the interpolating polynomial is guaranteed if the input values are in the so-called NCA-configuration, known from multivariate-polynomial interpolation theory. The constructed interpolating polynomial presumably bounds the dependency of the number of loop iterations on arbitrary values of the program variables. This hypothesis is verified by a third-party proof assistant.A prototype tool has been developed which implements this method. This prototype can infer piecewise polynomial loop-bound functions for a large class of loops in Java programs. Applicability of the prototype has been tested on a series of safety-critical case studies. For most of the loops in the case studies, loop-bound functions could be inferred (and verified using a proof assistant).

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Abstract. We present a size-aware type system for first-order shapely function definitions. Here, a function definition is called shapely when the size of the result is determined exactly by a polynomial in the sizes of the arguments. Examples of shapely function definitions may be implementations of matrix multiplication and the Cartesian product of two lists.The type system is proved to be sound w.r.t. the operational semantics of the language. The type checking problem is shown to be undecidable in general. We define a natural syntactic restriction such that the type checking becomes decidable, even though size polynomials are not necessarily linear or monotonic.Furthermore, we have shown that the type-inference problem is at least semi-decidable (under this restriction). We have implemented a procedure that combines run-time testing and type-checking to automatically obtain size dependencies. It terminates on total typable function definitions.

For real-time and embedded systems limiting the consumption of time and memory resources is often an important part of the requirements. Being able to predict bounds on the consumption of these resources during the development process of the code can be of great value.Recent research results have advanced the state of the art of resource consumption analysis. In this paper we present a tool that makes it possible to apply these research results in practice for real-time systems enabling Java developers to analyse loop bounds, bounds on heap size and bounds on stack size. We describe which theoretical additions were needed in order to achieve this.We give an overview of the capabilities of the tool ResAna that is the result of this effort. The tool can not only perform generally applicable analyses, but it also contains a part of the analysis which is dedicated to the developers' (real-time) virtual machine, such that the results apply directly to the actual development environment that is used in practice.

This work introduces collected size semantics of strict functional programs over lists. The collected size semantics of a function definition is a multivalued size function that collects the dependencies between every possible output size and the corresponding input sizes. Such functions annotate standard types and are defined by conditional rewriting rules generated during type inference. We focus on the connection between the rewriting rules and lower and upper bounds on the multivalued size functions, when the bounds are given by piecewise polynomials. We show how, given a set of conditional rewriting rules, one can infer bounds that define an indexed family of polynomials that approximates the multivalued size function. Using collected size semantics we are able to infer non-monotonic and non-linear lower and upper polynomial bounds for many functional programs. As a feasibility study, we use the procedure to infer lower and upper polynomial size-bounds on typical functions of a list library.