Amortised Memory Analysis Using the Depth of Data Structures

Abstract

Abstract. Hofmann and Jost have presented a heap space analysis [1] that finds linear space bounds for many functional programs. It uses an amortised analysis: assigning hypothetical amounts of free space (called potential) to data structures in proportion to their sizes using type annotations. Constraints on these annotations in the type system ensure that the total potential assigned to the input is an upper bound on the total memory required to satisfy all allocations. We describe a related system for bounding the stack space requirements which uses the depth of data structures, by expressing potential in terms of maxima as well as sums. This is achieved by adding extra structure to typing contexts (inspired by O'Hearn's bunched typing [2]) to describe the form of the bounds. We will also present the extra steps that must be taken to construct a typing during the analysis.Obtaining bounds on the resource requirements of programs can be crucial for ensuring that they enjoy reliability and security properties, particularly for use in constrained systems such as mobile phones, smartcards and embedded systems. Hofmann and Jost have presented a type-based amortised analysis for finding upper bounds on the heap memory required for programs in a simple functional programming language [1]. The form of these bounds is limited to linear functions with respect to the size of the program's input. Fortunately, this is sufficient for a wide variety of interesting programs. Moreover, the analysis was successfully used to certify such bounds in a Proof Carrying Code system [3].However, it is also important to bound the stack space requirements, especially for functional programs where it is easy to cause excessive stack usage by accident. The Hofmann-Jost analysis has previously been adapted to measure stack space [4,5], but the form of the bounds was again limited to linear functions in terms of the total size of the input.In this work we present a similar analysis where bounds are given as maxplus expressions on the depth of data structures. This is far more precise for programs operating on tree-structured data.Like Hofmann-Jost, our analysis consists of two parts: a type-system which certifies that a given bound really is an upper bound on the stack memory

Amortized Resource Analysis with Polynomial Potential

“…Essentially, the inference is done by a conventional type checking that produces linear inequalities which can be solved with linear programming. Furthermore, it has been shown [2] that the same potential-based approach can be similarly applied to a wide range of resources such as time and stack space [5] as well as to polymorphic, higher-order programs [6]. Now consider the function pairs:L(int)→L(int, int) that computes the twoelement sets of a given set (if one views the input list as a set).…”

Automatic amortised analysis of dynamic memory allocation for lazy functional programs

“…While there has subsequently been significant interest in the use of amortised analysis for automatic resource usage analysis, using an advanced per-reference potential, none of this newer work, however, considers lazy evaluation. Hofmann and Jost [19] were the first to develop an automatic amortised analysis for heap consumption, exploiting a difference metric similar to that used by Crary and Weirich [11] (the latter, however, only check bounds, and therefore do not perform an automatic static analysis of the kind we require); Hofmann et al have extended their method to cover a comprehensive subset of Java, including imperative updates, inheritance and type casts [20,21]; Shkaravska et al [41] subsequently considered heap consumption inference for first-order polymorphic lists; and Campbell [9] has developed the ideas of depth-based and temporary credit uses to give better results for stack usage. Hoffmann et al [18] achieved another breakthrough by extending the technique to infer multivariate polynomial cost functions, still only requiring efficient LP solving.…”

Stack Bound Inference for Abstract Java Bytecode

“…The work [13] does a similar analysis where the stack bounds are given as the max-plus expressions on the depth of data structures. This analysis is still for functional programs.…”