Polynomial Size Analysis of First-Order Shapely Functions

Abstract

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.

Collected Size Semantics for Strict Functional Programs over General Polymorphic Lists

“…This research extends our work [17][18][19]21] about shapely function definitions that have a single-valued, exact input-output polynomial size functions. Our non-monotonic framework resembles [1] in which the authors describe monotonic resource consumption for Java bytecode by means of Cost Equation Systems (CESs), which are similar to, but more general than recurrence equations.…”

“…In many cases, analysis of size behavior is an important part of resource analysis, e.g. using lower and upper bounds of size dependencies to check and infer non-linear bounds on heap consumption [17,20]. Here, we consider size analysis of strict first-order functional programs over polymorphic non-cyclic lists.…”

Test-based inference of polynomial loop-bound functions

“…This polynomial interpretation method was already applied to Size Analysis in previous work [17,18]. The main challenge we face when we adjust the interpolation theory to inferring imperative loop-bound functions is that test data must not only lie on a grid (or more generally, be in NCA), but also satisfy the loop condition C. In Figure 1 we show the set of points satisfying the (in)equalities i < x, i > 0 and x > 0.…”

“…This work builds on our previous research on size analysis [17], where we studied polynomial dependencies of the sizes of output data structures (e.g. the length of a linked list) on the sizes of input data structures.…”

Making resource analysis practical for real-time Java

“…The polynomial interpolation based technique was successfully applied in the analysis of output-on-input datastructure size relations for functions in a functional language, in [29], [31] and [28]. This method can for instance be used to determine that if the append function gets two lists of lengths n and m as input, it will return a list of length n + m.…”