Templates and recurrences: better together

Abstract

This paper is the confluence of two streams of ideas in the literature on generating numerical invariants, namely: (1) template-based methods, and (2) recurrence-based methods.A template-based method begins with a template that contains unknown quantities, and finds invariants that match the template by extracting and solving constraints on the unknowns. A disadvantage of template-based methods is that they require fixing the set of terms that may appear in an invariant in advance. This disadvantage is particularly prominent for non-linear invariant generation, because the user must supply maximum degrees on polynomials, bases for exponents, etc.On the other hand, recurrence-based methods are able to find sophisticated non-linear mathematical relations, including polynomials, exponentials, and logarithms, because such relations arise as the solutions to recurrences. However, a disadvantage of past recurrence-based invariant-generation methods is that they are primarily loop-based analyses: they use recurrences to relate the pre-state and post-state of a loop, so it is not obvious how to apply them to a recursive procedure, especially if the procedure is non-linearly recursive (e.g., a tree-traversal algorithm).In this paper, we combine these two approaches and obtain a technique that uses templates in which the unknowns are functions rather than numbers, and the constraints on the unknowns are recurrences. The technique synthesizes invariants involving polynomials, exponentials, and logarithms, even in the presence of arbitrary control-flow, including any

Synthesis with Asymptotic Resource Bounds

“…In this section, we evaluate the effectiveness and performance of SynPlexity, and compare it to existing tools. 4 We implemented SynPlexity in Haskell on top of Synquid by extending its type system with recurrence annotations as presented in Sect. 3.…”

“…Within the literature on automated resource-bound analysis, there are methods that extract and solve recurrence relations for imperative code [2,4,7,15]. However, these methods are unlike the type system presented in this work because they extract concrete complexity bounds as recurrence relations, and then solve the recurrences to find a concrete upper bound on resource usage.…”

Algebraic Program Analysis

“…Section 4.2 shows how the algebraic approach can be applied to summarize linearly recursive procedures. But to compute summaries for generally recursive procedures, current-generation algebraic-programanalysis tools fall back on another non-algebraic scheme (such as hybrid iterative/algebraic, like Kleene or Newton iteration [40,53], or the template-based approach of [16]). This raises the question: is there a practical algebraic method for analyzing general recursion?…”

Synthesis with Asymptotic Resource Bounds

“…4 how the algorithm synthesizes the O(log x) implementation of prod presented in Eqn. (4). We omit the type contexts in the example.…”