TiML: a functional language for practical complexity analysis with invariants

Abstract

We present TiML (Timed ML), an ML-like functional language with time-complexity annotations in types. It uses indexed types to express sizes of data structures and upper bounds on running time of functions; and refinement kinds to constrain these indices, expressing data-structure invariants and pre/post-conditions. Indexed types are flexible enough that TiML avoids a built-in notion of "size, " and the programmer can choose to index user-defined datatypes in any way that helps her analysis. TiML's distinguishing characteristic is supporting highly automated time-bound verification applicable to data structures with nontrivial invariants. The programmer provides type annotations, and the typechecker generates verification conditions that are discharged by an SMT solver. Type and index inference are supported to lower annotation burden, and, furthermore, big-O complexity can be inferred from recurrences generated during typechecking by a recurrence solver based on heuristic pattern matching (e.g. using the Master Theorem to handle divide-and-conquerlike recurrences). We have evaluated TiML's usability by implementing a broad suite of case-study modules, demonstrating that TiML, though lacking full automation and theoretical completeness, is versatile enough to verify worst-case and/or amortized complexities for algorithms and data structures like classic list operations, merge sort, Dijkstra's shortest-path algorithm, red-black trees, Braun trees, functional queues, and dynamic tables with bounds like mn log n. The learning curve and annotation burden are reasonable, as we argue with empirical results on our case studies. We formalized TiML's type-soundness proof in Coq.

Liquidate your assets: reasoning about resource usage in liquid Haskell

“…However, none of these approaches can express correctness properties, which as we have seen, allow for a more precise analysis. Recent work [McCarthy et al 2017;Wang et al 2017] combines indexed types with functional correctness. McCarthy et al [2017] develop a Coq library that uses a monad indexed by a predicate to measure runtimes.…”

“…Similarly to Danielsson [2008], relational cost analysis is not supported. TiML [Wang et al 2017] indexes the types of functions with their time bounds. A significant feature of this system is that it provides automated support for solving recurrence relations, by heuristically matching against cases of the Master Theorem.…”

“…Unfortunately, however, performance bugs are as difficult to detect as they are common [Jin et al 2012]. As a result, the problem of statically analysing the resource usage, or execution cost, of programs has been subject to much research in which a broad range of techniques have been studied, including resource-aware type systems [Çiçek et al 2017;Hoffmann et al 2012;Hofmann and Jost 2003;Jost et al 2017;Wang et al 2017], program and separation logics [Aspinall et al 2007;Atkey 2010], and sized types [Vasconcelos 2008].…”

Verifying Asymptotic Time Complexity of Imperative Programs in Isabelle

“…Wang et al [20] present TiML, a functional programming language which can be annotated by invariants and specifically also with time complexity annotations in types. The type checker extracts verification conditions from these programs, which are handled by an SMT solver.…”

Automatic Space Bound Analysis for Functional Programs with Garbage Collection

“…To assist programmers with deriving memory bounds, the programming language community has developed automatic and semi-automatic analysis techniques [24,12,2]. These systems are often special cases of more general resource bound analyses that are based on abstract interpretation [18,7,37], recurrence solving [16,1,14,28], type systems [27,22,29,43,42,15], program logics [5,10,9,35], proof assistants [33,11], and term rewriting [6,34,17].…”