A Fistful of Dollars: Formalizing Asymptotic Complexity Claims via Deductive Program Verification

Abstract

We present a framework for simultaneously verifying the functional correctness and the worst-case asymptotic time complexity of higher-order imperative programs. We build on top of Separation Logic with Time Credits, embedded in an interactive proof assistant. We formalize the O notation, which is key to enabling modular specifications and proofs. We cover the subtleties of the multivariate case, where the complexity of a program fragment depends on multiple parameters. We propose a way of integrating complexity bounds into specifications, present lemmas and tactics that support a natural reasoning style, and illustrate their use with a collection of examples.

Verifying Asymptotic Time Complexity of Imperative Programs in Isabelle

“…The most closely-related is the impressive work by Guéneau et al [12] for asymptotic time complexity analysis in Coq. We now take a closer look at the similarities and differences: -Guéneau et al give a structured overview of different problems that arise when working informally with asymptotic complexity in several variables, then solve this problem by rigorously defining asymptotic domination (which is essentially f ∈ O(g)) with filters and develop automation for reasoning about it.…”

“…Interactive theorem provers are useful tools for performing such a verification, as their soundness is based on a small trusted kernel, hence long derivations can be made with a very high level of confidence. So far, the work of Guéneau et al [12,6] appears to be the only general framework for asympotic time complexity analysis of imperative programs in an interactive theorem prover. The framework is built in Coq, based on Charguéraud's CFML package [5] for verifying imperative programs using characteristic formulas.…”

“…The framework is an extension of Imperative HOL [2], which represents imperative programs as monads. Compared to [12], we go further in two directions. First, we incorporate the work of Eberl [11] on the Akra-Bazzi theorem to analyze several divide-and-conquer algorithms.…”

Time Credits and Time Receipts in Iris

“…Guéneau et al [7] use time credits to reason about asymptotic complexity, that is, about the manner in which a program's complexity grows as the size of its input grows towards infinity. Does such asymptotic reasoning make sense in Iris $ȯ , where no program is ever executed for N time steps or beyond?…”

Hoare Logics for Time Bounds

“…Similar ideas were used by Hoffmann et al [10] to prove lock-freedom of concurrent programs, and by Charguéraud and Pottier [7] to verify the amortised running time of the Union-Find data structure in Coq. Guéneau et al [8] recently extended their framework to also obtain O results for the running time of programs. None of these works include verified VCGs.…”