arXiv subjects

Bohua Zhan

Publications and source records attributed to Bohua Zhan.

24 records · Page 2Linked to original sources

arXiv2018

Verifying Asymptotic Time Complexity of Imperative Programs in Isabelle

We present a framework in Isabelle for verifying asymptotic time complexity of imperative programs. We build upon an extension of Imperative HOL and its separation logic to include running time. In addition to the basic arguments, our framework is able to handle advanced techniques for time complexity analysis, such as the use of the Akra-Bazzi theorem and amortized analysis. Various automation is built and incorporated into the auto2 prover to reason about separation logic with time credits, and to derive asymptotic behavior of functions. As case studies, we verify the asymptotic time complexity (in addition to functional correctness) of imperative algorithms and data structures such as median of medians selection, Karatsuba's algorithm, and splay trees.

cs.LO↗

arXiv2017

Formalization of the fundamental group in untyped set theory using auto2

We present a new framework for formalizing mathematics in untyped set theory using auto2. Using this framework, we formalize in Isabelle/FOL the entire chain of development from the axioms of set theory to the definition of the fundamental group for an arbitrary topological space. The auto2 prover is used as the sole automation tool, and enables succinct proof scripts throughout the project.

cs.LO↗

arXiv2016

Explicit Koszul-dualizing bimodules in bordered Heegaard Floer homology

We give a combinatorial proof of the quasi-invertibility of $\widehat{CFDD}(\mathbb{I}_\mathcal{Z})$ in bordered Heegaard Floer homology, which implies a Koszul self-duality on the dg-algebra $\mathcal{A}(\mathcal{Z})$, for each pointed matched circle $\mathcal{Z}$. This is done by giving an explicit description of a rank 1 model for $\widehat{CFAA}(\mathbb{I}_\mathcal{Z})$, the quasi-inverse of $\widehat{CFDD}(\mathbb{I}_\mathcal{Z})$. This description is obtained by applying homological perturbation theory to a larger, previously known model of $\widehat{CFAA}(\mathbb{I}_\mathcal{Z})$.

math.GT↗

arXiv2016

Combinatorial Proofs in Bordered Heegaard Floer homology

Using bordered Floer theory, we give a combinatorial construction and proof of invariance for the hat version of Heegaard Floer homology. As a part of the proof, we also establish combinatorially the invariance of the linear-categorical representation of the strongly-based mapping class groupoid given by the same theory.

math.GT↗

arXiv2016

AUTO2, a saturation-based heuristic prover for higher-order logic

We introduce a new theorem prover for classical higher-order logic named auto2. The prover is designed to make use of human-specified heuristics when searching for proofs. The core algorithm is a best-first search through the space of propositions derivable from the initial assumptions, where new propositions are added by user-defined functions called proof steps. We implemented the prover in Isabelle/HOL, and applied it to several formalization projects in mathematics and computer science, demonstrating the high level of automation it can provide in a variety of possible proof tasks.

cs.LO↗

arXiv2012

Super-Polynomial Quantum Speed-ups for Boolean Evaluation Trees with Hidden Structure

We give a quantum algorithm for evaluating a class of boolean formulas (such as NAND trees and 3-majority trees) on a restricted set of inputs. Due to the structure of the allowed inputs, our algorithm can evaluate a depth $n$ tree using $O(n^{2+\logω})$ queries, where $ω$ is independent of $n$ and depends only on the type of subformulas within the tree. We also prove a classical lower bound of $n^{Ω(\log\log n)}$ queries, thus showing a (small) super-polynomial speed-up.

quant-ph↗