# Praktikum: Spezifikation und Verifikation, Summer 2018

## Overview

 Title Spezifikation und Verifikation (Specification and Verification) Term Summer 2018 Module Type Bachelor-Praktikum (Practical Course for BSc students, IN0012) Master-Praktikum (Practical Course for MSc students, IN2106) Preliminaries Basic knowledge of Isabelle (e.g. Functional Data Structures (IN2347), Semantics (IN2055), Interactive Software Verification (IN3350)) ECTS 10 Organisation Julian Brunner, Tobias Nipkow

## Content

Participants will work on a project by themselves using the interactive theorem prover Isabelle. The practical course will run throughout the semester.

## Application

The application will be through the Matching Platform. There will be no kick-off meeting; instead contact Julian Brunner via email in advance and indicate what prior experience you have with Isabelle (e.g. through one of the above-mentioned lectures) and possibly what particular topics you are interested in.

Note that prior experience with Isabelle is mandatory.

## Topics

### Parser Combinators

Parser combinators are a very common technique in each functional programmer's toolbox to parse text. They are usually implemented as functions mapping input to an optional result, together with higher-order functions for sequential composition, alternatives, repetition and others. Implement parser combinators in Isabelle and if necessary, provide setup for other Isabelle packages.

### Formal Global Optimization (previously Parameterized Branch-and-Bound Method with Applications to Numerical Problems)

Nonlinear optimization problems can be solved with numerical branch-and-bound methods. For formal verification, it suffices to check certificates that encode branches and bounds. HOL-Light [1] directly produces Theorems from such certificates. With the existing infrastructure in Isabelle, it is more convenient to verify an algorithm that checks the certificates. We have such an algorithm implemented in Isabelle/HOL, but not verified. The task is to verify this algorithm.

References: [1]

### Decision Procedures for Metric Spaces

Solovay et al. [1] present decidability results for various theories, including metric spaces, normed spaces, inner product spaces. Isabelle/HOL and HOL-Light feature a tactic for reasoning in normed spaces. Recently [2], metric spaces and a dedicated tactic were introduced in HOL Light. The task is to port (or re-implement) the tactic for metric spaces in Isabelle/HOL and develop one for inner product spaces.

References: [1] [2]

### Newton Iteration

Newton's method is a simple method for finding approximations of roots of non-linear real functions. The goal is to develop a generic framework for this that can be instantiated for particular functions and connecting it with Isabelle's existing packages for interval arithmetic and Taylor models.

### Verification of an Interesting Algorithm or Data Structure

You are welcome to propose an algorithm or data structure and discuss the realizability with your advisor. Some examples of algorithms and data structures that were verified in past lab courses: Finger Trees, Skew Binomial Queues, Dijkstra's Algorithm, Conversion Between Regular Expressions and Finite Automata.

Ideas: String Search Algorithms (Boyer-Moore, Knuth-Morris-Pratt), Graph Algorithms (A*, Bellman-Ford, Kruskal)

### Verification of compositional algorithms for factored transition systems

Factored transition systems succinctly represent state spaces in applications such as Artificial Intelligence (AI) planning and model checking. Many problems defined on such systems are graph theoretic problems on their state space, such as computing reachability or the diameter of the state space. A problem with naively using state-of-the-art graph theoretic algorithms is that they would require the construction of the state space, which can be exponentially bigger than the input factored system, a problem referred to as the state space explosion problem. Compositional algorithms are one approach to alleviate state space explosion, where only state spaces of abstractions are constructed. This project concerns formalising some aspects of compositional algorithms from existing AI planning or model checking literature in Isabelle. Example from the literature discussing compositional algorithms are given below.

### Verification of an approximation algorithm for a graph theoretic problem

Many basic graph theoretic problems are either NP-hard or cannot be solved in better than polynomial time. This makes solving those problems prohibitive if not impossible for real-world graphs. Approximation algorithms circumvent that by using less resources than exact algorithms, at the expense of providing only approximate solutions. In this project the student would formally verify that 1) the approximate solutions of those algorithms meet a certain quality citerion, 2) the upper bounds on their runtimes are correct. A particularly interesting algorithm is the algorithm described below for approximating the diameter of an undirected graph due to Aingworth et al..

### Program Analysis for an Assembler-Style Language

This project would formalize a number of simple program analyses on UPPAAL byte code. The goal is to establish a number of properties that are relevant for model checking (no knowledge on this part is needed). The project can start from an existing formalization of the semantics of the byte code.

### Formalization of elementary number theory using auto2

Auto2 is a recently developed proof automation tool in Isabelle. It is intended to emulate human reasoning when searching for a proof, and contains a mixture of ideas from existing approaches to automation: tactics and the use of automated theorem provers. It has been successfully applied to several case studies in mathematics and computer science, including the formalization of mathematics from the foundations up to the definition of the fundamental group (using set theory), and the verification of functional and imperative data structures (using higher-order logic).

The course consists of learning how to use the auto2 prover, and applying it in a small formalization project. The suggested topic is elementary number theory, based on existing formalization of basic mathematics in set theory. Through this project, the student can learn about some recent ideas on proof automation, and potentially contribute to the development of the still evolving auto2 prover.