Verification of Graph Programs

This thesis is concerned with verifying the correctness of programs written in GP 2 (for Graph Programs), an experimental, nondeterministic graph manipulation language, in which program states are graphs, and computational steps are applications of graph transformation rules. GP 2 allows for visual programming at a high level of abstraction, with the programmer freed from manipulating low-level data structures and instead solving graph-based problems in a direct, declarative, and rule-based way. To verify that a graph program meets some specification, however, has been -- prior to the work described in this thesis -- an ad hoc task, detracting from the appeal of using GP 2 to reason about graph algorithms, high-level system specifications, pointer structures, and the many other practical problems in software engineering and programming languages that can be modelled as graph problems. This thesis describes some contributions towards the challenge of verifying graph programs, in particular, Hoare logics with which correctness specifications can be proven in a syntax-directed and compositional manner. We contribute calculi of proof rules for GP 2 that allow for rigorous reasoning about both partial correctness and termination of graph programs. These are given in an extensional style, i.e. independent of fixed assertion languages. This approach allows for the re-use of proof rules with different assertion languages for graphs, and moreover, allows for properties of the calculi to be inherited: soundness, completeness for termination, and relative completeness (for sufficiently expressive assertion languages). We propose E-conditions as a graphical, intuitive assertion language for expressing properties of graphs -- both about their structure and labelling -- generalising the nested conditions of Habel, Pennemann, and Rensink. We instantiate our calculi with this language, explore the relationship between the decidability of the model checking problem and the existence of effective constructions for the extensional assertions, and fix a subclass of graph programs for which we have both. The calculi are then demonstrated by verifying a number of data- and structure-manipulating programs. We explore the relationship between E-conditions and classical logic, defining translations between the former and a many-sorted predicate logic over graphs; the logic being a potential front end to an implementation of our work in a proof assistant. Finally, we speculate on several avenues of interesting future work; in particular, a possible extension of E-conditions with transitive closure, for proving specifications involving properties about arbitrary-length paths

Dagstuhl Reports : Volume 1, Issue 2, February 2011

Online Privacy: Towards Informational Self-Determination on the Internet (Dagstuhl Perspectives Workshop 11061) : Simone Fischer-Hübner, Chris Hoofnagle, Kai Rannenberg, Michael Waidner, Ioannis Krontiris and Michael Marhöfer Self-Repairing Programs (Dagstuhl Seminar 11062) : Mauro Pezzé, Martin C. Rinard, Westley Weimer and Andreas Zeller Theory and Applications of Graph Searching Problems (Dagstuhl Seminar 11071) : Fedor V. Fomin, Pierre Fraigniaud, Stephan Kreutzer and Dimitrios M. Thilikos Combinatorial and Algorithmic Aspects of Sequence Processing (Dagstuhl Seminar 11081) : Maxime Crochemore, Lila Kari, Mehryar Mohri and Dirk Nowotka Packing and Scheduling Algorithms for Information and Communication Services (Dagstuhl Seminar 11091) Klaus Jansen, Claire Mathieu, Hadas Shachnai and Neal E. Youn

Learning neural algorithms with graph structures

Graph structures, like syntax trees, social networks, and programs, are ubiquitous in many real world applications including knowledge graph inference, chemistry and social network analysis. Over the past several decades, many expert-designed algorithms on graphs have been proposed with nice theoretical properties. However most of them are not data-driven, and will not benefit from the growing scale of available data. Recent advances in deep learning have shown strong empirical performances for images, texts and signals, typically with little domain knowledge. However the combinatorial and discrete nature of the graph data makes it non-trivial to apply neural networks in this domain. Based on the pros and cons of these two, this thesis will discuss several aspects on how to build a tight connection between neural networks and the classical algorithms for graphs. Specifically: - Algorithm inspired deep graph learning: The existing algorithms provide an inspiration of deep architecture design, for both the discriminative learning and generative modeling of graphs. Regarding the discriminative representation learning, we show how the graphical model inference algorithms can inspire the design of graph neural networks for chemistry and bioinformatics applications, and how to scale it up with the idea borrowed from steady states algorithms like PageRank; for generative modeling, we build an HSMM inspired neural segmental generative modeling for signal sequences; and for a class of graphs, we leverage the idea of attribute grammar for syntax trees to help regulate the deep networks. - Deep learning enhanced graph algorithms: the algorithm framework has procedures that can be enhanced by learnable deep network components. We demonstrate by learning the heuristic function in greedy algorithms with reinforcement learning for combinatorial optimization problems over graphs, such as vertex cover and max cut, and optimal touring problem for real world applications like fuzzing. - Towards Inductive reasoning with graph structures: As the algorithm structure generally provides a good inductive bias for the problem, we take an initial step towards inductive reasoning for such structure, where we make attempts to reason about the loop invariant for program verification and the reaction templates for retrosynthesis structured prediction.Ph.D

An Interactive Graph Theory System

The medium of computer graphics provides a capability for dealing with pictures in man-machine communication. Graph Theory is used to model relationships which are represented by pictures and is therefore an appropriate discipline for the application of an interactive computer graphics system. Previous efforts to solve Graph Theoretic problems by computer have usually involved specialized programs written in a symbolic assembly language or algebraic compiler language. In recent years, graphics equipment with processing power has been commercially available for use as a remote terminal to a large central computer. Although these terminals typically include a small general purpose computer, the potential of using one as programmable subsystem has received little attention. These motivations have led to the design and implementation of an interactive graphics system for solving Graph Theoretic problems. The system operates on an IBM 7040 with a DEC-338 graphics terminal connected by voice-grade telephone line. To provide effective response times, computing power is appropriately divided between the two machines. The remote computer graphics terminal is controlled by a special-purpose executive program. This executive includes an interpreter of a command language oriented towards the control of existence and display of graphs. Several interactive functions such as graph drawing and editing are available to a user through light button and pushbutton selection. These functions which are local to the terminal are programmed in a mixture of the terminal computer\u27s machine language and the interpreted command language. For more significant computational requirements the central computer is used, but response time for interactive operation is then diminished. In order to overcome the speed of the telephone link, the central computer may call upon a program at the terminal as a subroutine. Based on the mathematical terminology used to define graphs, a high level language was developed for the specification of interactive algorithms. A growing library of these algorithms provides routines to aid in the construction and recognition of various types of graphs. Other routines are used for computing certain properties of graphs. Graphs may be transformed by some routines with respect to both connectivity and layout. Any number of graphs my be saved and later restored. A programmer using the terminal as an alphanumeric console may call upon the programming features of the system to develop new interactive algorithms and add them to the library. Programs may also be created for the display terminal, using the central computer for assembly. Examples of system use which are presented include finding a shortest path between any pair of vertices in a weighted directed graph, determining the maximally complete subgraphs of an arbitrary graph, interpreting a graph as a Mealy model of a finite state machine, and laying out a tree for aesthetic presentation

Program slicing by calculation

Program slicing is a well known family of techniques used to identify code fragments which depend on or are depended upon specific program entities. They are particularly useful in the areas of reverse engineering, program understanding, testing and software maintenance. Most slicing methods, usually oriented towards the imperatice or object paradigms, are based on some sort of graph structure representing program dependencies. Slicing techniques amount, therefore, to (sophisticated) graph transversal algorithms. This paper proposes a completely different approach to the slicing problem for functional programs. Instead of extracting program information to build an underlying dependencies' structure, we resort to standard program calculation strategies, based on the so-called Bird-Meertens formalism. The slicing criterion is specified either as a projection or a hiding function which, once composed with the original program, leads to the identification of the intended slice. Going through a number of examples, the paper suggests this approach may be an interesting, even if not completely general, alternative to slicing functional programsFundação para a Ciência e a Tecnologia (FCT

New Classes of Distributed Time Complexity

A number of recent papers -- e.g. Brandt et al. (STOC 2016), Chang et al. (FOCS 2016), Ghaffari & Su (SODA 2017), Brandt et al. (PODC 2017), and Chang & Pettie (FOCS 2017) -- have advanced our understanding of one of the most fundamental questions in theory of distributed computing: what are the possible time complexity classes of LCL problems in the LOCAL model? In essence, we have a graph problem $\Pi$ in which a solution can be verified by checking all radius-$O(1)$ neighbourhoods, and the question is what is the smallest $T$ such that a solution can be computed so that each node chooses its own output based on its radius-$T$ neighbourhood. Here $T$ is the distributed time complexity of $\Pi$. The time complexity classes for deterministic algorithms in bounded-degree graphs that are known to exist by prior work are $\Theta(1)$, $\Theta(\log^* n)$, $\Theta(\log n)$, $\Theta(n^{1/k})$, and $\Theta(n)$. It is also known that there are two gaps: one between $\omega(1)$ and $o(\log \log^* n)$, and another between $\omega(\log^* n)$ and $o(\log n)$. It has been conjectured that many more gaps exist, and that the overall time hierarchy is relatively simple -- indeed, this is known to be the case in restricted graph families such as cycles and grids. We show that the picture is much more diverse than previously expected. We present a general technique for engineering LCL problems with numerous different deterministic time complexities, including $\Theta(\log^{\alpha}n)$ for any $\alpha\ge1$, $2^{\Theta(\log^{\alpha}n)}$ for any $\alpha\le 1$, and $\Theta(n^{\alpha})$ for any $\alpha <1/2$ in the high end of the complexity spectrum, and $\Theta(\log^{\alpha}\log^* n)$ for any $\alpha\ge 1$, $\smash{2^{\Theta(\log^{\alpha}\log^* n)}}$ for any $\alpha\le 1$, and $\Theta((\log^* n)^{\alpha})$ for any $\alpha \le 1$ in the low end; here $\alpha$ is a positive rational number

Towards Practical Graph-Based Verification for an Object-Oriented Concurrency Model

To harness the power of multi-core and distributed platforms, and to make the development of concurrent software more accessible to software engineers, different object-oriented concurrency models such as SCOOP have been proposed. Despite the practical importance of analysing SCOOP programs, there are currently no general verification approaches that operate directly on program code without additional annotations. One reason for this is the multitude of partially conflicting semantic formalisations for SCOOP (either in theory or by-implementation). Here, we propose a simple graph transformation system (GTS) based run-time semantics for SCOOP that grasps the most common features of all known semantics of the language. This run-time model is implemented in the state-of-the-art GTS tool GROOVE, which allows us to simulate, analyse, and verify a subset of SCOOP programs with respect to deadlocks and other behavioural properties. Besides proposing the first approach to verify SCOOP programs by automatic translation to GTS, we also highlight our experiences of applying GTS (and especially GROOVE) for specifying semantics in the form of a run-time model, which should be transferable to GTS models for other concurrent languages and libraries.Comment: In Proceedings GaM 2015, arXiv:1504.0244