Widths of regular and context-free languages

Given a partially-ordered finite alphabet $\Sigma$ and a language $L\subseteq \Sigma^*$, how large can an antichain in $L$ be (where $L$ is given the lexicographic ordering)? More precisely, since $L$ will in general be infinite, we should ask about the rate of growth of maximum antichains consisting of words of length $n$. This fundamental property of partial orders is known as the width, and in a companion work we show that the problem of computing the information leakage permitted by a deterministic interactive system modeled as a finite-state transducer can be reduced to the problem of computing the width of a certain regular language. In this paper, we show that if $L$ is regular then there is a dichotomy between polynomial and exponential antichain growth. We give a polynomial-time algorithm to distinguish the two cases, and to compute the order of polynomial growth, with the language specified as an NFA. For context-free languages we show that there is a similar dichotomy, but now the problem of distinguishing the two cases is undecidable. Finally, we generalise the lexicographic order to tree languages, and show that for regular tree languages there is a trichotomy between polynomial, exponential and doubly exponential antichain growth.Comment: 22 page

Path-Based Program Repair

We propose a path-based approach to program repair for imperative programs. Our repair framework takes as input a faulty program, a logic specification that is refuted, and a hint where the fault may be located. An iterative abstraction refinement loop is then used to repair the program: in each iteration, the faulty program part is re-synthesized considering a symbolic counterexample, where the control-flow is kept concrete but the data-flow is symbolic. The appeal of the idea is two-fold: 1) the approach lazily considers candidate repairs and 2) the repairs are directly derived from the logic specification. In contrast to prior work, our approach is complete for programs with finitely many control-flow paths, i.e., the program is repaired if and only if it can be repaired at the specified fault location. Initial results for small programs indicate that the approach is useful for debugging programs in practice.Comment: In Proceedings FESCA 2015, arXiv:1503.0437

Inducing Probabilistic Grammars by Bayesian Model Merging

We describe a framework for inducing probabilistic grammars from corpora of positive samples. First, samples are {\em incorporated} by adding ad-hoc rules to a working grammar; subsequently, elements of the model (such as states or nonterminals) are {\em merged} to achieve generalization and a more compact representation. The choice of what to merge and when to stop is governed by the Bayesian posterior probability of the grammar given the data, which formalizes a trade-off between a close fit to the data and a default preference for simpler models (`Occam's Razor'). The general scheme is illustrated using three types of probabilistic grammars: Hidden Markov models, class-based $n$-grams, and stochastic context-free grammars.Comment: To appear in Grammatical Inference and Applications, Second International Colloquium on Grammatical Inference; Springer Verlag, 1994. 13 page

Coarse-grained reconfigurable array architectures

Coarse-Grained Reconfigurable Array (CGRA) architectures accelerate the same inner loops that benefit from the high ILP support in VLIW architectures. By executing non-loop code on other cores, however, CGRAs can focus on such loops to execute them more efficiently. This chapter discusses the basic principles of CGRAs, and the wide range of design options available to a CGRA designer, covering a large number of existing CGRA designs. The impact of different options on flexibility, performance, and power-efficiency is discussed, as well as the need for compiler support. The ADRES CGRA design template is studied in more detail as a use case to illustrate the need for design space exploration, for compiler support and for the manual fine-tuning of source code