Verification of PCP-Related Computational Reductions in Coq

We formally verify several computational reductions concerning the Post correspondence problem (PCP) using the proof assistant Coq. Our verifications include a reduction of a string rewriting problem generalising the halting problem for Turing machines to PCP, and reductions of PCP to the intersection problem and the palindrome problem for context-free grammars. Interestingly, rigorous correctness proofs for some of the reductions are missing in the literature

Cover results and normal forms

The purpose of this paper was to sketch an area of problems for the concept of cover. We showed that in spite of some remarks in the literature the problem of covering (unambiguous and -free) cfg's with cfg's in GNF is open. Moreover we gave some properties of covers and we showed a relation between covers and parsability

A survey of normal form covers for context-free grammars

An overview is given of cover results for normal forms of context-free grammars. The emphasis in this paper is on the possibility of constructing ɛ-free grammars, non-left-recursive grammars and grammars in Greibach normal form. Among others it is proved that any ɛ-free context-free grammar can be right covered with a context-free grammar in Greibach normal form. All the cover results concerning the ɛ-free grammars, the non-left-recursive grammars and the grammars in Greibach normal form are listed, with respect to several types of covers, in a cover-table

Empirical Risk Minimization for Probabilistic Grammars: Sample Complexity and Hardness of Learning

Probabilistic grammars are generative statistical models that are useful for compositional and sequential structures. They are used ubiquitously in computational linguistics. We present a framework, reminiscent of structural risk minimization, for empirical risk minimization of probabilistic grammars using the log-loss. We derive sample complexity bounds in this framework that apply both to the supervised setting and the unsupervised setting. By making assumptions about the underlying distribution that are appropriate for natural language scenarios, we are able to derive distribution-dependent sample complexity bounds for probabilistic grammars. We also give simple algorithms for carrying out empirical risk minimization using this framework in both the supervised and unsupervised settings. In the unsupervised case, we show that the problem of minimizing empirical risk is NP-hard. We therefore suggest an approximate algorithm, similar to expectation-maximization, to minimize the empirical risk. Learning from data is central to contemporary computational linguistics. It is in common in such learning to estimate a model in a parametric family using the maximum likelihood principle. This principle applies in the supervised case (i.e., using annotate

Structure preserving transformations on non-left-recursive grammars

We will be concerned with grammar covers, The first part of this paper presents a general framework for covers. The second part introduces a transformation from nonleft-recursive grammars to grammars in Greibach normal form. An investigation of the structure preserving properties of this transformation, which serves also as an illustration of our framework for covers, is presented

On the covering of left recursive grammars

In this paper we show that some prevailing ideas on the elimination of left recursion in a context-free grammar are not valid. An algorithm and a proof are given to show that every proper context-free grammar is covered by a non-left-recursive grammar

Global Thresholding and Multiple Pass Parsing

We present a variation on classic beam thresholding techniques that is up to an order of magnitude faster than the traditional method, at the same performance level. We also present a new thresholding technique, global thresholding, which, combined with the new beam thresholding, gives an additional factor of two improvement, and a novel technique, multiple pass parsing, that can be combined with the others to yield yet another 50% improvement. We use a new search algorithm to simultaneously optimize the thresholding parameters of the various algorithms.Comment: Fixed latex errors; fixed minor errors in published versio

Interpretation and reduction of attribute grammars

An attribute grammar (AG) is in reduced form if in all its derivation trees every attribute contributes to the translation. We prove that, eventhough AG are generally not in reduced form, they can be reduced, i.e., put into reduced form, without modifying their translations. This is shown first for noncircular AG and then for arbitrary AG. In both cases the reduction consists of easy (almost syntactic) transformations which do not change the semantic domain of the AG. These easy transformations are formalized by introducing the notion of AG interpretation as an extension to AG of the concept of context-free grammar form. Finally we prove that any general algorithm for reducing even the simple class of L-AG needs exponential time (in the size of the input AG) infinitely often