Separation and the Successor Relation

We investigate two problems for a class C of regular word languages. The C-membership problem asks for an algorithm to decide whether an input language belongs to C. The C-separation problem asks for an algorithm that, given as input two regular languages, decides whether there exists a third language in C containing the first language, while being disjoint from the second. These problems are considered as means to obtain a deep understanding of the class C. It is usual for such classes to be defined by logical formalisms. Logics are often built on top of each other, by adding new predicates. A natural construction is to enrich a logic with the successor relation. In this paper, we obtain new and simple proofs of two transfer results: we show that for suitable logically defined classes, the membership, resp. the separation problem for a class enriched with the successor relation reduces to the same problem for the original class. Our reductions work both for languages of finite words and infinite words. The proofs are mostly self-contained, and only require a basic background on regular languages. This paper therefore gives simple proofs of results that were considered as difficult, such as the decidability of the membership problem for the levels 1, 3/2, 2 and 5/2 of the dot-depth hierarchy

A Generic Polynomial Time Approach to Separation by First-Order Logic Without Quantifier Alternation

We look at classes of languages associated to fragments of first-order logic B??, in which quantifier alternations are disallowed. Each such fragment is fully determined by choosing the set of predicates on positions that may be used. Equipping first-order logic with the linear ordering and possibly the successor relation as predicates yields two natural fragments, which were investigated by Simon and Knast, who proved that these two variants have decidable membership: "does an input regular language belong to the class ?". We extend their results in two orthogonal directions. - First, instead of membership, we explore the more general separation problem: decide if two regular languages can be separated by a language from the class under study. - Second, we use more general inputs: classes ? of group languages (i.e., recognized by a DFA in which each letter induces a permutation of the states) and extensions thereof, written ?^+. We rely on a characterization of B?? by the operator BPol: given an input class ?, it outputs a class BPol(?) that corresponds to a variant of B?? equipped with special predicates associated to ?. The classes BPol(?) and BPol(?^+) capture many natural variants of B?? which use predicates such as the linear ordering, the successor, the modular predicates or the alphabetic modular predicates. We show that separation is decidable for BPol(?) and BPol(?^+) when this is the case for ?. This was already known for BPol(?) and for two particular classes of the form BPol(?^+). Yet, the algorithms were indirect and relied on involved frameworks, yielding poor upper complexity bounds. In contrast, our approach is direct. We work only with elementary concepts (mainly, finite automata). Our main contribution consists in polynomial time Turing reductions from both BPol(?)- and BPol(?^+)-separation to ?-separation. This yields polynomial time algorithms for several key variants of B??, including those equipped with the linear ordering and possibly the successor and/or the modular predicates

Complexity Analysis of Precedence Terminating Infinite Graph Rewrite Systems

The general form of safe recursion (or ramified recurrence) can be expressed by an infinite graph rewrite system including unfolding graph rewrite rules introduced by Dal Lago, Martini and Zorzi, in which the size of every normal form by innermost rewriting is polynomially bounded. Every unfolding graph rewrite rule is precedence terminating in the sense of Middeldorp, Ohsaki and Zantema. Although precedence terminating infinite rewrite systems cover all the primitive recursive functions, in this paper we consider graph rewrite systems precedence terminating with argument separation, which form a subclass of precedence terminating graph rewrite systems. We show that for any precedence terminating infinite graph rewrite system G with a specific argument separation, both the runtime complexity of G and the size of every normal form in G can be polynomially bounded. As a corollary, we obtain an alternative proof of the original result by Dal Lago et al.Comment: In Proceedings TERMGRAPH 2014, arXiv:1505.06818. arXiv admin note: text overlap with arXiv:1404.619

The Foundations of Mathematics in the Physical Reality

In this article we present an axiomatic definition of sets with individuals and a definition of natural numbers (finite ordinal numbers). We use the axioms pairs, union, regularity and separation of the standard set theory ZF. The equality of sets should be defined thus the axiom of extensionality is not used. And there are individuals thus there is no empty set. The principle of mathematical induction is proved for natural numbers. Then we define ordinal numbers and postulate the union set of all natural numbers and define transfinite ordinal numbers.Comment: 11 page

State succession to investment treaties: mapping the issues

Following recent decisions in Sanum v Laos and World Wide Minerals v Kazakhstan, investment lawyers have begun to engage with the legal rules governing State succession to treaties. As State succession is one of the more technical and controversial areas of general international law, this engagement can present challenges; however, the issues are too important to be ignored. This article maps out the most pressing questions of State succession that investment lawyers have faced, or are likely to face in the future. It identifies the three most salient problems — viz the succession of new States to ICSID membership and to old BITs, and the impact of cession of territory on investment protection. With respect to each of these three problems, the article analyses the general regime of State succession and its application to the investment law context, highlighting uncertainties in the law and proposing ways of dealing with them

Topological Semantics and Decidability

It is well-known that the basic modal logic of all topological spaces is $S4$. However, the structure of basic modal and hybrid logics of classes of spaces satisfying various separation axioms was until present unclear. We prove that modal logics of $T_0$, $T_1$ and $T_2$ topological spaces coincide and are S4$. We also examine basic hybrid logics of these classes and prove their decidability; as part of this, we find out that the hybrid logics of$T_1$and T_2$ spaces coincide.Comment: presentation changes, results about concrete structure adde

Adjusting Imperfect Data: Overview and Case Studies

[Excerpt] In this chapter, instead of using the similarity in the cleaned datasets to investigate economic fundamentals, we focus on the differences in the underlying ‘dirty’ data. We describe two data elements that remain fundamentally different across countries, and the extent to which they differ. We then proceed to document some of the problems that affect longitudinally linked administrative data in general, and we describe some of the solutions analysts and statistical agencies have implemented, and some that they did not implement. In each case, we explain the reasons for and against implementing a particular adjustment, and explore, through a select set of case studies, how each adjustment or absence thereof might affect the data. By giving the reader a look behind the scenes, we intend to strengthen the reader’s understanding of the data. Thus equipped, the reader can form his or her own opinion as to the degree of comparability of the findings across the different countries

Dynamic Complexity of Formal Languages

The paper investigates the power of the dynamic complexity classes DynFO, DynQF and DynPROP over string languages. The latter two classes contain problems that can be maintained using quantifier-free first-order updates, with and without auxiliary functions, respectively. It is shown that the languages maintainable in DynPROP exactly are the regular languages, even when allowing arbitrary precomputation. This enables lower bounds for DynPROP and separates DynPROP from DynQF and DynFO. Further, it is shown that any context-free language can be maintained in DynFO and a number of specific context-free languages, for example all Dyck-languages, are maintainable in DynQF. Furthermore, the dynamic complexity of regular tree languages is investigated and some results concerning arbitrary structures are obtained: there exist first-order definable properties which are not maintainable in DynPROP. On the other hand any existential first-order property can be maintained in DynQF when allowing precomputation.Comment: Contains the material presenten at STACS 2009, extendes with proofs and examples which were omitted due lack of spac