On the State Complexity of Partial Derivative Automata For Regular Expressions with Intersection

Extended regular expressions (with complement and intersection) are used in many applications due to their succinctness. In particular, regular expressions extended with intersection only (also called semi-extended) can already be exponentially smaller than standard regular expressions or equivalent nondeterministic finite automata (NFA). For practical purposes it is important to study the average behaviour of conversions between these models. In this paper, we focus on the conversion of regular expressions with intersection to nondeterministic finite automata, using partial derivatives and the notion of support. First, we give a tight upper bound of 2O(n) for the worst-case number of states of the resulting partial derivative automaton, where n is the size of the expression. Using the framework of analytic combinatorics, we then establish an upper bound of (1.056 + o(1))n for its asymptotic average-state complexity, which is significantly smaller than the one for the worst case. (c) IFIP International Federation for Information Processing 2016

Global Numerical Constraints on Trees

We introduce a logical foundation to reason on tree structures with constraints on the number of node occurrences. Related formalisms are limited to express occurrence constraints on particular tree regions, as for instance the children of a given node. By contrast, the logic introduced in the present work can concisely express numerical bounds on any region, descendants or ancestors for instance. We prove that the logic is decidable in single exponential time even if the numerical constraints are in binary form. We also illustrate the usage of the logic in the description of numerical constraints on multi-directional path queries on XML documents. Furthermore, numerical restrictions on regular languages (XML schemas) can also be concisely described by the logic. This implies a characterization of decidable counting extensions of XPath queries and XML schemas. Moreover, as the logic is closed under negation, it can thus be used as an optimal reasoning framework for testing emptiness, containment and equivalence

Extended Regular Expressions: Succinctness and Decidability

Most modern implementations of regular expression engines allow the use of variables (also called back references). The resulting extended regular expressions (which, in the literature, are also called practical regular expressions, rewbr, or regex) are able to express non-regular languages. The present paper demonstrates that extended regular-expressions cannot be minimized effectively (neither with respect to length, nor number of variables), and that the tradeoff in size between extended and ``classical\u27\u27 regular expressions is not bounded by any recursive function. In addition to this, we prove the undecidability of several decision problems (universality, equivalence, inclusion, regularity, and cofiniteness) for extended regular expressions. Furthermore, we show that all these results hold even if the extended regular expressions contain only a single variable

Extended regular expressions: succinctness and decidability

Most modern implementations of regular expression engines allow the use of variables (also called backreferences). The resulting extended regular expressions (which, in the literature, are also called practical regular expressions, rewbr, or regex) are able to express non-regular languages. The present paper demonstrates that extended regular-expressions cannot be minimized effectively (neither with respect to length, nor number of variables), and that the tradeoff in size between extended and "classical" regular expressions is not bounded by any recursive function. In addition to this, we prove the undecidability of several decision problems (universality, regularity, and cofiniteness) for extended regular expressions. Furthermore, we show that all these results hold even if the extended regular expressions contain only a single variable. © 2012 Springer Science+Business Media, LLC

On regular temporal logics with past

The IEEE standardized Property Specification Language, PSL for short, extends the well-known linear-time temporal logic LTL with so-called semi-extended regular expressions. PSL and the closely related SystemVerilog Assertions, SVA for short, are increasingly used in many phases of the hardware design cycle, from specification to verification. In this article, we extend the common core of these specification languages with past operators. We name this extension PPSL. Although all ω-regular properties are expressible in PSL, SVA, and PPSL, past operators often allow one to specify properties more naturally and concisely. In fact, we show that PPSL is exponentially more succinct than the cores of PSL and SVA. On the star-free properties, PPSL is double exponentially more succinct than LTL. Furthermore, we present a translation of PPSL into language-equivalent nondeterministic Büchi automata, which is based on novel constructions for 2-way alternating automata. The upper bound on the size of the resulting nondeterministic Büchi automata obtained by our translation is almost the same as the upper bound for the nondeterministic Büchi automata obtained from existing translations for PSL and SVA. Consequently, the satisfiability problem and the model-checking problem for PPSL fall into the same complexity classes as the corresponding problems for PSL and SV

From Finite Automata to Regular Expressions and Back--A Summary on Descriptional Complexity

The equivalence of finite automata and regular expressions dates back to the seminal paper of Kleene on events in nerve nets and finite automata from 1956. In the present paper we tour a fragment of the literature and summarize results on upper and lower bounds on the conversion of finite automata to regular expressions and vice versa. We also briefly recall the known bounds for the removal of spontaneous transitions (epsilon-transitions) on non-epsilon-free nondeterministic devices. Moreover, we report on recent results on the average case descriptional complexity bounds for the conversion of regular expressions to finite automata and brand new developments on the state elimination algorithm that converts finite automata to regular expressions.Comment: In Proceedings AFL 2014, arXiv:1405.527

Expressiveness and static analysis of extended conjunctive regular path queries

We study the expressiveness and the complexity of static analysis of extended conjunctive regular path queries (ECRPQs), introduced by Barceló et al. (PODS '10). ECRPQs are an extension of con- junctive regular path queries (CRPQs), a well-studied language for querying graph structured databases. Our first main result shows that query containment and equivalence of a CRPQ in an ECRPQ is undecidable. This settles one of the main open problems posed by Barceló et al. As a second main result, we prove a non-recursive succinctness gap between CRPQs and the CRPQ-expressible fragment of ECRPQs. Apart from this, we develop a tool for proving inexpressibility results for CRPQs and ECRPQs. In particular, this enables us to show that there exist queries definable by regular expressions with backreferencing, but not expressible by ECRPQs