Regular expressions can be extended with lookarounds for contextual matching. This paper discusses a Coq formalization of the theory of regular expressions with lookarounds. We provide an efficient and purely functional algorithm for matching expressions with lookarounds and verify its correctness. The algorithm runs in time linear in both the size of the regular expression as well as the input string. Our experimental results provide empirical support to our complexity analysis. To the best of our knowledge, this is the first formalization of a linear-time matching algorithm for regular expressions with lookarounds.

Regular expressions have been extended with lookaround assertions, which are subdivided into lookahead and lookbehind assertions. These constructs are used to refine when a match for a pattern occurs in the input text based on the surrounding context. Current implementation techniques for lookaround involve backtracking search, which can give rise to running time that is super-linear in the length of input text. In this paper, we first consider a formal mathematical semantics for lookaround, which complements the commonly used operational understanding of lookaround in terms of a backtracking implementation. Our formal semantics allows us to establish several equational properties for simplifying lookaround assertions. Additionally, we propose a new algorithm for matching regular expressions with lookaround that has time complexity O m ⋅ n , where m is the size of the regular expression and n is the length of the input text. The algorithm works by evaluating lookaround assertions in a bottom-up manner. Our algorithm makes use of a new notion of nondeterministic finite automata (NFAs), which we call oracle-NFAs. These automata are augmented with epsilon-transitions that are guarded by oracle queries that provide the truth values of lookaround assertions at every position in the text. We provide an implementation of our algorithm that incorporates three performance optimizations for reducing the work performed and memory used. We present an experimental comparison against PCRE and Java’s regex library, which are state-of-the-art regex engines that support lookaround assertions. Our experimental results show that, in contrast to PCRE and Java, our implementation does not suffer from super-linear running time and is several times faster.

Regular expressions are commonly used for finding and extracting matches from sequence data. Due to the inherent ambiguity of regular expressions, a disambiguation policy must be considered for the match extraction problem, in order to uniquely determine the desired match out of the possibly many matches. The most common disambiguation policies are the POSIX policy and the greedy (PCRE) policy. The POSIX policy chooses the longest match out of the leftmost ones. The greedy policy chooses a leftmost match and further disambiguates using a greedy interpretation of Kleene iteration to match as many times as possible. The choice of disambiguation policy can affect the output of match extraction, which can be an issue for reusing regular expressions across regex engines. In this paper, we introduce and study the notion of disambiguation robustness for regular expressions. A regular expression is robust if its extraction semantics is indifferent to whether the POSIX or greedy disambiguation policy is chosen. This gives rise to a decision problem for regular expressions, which we prove to be PSPACE-complete. We propose a static analysis algorithm for checking the (non-)robustness of regular expressions and two performance optimizations. We have implemented the proposed algorithms and we have shown experimentally that they are practical for analyzing large datasets of regular expressions derived from various application domains.

Regular pattern matching is used in numerous application domains, including text processing, bioinformatics, and network security. Patterns are typically expressed with an extended syntax of regular expressions. This syntax includes the computationally challenging construct of bounded repetition or counting, which describes the repetition of a pattern a fixed number of times. We develop a specialized in-memory hardware architecture that integrates counter and bit vector modules into a state-of-the-art in-memory NFA accelerator. The design is inspired by the theoretical model of nondeterministic counter automata (NCA). A key feature of our approach is that we statically analyze regular expressions to determine bounds on the amount of memory needed for the occurrences of bounded repetition. The results of this analysis are used by a regex-to-hardware compiler in order to make an appropriate selection of counter or bit vector modules. We evaluate our hardware implementation using a simulator based on circuit parameters collected by SPICE simulation in TSMC 28nm CMOS process. We find that the use of counter and bit vector modules outperforms unfolding solutions by orders of magnitude. Experiments concerning realistic workloads show up to 76% energy reduction and 58% area reduction in comparison to CAMA, a recently proposed in-memory NFA accelerator.

Abstract We investigate efficient algorithms for the online monitoring of properties written in metric temporal logic (MTL). We employ an abstract algebraic semantics based on semirings. It encompasses the Boolean semantics and a quantitative semantics capturing the robustness of satisfaction, which is based on the max-min semiring over the extended real numbers. We provide a precise equational characterization of the class of semirings for which our semantics can be viewed as an approximation to an alternative semantics that quantifies the distance of a system trace from the set of all traces that satisfy the desired property.

We present pumping lemmas for five classes of functions definable by fragments of weighted automata over the min-plus semiring, the max-plus semiring and the semiring of natural numbers. As a corollary we show that the hierarchy of functions definable by unambiguous, finitely-ambiguous, polynomially-ambiguous weighted automata, and the full class of weighted automata is strict for the min-plus and max-plus semirings.

Watermarking is a way of embedding information in digital documents. Much research has been done on techniques for watermarking relational databases and XML documents, where the process of embedding information shouldn't distort query outputs too much. Recently, techniques have been proposed to watermark some classes of relational structures preserving first-order and monadic second order queries. For relational structures whose Gaifman graphs have bounded degree, watermarking can be done preserving first-order queries. We extend this line of work and study watermarking schemes for other classes of structures. We prove that for relational structures whose Gaifman graphs belong to a class of graphs that have locally bounded tree-width and is closed under minors, watermarking schemes exist that preserve first-order queries. We use previously known properties of logical formulas and graphs, and build on them with some technical work to make them work in our context. This constitutes a part of the first steps to understand the extent to which techniques from algorithm design and computational learning theory can be adapted for watermarking.