Sofic-Dyck shifts

We define the class of sofic-Dyck shifts which extends the class of Markov-Dyck shifts introduced by Inoue, Krieger and Matsumoto. Sofic-Dyck shifts are shifts of sequences whose finite factors form unambiguous context-free languages. We show that they correspond exactly to the class of shifts of sequences whose sets of factors are visibly pushdown languages. We give an expression of the zeta function of a sofic-Dyck shift

Recognizing well-parenthesized expressions in the streaming model

Motivated by a concrete problem and with the goal of understanding the sense in which the complexity of streaming algorithms is related to the complexity of formal languages, we investigate the problem Dyck(s) of checking matching parentheses, with $s$ different types of parenthesis. We present a one-pass randomized streaming algorithm for Dyck(2) with space \Order(\sqrt{n}\log n), time per letter \polylog (n), and one-sided error. We prove that this one-pass algorithm is optimal, up to a \polylog n factor, even when two-sided error is allowed. For the lower bound, we prove a direct sum result on hard instances by following the "information cost" approach, but with a few twists. Indeed, we play a subtle game between public and private coins. This mixture between public and private coins results from a balancing act between the direct sum result and a combinatorial lower bound for the base case. Surprisingly, the space requirement shrinks drastically if we have access to the input stream in reverse. We present a two-pass randomized streaming algorithm for Dyck(2) with space \Order((\log n)^2), time \polylog (n) and one-sided error, where the second pass is in the reverse direction. Both algorithms can be extended to Dyck(s) since this problem is reducible to Dyck(2) for a suitable notion of reduction in the streaming model.Comment: 20 pages, 5 figure

The descriptive complexity approach to LOGCFL

Building upon the known generalized-quantifier-based first-order characterization of LOGCFL, we lay the groundwork for a deeper investigation. Specifically, we examine subclasses of LOGCFL arising from varying the arity and nesting of groupoidal quantifiers. Our work extends the elaborate theory relating monoidal quantifiers to NC1 and its subclasses. In the absence of the BIT predicate, we resolve the main issues: we show in particular that no single outermost unary groupoidal quantifier with FO can capture all the context-free languages, and we obtain the surprising result that a variant of Greibach's ``hardest context-free language'' is LOGCFL-complete under quantifier-free BIT-free projections. We then prove that FO with unary groupoidal quantifiers is strictly more expressive with the BIT predicate than without. Considering a particular groupoidal quantifier, we prove that first-order logic with majority of pairs is strictly more expressive than first-order with majority of individuals. As a technical tool of independent interest, we define the notion of an aperiodic nondeterministic finite automaton and prove that FO translations are precisely the mappings computed by single-valued aperiodic nondeterministic finite transducers.Comment: 10 pages, 1 figur

Improved Streaming Algorithm for Dyck(s) Recognition

Keeping in mind, that any context free language can be mapped to a subset of Dyck languages and by seeing various database applications of Dyck, mainly verifying the well-formedness of XML file, we study the randomized streaming algorithms for the recognition of Dyck(s) languages, with s different types of parenthesis. The main motivation of this work is well known space bound for any T-pass streaming algorithm is (√n/T). Let x be the input stream of length n with maximum height hmax. Here we present a single-pass randomized streaming algorithms to decide the membership of x in Dyck(s) using Counting Bloomfilter (CBF) with space O (hmax) bits, ploylog(n) time per letter with two-sided error probability. Two-sided error is because of the false negative and false positives of counting bloomfilter. This algorithms denies the necessity of streaming reduction of Dyck(s) into Dyck(2), that reduces the space even further by the factor of O (log s), compared to those uses streaming reduction. We also present an improved single-pass randomized streaming algorithm for recognizing Dyck(2) with space O (√n) bits, which is the proven lower bound. Time bound is same polylog(n), as other existing algorithms and error is one-sided. In this algorithm, we extended the existing approach of periodically compressing stack information. Existing approach uses two stacks and a linear hash function, instead of this we are using three stacks and same linear hash function to achieve space lower bound of O (√n). We also present another single-pass streaming algorithm with O (hmax) space that uses counting bloomfilter and directly acts on Dyck(s

Low-Latency Sliding Window Algorithms for Formal Languages

Low-latency sliding window algorithms for regular and context-free languages are studied, where latency refers to the worst-case time spent for a single window update or query. For every regular language $L$ it is shown that there exists a constant-latency solution that supports adding and removing symbols independently on both ends of the window (the so-called two-way variable-size model). We prove that this result extends to all visibly pushdown languages. For deterministic 1-counter languages we present a $\mathcal{O}(\log n)$ latency sliding window algorithm for the two-way variable-size model where $n$ refers to the window size. We complement these results with a conditional lower bound: there exists a fixed real-time deterministic context-free language $L$ such that, assuming the OMV (online matrix vector multiplication) conjecture, there is no sliding window algorithm for $L$ with latency $n^{1/2-\epsilon}$ for any $\epsilon>0$, even in the most restricted sliding window model (one-way fixed-size model). The above mentioned results all refer to the unit-cost RAM model with logarithmic word size. For regular languages we also present a refined picture using word sizes $\mathcal{O}(1)$, $\mathcal{O}(\log\log n)$, and $\mathcal{O}(\log n)$.Comment: A short version will be presented at the conference FSTTCS 202