On a temporal logic of prefixes and infixes

A classic result by Stockmeyer [16] gives a non-elementary lower bound to the emptiness problem for star-free generalized regular expressions. This result is intimately connected to the satisfiability problem for interval temporal logic, notably for formulas that make use of the so-called chop operator. Such an operator can indeed be interpreted as the inverse of the concatenation operation on regular languages, and this correspondence enables reductions between non-emptiness of star-free generalized regular expressions and satisfiability of formulas of the interval temporal logic of the chop operator under the homogeneity assumption [5]. In this paper, we study the complexity of the satisfiability problem for a suitable weakening of the chop interval temporal logic, that can be equivalently viewed as a fragment of Halpern and Shoham interval logic featuring the operators B, for \u201cbegins\u201d, corresponding to the prefix relation on pairs of intervals, and D, for \u201cduring\u201d, corresponding to the infix relation. The homogeneous models of the considered logic naturally correspond to languages defined by restricted forms of regular expressions, that use union, complementation, and the inverses of the prefix and infix relations

The Quantifier Alternation Hierarchy of Synchronous Relations

The class of synchronous relations, also known as automatic or regular, is one of the most studied subclasses of rational relations. It enjoys many desirable closure properties and is known to be logically characterized: the synchronous relations are exactly those that are defined by a first-order formula on the structure of all finite words, with the prefix, equal-length and last-letter predicates. Here, we study the quantifier alternation hierarchy of this logic. We show that it collapses at level Sigma_3 and that all levels below admit decidable characterizations. Our results reveal the connections between this hierarchy and the well-known hierarchy of first-order defined languages of finite words

Decision Problems For Convex Languages

In this paper we examine decision problems associated with various classes of convex languages, studied by Ang and Brzozowski (under the name "continuous languages"). We show that we can decide whether a given language L is prefix-, suffix-, factor-, or subword-convex in polynomial time if L is represented by a DFA, but that the problem is PSPACE-hard if L is represented by an NFA. In the case that a regular language is not convex, we prove tight upper bounds on the length of the shortest words demonstrating this fact, in terms of the number of states of an accepting DFA. Similar results are proved for some subclasses of convex languages: the prefix-, suffix-, factor-, and subword-closed languages, and the prefix-, suffix-, factor-, and subword-free languages.Comment: preliminary version. This version corrected one typo in Section 2.1.1, line

Comparator automata in quantitative verification

The notion of comparison between system runs is fundamental in formal verification. This concept is implicitly present in the verification of qualitative systems, and is more pronounced in the verification of quantitative systems. In this work, we identify a novel mode of comparison in quantitative systems: the online comparison of the aggregate values of two sequences of quantitative weights. This notion is embodied by {\em comparator automata} ({\em comparators}, in short), a new class of automata that read two infinite sequences of weights synchronously and relate their aggregate values. We show that {aggregate functions} that can be represented with B\"uchi automaton result in comparators that are finite-state and accept by the B\"uchi condition as well. Such {\em $\omega$-regular comparators} further lead to generic algorithms for a number of well-studied problems, including the quantitative inclusion and winning strategies in quantitative graph games with incomplete information, as well as related non-decision problems, such as obtaining a finite representation of all counterexamples in the quantitative inclusion problem. We study comparators for two aggregate functions: discounted-sum and limit-average. We prove that the discounted-sum comparator is $\omega$-regular iff the discount-factor is an integer. Not every aggregate function, however, has an $\omega$-regular comparator. Specifically, we show that the language of sequence-pairs for which limit-average aggregates exist is neither $\omega$-regular nor $\omega$-context-free. Given this result, we introduce the notion of {\em prefix-average} as a relaxation of limit-average aggregation, and show that it admits $\omega$-context-free comparators

Morphological word structure in English and Swedish : the evidence from prosody

Trubetzkoy's recognition of a delimitative function of phonology, serving to signal boundaries between morphological units, is expressed in terms of alignment constraints in Optimality Theory, where the relevant constraints require specific morphological boundaries to coincide with phonological structure (Trubetzkoy 1936, 1939, McCarthy & Prince 1993). The approach pursued in the present article is to investigate the distribution of phonological boundary signals to gain insight into the criteria underlying morphological analysis. The evidence from English and Swedish suggests that necessary and sufficient conditions for word-internal morphological analysis concern the recognizability of head constituents, which include the rightmost members of compounds and head affixes. The claim is that the stability of word-internal boundary effects in historical perspective cannot in general be sufficiently explained in terms of memorization and imitation of phonological word form. Rather, these effects indicate a morphological parsing mechanism based on the recognition of word-internal head constituents. Head affixes can be shown to contrast systematically with modifying affixes with respect to syntactic function, semantic content, and prosodic properties. That is, head affixes, which cannot be omitted, often lack inherent meaning and have relatively unmarked boundaries, which can be obscured entirely under specific phonological conditions. By contrast, modifying affixes, which can be omitted, consistently have inherent meaning and have stronger boundaries, which resist prosodic fusion in all phonological contexts. While these correlations are hardly specific to English and Swedish it remains to be investigated to which extent they hold cross-linguistically. The observation that some of the constituents identified on the basis of prosodic evidence lack inherent meaning raises the issue of compositionality. I will argue that certain systematic aspects of word meaning cannot be captured with reference to the syntagmatic level, but require reference to the paradigmatic level instead. The assumption is then that there are two dimensions of morphological analysis: syntagmatic analysis, which centers on the criteria for decomposing words in terms of labelled constituents, and paradigmatic analysis, which centers on the criteria for establishing relations among (whole) words in the mental lexicon. While meaning is intrinsically connected with paradigmatic analysis (e.g. base relations, oppositeness) it is not essential to syntagmatic analysis

A new approach to the 22-regularity of the ℓ\ell-abelian complexity of 22-automatic sequences

We prove that a sequence satisfying a certain symmetry property is $2$-regular in the sense of Allouche and Shallit, i.e., the $\mathbb{Z}$-module generated by its $2$-kernel is finitely generated. We apply this theorem to develop a general approach for studying the $\ell$-abelian complexity of $2$-automatic sequences. In particular, we prove that the period-doubling word and the Thue--Morse word have $2$-abelian complexity sequences that are $2$-regular. Along the way, we also prove that the $2$-block codings of these two words have $1$-abelian complexity sequences that are $2$-regular.Comment: 44 pages, 2 figures; publication versio