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Amaldev Manuel

Publications and source records attributed to Amaldev Manuel.

7 recordsLinked to original sources
arXiv2026

Set Automata and Limits of Decidability of Two-Variable Logic on Data Words

We extend the two-variable logic on data words with guarded regular binary predicates of the form $\widetilde{L}(x,y)$ that is true if positions $x$ and $y$ are in the same class and the factor strictly between $x$ and $y$ is in the regular language $L$. We characterise the class of monoids for which the extension of the two-variable logic with guarded predicates recognised by the monoid is decidable, namely the class of idempotent monoids whose two-sided ideals are linearly ordered. For this, we introduce an automata formalism, set automata, that is equivalent to the class automata of Boja\'nczyk and Lasota and thus has an undecidable emptiness problem. We identify a subclass of set automata called ordered quasi-normal set automata that has a decidable emptiness problem by reduction to the emptiness problem of ordered multicounter automata. We show that the two-variable logic extended with guarded regular predicates recognised by a semigroup $S$ is expressively equivalent to a quasi-normal set automaton with the semigroup of transformations $S$. In particular, if $S$ is a linear band monoid then the resulting automaton is ordered, and the decidability result follows.

cs.LO↗
arXiv2024

Edit Distance of Finite State Transducers

We lift metrics over words to metrics over word-to-word transductions, by defining the distance between two transductions as the supremum of the distances of their respective outputs over all inputs. This allows to compare transducers beyond equivalence. Two transducers are close (resp. $k$-close) with respect to a metric if their distance is finite (resp. at most $k$). Over integer-valued metrics computing the distance between transducers is equivalent to deciding the closeness and $k$-closeness problems. For common integer-valued edit distances such as, Hamming, transposition, conjugacy and Levenshtein family of distances, we show that the closeness and the $k$-closeness problems are decidable for functional transducers. Hence, the distance with respect to these metrics is also computable. Finally, we relate the notion of distance between functions to the notions of diameter of a relation and index of a relation in another. We show that computing edit distance between functional transducers is equivalent to computing diameter of a rational relation and both are a specific instance of the index problem of rational relations.

cs.FL↗
arXiv2023

Deciding Conjugacy of a Rational Relation

The study of rational relations is fundamental to the study of formal languages and automata theory. A rational relation is conjugate if each pair of words in the relation is conjugate (or cyclic shifts of each other). The notion of conjugacy has been central in addressing many important algorithmic questions about rational relations. We address the problem of checking whether a rational relation is conjugate and show that it is decidable. Towards our decision procedure, we establish a new result that is of independent interest to word combinatorics. We identify a necessary and sufficient condition for the set of pairs given by $(a_0,b_0) G_1^* (a_1,b_1) \cdots G_k^*(a_k,b_k), k \geq 0$ to be conjugate, where $G_i$ is a (not necessarily rational) conjugate relation and $a_i, b_i$ are arbitrary words. This is similar to, and a nontrivial generalisation of, a characterisation given by Lyndon and Sch\"utzenberger in 1962 for the conjugacy of a pair of words. Furthermore, our condition can be evaluated in polynomial time, yielding a PTIME procedure for deciding the conjugacy of a rational relation given as a sumfree expression. Since any arbitrary rational expression can be expressed as a sum of sumfree expressions (with an exponential blow-up), decidability of conjugacy of rational relations follows.

cs.FL↗
arXiv2021

An Algebraic Characterisation of First-Order Logic with Neighbour

We give an algebraic characterisation of first-order logic with the neighbour relation, on finite words. For this, we consider languages of finite words over alphabets with an involution on them. The natural algebras for such languages are involution semigroups. To characterise the logic, we define a special kind of semidirect product of involution semigroups, called the locally hermitian product. The characterisation theorem for FO with neighbour states that a language is definable in the logic if and only if it is recognised by a locally hermitian product of an aperiodic commutative involution semigroup, and a locally trivial involution semigroup. We then define the notion of involution varieties of languages, namely classes of languages closed under Boolean operations, quotients, involution, and inverse images of involutory morphisms. An Eilenberg-type correspondence is established between involution varieties of languages and pseudovarieties of involution semigroups.

cs.LO↗
arXiv2019

Logics for Reversible Regular Languages and Semigroups with Involution

We present MSO and FO logics with predicates `between' and `neighbour' that characterise various fragments of the class of regular languages that are closed under the reverse operation. The standard connections that exist between MSO and FO logics and varieties of finite semigroups extend to this setting with semigroups extended with an involution. The case is different for FO with neighbour relation where we show that one needs additional equations to characterise the class.

cs.FL↗
arXiv2014

$μ$-calculus on data words

We study the decidability and expressiveness issues of $μ$-calculus on data words and data $ω$-words. It is shown that the full logic as well as the fragment which uses only the least fixpoints are undecidable, while the fragment containing only greatest fixpoints is decidable. Two subclasses, namely BMA and BR, obtained by limiting the compositions of formulas and their automata characterizations are exhibited. Furthermore, Data-LTL and two-variable first-order logic are expressed as unary alternation-free fragment of BMA. Finally basic inclusions of the fragments are discussed.

cs.LO↗