arXiv subjects

Antoni Puch

Publications and source records attributed to Antoni Puch.

3 recordsLinked to original sources

arXiv2026

Reachability in VASS Extended with Integer Counters

We consider a variant of VASS extended with integer counters, denoted VASS+Z. These are automata equipped with N and Z counters; the N-counters are required to remain nonnegative and the Z-counters do not have this restriction. We study the complexity of the reachability problem for VASS+Z when the number of N-counters is fixed. We show that reachability is NP-complete in 1-VASS+Z (i.e. when there is only one N-counter) regardless of unary or binary encoding. For $d \geq 2$, using a KLMST-based algorithm, we prove that reachability in d-VASS+Z lies in the complexity class $\mathcal{F}_{d+2}$. Our upper bound improves on the naively obtained Ackermannian complexity by simulating the Z-counters with N-counters. To complement our upper bounds, we show that extending VASS with integer counters significantly lowers the number of N-counters needed to exhibit hardness. We prove that reachability in unary 2-VASS+Z is PSPACE-hard; without Z-counters this lower bound is only known in dimension 5. We also prove that reachability in unary 3-VASS+Z is TOWER-hard. Without Z-counters, reachability in 3-VASS has elementary complexity and TOWER-hardness is only known in dimension 8.

cs.FL↗

arXiv2025

Pumping-Like Results for Copyless Cost Register Automata and Polynomially Ambiguous Weighted Automata

In this work we consider two rich subclasses of weighted automata over fields: polynomially ambiguous weighted automata and copyless cost register automata. Primarily we are interested in understanding their expressiveness power. Over the field of rationals and $1$-letter alphabets, it is known that the two classes coincide; they are equivalent to linear recurrence sequences (LRS) whose exponential bases are roots of rationals. We develop a tool we call Pumping Sequence Families, which, by exploiting the simple single-letter behaviour of the models, yields two pumping-like results over arbitrary fields with unrestricted alphabets, one for each class. As a corollary of these results, we present examples proving that the two classes become incomparable over the field of rationals with unrestricted alphabets. We complement the results by analysing the zeroness and equivalence problems. For weighted automata (even unrestricted) these problems are well understood: there are polynomial time, and even NC$^2$ algorithms. For copyless cost register automata we show that the two problems are \textsc{PSpace}-complete, where the difficulty is to show the lower bound.

cs.FL↗

arXiv2024

Factoring through monomial representations: arithmetic characterizations and ambiguity of weighted automata

We characterize group representations that factor through monomial representations, respectively, block-triangular representations with monomial diagonal blocks, by arithmetic properties. Similar results are obtained for semigroup representations by invertible transformations. The characterizations use results on unit equations from Diophantine number theory (by Evertse, van der Poorten, and Schlickewei in characteristic zero, and by Derksen and Masser in positive characteristic). Specialized to finitely generated groups in characteristic zero, one of our main theorems recovers an improvement of a very recent similar characterization by Corvaja, Demeio, Rapinchuk, Ren, and Zannier that was motivated by the study of the bounded generation (BG) property. In positive characteristic, we get a characterization of linear BG groups, recovering a theorem of Ab\'ert, Lubotzky, and Pyber from 2003. Our motivation comes from weighted finite automata (WFA) over a field. For invertible WFA we show that $M$-ambiguity, finite ambiguity, and polynomial ambiguity are characterized by arithmetic properties. We discover a full correspondence between arithmetic properties and a complexity hierarchy for WFA based on ambiguity. In the invertible case, this is a far-reaching generalization of a recent result by Bell and the second author, characterizing unambiguous WFA, that resolved a 1979 conjecture of Reutenauer. As a consequence, using the computability of the (linear) Zariski closure of a finitely generated matrix semigroup, the $M$-ambiguity, finite ambiguity, and polynomial ambiguity properties are algorithmically decidable for invertible WFA.

math.GR↗