Related papers: The complexity of classification problems for mode…
We assess the descriptive complexity of *bisimilarity* or "equality of behavior" on a family of Markov decision processes over uncountable standard Borel spaces, namely *nondeterministic labelled Markov processes* (NLMP). We show that…
Ong has shown that the modal mu-calculus model checking problem (equivalently, the alternating parity tree automaton (APT) acceptance problem) of possibly-infinite ranked trees generated by order-n recursion schemes is n-EXPTIME complete.…
Recurrence properties of systems and associated sets of integers that suffice for recurrence are classical objects in topological dynamics. We describe relations between recurrence in different sorts of systems, study ways to formulate…
Theories of classification distinguish classes with some good structure theorem from those for which none is possible. Some classes (dense linear orders, for instance) are non-classifiable in general, but are classifiable when we consider…
computable functions are defined by abstract finite deterministic algorithms on many-sorted algebras. We show that there exist finite universal algebraic specifications that specify uniquely (up to isomorphism) (i) all abstract computable…
A Borel equivalence relation on a Polish space is said to be countable if all of its equivalence classes are countable. Standard examples of countable Borel equivalence relations (on the space of subsets of the integers) that occur in…
What is computable with limited resources? How can we verify the correctness of computations? How to measure computational power with precision? Despite the immense scientific and engineering progress in computing, we still have only…
In representation theory, the problem of classifying pairs of matrices up to simultaneous similarity is used as a measure of complexity; classification problems containing it are called wild problems. We show in an explicit form that this…
This article presents a general solution to the problem of computational complexity. First, it gives a historical introduction to the problem since the revival of the foundational problems of mathematics at the end of the 19th century.…
We classify essential algebras whose irredundant non-refinable covers consist of primal algebras. The proof is obtained by constructing one to one correspondence between such algebras and partial orders on finite sets. Further, we prove…
We investigate enumerability properties for classes of sets which permit recursive, lexicographically increasing approximations, or left-r.e. sets. In addition to pinpointing the complexity of left-r.e. Martin-L\"{o}f, computably, Schnorr,…
Consider a multi-class labelling problem, where the labels can take values in $[k]$, and a predictor predicts a distribution over the labels. In this work, we study the following foundational question: Are there notions of multi-class…
A new syntactic characterization of problems complete via Turing reductions is presented. General canonical forms are developed in order to define such problems. One of these forms allows us to define complete problems on ordered…
This work connects two mathematical fields - computational complexity and interval linear algebra. It introduces the basic topics of interval linear algebra - regularity and singularity, full column rank, solving a linear system, deciding…
Completion problems, of recovering a point from a set of observed coordinates, are abundant in applications to image reconstruction, phylogenetics, and data science. We consider a completion problem coming from algebraic statistics: to…
We reduce the isomorphism problem for undirected graphs without loops to the isomorphism problems for a class of finite dimensional $2$-step nilpotent Lie algebras over a field and for a class of finite $p$-groups. We show that the…
In the paper we define three new complexity classes for Turing Machine undecidable problems inspired by the famous Cook/Levin's NP-complete complexity class for intractable problems. These are U-complete (Universal complete), D-complete…
We examine the computational complexity of testing and finding small plans in probabilistic planning domains with both flat and propositional representations. The complexity of plan evaluation and existence varies with the plan type sought;…
Automated theorem provers (ATPs) can disprove conjectures by saturating a set of clauses, but the resulting saturated sets are opaque certificates. In the unit equational fragment, a saturated set can in fact be read as a convergent rewrite…
Larrauri and \v{Z}ivn\'y [ICALP'25/ACM ToCL'24] recently established a complete complexity classification of the problem of solving a system of equations over a monoid $N$ assuming that a solution exists over a monoid $M$, where both…