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We develop some basic results about full amalgamation classes with intrinsic trascendentals. These classes have generics whose models may have finite subsets whose intrinsic closure is not contained in its algebraic closure. We will show…

The polynomial multiplication problem has attracted considerable attention since the early days of computer algebra, and several algorithms have been designed to achieve the best possible time complexity. More recently, efforts have been…

For sets $A, B\subset \mathbb N$, their sumset is $A + B := \{a+b: a\in A, b\in B\}$. If we cannot write a set $C$ as $C = A+B$ with $|A|, |B|\geq 2$, then we say that $C$ is $\textit{irreducible}$. The question of whether a given set $C$…

We introduce a flexible class of well-quasi-orderings (WQOs) on words that generalizes the ordering of (not necessarily contiguous) subwords. Each such WQO induces a class of piecewise testable languages (PTLs) as Boolean combinations of…

Classical complexity theory measures the cost of computing a function, but many computational tasks require committing to one valid output among several. We introduce determination depth -- the minimum number of sequential layers of…

Continuing the study of complexity theory of Koepke's Ordinal Turing Machines (OTMs) that was started by Rin, L\"owe and the author, we prove the following results: (1) An analogue of Ladner's theorem for OTMs holds: That is, there are…

We investigate replicable learning algorithms. Ideally, we would like to design algorithms that output the same canonical model over multiple runs, even when different runs observe a different set of samples from the unknown data…

Computability on uncountable sets has no standard formalization, unlike that on countable sets, which is given by Turing machines. Some of the approaches to define computability in these sets rely on order-theoretic structures to translate…

Attribute grammars (AGs) are a formal technique for defining semantics of programming languages. Existing complexity proofs on the circularity problem of AGs are based on automata theory, such as writing pushdown acceptor and alternating…

In 1982, Courcelle and Franchi-Zannettacci showed that the equivalence problem of separated non-nested attribute systems can be reduced to the equivalence problem of total deterministic separated basic macro tree transducers. They also gave…

Constructor rewriting systems are said to be cons-free if any constructor term occurring in the rhs of a rule must be a subterm of the lhs of the rule. Roughly, such systems cannot build new data structures during their evaluation. In…

There is an important and interesting open question in computational complexity on the relation between the complexity classes $\mathcal{NP}$ and $\mathcal{PSPACE}$. It is a widespread belief that $\mathcal{NP}\ne\mathcal{PSPACE}$. In this…

Constructor rewriting systems are said to be cons-free if, roughly, constructor terms in the right-hand sides of rules are subterms of constructor terms in the left-hand side; the computational intuition is that rules cannot build new data…

It is known that for any class C closed under union and intersection, the Boolean closure of C, the Boolean hierarchy over C, and the symmetric difference hierarchy over C all are equal. We prove that these equalities hold for any…

A bisimulation for a coalgebra of a functor on the category of sets can be described via a coalgebra in the category of relations, of a lifted functor. A final coalgebra then gives rise to the coinduction principle, which states that two…

We provide efficient replicable algorithms for the problem of learning large-margin halfspaces. Our results improve upon the algorithms provided by Impagliazzo, Lei, Pitassi, and Sorrell [STOC, 2022]. We design the first…

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…

The relationship between the complexity classes P and NP is an unsolved question in the field of theoretical computer science. In this paper, we look at the link between the P - NP question and the "Deterministic" versus "Non Deterministic"…

We show that the common theory of all modules over a tubular algebra (over a recursive algebraically closed field) is decidable. This result supports a long standing conjecture of Mike Prest which says that a finite-dimensional algebra…

In this paper we investigate the complexity of abduction, a fundamental and important form of non-monotonic reasoning. Given a knowledge base explaining the world's behavior it aims at finding an explanation for some observed manifestation.…