Related papers: Logical Characterizations of Weighted Complexity C…
In a recent article, we introduced and studied a precise class of dynamical systems called solvable systems. These systems present a dynamic ruled by discontinuous ordinary differential equations with solvable right-hand terms and unique…
In this paper we provide purely model-theoretic (algebraic) characterisations for classes definable in second-order logic and for pseudo-elementary classes (including PC and PC_{\Delta} classes). Classical results of this flavour include…
We study the collection of first-order logical schemata all of whose instances are theorems of a given theory $T$; we call these the validities of $T$ ($\mathsf{V}(T)$). It is easy to see that if $T$ is a decidable theory, then…
Since the 1970s with the work of McNaughton, Papert and Sch\"utzenberger, a regular language is known to be definable in the first-order logic if and only if its syntactic monoid is aperiodic. This algebraic characterisation of a…
In automated complexity analysis, noninterference-based type systems statically guarantee, via soundness, the property that well-typed programs compute functions of a given complexity class, e.g., the class FP of functions computable in…
In this thesis we study inductive definitions over finite structures, particularly, the depth of inductive definitions. We also study infinitary finite variable logic which contains fixed-point logic and we introduce a new complexity…
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…
In this note, given a pair $(\mathfrak{g}, \lambda)$, where $\mathfrak{g}$ is a complex semisimple Lie algebra and $\lambda \in \mathfrak{h}^*$ is a dominant integral weight of $\mathfrak{g}$, where $\mathfrak{h} \subset \mathfrak{g}$ is…
We investigate different set-theoretic constructions in Residuated Logic based on Fitting's work on Intuitionistic Set Theory. We start by stating some results concerning constructible sets within valued models of Set Theory. We present two…
We develop a novel formal theory of finite structures, based on a view of finite structures as a fundamental artifact of computing and programming, forming a common platform for computing both within particular finite structures, and in the…
An explicit invariant-theoretic description of the moduli space $\mathcal{M}_3^1$ of degree-three rational maps on $\mathbb{P}^1$ is developed. A cubic map $\phi$ is represented, up to conjugation, by the pair of binary forms $(f, g) \in…
Parameterized complexity theory offers a framework for a refined analysis of hard algorithmic problems. Instead of expressing the running time of an algorithm as a function of the input size only, running times are expressed with respect to…
We propose logical characterizations of problems solvable in deterministic polylogarithmic time (PolylogTime) and polylogarithmic space (PolylogSpace). We introduce a novel two-sorted logic that separates the elements of the input domain…
NTP2 is a large class of first-order theories defined by Shelah and generalizing simple and NIP theories. Algebraic examples of NTP2 structures are given by ultra-products of p-adics and certain valued difference fields (such as a…
We show that a semi-classical quantum field theory comes with a versal family with the property that the corresponding partition function generates all path integrals and satisfies a system of 2nd order differential equations determined by…
In this paper, I discuss two logics for weighted finite structures: first-order logic with summation (FO(SUM)) and its recursive extension IFP(SUM). These logics originate from foundational work by Gr\"adel, Gurevich, and Meer in the 1990s.…
We recently described a formalism for reasoning with if-then rules that re expressed with different levels of firmness [18]. The formalism interprets these rules as extreme conditional probability statements, specifying orders of magnitude…
We prove that for any $\varepsilon>0$, a non-deterministic Turing machine $\mathcal{T}$ with time complexity $T(n)$ can be emulated by an $S$-machine with time and space complexities at most $T(n)^{1+\varepsilon}$ and $T(n)$, respectively.…
We give new proofs of soundness (all representable functions on base types lies in certain complexity classes) for Elementary Affine Logic, LFPL (a language for polytime computation close to realistic functional programming introduced by…
A theoretical development is carried to establish fundamental results about rank-initial embeddings and automorphisms of countable non-standard models of set theory, with a keen eye for their sets of fixed points. These results are then…