Inclusion Diagrams for Classes of Deterministic Bottom-up Tree-to-Tree-Series Transformations

In this paper we investigate the relationship between classes of tree-to-tree-series (for short: t-ts) and o-tree-to-tree-series (for short: o-t-ts) transformations computed by restricted deterministic bottom-up weighted tree transducers (for short: deterministic bu-w-tt). Essentially, deterministic bu-w-tt are deterministic bottom-up tree series transducers [EFV02, FV03, ful, FGV04], but the former are de ned over monoids whereas the latter are de ned over semirings and only use the multiplicative monoid thereof. In particular, the common restrictions of non-deletion, linearity, totality, and homomorphism [Eng75] can equivalently be de ned for deterministic bu-w-tt. Using well-known results of classical tree transducer theory (cf., e.g., [Eng75, Fül91]) and also new results on deterministic bu-w-tt, we order classes of t-ts and o-t-ts transformations computed by restricted deterministic bu-w-tt by set inclusion. More precisely, for every commutative monoid we completely specify the inclusion relation of the classes of t-ts and o-t-ts transformations for all sensible combinations of restrictions by means of inclusion diagrams

Hierarchies of tree series transformations

AbstractWe study bottom-up and top-down tree series transducers over a semiring A and denote the tree series transformation classes computed by them by BOTt−ts(A) and TOPt−ts(A), respectively. We present the inclusion diagram of the classes p-BOTt−tsn(A), p-TOPt−tsn(A), p-BOTt−tsn+1(A), and p-TOPt−tsn+1(A) and prove its correctness, where A is a commutative izz-semiring (izz=idempotent, zero-divisor free, and zero-sum free) and the prefix p stands for polynomial. This inclusion diagram implies the properness of the following four hierarchies: p-TOPt−ts(A)⊆p-TOPt−ts2(A)⊆p-TOPt−ts3(A)⊆⋯,p-BOTt−ts(A)⊆p-BOTt−ts2(A)⊆p-BOTt−ts3(A)⊆⋯,p-TOPt−ts(A)⊆p-BOTt−ts2(A)⊆p-TOPt−ts3(A)⊆p-BOTt−ts4(A)⊆⋯,p-BOTt−ts(A)⊆p-TOPt−ts2(A)⊆p-BOTt−ts3(A)⊆p-TOPt−ts4(A)⊆⋯,where the first hierarchy generalizes the famous top-down tree transformation hierarchy of Engelfriet (Math. Systems Theory 15 (1982) 95–125). As the second main result we prove that the first two hierarchies are proper even for arbitrary (i.e., not necessarily commutative) izz-semirings

A counterexample to Thiagarajan's conjecture on regular event structures

We provide a counterexample to a conjecture by Thiagarajan (1996 and 2002) that regular event structures correspond exactly to event structures obtained as unfoldings of finite 1-safe Petri nets. The same counterexample is used to disprove a closely related conjecture by Badouel, Darondeau, and Raoult (1999) that domains of regular event structures with bounded $\natural$-cliques are recognizable by finite trace automata. Event structures, trace automata, and Petri nets are fundamental models in concurrency theory. There exist nice interpretations of these structures as combinatorial and geometric objects. Namely, from a graph theoretical point of view, the domains of prime event structures correspond exactly to median graphs; from a geometric point of view, these domains are in bijection with CAT(0) cube complexes. A necessary condition for both conjectures to be true is that domains of regular event structures (with bounded $\natural$-cliques) admit a regular nice labeling. To disprove these conjectures, we describe a regular event domain (with bounded $\natural$-cliques) that does not admit a regular nice labeling. Our counterexample is derived from an example by Wise (1996 and 2007) of a nonpositively curved square complex whose universal cover is a CAT(0) square complex containing a particular plane with an aperiodic tiling. We prove that other counterexamples to Thiagarajan's conjecture arise from aperiodic 4-way deterministic tile sets of Kari and Papasoglu (1999) and Lukkarila (2009). On the positive side, using breakthrough results by Agol (2013) and Haglund and Wise (2008, 2012) from geometric group theory, we prove that Thiagarajan's conjecture is true for regular event structures whose domains occur as principal filters of hyperbolic CAT(0) cube complexes which are universal covers of finite nonpositively curved cube complexes

Game Comonads & Generalised Quantifiers

Game comonads, introduced by Abramsky, Dawar and Wang and developed by Abramsky and Shah, give an interesting categorical semantics to some Spoiler-Duplicator games that are common in finite model theory. In particular they expose connections between one-sided and two-sided games, and parameters such as treewidth and treedepth and corresponding notions of decomposition. In the present paper, we expand the realm of game comonads to logics with generalised quantifiers. In particular, we introduce a comonad graded by two parameter $n \leq k$ such that isomorphisms in the resulting Kleisli category are exactly Duplicator winning strategies in Hella's $n$-bijection game with $k$ pebbles. We define a one-sided version of this game which allows us to provide a categorical semantics for a number of logics with generalised quantifiers. We also give a novel notion of tree decomposition that emerges from the construction

Multioperator Weighted Monadic Datalog

In this thesis we will introduce multioperator weighted monadic datalog (mwmd), a formal model for specifying tree series, tree transformations, and tree languages. This model combines aspects of multioperator weighted tree automata (wmta), weighted monadic datalog (wmd), and monadic datalog tree transducers (mdtt). In order to develop a rich theory we will define multiple versions of semantics for mwmd and compare their expressiveness. We will study normal forms and decidability results of mwmd and show (by employing particular semantic domains) that the theory of mwmd subsumes the theory of both wmd and mdtt. We conclude this thesis by showing that mwmd even contain wmta as a syntactic subclass and present results concerning this subclass

The momentum amplituhedron

Diese Dissertation befasst sich mit einigen der jüngeren theoretischen Entwicklungen auf dem Gebiet der Streuamplituden. In den letzten Jahren wurde immer mehr der traditionelle Ansatz der Extraktion von Streuamplituden aus Feynman-Diagrammen zugunsten von Techniken, die als On-Shell-Methoden bekannt sind, aufgegeben. Diese Methoden offenbaren eine interessante Beziehung zwischen Streuamplituden und einer Geometrie, die als positive Grassmannsche Geometrie bekannt ist und zu einer radikalen Neuformulierung von Streuamplituden durch so genannte positiven Geometrien geführt hat. Positive Geometrien sind Geometrien mit Rändern aller Kodimensionen und gewissen zugehörigen \emph{kanonischen Formen}, aus denen Streuamplitude extrahiert werden können. Der zentrale Akteur dieser Dissertation ist das Impulsamplituhedron, welches durch die Positive Geometrie gegeben ist und die on-shell Amplituden auf Baumniveau in der maximal supersymmetrischen Yang-Mills-Theorie kodiert, die im Raum der Spinor-Helizitätsvariablen definiert ist. Die canonical Form das Impulsamplituhedron verfügt über eine besondere Singularitätsstruktur, die die physikalischen Singularitäten der Streuamplituden in allen Helizitätssektoren auf Baumniveau kodiert, aus denen die Streuamplituden extrahiert werden können. Dies ermöglicht es, Streuamplituden in maximal supersymmetrischen Yang-Mills Theorie zu bestimmen ohne Bezug auf Felder, Lagrangedichten, Raumzeit oder Feynman-Diagramme zu nehmen. In neueren Arbeiten über das Impulsamplituhedron konnten wir sehen, das seine kanonische Form mit der kanonischen Form - die mit einer Geometrie assoziiert ist, welche die Streuamplituden für bi-adjungierte Skalare - dem kinematischen Associahedron kodiert, in Verbindung gebracht werden kann. Die Definition des Impusamplituhedron auf dem Raum der Spinor-Helizitäts-Variablen ermöglicht einen direkten Vergleich von Geometrien, mit unterschiedlich Farb-geordneten Streuamplituden im selben Raum verbunden sind. Die wird genutzt, um die Kleiss-Kuijf-Relationen -- eine Reihe von Beziehungen zwischen Streuamplituden verschiedener Farbordnungen, wiederherzustellen, die sich aus der Farbzerlegung von Streuamplituden ergeben. Die Kleiss-Kuijf-Relationen manifestieren sich als orientierte Summen von Impulsamplituhedronen verschiedener Farbordnungen ohne Vertices in ihren Rändern. Wir leiten einen homologischen Algorithmus ab, der auf diesem Prinzip basiert, um Kleiss-Kuijf-Beziehungen für Impulsamplituhedronen zu finden.This dissertation focus on some of the modern theoretical developments in the field of scattering amplitudes. Recent years have seen a departure from the traditional approach of extracting scattering amplitudes in terms of Feynman diagrams in favor of techniques known as on-shell methods. These methods reveal a striking relationship between scattering amplitudes and a geometry known as the positive Grassmannian, leading to a radical reformulation of scattering amplitudes in terms of so-called positive geometries. Positive geometries are geometries with boundaries of all codimensions and have a certain associated canonical form. In some special cases, physical observables can be extracted from the canonical forms of positive geometries. The central player in this dissertation is the \emph{momentum amplituhedron} which is the positive geometry encoding on-shell tree-level amplitudes in maximally supersymmetric Yang-Mills theory defined on the space of spinor helicity variables. The momentum amplituhedron is equipped with a canonical form with a particular singularity structure, encoding the physical singularities of scattering amplitudes in all helicity sectors at tree-level, from which scattering amplitudes can be extracted. This allows us to determine scattering amplitudes in maximally supersymmetric Yang-Mills without reference to fields, Lagrangians, space-time, or Feynman diagrams. We will in this dissertation report on the most recent results for the momentum amplituhedron obtained in collaboration with other authors. In particular, we will see that its canonical form can be related to the canonical form associated with a geometry encoding scattering amplitudes for bi-adjoint scalars -- the kinematic associahedron. Furthermore, since we can define the momentum amplituhedron on the space of spinor helicity variables, it allows for a direct comparison of geometries associated with differently color-ordered scattering amplitudes in the same space. This ability to compare momentum amplituhedra of different color orderings will be employed to rederive the Kleiss-Kuijf relations, a set of relations between scattering amplitudes of different color orderings stemming from the color decomposition of scattering amplitudes. The Kleiss-Kuijf relations will appear as oriented sums of momentum amplituhedra of different color orderings with no vertices in their boundary stratifications. We will use this fact to derive a homological algorithm based on this principle to find Kleiss-Kuijf relations for momentum amplituhedra

1-Safe Petri nets and special cube complexes: equivalence and applications

Nielsen, Plotkin, and Winskel (1981) proved that every 1-safe Petri net $N$ unfolds into an event structure $\mathcal{E}_N$. By a result of Thiagarajan (1996 and 2002), these unfoldings are exactly the trace regular event structures. Thiagarajan (1996 and 2002) conjectured that regular event structures correspond exactly to trace regular event structures. In a recent paper (Chalopin and Chepoi, 2017, 2018), we disproved this conjecture, based on the striking bijection between domains of event structures, median graphs, and CAT(0) cube complexes. On the other hand, in Chalopin and Chepoi (2018) we proved that Thiagarajan's conjecture is true for regular event structures whose domains are principal filters of universal covers of (virtually) finite special cube complexes. In the current paper, we prove the converse: to any finite 1-safe Petri net $N$ one can associate a finite special cube complex ${X}_N$ such that the domain of the event structure $\mathcal{E}_N$ (obtained as the unfolding of $N$) is a principal filter of the universal cover $\widetilde{X}_N$ of $X_N$. This establishes a bijection between 1-safe Petri nets and finite special cube complexes and provides a combinatorial characterization of trace regular event structures. Using this bijection and techniques from graph theory and geometry (MSO theory of graphs, bounded treewidth, and bounded hyperbolicity) we disprove yet another conjecture by Thiagarajan (from the paper with S. Yang from 2014) that the monadic second order logic of a 1-safe Petri net is decidable if and only if its unfolding is grid-free. Our counterexample is the trace regular event structure $\mathcal{\dot E}_Z$ which arises from a virtually special square complex $\dot Z$. The domain of $\mathcal{\dot E}_Z$ is grid-free (because it is hyperbolic), but the MSO theory of the event structure $\mathcal{\dot E}_Z$ is undecidable