IA Scholar Query: Ready to preorder: an algebraic and general proof.
https://scholar.archive.org/
Internet Archive Scholar query results feedeninfo@archive.orgTue, 29 Nov 2022 00:00:00 GMTfatcat-scholarhttps://scholar.archive.org/help1440Converse extensionality and apartness
https://scholar.archive.org/work/f7ehdnhjzfajtoek4o3rbysnwy
In this paper we try to find a computational interpretation for a strong form of extensionality, which we call "converse extensionality". Converse extensionality principles, which arise as the Dialectica interpretation of the axiom of extensionality, were first studied by Howard. In order to give a computational interpretation to these principles, we reconsider Brouwer's apartness relation, a strong constructive form of inequality. Formally, we provide a categorical construction to endow every typed combinatory algebra with an apartness relation. We then exploit that functions reflect apartness, in addition to preserving equality, to prove that the resulting categories of assemblies model a converse extensionality principle.Benno van den Berg, Robert Passmannwork_f7ehdnhjzfajtoek4o3rbysnwyTue, 29 Nov 2022 00:00:00 GMTSimulations for Event-Clock Automata
https://scholar.archive.org/work/efqfqlg6jjhm7f7vvfer7dnimm
Event-clock automata are a well-known subclass of timed automata which enjoy admirable theoretical properties, e.g., determinizability, and are practically useful to capture timed specifications. However, unlike for timed automata, there exist no implementations for event-clock automata. A main reason for this is the difficulty in adapting zone-based algorithms, critical in the timed automata setting, to the event-clock automata setting. This difficulty was recently studied in [Geeraerts et al 2011,2014], where the authors proposed a solution using zone extrapolations. In this article, we propose a different zone-based algorithm to solve the reachability problem for event-clock automata, using simulations for finiteness. A surprising consequence of our result is that for event-predicting automata, the subclass of event-clock automata that only use prophecy clocks, we obtain finiteness even without any simulations. For general event-clock automata, our new algorithm exploits the G-simulation framework, which is the coarsest known simulation relation for reachability, and has been recently used for advances in other extensions of timed automata.S Akshay, Paul Gastin, R Govind, B Srivathsanwork_efqfqlg6jjhm7f7vvfer7dnimmTue, 29 Nov 2022 00:00:00 GMT2019
https://scholar.archive.org/work/wcy47hfvvvdwvfgnwx2cuak4ze
On completion of this course, students will have knowledge in: • CO1.Basics of electrochemistry. Classical & modern batteries and fuel cells. CO2. Causes & effects of corrosion of metals and control of corrosion. Modification of surface properties of metals to develop resistance to corrosion, wear, tear, impact etc. by electroplating and electroless plating. CO3. Production & consumption of energy for industrialization of country and living standards of people. Utilization of solar energy for different useful forms of energy. CO4. Understanding Phase rule and instrumental techniques and its applications. CO5.Over viewing of synthesis, properties and applications of nanomaterials.BTECH.CSwork_wcy47hfvvvdwvfgnwx2cuak4zeMon, 28 Nov 2022 00:00:00 GMT2021
https://scholar.archive.org/work/n7rhmaerpvfrhha4draeqwscs4
Course Objectives: 1. Learn and understand basic concepts and principles of Physics. 2. Make students familiar with latest trends in material science research and learn about novel materials and its applications. 3. Make students confident in analyzing engineering problems and apply its solutions effectively and meaningfully. 4. Gain knowledge in interference and diffraction of light and its applications in new technology. Course Outcomes: CO1: Learn and understand more about basic principles and to develop problem solving skills and implementation in technology. CO2: Study material properties and their application and its use in engineering applications and studies. CO3: Understand crystal structure and applications to boost the technical skills and its applications. CO4: Apply light phenomena in new technology. Module 1 Classical free electron theory-Free-electron concept (Drift velocity, Thermal velocity, Mean collision time, Mean free path, relaxation time) -Expression for electrical conductivity-Failure of classical free electron theory. Quantum free electron theory, Assumptions, Fermi factor, Fermi-Dirac Statistics. Expression for electrical conductivity based on quantum free electron theory. Merits of quantum free electron theory. Temperature dependence of electrical resistivity -Specific heat -Thermionic emission. Hall effect (Qualitative) -Wiedemann-Franz law. Teaching Methodology: Chalk and talk method: Classical free electron theory-Free-electron concept (Drift velocity, Thermal velocity, Mean collision time, Mean free path, relaxation time) -Expression for electrical conductivity-Failure of classical free electron theory. Powerpoint presentation: Quantum free electron theory, Assumptions, Fermi factor, Fermi-Dirac Statistics. Expression for electrical conductivity based on quantum free electron theory. Merits of quantum free electron theory. Temperature dependence of electrical resistivity -Specific heat -Thermionic emission. Wiedemann-Franz law. Self-study material: Hall effect (Qualitative) 9 Hours Module 2 Interaction of radiation with matter -Absorption-Spontaneous emission -Stimulated emission-Einstein's coefficients (expression for energy density). Requisites of a Laser system. Condition for laser action. Principle, Construction and working of He-Ne laser. Propagation mechanism in optical fibers. Angle of acceptance. Numerical aperture. Types of optical fibers-Step index and Graded index fiber. Modes of propagation-Single mode and Multimode fibers. Attenuation-Attenuation mechanisms. Teaching Methodology: Chalk and talk method: Interaction of radiation with matter -Absorption-Spontaneous emission -Stimulated emission-Einstein's coefficients (expression for energy density). Requisites of a Laser system. Condition for laser action. Propagation mechanism in optical fibers. Angle of acceptance. Numerical aperture. Powerpoint presentation: Types of optical fibers-Step index and Graded index fiber. Modes of propagation-Single mode and Multimode fibers. Video: Construction and working of He-Ne laser. Self-study material: Attenuation-Attenuation mechanisms. 9 Hours Module 3 Temperature dependence of resistivity in metals and superconducting materials. Effect of magnetic field (Meissner effect). Isotope effect -Type I and Type II superconductors-Temperature dependence of critical field. BCS theory (qualitative). High temperature superconductors-Josephson effect -SQUID-Applications of superconductors-Maglev vehicles (qualitative). Magnetic dipole-dipole moment-flux density-magnetic field intensity-Intensity of magnetization-magnetic permeability-susceptibility-relation between permeability and susceptibility. Classification of magnetic materials-Dia, Para, Ferromagnetism. Hysteresis-soft and hard magnetic materials. Teaching Methodology: Chalk and talk method: Temperature dependence of resistivity in metals and superconducting materials. Effect of magnetic field (Meissner effect). Isotope effect -Type I and Type II superconductors-Temperature dependence of critical field. BCS theory (qualitative). High temperature superconductors-Powerpoint presentation: Josephson effect -SQUID-Applications of superconductors. Magnetic dipole-dipole moment-flux density-magnetic field intensity-Intensity of magnetization-magnetic permeability-susceptibility-relation between permeability and susceptibility. Hysteresis-soft and hard magnetic materials. Video: Maglev vehicles (qualitative). Self-study material: Classification of magnetic materials-Dia, Para, Ferromagnetism 9 Hours Module 4 Amorphous and crystalline materials-Space lattice, Bravais lattice-Unit cell, primitive cell. Lattice parameters. Crystal systems. Direction and planes in a crystal. Miller indices -Determination of Miller indices of a plane. Expression for interplanar spacing. Atoms per unit cell -Co-ordination number. Relation between atomic radius and lattice constant -Atomic packing factors (SC, FCC, BCC). Bragg's law. Determination of crystal structure using Bragg's X-ray diffractometer -X-ray spectrum. Teaching Methodology: Chalk and talk method: Direction and planes in a crystal. Miller indices -Determination of Miller indices of a plane. Powerpoint presentation: Atoms per unit cell -Co-ordination number. Relation between atomic radius and lattice constant -Atomic packing factors (SC, FCC, BCC). Bragg's law. Determination of crystal structure using Bragg's X-ray diffractometer -X-ray spectrum. Self-study material: Amorphous and crystalline materials-Space lattice, Bravais lattice-Unit cell, primitive cell. Lattice parameters. Crystal systems. 9 Hours Module 5 Interference of light -Superposition of two coherent waves-Constructive and destructive interference. Interference in thin films -Wedge shaped thin film-Air wedge -Application to find the diameter of a thin wire. Newton's rings -Application to find the refractive index of a liquid. Diffraction of light -Classes of diffraction -Fresnel and Fraunhofer diffraction. Fresnel theory of half period zone -Zone plate.BTECH.CSwork_n7rhmaerpvfrhha4draeqwscs4Mon, 28 Nov 2022 00:00:00 GMTDescent modulus and applications
https://scholar.archive.org/work/hcdbtlfasrblxfr4a6ksnl3kle
The norm of the gradient ∇f (x) measures the maximum descent of a real-valued smooth function f at x. For (nonsmooth) convex functions, this is expressed by the distance dist(0, ∂f (x)) of the subdifferential to the origin, while for general real-valued functions defined on metric spaces by the notion of metric slope |∇f |(x). In this work we propose an axiomatic definition of descent modulus T [f ](x) of a real-valued function f at every point x, defined on a general (not necessarily metric) space. The definition encompasses all above instances as well as average descents for functions defined on probability spaces. We show that a large class of functions are completely determined by their descent modulus and corresponding critical values. This result is already surprising in the smooth case: a one-dimensional information (norm of the gradient) turns out to be almost as powerful as the knowledge of the full gradient mapping. In the nonsmooth case, the key element for this determination result is the break of symmetry induced by a downhill orientation, in the spirit of the definition of the metric slope. The particular case of functions defined on finite spaces is studied in the last section. In this case, we obtain an explicit classification of descent operators that are, in some sense, typical.Aris Daniilidis, Laurent Miclo, David Salaswork_hcdbtlfasrblxfr4a6ksnl3kleMon, 21 Nov 2022 00:00:00 GMTOn posetal and complete partial applicative structures
https://scholar.archive.org/work/c5rcu2qthfctvcprgpvvwqwewa
Every partial applicative structure gives rise to an indexed binary relation, that is a contravariant functor from the category of sets to the category of sets endowed with binary relations and maps preserving them. In this paper we characterize those partial applicative structures giving rise to indexed relations satisfying certain elementary properties in terms of algebraic or computational properties. We will then provide a characterization of those partial applicative structures giving rise to indexed preorders and indexed posets, and we will relate the latter ones to some particular classes of unary partial endofunctions. We will analyze the relation between a series of computational and algebraic properties in the posetal case. Finally, we will study the problem of existence of suprema in the case of partial applicative structures giving rise to indexed preorders, by providing some necessary conditions for a partial applicative structure to be complete.Samuele Maschiowork_c5rcu2qthfctvcprgpvvwqwewaMon, 21 Nov 2022 00:00:00 GMTTropical Extensions and Baker-Lorscheid Multiplicities for Idylls
https://scholar.archive.org/work/d5ikgcd4ofccvkff3lrvou5zji
In a recent paper, Matt Baker and Oliver Lorscheid showed that Descartes's Rule of Signs and Newton's Polygon Rule can both be interpreted as multiplicities of polynomials over hyperfields. Hyperfields are a generalization of fields which encode things like the arithmetic of signs or of absolute values. By looking at multiplicities of polynomials over such algebras, Baker and Lorscheid showed that you can recover the rules of Descartes and Newton. In this paper, we define tropical extensions which are a generalization of the hyperfield semidirect product described by Nathan Bowler and Ting Su. Examples of tropical extensions are extending the tropical hyperfield to higher ranks, or extending the hyperfield of signs to the tropical real hyperfield by including a valuation. We work in the larger categories of ordered blueprints or of idylls to generalize the theory and to work with simpler axioms. The results of this paper concern the interaction of multiplicities and tropical extensions. First, there is a lifting theorem from initial forms to the entire polynomial from which we will show that multiplicities for a polynomial are equal to the corresponding multiplicity for some initial form. Second, we show that tropical extensions preserve the property that the sum of all multiplicities is bounded by the degree. Consequentially, we have this degree bound for every stringent hyperfield. This gives a partial answer to a question posed by Baker and Lorscheid about which hyperfields have this property.Trevor Gunnwork_d5ikgcd4ofccvkff3lrvou5zjiFri, 11 Nov 2022 00:00:00 GMTOrders On Free Metabelian Groups
https://scholar.archive.org/work/ptajysfkpfd3dlxp6rp6wjgjq4
A bi-order on a group G is a total, bi-multiplication invariant order. Such an order is regular if the positive cone associated to the order can be recognised by a regular language. A subset S in an orderable group (G,⩽) is convex if for all f⩽ g in S, every element h∈ G satisfying f⩽ h ⩽ g belongs to S. In this paper, we study the convex hull of the derived subgroup of a free metabelian group with respect to a bi-order. As an application, we prove that non-abelian free metabelian groups of finite rank do not admit a regular bi-order while they are computably bi-orderable.Wenhao Wangwork_ptajysfkpfd3dlxp6rp6wjgjq4Wed, 09 Nov 2022 00:00:00 GMTA deterministic near-linear time approximation scheme for geometric transportation
https://scholar.archive.org/work/tk7ipg4xx5crxa2cspgcn4prce
Given a set of points P = (P^+ ⊔ P^-) ⊂ℝ^d for some constant d and a supply function μ:P→ℝ such that μ(p) > 0 ∀ p ∈ P^+, μ(p) < 0 ∀ p ∈ P^-, and ∑_p∈ Pμ(p) = 0, the geometric transportation problem asks one to find a transportation map τ: P^+× P^-→ℝ_≥ 0 such that ∑_q∈ P^-τ(p, q) = μ(p) ∀ p ∈ P^+, ∑_p∈ P^+τ(p, q) = -μ(q) ∀ q ∈ P^-, and the weighted sum of Euclidean distances for the pairs ∑_(p,q)∈ P^+× P^-τ(p, q)· ||q-p||_2 is minimized. We present the first deterministic algorithm that computes, in near-linear time, a transportation map whose cost is within a (1 + ε) factor of optimal. More precisely, our algorithm runs in O(nε^-(d+2)log^5nloglogn) time for any constant ε > 0. While a randomized nε^-O(d)log^O(d)n time algorithm was discovered in the last few years, all previously known deterministic (1 + ε)-approximation algorithms run in Ω(n^3/2) time. A similar situation existed for geometric bipartite matching, the special case of geometric transportation where all supplies are unit, until a deterministic nε^-O(d)log^O(d)n time (1 + ε)-approximation algorithm was presented at STOC 2022. Surprisingly, our result is not only a generalization of the bipartite matching one to arbitrary instances of geometric transportation, but it also reduces the running time for all previously known (1 + ε)-approximation algorithms, randomized or deterministic, even for geometric bipartite matching, by removing the dependence on the dimension d from the exponent in the running time's polylog.Kyle Foxwork_tk7ipg4xx5crxa2cspgcn4prceMon, 07 Nov 2022 00:00:00 GMTSigned permutohedra, delta-matroids, and beyond
https://scholar.archive.org/work/2yzqr44rpvgeheytlc7pyjivwi
We establish a connection between the algebraic geometry of the type B permutohedral toric variety and the combinatorics of delta-matroids. Using this connection, we compute the volume and lattice point counts of type B generalized permutohedra. Applying tropical Hodge theory to a new framework of "tautological classes of delta-matroids," modeled after certain vector bundles associated to realizable delta-matroids, we establish the log-concavity of a Tutte-like invariant for a broad family of delta-matroids that includes all realizable delta-matroids. Our results include new log-concavity statements for all (ordinary) matroids as special cases.Christopher Eur, Alex Fink, Matt Larson, Hunter Spinkwork_2yzqr44rpvgeheytlc7pyjivwiSun, 06 Nov 2022 00:00:00 GMTVarieties of unary-determined distributive ℓ-magmas and bunched implication algebras
https://scholar.archive.org/work/qh6fgnckk5heha6bibjc34mx3e
A distributive lattice-ordered magma (dℓ-magma) (A,∧,∨,·) is a distributive lattice with a binary operation · that preserves joins in both arguments, and when · is associative then (A,∨,·) is an idempotent semiring. A dℓ-magma with a top ⊤ is unary-determined if x· y=(x·⊤∧ y) ∨(x∧⊤·y). These algebras are term-equivalent to a subvariety of distributive lattices with ⊤ and two join-preserving unary operations p,q. We obtain simple conditions on p,q such that x· y=(px∧ y)∨(x∧ qy) is associative, commutative, idempotent and/or has an identity element. This generalizes previous results on the structure of doubly idempotent semirings and, in the case when the distributive lattice is a Heyting algebra, it provides structural insight into unary-determined algebraic models of bunched implication logic. We also provide Kripke semantics for the algebras under consideration, which leads to more efficient algorithms for constructing finite models. We find all subdirectly irreducible algebras up to cardinality eight in which p=q is a closure operator, as well as all finite unary-determined bunched implication chains and map out the poset of join-irreducible varieties generated by them.Natanael Alpay, Peter Jipsen, Melissa Sugimotowork_qh6fgnckk5heha6bibjc34mx3eSat, 05 Nov 2022 00:00:00 GMTHigher-Order MSL Horn Constraints
https://scholar.archive.org/work/c3hrva5dqnbh7hc5x4b2mh2nhu
The monadic shallow linear (MSL) class is a decidable fragment of first-order Horn clauses that was discovered and rediscovered around the turn of the century, with applications in static analysis and verification. We propose a new class of higher-order Horn constraints which extend MSL to higher-order logic and develop a resolution-based decision procedure. Higher-order MSL Horn constraints can quite naturally capture the complex patterns of call and return that are possible in higher-order programs, which make them well suited to higher-order program verification. In fact, we show that the higher-order MSL satisfiability problem and the HORS model checking problem are interreducible, so that higher-order MSL can be seen as a constraint-based approach to higher-order model checking. Finally, we describe an implementation of our decision procedure and its application to verified socket programming.Jerome Jochems and Eddie Jones and Steven Ramsaywork_c3hrva5dqnbh7hc5x4b2mh2nhuWed, 26 Oct 2022 00:00:00 GMTInstantons and rational homology spheres
https://scholar.archive.org/work/oa3toarglvdnbgfoifl6x374lu
In previous work, the second author defined 'equivariant instanton homology groups' I^∙(Y,π;R) for a rational homology 3-sphere Y, a set of auxiliary data π, and a PID R. These objects are modules over the cohomology ring H^-*(BSO_3;R). We prove that the equivariant instanton homology groups I^∙(Y;R) are independent of the auxiliary data π, and thus define topological invariants of rational homology spheres. Further, we prove that these invariants are functorial under cobordisms of 3-manifolds with a path between the boundary components. For any rational homology sphere Y, we may also define an analogue of Floer's irreducible instanton homology group of integer homology spheres I_*(Y, π; R) which now depends on the auxiliary data π, unlike the equivariant instanton homology groups. However, our methods allow us to prove a precise "wall-crossing formula" for I_*(Y, π; R) as the auxiliary data π moves between adjacent chambers. We use this to define an instanton invariant λ_I(Y) ∈ Q of rational homology spheres, conjecturally equal to the Casson-Walker invariant. Our approach to invariance uses a novel technique known as a suspended flow category. Given an obstructed cobordism W: Y → Y', which supports reducible instantons which can neither be cut out transversely nor be removed by a small change to the perturbation, we remove and replace a neighborhood of obstructed solutions in the moduli space of instantons. The resulting moduli spaces have a new type of boundary component, so do not define a chain map between the instanton chain complexes of Y and Y'. However, it does define a chain map between the instanton chain complex of Y and a sort of suspension of the instanton chain complex of Y'.Aliakbar Daemi, Mike Miller Eismeierwork_oa3toarglvdnbgfoifl6x374luTue, 25 Oct 2022 00:00:00 GMTTopics in topological combinatorics: Simplicial complexes, finite geometry, and the topology of circle-valued maps
https://scholar.archive.org/work/hnoqjmnqovdi3m6a4giyspr4yy
Combinatorics is a loosely-organized area of mathematics having to do with counting and optimizing discrete mathematical structures. Topology is an area of mathematics having to do with deforming and distinguishing continuous mathematical structures. Topological combinatorics refers to the application of ideas from topology to problems in combinatorics. This thesis consists of an introduction, followed by five papers on various topics in topological combinatorics. Each paper is given in its original form, as its own chapter, with co-authors and current publication status indicated at the beginning of the chapter. Here we give definitions and summarize the results of the later chapters, including directions for further research.Matthew Superdockwork_hnoqjmnqovdi3m6a4giyspr4yyMon, 24 Oct 2022 00:00:00 GMTDifferential privacy for metric spaces: information-theoretic models for privacy and utility with new applications to metric domains
https://scholar.archive.org/work/k7ua6tg73zhgfcumtuf5g56s2m
The problem of data privacy – protecting sensitive or personal data from discovery – has been a long-standing research issue. In this regard, differential privacy, introduced in 2006, is considered to be the gold standard. Differential privacy was designed to protect the privacy of individuals in statistical datasets such as census datasets. Its widespread popularity has led to interest in applying differential privacy to new domains for which it was not originally designed, such as text documents. This raises questions regarding the interpretability of differential privacy's guarantees, which are usually expressed in the language of statistical disclosure control. In addition, it escalates the need for answers to core issues currently debated in the differential privacy community: how does the application of differential privacy protect against inference attacks? How can the use of noise-adding mechanisms guarantee the release of useful information? And how can this privacy-utility balance be achieved? The goal of this thesis is to address these foundational questions. Firstly, we approach the problem of interpretability by exploring a generalisation of differential privacy for metric domains known as metric differential privacy or d-privacy. Metric differential privacy abstracts away from the particulars of statistical databases and permits reasoning about privacy on more general domains endowed with a metric. This allows differential privacy's guarantees to be understood in more general terms which can be applied to arbitrary domains of interest, including text documents. Secondly, we propose to study the key questions surrounding privacy and utility in differential privacy using the Quantitative Information Flow (QIF) framework, an information-theoretic framework previously used to analyse threats to secure systems. In this thesis, we repurpose QIF to analyse the privacy and utility guarantees provided by differentially private systems modelled as probabilistic channels. Using information flo [...]Natasha Fernandeswork_k7ua6tg73zhgfcumtuf5g56s2mWed, 19 Oct 2022 00:00:00 GMTDynamic Probabilistic Input Output Automata
https://scholar.archive.org/work/fdbrmxcaeffjbgi6ssfie37pvu
We present probabilistic dynamic I/O automata, a framework to model dynamic probabilistic systems. Our work extends dynamic I/O Automata formalism of Attie & Lynch [Paul C. Attie and Nancy A. Lynch, 2016] to the probabilistic setting. The original dynamic I/O Automata formalism included operators for parallel composition, action hiding, action renaming, automaton creation, and behavioral sub-typing by means of trace inclusion. They can model mobility by using signature modification. They are also hierarchical: a dynamically changing system of interacting automata is itself modeled as a single automaton. Our work extends all these features to the probabilistic setting. Furthermore, we prove necessary and sufficient conditions to obtain the monotonicity of automata creation/destruction with implementation preorder. Our construction uses a novel proof technique based on homomorphism that can be of independent interest. Our work lays down the foundations for extending composable secure-emulation of Canetti et al. [Ran Canetti et al., 2007] to dynamic settings, an important tool towards the formal verification of protocols combining probabilistic distributed systems and cryptography in dynamic settings (e.g. blockchains, secure distributed computation, cybersecure distributed protocols, etc).Pierre Civit, Maria Potop-Butucaru, Christian Scheidelerwork_fdbrmxcaeffjbgi6ssfie37pvuMon, 17 Oct 2022 00:00:00 GMTA tradeoff between universality of equivariant models and learnability of symmetries
https://scholar.archive.org/work/f4okgc3nzffjpfjg675p4olgti
We prove an impossibility result, which in the context of function learning says the following: under certain conditions, it is impossible to simultaneously learn symmetries and functions equivariant under them using an ansatz consisting of equivariant functions. To formalize this statement, we carefully study notions of approximation for groups and semigroups. We analyze certain families of neural networks for whether they satisfy the conditions of the impossibility result: what we call "linearly equivariant" networks, and group-convolutional networks. A lot can be said precisely about linearly equivariant networks, making them theoretically useful. On the practical side, our analysis of group-convolutional neural networks allows us generalize the well-known "convolution is all you need" theorem to non-homogeneous spaces. We additionally find an important difference between group convolution and semigroup convolution.Vasco Portilheirowork_f4okgc3nzffjpfjg675p4olgtiMon, 17 Oct 2022 00:00:00 GMTFixed Points and Noetherian Topologies
https://scholar.archive.org/work/fwem4dej5bcabmj7orhwvbm5ma
This paper provides a canonical construction of a Noetherian least fixed point topology. While such least fixed point are not Noetherian in general, we prove that under a mild assumption, one can use a topological minimal bad sequence argument to prove that they are. We then apply this fixed point theorem to rebuild known Noetherian topologies with a uniform proof. In the case of spaces that are defined inductively (such as finite words and finite trees), we provide a uniform definition of a divisibility topology using our fixed point theorem. We then prove that the divisibility topology is a generalisation of the divisibility preorder introduced by Hasegawa in the case of well-quasi-orders.Aliaume Lopezwork_fwem4dej5bcabmj7orhwvbm5maMon, 17 Oct 2022 00:00:00 GMTWitnessed Symmetric Choice and Interpretations in Fixed-Point Logic with Counting
https://scholar.archive.org/work/kv2cfbxp3bbnnb5wwmyey5huoq
At the core of the quest for a logic for PTime is a mismatch between algorithms making arbitrary choices and isomorphism-invariant logics. One approach to overcome this problem is witnessed symmetric choice. It allows for choices from definable orbits which are certified by definable witnessing automorphisms. We consider the extension of fixed-point logic with counting (IFPC) with witnessed symmetric choice (IFPC+WSC) and a further extension with an interpretation operator (IFP+WSC+I). The latter operator evaluates a subformula in the structure defined by an interpretation. This structure possibly has other automorphisms exploitable by the WSC-operator. For similar extensions of pure fixed-point logic (IFP) it is known that IFP+WSC+I simulates counting which IFP+WSC fails to do. For IFPC it is unknown whether the interpretation operator increases expressiveness and thus allows studying the relation between WSC and interpretations beyond counting. In this paper, we prove that if IFPC+WSC+I canonizes a particular class of base graphs, then it also canonizes the corresponding CFI graphs. This differs from various other logics, where CFI graphs provide difficult instances. To canonize CFI graphs, we nest WSC and interpretation operators. We show that for CFI graphs this deeper nesting is indeed necessary. Lastly, we separate IFPC+WSC from IFPC+WSC+I, so for IFPC the interpretation operator increases expressiveness, too. In particular, IFPC+WSC is not closed under FO-reductions.Moritz Lichterwork_kv2cfbxp3bbnnb5wwmyey5huoqFri, 14 Oct 2022 00:00:00 GMTNotes on CSPs and Polymorphisms
https://scholar.archive.org/work/kouwgol6o5h55lxjkqyjupnv2i
These are notes from a multi-year learning seminar on the algebraic approach to Constraint Satisfaction Problems (CSPs). The main topics covered are the theory of algebraic structures with few subpowers, the theory of absorbing subalgebras and its applications to studying CSP templates which can be solved by local consistency methods, and the dichotomy theorem for conservative CSP templates. Subsections and appendices cover supplementary material.Zarathustra Bradywork_kouwgol6o5h55lxjkqyjupnv2iThu, 13 Oct 2022 00:00:00 GMT