Advanced Dynamic Programming in Semiring and Hypergraph Frameworks ∗
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Scott Spaces and the Dcpo Category
SCOTT SPACES AND THE DCPO CATEGORY JORDAN BROWN Abstract. Directed-complete partial orders (dcpo’s) arise often in the study of λ-calculus. Here we investigate certain properties of dcpo’s and the Scott spaces they induce. We introduce a new construction which allows for the canonical extension of a partial order to a dcpo and give a proof that the dcpo introduced by Zhao, Xi, and Chen is well-filtered. Contents 1. Introduction 1 2. General Definitions and the Finite Case 2 3. Connectedness of Scott Spaces 5 4. The Categorical Structure of DCPO 6 5. Suprema and the Waybelow Relation 7 6. Hofmann-Mislove Theorem 9 7. Ordinal-Based DCPOs 11 8. Acknowledgments 13 References 13 1. Introduction Directed-complete partially ordered sets (dcpo’s) often arise in the study of λ-calculus. Namely, they are often used to construct models for λ theories. There are several versions of the λ-calculus, all of which attempt to describe the ‘computable’ functions. The first robust descriptions of λ-calculus appeared around the same time as the definition of Turing machines, and Turing’s paper introducing computing machines includes a proof that his computable functions are precisely the λ-definable ones [5] [8]. Though we do not address the λ-calculus directly here, an exposition of certain λ theories and the construction of Scott space models for them can be found in [1]. In these models, computable functions correspond to continuous functions with respect to the Scott topology. It is thus with an eye to the application of topological tools in the study of computability that we investigate the Scott topology. -
Section 8.6 What Is a Partial Order?
Announcements ICS 6B } Regrades for Quiz #3 and Homeworks #4 & Boolean Algebra & Logic 5 are due Today Lecture Notes for Summer Quarter, 2008 Michele Rousseau Set 8 – Ch. 8.6, 11.6 Lecture Set 8 - Chpts 8.6, 11.1 2 Today’s Lecture } Chapter 8 8.6, Chapter 11 11.1 ● Partial Orderings 8.6 Chapter 8: Section 8.6 ● Boolean Functions 11.1 Partial Orderings (Continued) Lecture Set 8 - Chpts 8.6, 11.1 3 What is a Partial Order? Some more defintions Let R be a relation on A. The R is a partial order iff R is: } If A,R is a poset and a,b are A, reflexive, antisymmetric, & transitive we say that: } A,R is called a partially ordered set or “poset” ● “a and b are comparable” if ab or ba } Notation: ◘ i.e. if a,bR and b,aR ● If A, R is a poset and a and b are 2 elements of A ● “a and b are incomparable” if neither ab nor ba such that a,bR, we write a b instead of aRb ◘ i.e if a,bR and b,aR } If two objects are always related in a poset it is called a total order, linear order or simple order. NOTE: it is not required that two things be related under a partial order. ● In this case A,R is called a chain. ● i.e if any two elements of A are comparable ● That’s the “partial” of it. ● So for all a,b A, it is true that a,bR or b,aR 5 Lecture Set 8 - Chpts 8.6, 11.1 6 1 Now onto more examples… More Examples Let Aa,b,c,d and let R be the relation Let A0,1,2,3 and on A represented by the diagraph Let R0,01,1, 2,0,2,2,2,33,3 The R is reflexive, but We draw the associated digraph: a b not antisymmetric a,c &c,a It is easy to check that R is 0 and not 1 c d transitive d,cc,a, but not d,a Refl. -
Semiring Frameworks and Algorithms for Shortest-Distance Problems
Journal of Automata, Languages and Combinatorics u (v) w, x–y c Otto-von-Guericke-Universit¨at Magdeburg Semiring Frameworks and Algorithms for Shortest-Distance Problems Mehryar Mohri AT&T Labs – Research 180 Park Avenue, Rm E135 Florham Park, NJ 07932 e-mail: [email protected] ABSTRACT We define general algebraic frameworks for shortest-distance problems based on the structure of semirings. We give a generic algorithm for finding single-source shortest distances in a weighted directed graph when the weights satisfy the conditions of our general semiring framework. The same algorithm can be used to solve efficiently clas- sical shortest paths problems or to find the k-shortest distances in a directed graph. It can be used to solve single-source shortest-distance problems in weighted directed acyclic graphs over any semiring. We examine several semirings and describe some specific instances of our generic algorithms to illustrate their use and compare them with existing methods and algorithms. The proof of the soundness of all algorithms is given in detail, including their pseudocode and a full analysis of their running time complexity. Keywords: semirings, finite automata, shortest-paths algorithms, rational power series Classical shortest-paths problems in a weighted directed graph arise in various contexts. The problems divide into two related categories: single-source shortest- paths problems and all-pairs shortest-paths problems. The single-source shortest-path problem in a directed graph consists of determining the shortest path from a fixed source vertex s to all other vertices. The all-pairs shortest-distance problem is that of finding the shortest paths between all pairs of vertices of a graph. -
[Math.NT] 1 Nov 2006
ADJOINING IDENTITIES AND ZEROS TO SEMIGROUPS MELVYN B. NATHANSON Abstract. This note shows how iteration of the standard process of adjoining identities and zeros to semigroups gives rise naturally to the lexicographical ordering on the additive semigroups of n-tuples of nonnegative integers and n-tuples of integers. 1. Semigroups with identities and zeros A binary operation ∗ on a set S is associative if (a ∗ b) ∗ c = a ∗ (b ∗ c) for all a,b,c ∈ S. A semigroup is a nonempty set with an associative binary operation ∗. The semigroup is abelian if a ∗ b = b ∗ a for all a,b ∈ S. The trivial semigroup S0 consists of a single element s0 such that s0 ∗ s0 = s0. Theorems about abstract semigroups are, in a sense, theorems about the pure process of multiplication. An element u in a semigroup S is an identity if u ∗ a = a ∗ u = a for all a ∈ S. If u and u′ are identities in a semigroup, then u = u ∗ u′ = u′ and so a semigroup contains at most one identity. A semigroup with an identity is called a monoid. If S is a semigroup that is not a monoid, that is, if S does not contain an identity element, there is a simple process to adjoin an identity to S. Let u be an element not in S and let I(S,u)= S ∪{u}. We extend the binary operation ∗ from S to I(S,u) by defining u ∗ a = a ∗ u = a for all a ∈ S, and u ∗ u = u. Then I(S,u) is a monoid with identity u. -
Estimation of Viterbi Path in Bayesian Hidden Markov Models
Estimation of Viterbi path in Bayesian hidden Markov models Jüri Lember1,∗ Dario Gasbarra2, Alexey Koloydenko3 and Kristi Kuljus1 May 14, 2019 1University of Tartu, Estonia; 2University of Helsinki, Finland; 3Royal Holloway, University of London, UK Abstract The article studies different methods for estimating the Viterbi path in the Bayesian framework. The Viterbi path is an estimate of the underlying state path in hidden Markov models (HMMs), which has a maximum joint posterior probability. Hence it is also called the maximum a posteriori (MAP) path. For an HMM with given parameters, the Viterbi path can be easily found with the Viterbi algorithm. In the Bayesian framework the Viterbi algorithm is not applicable and several iterative methods can be used instead. We introduce a new EM-type algorithm for finding the MAP path and compare it with various other methods for finding the MAP path, including the variational Bayes approach and MCMC methods. Examples with simulated data are used to compare the performance of the methods. The main focus is on non-stochastic iterative methods and our results show that the best of those methods work as well or better than the best MCMC methods. Our results demonstrate that when the primary goal is segmentation, then it is more reasonable to perform segmentation directly by considering the transition and emission parameters as nuisance parameters. Keywords: HMM, Bayes inference, MAP path, Viterbi algorithm, segmentation, EM, variational Bayes, simulated annealing 1 Introduction and preliminaries arXiv:1802.01630v2 [stat.CO] 11 May 2019 Hidden Markov models (HMMs) are widely used in application areas including speech recognition, com- putational linguistics, computational molecular biology, and many more. -
Arxiv:1811.03543V1 [Math.LO]
PREDICATIVE WELL-ORDERING NIK WEAVER Abstract. Confusion over the predicativist conception of well-ordering per- vades the literature and is responsible for widespread fundamental miscon- ceptions about the nature of predicative reasoning. This short note aims to explain the core fallacy, first noted in [9], and some of its consequences. 1. Predicativism Predicativism arose in the early 20th century as a response to the foundational crisis which resulted from the discovery of the classical paradoxes of naive set theory. It was initially developed in the writings of Poincar´e, Russell, and Weyl. Their central concern had to do with the avoidance of definitions they considered to be circular. Most importantly, they forbade any definition of a real number which involves quantification over all real numbers. This version of predicativism is sometimes called “predicativism given the nat- ural numbers” because there is no similar prohibition against defining a natural number by means of a condition which quantifies over all natural numbers. That is, one accepts N as being “already there” in some sense which is sufficient to void any danger of vicious circularity. In effect, predicativists of this type consider un- countable collections to be proper classes. On the other hand, they regard countable sets and constructions as unproblematic. 2. Second order arithmetic Second order arithmetic, in which one has distinct types of variables for natural numbers (a, b, ...) and for sets of natural numbers (A, B, ...), is thus a good setting for predicative reasoning — predicative given the natural numbers, but I will not keep repeating this. Here it becomes easier to frame the restriction mentioned above in terms of P(N), the power set of N, rather than in terms of R. -
A Guide to Self-Distributive Quasigroups, Or Latin Quandles
A GUIDE TO SELF-DISTRIBUTIVE QUASIGROUPS, OR LATIN QUANDLES DAVID STANOVSKY´ Abstract. We present an overview of the theory of self-distributive quasigroups, both in the two- sided and one-sided cases, and relate the older results to the modern theory of quandles, to which self-distributive quasigroups are a special case. Most attention is paid to the representation results (loop isotopy, linear representation, homogeneous representation), as the main tool to investigate self-distributive quasigroups. 1. Introduction 1.1. The origins of self-distributivity. Self-distributivity is such a natural concept: given a binary operation on a set A, fix one parameter, say the left one, and consider the mappings ∗ La(x) = a x, called left translations. If all such mappings are endomorphisms of the algebraic structure (A,∗ ), the operation is called left self-distributive (the prefix self- is usually omitted). Equationally,∗ the property says a (x y) = (a x) (a y) ∗ ∗ ∗ ∗ ∗ for every a, x, y A, and we see that distributes over itself. Self-distributivity∈ was pinpointed already∗ in the late 19th century works of logicians Peirce and Schr¨oder [69, 76], and ever since, it keeps appearing in a natural way throughout mathematics, perhaps most notably in low dimensional topology (knot and braid invariants) [12, 15, 63], in the theory of symmetric spaces [57] and in set theory (Laver’s groupoids of elementary embeddings) [15]. Recently, Moskovich expressed an interesting statement on his blog [60] that while associativity caters to the classical world of space and time, distributivity is, perhaps, the setting for the emerging world of information. -
Higher Braid Groups and Regular Semigroups from Polyadic-Binary Correspondence
mathematics Article Higher Braid Groups and Regular Semigroups from Polyadic-Binary Correspondence Steven Duplij Center for Information Technology (WWU IT), Universität Münster, Röntgenstrasse 7-13, D-48149 Münster, Germany; [email protected] Abstract: In this note, we first consider a ternary matrix group related to the von Neumann regular semigroups and to the Artin braid group (in an algebraic way). The product of a special kind of ternary matrices (idempotent and of finite order) reproduces the regular semigroups and braid groups with their binary multiplication of components. We then generalize the construction to the higher arity case, which allows us to obtain some higher degree versions (in our sense) of the regular semigroups and braid groups. The latter are connected with the generalized polyadic braid equation and R-matrix introduced by the author, which differ from any version of the well-known tetrahedron equation and higher-dimensional analogs of the Yang-Baxter equation, n-simplex equations. The higher degree (in our sense) Coxeter group and symmetry groups are then defined, and it is shown that these are connected only in the non-higher case. Keywords: regular semigroup; braid group; generator; relation; presentation; Coxeter group; symmetric group; polyadic matrix group; querelement; idempotence; finite order element MSC: 16T25; 17A42; 20B30; 20F36; 20M17; 20N15 Citation: Duplij, S. Higher Braid Groups and Regular Semigroups from Polyadic-Binary 1. Introduction Correspondence. Mathematics 2021, 9, We begin by observing that the defining relations of the von Neumann regular semi- 972. https://doi.org/10.3390/ groups (e.g., References [1–3]) and the Artin braid group [4,5] correspond to such properties math9090972 of ternary matrices (over the same set) as idempotence and the orders of elements (period). -
Learning Binary Relations and Total Orders
Learning Binary Relations and Total Orders Sally A Goldman Ronald L Rivest Department of Computer Science MIT Lab oratory for Computer Science Washington University Cambridge MA St Louis MO rivesttheorylcsmitedu sgcswustledu Rob ert E Schapire ATT Bell Lab oratories Murray Hill NJ schapireresearchattcom Abstract We study the problem of learning a binary relation b etween two sets of ob jects or b etween a set and itself We represent a binary relation b etween a set of size n and a set of size m as an n m matrix of bits whose i j entry is if and only if the relation holds b etween the corresp onding elements of the twosetsWe present p olynomial prediction algorithms for learning binary relations in an extended online learning mo del where the examples are drawn by the learner by a helpful teacher by an adversary or according to a uniform probability distribution on the instance space In the rst part of this pap er we present results for the case that the matrix of the relation has at most k rowtyp es We present upp er and lower b ounds on the number of prediction mistakes any prediction algorithm makes when learning such a matrix under the extended online learning mo del Furthermore we describ e a technique that simplies the pro of of exp ected mistake b ounds against a randomly chosen query sequence In the second part of this pap er we consider the problem of learning a binary re lation that is a total order on a set We describ e a general technique using a fully p olynomial randomized approximation scheme fpras to implement a randomized -
Propositional Fuzzy Logics: Tableaux and Strong Completeness Agnieszka Kulacka
Imperial College London Department of Computing Propositional Fuzzy Logics: Tableaux and Strong Completeness Agnieszka Kulacka A thesis submitted in partial fulfilment of the requirements for the degree of Doctor of Philosophy in Computing Research. London 2017 Acknowledgements I would like to thank my supervisor, Professor Ian Hodkinson, for his patient guidance and always responding to my questions promptly and helpfully. I am deeply grateful for his thorough explanations of difficult topics, in-depth discus- sions and for enlightening suggestions on the work at hand. Studying under the supervision of Professor Hodkinson also proved that research in logic is enjoyable. Two other people influenced the quality of this work, and these are my exam- iners, whose constructive comments shaped the thesis to much higher standards both in terms of the content as well as the presentation of it. This project would not have been completed without encouragement and sup- port of my husband, to whom I am deeply indebted for that. Abstract In his famous book Mathematical Fuzzy Logic, Petr H´ajekdefined a new fuzzy logic, which he called BL. It is weaker than the three fundamental fuzzy logics Product,Lukasiewicz and G¨odel,which are in turn weaker than classical logic, but axiomatic systems for each of them can be obtained by adding axioms to BL. Thus, H´ajek placed all these logics in a unifying axiomatic framework. In this dissertation, two problems concerning BL and other fuzzy logics have been considered and solved. One was to construct tableaux for BL and for BL with additional connectives. Tableaux are automatic systems to verify whether a given formula must have given truth values, or to build a model in which it does not have these specific truth values. -
Order Relations and Functions
Order Relations and Functions Problem Session Tonight 7:00PM – 7:50PM 380-380X Optional, but highly recommended! Recap from Last Time Relations ● A binary relation is a property that describes whether two objects are related in some way. ● Examples: ● Less-than: x < y ● Divisibility: x divides y evenly ● Friendship: x is a friend of y ● Tastiness: x is tastier than y ● Given binary relation R, we write aRb iff a is related to b by relation R. Order Relations “x is larger than y” “x is tastier than y” “x is faster than y” “x is a subset of y” “x divides y” “x is a part of y” Informally An order relation is a relation that ranks elements against one another. Do not use this definition in proofs! It's just an intuition! Properties of Order Relations x ≤ y Properties of Order Relations x ≤ y 1 ≤ 5 and 5 ≤ 8 Properties of Order Relations x ≤ y 1 ≤ 5 and 5 ≤ 8 1 ≤ 8 Properties of Order Relations x ≤ y 42 ≤ 99 and 99 ≤ 137 Properties of Order Relations x ≤ y 42 ≤ 99 and 99 ≤ 137 42 ≤ 137 Properties of Order Relations x ≤ y x ≤ y and y ≤ z Properties of Order Relations x ≤ y x ≤ y and y ≤ z x ≤ z Properties of Order Relations x ≤ y x ≤ y and y ≤ z x ≤ z Transitivity Properties of Order Relations x ≤ y Properties of Order Relations x ≤ y 1 ≤ 1 Properties of Order Relations x ≤ y 42 ≤ 42 Properties of Order Relations x ≤ y 137 ≤ 137 Properties of Order Relations x ≤ y x ≤ x Properties of Order Relations x ≤ y x ≤ x Reflexivity Properties of Order Relations x ≤ y Properties of Order Relations x ≤ y 19 ≤ 21 Properties of Order Relations x ≤ y 19 ≤ 21 21 ≤ 19? Properties of Order Relations x ≤ y 19 ≤ 21 21 ≤ 19? Properties of Order Relations x ≤ y 42 ≤ 137 Properties of Order Relations x ≤ y 42 ≤ 137 137 ≤ 42? Properties of Order Relations x ≤ y 42 ≤ 137 137 ≤ 42? Properties of Order Relations x ≤ y 137 ≤ 137 Properties of Order Relations x ≤ y 137 ≤ 137 137 ≤ 137? Properties of Order Relations x ≤ y 137 ≤ 137 137 ≤ 137 Antisymmetry A binary relation R over a set A is called antisymmetric iff For any x ∈ A and y ∈ A, If xRy and y ≠ x, then yRx. -
Sequence Models
Sequence Models Noah Smith Computer Science & Engineering University of Washington [email protected] Lecture Outline 1. Markov models 2. Hidden Markov models 3. Viterbi algorithm 4. Other inference algorithms for HMMs 5. Learning algorithms for HMMs Shameless Self-Promotion • Linguistic Structure Prediction (2011) • Links material in this lecture to many related ideas, including some in other LxMLS lectures. • Available in electronic and print form. MARKOV MODELS One View of Text • Sequence of symbols (bytes, letters, characters, morphemes, words, …) – Let Σ denote the set of symbols. • Lots of possible sequences. (Σ* is infinitely large.) • Probability distributions over Σ*? Pop QuiZ • Am I wearing a generative or discriminative hat right now? Pop QuiZ • Generative models tell a • Discriminative models mythical story to focus on tasks (like explain the data. sorting examples). Trivial Distributions over Σ* • Give probability 0 to sequences with length greater than B; uniform over the rest. • Use data: with N examples, give probability N-1 to each observed sequence, 0 to the rest. • What if we want every sequence to get some probability? – Need a probabilistic model family and algorithms for constructing the model from data. A History-Based Model n+1 p(start,w1,w2,...,wn, stop) = γ(wi w1,w2,...,wi 1) | − i=1 • Generate each word from left to right, conditioned on what came before it. Die / Dice one die two dice start one die per history: … … … start I one die per history: … … … history = start start I want one die per history: … … history = start I … start I want a one die per history: … … … history = start I want start I want a flight one die per history: … … … history = start I want a start I want a flight to one die per history: … … … history = start I want a flight start I want a flight to Lisbon one die per history: … … … history = start I want a flight to start I want a flight to Lisbon .