Descriptive Complexity: a Logician's Approach to Computation
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Infinitary Logic and Inductive Definability Over Finite Structures
University of Pennsylvania ScholarlyCommons Technical Reports (CIS) Department of Computer & Information Science November 1991 Infinitary Logic and Inductive Definability Over Finite Structures Anuj Dawar University of Pennsylvania Steven Lindell University of Pennsylvania Scott Weinstein University of Pennsylvania Follow this and additional works at: https://repository.upenn.edu/cis_reports Recommended Citation Anuj Dawar, Steven Lindell, and Scott Weinstein, "Infinitary Logic and Inductive Definability Over Finite Structures", . November 1991. University of Pennsylvania Department of Computer and Information Science Technical Report No. MS-CIS-91-97. This paper is posted at ScholarlyCommons. https://repository.upenn.edu/cis_reports/365 For more information, please contact [email protected]. Infinitary Logic and Inductive Definability Over Finite Structures Abstract The extensions of first-order logic with a least fixed point operators (FO + LFP) and with a partial fixed point operator (FO + PFP) are known to capture the complexity classes P and PSPACE respectively in the presence of an ordering relation over finite structures. Recently, Abiteboul and Vianu [AV91b] investigated the relation of these two logics in the absence of an ordering, using a mchine model of generic computation. In particular, they showed that the two languages have equivalent expressive power if and only if P = PSPACE. These languages can also be seen as fragments of an infinitary logic where each ω formula has a bounded number of variables, L ∞ω (see, for instance, [KV90]). We present a treatment of the results in [AV91b] from this point of view. In particular, we show that we can write a formula of FO + LFP and P from ordered structures to classes of structures where every element is definable. -
Relations 21
2. CLOSURE OF RELATIONS 21 2. Closure of Relations 2.1. Definition of the Closure of Relations. Definition 2.1.1. Given a relation R on a set A and a property P of relations, the closure of R with respect to property P , denoted ClP (R), is smallest relation on A that contains R and has property P . That is, ClP (R) is the relation obtained by adding the minimum number of ordered pairs to R necessary to obtain property P . Discussion To say that ClP (R) is the “smallest” relation on A containing R and having property P means that • R ⊆ ClP (R), • ClP (R) has property P , and • if S is another relation with property P and R ⊆ S, then ClP (R) ⊆ S. The following theorem gives an equivalent way to define the closure of a relation. \ Theorem 2.1.1. If R is a relation on a set A, then ClP (R) = S, where S∈P P = {S|R ⊆ S and S has property P }. Exercise 2.1.1. Prove Theorem 2.1.1. [Recall that one way to show two sets, A and B, are equal is to show A ⊆ B and B ⊆ A.] 2.2. Reflexive Closure. Definition 2.2.1. Let A be a set and let ∆ = {(x, x)|x ∈ A}. ∆ is called the diagonal relation on A. Theorem 2.2.1. Let R be a relation on A. The reflexive closure of R, denoted r(R), is the relation R ∪ ∆. Proof. Clearly, R ∪ ∆ is reflexive, since (a, a) ∈ ∆ ⊆ R ∪ ∆ for every a ∈ A. -
Scalability of Reflexive-Transitive Closure Tables As Hierarchical Data Solutions
Scalability of Reflexive- Transitive Closure Tables as Hierarchical Data Solutions Analysis via SQL Server 2008 R2 Brandon Berry, Engineer 12/8/2011 Reflexive-Transitive Closure (RTC) tables, when used to persist hierarchical data, present many advantages over alternative persistence strategies such as path enumeration. Such advantages include, but are not limited to, referential integrity and a simple structure for data queries. However, RTC tables grow exponentially and present a concern with scaling both operational performance of the RTC table and volume of the RTC data. Discovering these practical performance boundaries involves understanding the growth patterns of reflexive and transitive binary relations and observing sample data for large hierarchical models. Table of Contents 1 Introduction ......................................................................................................................................... 2 2 Reflexive-Transitive Closure Properties ................................................................................................. 3 2.1 Reflexivity ...................................................................................................................................... 3 2.2 Transitivity ..................................................................................................................................... 3 2.3 Closure .......................................................................................................................................... 4 3 Scalability -
The Complexity Zoo
The Complexity Zoo Scott Aaronson www.ScottAaronson.com LATEX Translation by Chris Bourke [email protected] 417 classes and counting 1 Contents 1 About This Document 3 2 Introductory Essay 4 2.1 Recommended Further Reading ......................... 4 2.2 Other Theory Compendia ............................ 5 2.3 Errors? ....................................... 5 3 Pronunciation Guide 6 4 Complexity Classes 10 5 Special Zoo Exhibit: Classes of Quantum States and Probability Distribu- tions 110 6 Acknowledgements 116 7 Bibliography 117 2 1 About This Document What is this? Well its a PDF version of the website www.ComplexityZoo.com typeset in LATEX using the complexity package. Well, what’s that? The original Complexity Zoo is a website created by Scott Aaronson which contains a (more or less) comprehensive list of Complexity Classes studied in the area of theoretical computer science known as Computa- tional Complexity. I took on the (mostly painless, thank god for regular expressions) task of translating the Zoo’s HTML code to LATEX for two reasons. First, as a regular Zoo patron, I thought, “what better way to honor such an endeavor than to spruce up the cages a bit and typeset them all in beautiful LATEX.” Second, I thought it would be a perfect project to develop complexity, a LATEX pack- age I’ve created that defines commands to typeset (almost) all of the complexity classes you’ll find here (along with some handy options that allow you to conveniently change the fonts with a single option parameters). To get the package, visit my own home page at http://www.cse.unl.edu/~cbourke/. -
Canonical Models and the Complexity of Modal Team Logic
On the Complexity of Team Logic and its Two-Variable Fragment Martin Lück Leibniz Universität Hannover, Germany [email protected] Abstract. We study the logic FO(∼), the extension of first-order logic with team semantics by unrestricted Boolean negation. It was recently shown axiomatizable, but otherwise has not yet received much attention in questions of computational complexity. In this paper, we consider its two-variable fragment FO2(∼) and prove that its satisfiability problem is decidable, and in fact complete for the recently introduced non-elementary class TOWER(poly). Moreover, we classify the complexity of model checking of FO(∼) with respect to the number of variables and the quantifier rank, and prove a di- chotomy between PSPACE- and ATIME-ALT(exp, poly)-completeness. To achieve the lower bounds, we propose a translation from modal team logic MTL to FO2(∼) that extends the well-known standard translation from modal logic ML to FO2. For the upper bounds, we translate to a fragment of second-order logic. Keywords: team semantics, two-variable logic, complexity, satisfiability, model checking 2012 ACM Subject Classification: Theory of computation → Complexity theory and logic; Logic; 1. Introduction In the last decades, the work of logicians has unearthed a plethora of decidable fragments of first-order logic FO. Many of these cases are restricted quantifier prefixes, such as the BSR-fragment which contains only ∃∗∀∗-sentences [30]. Others include the guarded arXiv:1804.04968v1 [cs.LO] 13 Apr 2018 fragment GF [1], the recently introduced separated fragment SF [32, 34], or the two-variable fragment FO2 [13, 27, 31]. -
MATH 213 Chapter 9: Relations
MATH 213 Chapter 9: Relations Dr. Eric Bancroft Fall 2013 9.1 - Relations Definition 1 (Relation). Let A and B be sets. A binary relation from A to B is a subset R ⊆ A×B, i.e., R is a set of ordered pairs where the first element from each pair is from A and the second element is from B. If (a; b) 2 R then we write a R b or a ∼ b (read \a relates/is related to b [by R]"). If (a; b) 2= R, then we write a R6 b or a b. We can represent relations graphically or with a chart in addition to a set description. Example 1. A = f0; 1; 2g;B = f1; 2; 3g;R = f(1; 1); (2; 1); (2; 2)g Example 2. (a) \Parent" (b) 8x; y 2 Z; x R y () x2 + y2 = 8 (c) A = f0; 1; 2g;B = f1; 2; 3g; a R b () a + b ≥ 3 1 Note: All functions are relations, but not all relations are functions. Definition 2. If A is a set, then a relation on A is a relation from A to A. Example 3. How many relations are there on a set with. (a) two elements? (b) n elements? (c) 14 elements? Properties of Relations Definition 3 (Reflexive). A relation R on a set A is said to be reflexive if and only if a R a for all a 2 A. Definition 4 (Symmetric). A relation R on a set A is said to be symmetric if and only if a R b =) b R a for all a; b 2 A. -
Properties of Binary Transitive Closure Logics Over Trees S K
8 Properties of binary transitive closure logics over trees S K Abstract Binary transitive closure logic (FO∗ for short) is the extension of first-order predicate logic by a transitive closure operator of binary relations. Deterministic binary transitive closure logic (FOD∗) is the restriction of FO∗ to deterministic transitive closures. It is known that these logics are more powerful than FO on arbitrary structures and on finite ordered trees. It is also known that they are at most as powerful as monadic second-order logic (MSO) on arbitrary structures and on finite trees. We will study the expressive power of FO∗ and FOD∗ on trees to show that several MSO properties can be expressed in FOD∗ (and hence FO∗). The following results will be shown. A linear order can be defined on the nodes of a tree. The class EVEN of trees with an even number of nodes can be defined. On arbitrary structures with a tree signature, the classes of trees and finite trees can be defined. There is a tree language definable in FOD∗ that cannot be recognized by any tree walking automaton. FO∗ is strictly more powerful than tree walking automata. These results imply that FOD∗ and FO∗ are neither compact nor do they have the L¨owenheim-Skolem-Upward property. Keywords B 8.1 Introduction The question aboutthe best suited logic for describing tree properties or defin- ing tree languages is an important one for model theoretic syntax as well as for querying treebanks. Model theoretic syntax is a research program in mathematical linguistics concerned with studying the descriptive complex- Proceedings of FG-2006. -
User's Guide for Complexity: a LATEX Package, Version 0.80
User’s Guide for complexity: a LATEX package, Version 0.80 Chris Bourke April 12, 2007 Contents 1 Introduction 2 1.1 What is complexity? ......................... 2 1.2 Why a complexity package? ..................... 2 2 Installation 2 3 Package Options 3 3.1 Mode Options .............................. 3 3.2 Font Options .............................. 4 3.2.1 The small Option ....................... 4 4 Using the Package 6 4.1 Overridden Commands ......................... 6 4.2 Special Commands ........................... 6 4.3 Function Commands .......................... 6 4.4 Language Commands .......................... 7 4.5 Complete List of Class Commands .................. 8 5 Customization 15 5.1 Class Commands ............................ 15 1 5.2 Language Commands .......................... 16 5.3 Function Commands .......................... 17 6 Extended Example 17 7 Feedback 18 7.1 Acknowledgements ........................... 19 1 Introduction 1.1 What is complexity? complexity is a LATEX package that typesets computational complexity classes such as P (deterministic polynomial time) and NP (nondeterministic polynomial time) as well as sets (languages) such as SAT (satisfiability). In all, over 350 commands are defined for helping you to typeset Computational Complexity con- structs. 1.2 Why a complexity package? A better question is why not? Complexity theory is a more recent, though mature area of Theoretical Computer Science. Each researcher seems to have his or her own preferences as to how to typeset Complexity Classes and has built up their own personal LATEX commands file. This can be frustrating, to say the least, when it comes to collaborations or when one has to go through an entire series of files changing commands for compatibility or to get exactly the look they want (or what may be required). -
Relations April 4
math 55 - Relations April 4 Relations Let A and B be two sets. A (binary) relation on A and B is a subset R ⊂ A × B. We often write aRb to mean (a; b) 2 R. The following are properties of relations on a set S (where above A = S and B = S are taken to be the same set S): 1. Reflexive: (a; a) 2 R for all a 2 S. 2. Symmetric: (a; b) 2 R () (b; a) 2 R for all a; b 2 S. 3. Antisymmetric: (a; b) 2 R and (b; a) 2 R =) a = b. 4. Transitive: (a; b) 2 R and (b; c) 2 R =) (a; c) 2 R for all a; b; c 2 S. Exercises For each of the following relations, which of the above properties do they have? 1. Let R be the relation on Z+ defined by R = f(a; b) j a divides bg. Reflexive, antisymmetric, and transitive 2. Let R be the relation on Z defined by R = f(a; b) j a ≡ b (mod 33)g. Reflexive, symmetric, and transitive 3. Let R be the relation on R defined by R = f(a; b) j a < bg. Symmetric and transitive 4. Let S be the set of all convergent sequences of rational numbers. Let R be the relation on S defined by f(a; b) j lim a = lim bg. Reflexive, symmetric, transitive 5. Let P be the set of all propositional statements. Let R be the relation on P defined by R = f(a; b) j a ! b is trueg. -
A Transitive Closure Algorithm for Test Generation
A Transitive Closure Algorithm for Test Generation Srimat T. Chakradhar, Member, IEEE, Vishwani D. Agrawal, Fellow, IEEE, and Steven G. Rothweiler Abstract-We present a transitive closure based test genera- Cox use reduction lists to quickly determine necessary as- tion algorithm. A test is obtained by determining signal values signments [5]. that satisfy a Boolean equation derived from the neural net- These algorithms solve two main problems: (1) they work model of the circuit incorporating necessary conditions for fault activation and path sensitization. The algorithm is a determine logical consequences of a partial set of signal sequence of two main steps that are repeatedly executed: tran- assignments and (2) they determine the order in which sitive closure computation and decision-making. To compute decisions on fixing signals should be made. the transitive closure, we start with an implication graph whose In spite of making extensive use of the circuit structure vertices are labeled as the true and false states of all signals. A directed edge (x, y) in this graph represents the controlling in- and function, most algorithms have several shortcomings. fluence of the true state of signal x on the true state of signal y First, they do not guarantee the identification of all logi- that are connected through a wire or a gate. Since the impli- cal consequences of a partial set of signal assignments. In cation graph only includes local pairwise (or binary) relations, other words, local conditions are easier to identify than it is a partial representation of the netlist. Higher-order signal the global ones. -
Accelerating Transitive Closure of Large-Scale Sparse Graphs
New Jersey Institute of Technology Digital Commons @ NJIT Theses Electronic Theses and Dissertations 12-31-2020 Accelerating transitive closure of large-scale sparse graphs Sanyamee Milindkumar Patel New Jersey Institute of Technology Follow this and additional works at: https://digitalcommons.njit.edu/theses Part of the Computer Sciences Commons Recommended Citation Patel, Sanyamee Milindkumar, "Accelerating transitive closure of large-scale sparse graphs" (2020). Theses. 1806. https://digitalcommons.njit.edu/theses/1806 This Thesis is brought to you for free and open access by the Electronic Theses and Dissertations at Digital Commons @ NJIT. It has been accepted for inclusion in Theses by an authorized administrator of Digital Commons @ NJIT. For more information, please contact [email protected]. Copyright Warning & Restrictions The copyright law of the United States (Title 17, United States Code) governs the making of photocopies or other reproductions of copyrighted material. Under certain conditions specified in the law, libraries and archives are authorized to furnish a photocopy or other reproduction. One of these specified conditions is that the photocopy or reproduction is not to be “used for any purpose other than private study, scholarship, or research.” If a, user makes a request for, or later uses, a photocopy or reproduction for purposes in excess of “fair use” that user may be liable for copyright infringement, This institution reserves the right to refuse to accept a copying order if, in its judgment, fulfillment -
Efficient Transitive Closure Algorithms
Efficient Transitive Closure Algorithms Yards E. Ioannidis t Raghu Rantakrishnan ComputerSciences Department Universityof Wisconsin Madison.WJ 53706 Abstract closure (e.g. to find the shortestpaths between pairs of We havedeveloped some efficient algorithms for nodes) since it loses path information by merging all computing the transitive closure of a directed graph. nodes in a strongly connectedcomponent. Only the This paper presentsthe algorithmsfor the problem of Seminaive algorithm computes selection queries reachability. The algorithms,however, can be adapted efficiently. Thus, if we wish to find all nodesreachable to deal with path computationsand a signitkantJy from a given node, or to find the longest path from a broaderclass of queriesbased on onesided recursions. given node, with the exceptionof the Seminaivealgo- We analyze these algorithms and compare them to ritbm, we must essentiallycompute the entire transitive algorithms in the literature. The resulti indicate that closure(or find the longestpath from every node in the thesealgorithms, in addition to their ability to deal with graph)first and then perform a selection. queries that am generakations of transitive closure, We presentnew algorithmsbased on depth-first also perform very efficiently, in particular, in the con- searchand a schemeof marking nodes (to record ear- text of a dish-baseddatabase environment. lier computation implicitly) that computes transitive closure efficiently. They can also be adaptedto deal 1. Introduction with selection queries ‘and path computations Severaltransitive closure algoritluus have been efficiently. Jn particular, in the context of databasesan presentedin the literature.These include the Warshall importantconsideration is J/O cost, since it is expected and Warren algorithms, which use a bit-matrix that relations will not fit in main memory.