Domain Theory: an Introduction
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CTL Model Checking
CTL Model Checking Prof. P.H. Schmitt Institut fur¨ Theoretische Informatik Fakult¨at fur¨ Informatik Universit¨at Karlsruhe (TH) Formal Systems II Prof. P.H. Schmitt CTLMC Summer 2009 1 / 26 Fixed Point Theory Prof. P.H. Schmitt CTLMC Summer 2009 2 / 26 Fixed Points Definition Definition Let f : P(G) → P(G) be a set valued function and Z a subset of G. 1. Z is called a fixed point of f if f(Z) = Z . 2. Z is called the least fixed point of f is Z is a fixed point and for all other fixed points U of f the relation Z ⊆ U is true. 3. Z is called the greatest fixed point of f is Z is a fixed point and for all other fixed points U of f the relation U ⊆ Z is true. Prof. P.H. Schmitt CTLMC Summer 2009 3 / 26 Finite Fixed Point Lemma Let G be an arbitrary set, Let P(G) denote the power set of a set G. A function f : P(G) → P(G) is called monotone of for all X, Y ⊆ G X ⊆ Y ⇒ f(X) ⊆ f(Y ) Let f : P(G) → P(G) be a monotone function on a finite set G. 1. There is a least and a greatest fixed point of f. S n 2. n≥1 f (∅) is the least fixed point of f. T n 3. n≥1 f (G) is the greatest fixed point of f. Prof. P.H. Schmitt CTLMC Summer 2009 4 / 26 Proof of part (2) of the finite fixed point Lemma Monotonicity of f yields ∅ ⊆ f(∅) ⊆ f 2(∅) ⊆ .. -
Categories and Types for Axiomatic Domain Theory Adam Eppendahl
Department of Computer Science Research Report No. RR-03-05 ISSN 1470-5559 December 2003 Categories and Types for Axiomatic Domain Theory Adam Eppendahl 1 Categories and Types for Axiomatic Domain Theory Adam Eppendahl Submitted for the degree of Doctor of Philosophy University of London 2003 2 3 Categories and Types for Axiomatic Domain Theory Adam Eppendahl Abstract Domain Theory provides a denotational semantics for programming languages and calculi con- taining fixed point combinators and other so-called paradoxical combinators. This dissertation presents results in the category theory and type theory of Axiomatic Domain Theory. Prompted by the adjunctions of Domain Theory, we extend Benton’s linear/nonlinear dual- sequent calculus to include recursive linear types and define a class of models by adding Freyd’s notion of algebraic compactness to the monoidal adjunctions that model Benton’s calculus. We observe that algebraic compactness is better behaved in the context of categories with structural actions than in the usual context of enriched categories. We establish a theory of structural algebraic compactness that allows us to describe our models without reference to en- richment. We develop a 2-categorical perspective on structural actions, including a presentation of monoidal categories that leads directly to Kelly’s reduced coherence conditions. We observe that Benton’s adjoint type constructors can be treated individually, semantically as well as syntactically, using free representations of distributors. We type various of fixed point combinators using recursive types and function types, which we consider the core types of such calculi, together with the adjoint types. We use the idioms of these typings, which include oblique function spaces, to give a translation of the core of Levy’s Call-By-Push-Value. -
Slides by Akihisa
Complete non-orders and fixed points Akihisa Yamada1, Jérémy Dubut1,2 1 National Institute of Informatics, Tokyo, Japan 2 Japanese-French Laboratory for Informatics, Tokyo, Japan Supported by ERATO HASUO Metamathematics for Systems Design Project (No. JPMJER1603), JST Introduction • Interactive Theorem Proving is appreciated for reliability • But it's also engineering tool for mathematics (esp. Isabelle/jEdit) • refactoring proofs and claims • sledgehammer • quickcheck/nitpick(/nunchaku) • We develop an Isabelle library of order theory (as a case study) ⇒ we could generalize many known results, like: • completeness conditions: duality and relationships • Knaster-Tarski fixed-point theorem • Kleene's fixed-point theorem Order A binary relation ⊑ • reflexive ⟺ � ⊑ � • transitive ⟺ � ⊑ � and � ⊑ � implies � ⊑ � • antisymmetric ⟺ � ⊑ � and � ⊑ � implies � = � • partial order ⟺ reflexive + transitive + antisymmetric Order A binary relation ⊑ locale less_eq_syntax = fixes less_eq :: 'a ⇒ 'a ⇒ bool (infix "⊑" 50) • reflexive ⟺ � ⊑ � locale reflexive = ... assumes "x ⊑ x" • transitive ⟺ � ⊑ � and � ⊑ � implies � ⊑ � locale transitive = ... assumes "x ⊑ y ⟹ y ⊑ z ⟹ x ⊑ z" • antisymmetric ⟺ � ⊑ � and � ⊑ � implies � = � locale antisymmetric = ... assumes "x ⊑ y ⟹ y ⊑ x ⟹ x = y" • partial order ⟺ reflexive + transitive + antisymmetric locale partial_order = reflexive + transitive + antisymmetric Quasi-order A binary relation ⊑ locale less_eq_syntax = fixes less_eq :: 'a ⇒ 'a ⇒ bool (infix "⊑" 50) • reflexive ⟺ � ⊑ � locale reflexive = ... assumes -
Topology, Domain Theory and Theoretical Computer Science
Draft URL httpwwwmathtulaneedumwmtopandcspsgz pages Top ology Domain Theory and Theoretical Computer Science Michael W Mislove Department of Mathematics Tulane University New Orleans LA Abstract In this pap er we survey the use of ordertheoretic top ology in theoretical computer science with an emphasis on applications of domain theory Our fo cus is on the uses of ordertheoretic top ology in programming language semantics and on problems of p otential interest to top ologists that stem from concerns that semantics generates Keywords Domain theory Scott top ology p ower domains untyp ed lamb da calculus Subject Classication BFBNQ Intro duction Top ology has proved to b e an essential to ol for certain asp ects of theoretical computer science Conversely the problems that arise in the computational setting have provided new and interesting stimuli for top ology These prob lems also have increased the interaction b etween top ology and related areas of mathematics such as order theory and top ological algebra In this pap er we outline some of these interactions b etween top ology and theoretical computer science fo cusing on those asp ects that have b een most useful to one particular area of theoretical computation denotational semantics This pap er b egan with the goal of highlighting how the interaction of order and top ology plays a fundamental role in programming semantics and related areas It also started with the viewp oint that there are many purely top o logical notions that are useful in theoretical computer science that -
Mathematical Tools MPRI 2–6: Abstract Interpretation, Application to Verification and Static Analysis
Mathematical Tools MPRI 2{6: Abstract Interpretation, application to verification and static analysis Antoine Min´e CNRS, Ecole´ normale sup´erieure course 1, 2012{2013 course 1, 2012{2013 Mathematical Tools Antoine Min´e p. 1 / 44 Order theory course 1, 2012{2013 Mathematical Tools Antoine Min´e p. 2 / 44 Partial orders Partial orders course 1, 2012{2013 Mathematical Tools Antoine Min´e p. 3 / 44 Partial orders Partial orders Given a set X , a relation v2 X × X is a partial order if it is: 1 reflexive: 8x 2 X ; x v x 2 antisymmetric: 8x; y 2 X ; x v y ^ y v x =) x = y 3 transitive: 8x; y; z 2 X ; x v y ^ y v z =) x v z. (X ; v) is a poset (partially ordered set). If we drop antisymmetry, we have a preorder instead. course 1, 2012{2013 Mathematical Tools Antoine Min´e p. 4 / 44 Partial orders Examples of posets (Z; ≤) is a poset (in fact, completely ordered) (P(X ); ⊆) is a poset (not completely ordered) (S; =) is a poset for any S course 1, 2012{2013 Mathematical Tools Antoine Min´e p. 5 / 44 Partial orders Examples of posets (cont.) Given by a Hasse diagram, e.g.: g e f g v g c d f v f ; g e v e; g b d v d; f ; g c v c; e; f ; g b v b; c; d; e; f ; g a a v a; b; c; d; e; f ; g course 1, 2012{2013 Mathematical Tools Antoine Min´e p. -
The Poset of Closure Systems on an Infinite Poset: Detachability and Semimodularity Christian Ronse
The poset of closure systems on an infinite poset: Detachability and semimodularity Christian Ronse To cite this version: Christian Ronse. The poset of closure systems on an infinite poset: Detachability and semimodularity. Portugaliae Mathematica, European Mathematical Society Publishing House, 2010, 67 (4), pp.437-452. 10.4171/PM/1872. hal-02882874 HAL Id: hal-02882874 https://hal.archives-ouvertes.fr/hal-02882874 Submitted on 31 Aug 2020 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Portugal. Math. (N.S.) Portugaliae Mathematica Vol. xx, Fasc. , 200x, xxx–xxx c European Mathematical Society The poset of closure systems on an infinite poset: detachability and semimodularity Christian Ronse Abstract. Closure operators on a poset can be characterized by the corresponding clo- sure systems. It is known that in a directed complete partial order (DCPO), in particular in any finite poset, the collection of all closure systems is closed under arbitrary inter- section and has a “detachability” or “anti-matroid” property, which implies that the collection of all closure systems is a lower semimodular complete lattice (and dually, the closure operators form an upper semimodular complete lattice). After reviewing the history of the problem, we generalize these results to the case of an infinite poset where closure systems do not necessarily constitute a complete lattice; thus the notions of lower semimodularity and detachability are extended accordingly. -
Programming Language Semantics a Rewriting Approach
Programming Language Semantics A Rewriting Approach Grigore Ros, u University of Illinois at Urbana-Champaign Chapter 2 Background and Preliminaries 15 2.6 Complete Partial Orders and the Fixed-Point Theorem This section introduces a fixed-point theorem for complete partial orders. Our main use of this theorem is to give denotational semantics to iterative (in Section 3.4) and to recursive (in Section 4.8) language constructs. Complete partial orders with bottom elements are at the core of both domain theory and denotational semantics, often identified with the notion of “domain” itself. 2.6.1 Posets, Upper Bounds and Least Upper Bounds Here we recall basic notions of partial and total orders, such as upper bounds and least upper bounds, and discuss several examples and counter-examples. Definition 14. A partial order, or a poset (from partial order set) (D, ) consists of a set D and a binary v relation on D, written as an infix operation, which is v reflexive, that is, x x for any x D; • v ∈ transitive, that is, for any x, y, z D, x y and y z imply x z; and • ∈ v v v anti-symmetric, that is, if x y and y x for some x, y D then x = y. • v v ∈ A partial order (D, ) is called a total order when x y or y x for any x, y D. v v v ∈ Here are some examples of partial or total orders: ( (S ), ) is a partial order, where (S ) is the powerset of a set S , that is, the set of subsets of S . -
Partial Orders — Basics
Partial Orders — Basics Edward A. Lee UC Berkeley — EECS EECS 219D — Concurrent Models of Computation Last updated: January 23, 2014 Outline Sets Join (Least Upper Bound) Relations and Functions Meet (Greatest Lower Bound) Notation Example of Join and Meet Directed Sets, Bottom Partial Order What is Order? Complete Partial Order Strict Partial Order Complete Partial Order Chains and Total Orders Alternative Definition Quiz Example Partial Orders — Basics Sets Frequently used sets: • B = {0, 1}, the set of binary digits. • T = {false, true}, the set of truth values. • N = {0, 1, 2, ···}, the set of natural numbers. • Z = {· · · , −1, 0, 1, 2, ···}, the set of integers. • R, the set of real numbers. • R+, the set of non-negative real numbers. Edward A. Lee | UC Berkeley — EECS3/32 Partial Orders — Basics Relations and Functions • A binary relation from A to B is a subset of A × B. • A partial function f from A to B is a relation where (a, b) ∈ f and (a, b0) ∈ f =⇒ b = b0. Such a partial function is written f : A*B. • A total function or just function f from A to B is a partial function where for all a ∈ A, there is a b ∈ B such that (a, b) ∈ f. Edward A. Lee | UC Berkeley — EECS4/32 Partial Orders — Basics Notation • A binary relation: R ⊆ A × B. • Infix notation: (a, b) ∈ R is written aRb. • A symbol for a relation: • ≤⊂ N × N • (a, b) ∈≤ is written a ≤ b. • A function is written f : A → B, and the A is called its domain and the B its codomain. -
Least Fixed Points Revisited
LEAST FIXED POINTS REVISITED J.W. de Bakker Mathematical Centre, Amsterdam ABSTRACT Parameter mechanisms for recursive procedures are investigated. Con- trary to the view of Manna et al., it is argued that both call-by-value and call-by-name mechanisms yield the least fixed points of the functionals de- termined by the bodies of the procedures concerned. These functionals dif- fer, however, according to the mechanism chosen. A careful and detailed pre- sentation of this result is given, along the lines of a simple typed lambda calculus, with interpretation rules modelling program execution in such a way that call-by-value determines a change in the environment and call-by- name a textual substitution in the procedure body. KEY WORDS AND PHRASES: Semantics, recursion, least fixed points, parameter mechanisms, call-by-value, call-by-name, lambda calculus. Author's address Mathematical Centre 2e Boerhaavestraat 49 Amsterdam Netherlands 28 NOTATION Section 2 s,t,ti,t',... individual } function terms S,T,... functional p,p',... !~individual} boolean .J function terms P,... functional a~A, aeA} constants b r B, IBeB x,y,z,u X, ~,~ X] variables qcQ, x~0. J r149149 procedure symbols T(t I ..... tn), ~(tl,.-.,tn) ~ application T(TI' '~r)' P(~I' '~r)J ~x I . .xs I . .Ym.t, ~x] . .xs I . .ym.p, abstraction h~l ...~r.T, hX1 ..Xr.~ if p then t' else t"~ [ selection if p then p' else p"J n(T), <n(T),r(r)> rank 1 -< i <- n, 1 -< j -< r,) indices 1 -< h -< l, 1 _< k <- t,T,x,~, . -
Formal Programming Language Semantics Note 12 Complete Partial
Formal Programming Language Semantics note 12 CS4/MSc/TPG 18.11.02 Formal Programming Language Semantics note 12 Complete partial orders (CPOs) In this note we re-examine some of the ideas in the previous two notes from a slightly more abstract perspective. In particular, we are going to concentrate on the ideas behind the denotational semantics of while commands. By isolating the aspects of the situation which really made the semantics work, we will arrive at the basic concepts of a subject known as domain theory, which provides a general mathematical framework for much of denotational semantics. Complete partial orders. Let us write D for the set DCom = (S * S) from Note 9. Recall that we defined an element h ∈ D as the union of an increasing chain h0 ⊆ h1 ⊆ · · · . We now ask: what are the essential features of D that made this construction work? One reasonable answer (though not the only one!) is given by the following definitions: Definition 1 A partially ordered set, or poset, is a set D equipped with a binary relation v (called an order relation) such that the following hold for all x, y, z ∈ D: • x v x ( v is reflexive); • if x v y and y v z then x v z ( v is transitive); • if x v y and y v x then x = y ( v is antisymmetric). Definition 2 A complete partial order, or CPO, is a poset with the following prop- erty: every sequence of elements x0 v x1 v · · · in D has a limit or least upper F bound in D: that is, there is an element x ∈ D (written as i xi) such that • xi v x for all i (i.e. -
Arxiv:1301.2793V2 [Math.LO] 18 Jul 2014 Uaeacnetof Concept a Mulate Fpwrbet Uetepwre Xo) N/Rmk S Ff of Use Make And/Or Axiom), Presupp Principles
ON TARSKI’S FIXED POINT THEOREM GIOVANNI CURI To Orsola Abstract. A concept of abstract inductive definition on a complete lattice is formulated and studied. As an application, a constructive version of Tarski’s fixed point theorem is obtained. Introduction The fixed point theorem referred to in this paper is the one asserting that every monotone mapping on a complete lattice L has a least fixed point. The proof, due to A. Tarski, of this result, is a simple and most significant example of a proof that can be carried out on the base of intuitionistic logic (e.g. in the intuitionistic set theory IZF, or in topos logic), and that yet is widely regarded as essentially non-constructive. The reason for this fact is that Tarski’s construction of the fixed point is highly impredicative: if f : L → L is a monotone map, its least fixed point is given by V P , with P ≡ {x ∈ L | f(x) ≤ x}. Impredicativity here is found in the fact that the fixed point, call it p, appears in its own construction (p belongs to P ), and, indirectly, in the fact that the complete lattice L (and, as a consequence, the collection P over which the infimum is taken) is assumed to form a set, an assumption that seems only reasonable in an intuitionistic setting in the presence of strong impredicative principles (cf. Section 2 below). In concrete applications (e.g. in computer science and numerical analysis) the monotone operator f is often also continuous, in particular it preserves suprema of non-empty chains; in this situation, the least fixed point can be constructed taking the supremum of the ascending chain ⊥,f(⊥),f(f(⊥)), ..., given by the set of finite iterations of f on the least element ⊥. -
Two Counterexamples Concerning the Scott Topology on a Partial Order
TWO COUNTEREXAMPLES CONCERNING THE SCOTT TOPOLOGY ON A PARTIAL ORDER PETER HERTLING Fakult¨atf¨urInformatik, Universit¨atder Bundeswehr M¨unchen, 85577 Neubiberg, Germany e-mail address: [email protected] Abstract. We construct a complete lattice Z such that the binary supremum function sup : Z × Z ! Z is discontinuous with respect to the product topology on Z × Z of the Scott topologies on each copy of Z. In addition, we show that bounded completeness of a complete lattice Z is in general not inherited by the dcpo C(X; Z) of continuous functions from X to Z where X may be any topological space and where on Z the Scott topology is considered. 1. Introduction In this note two counterexamples in the theory of partial orders are constructed. The first counterexample concerns the question whether the binary supremum function on a sup semilattice is continuous with respect to the product topology of the Scott topology on each copy of the sup semilattice. The second counterexample concerns the question whether bounded completeness of a complete lattice is inherited by the dcpo of continuous functions from some topological space to the complete lattice. The Scott topology on a partially ordered set (short: poset)(X; ) has turned out to v be important in many areas of computer science. Given two posets (X1; 1) and (X2; 2), v v one can in a natural way define a partial order relation × on the product X1 X2. This v × gives us two natural topologies on the product Z = X1 X2: × The product of the Scott topology on (X1; 1) and of the Scott topology on (X2; 2), • v v the Scott topology on (Z; ×).