Algebraic Trace Theory
|
|
- Joseph Bertram Goodman
- 5 years ago
- Views:
Transcription
1 Algebraic Trace Theory EE249 Roberto Passerone Material from: Jerry R. Burch, Trace Theory for Automatic Verification of Real-Time Concurrent Systems, PhD thesis, CMU, August 1992 October 21, 2002 ee249 1
2 Outline Introduction Concurrency Algebras Trace Algebras Trace Structure Algebras Conservative Approximations Homomorphisms October 21, 2002 ee249 2
3 Abstract Algebra x + 3 = 5 ( x + 3 ) + ( -3 ) = 5 + ( -3 ) x + ( 3 + ( -3 ) ) = 2 x + 0 = 2 x = 2 When is it that the solution of a linear equation is unique? October 21, 2002 ee249 3
4 Uniqueness of Solution Addition is associative There is an identity element There is an inverse for each element No need for commutativity! Axiomatic approach: don t define the objects, define their properties Group = (D, f, id, inv) with the properties above Many possible examples: Integers under addition (Z, +, 0, -) Positive reals under multiplication (R +, x, 1, -1 ) Permutations under functional composition (F,, Id, -1 ) October 21, 2002 ee249 4
5 Algebraic Approach The properties of the functions must be true in any implementation of the algebra Hence they allow you to prove general facts that are independent of the particular implementation of the algebra For example, uniqueness of solutions for linear equations We will represent models of computation as instances of an algebra and prove general facts The algebra gives as a tradeoff between generality and structure October 21, 2002 ee249 5
6 Outline Introduction Concurrency Algebras Trace Algebras Trace Structure Algebras Conservative Approximations Homomorphisms October 21, 2002 ee249 6
7 Objective Create models for agent behaviors Study the relationships between different models October 21, 2002 ee249 7
8 A Concrete Model Define a domain of agents The set of finite state automata Defines an interface for each agent A set of signal names Defines a mechanism for instantiating agents A signal renaming function Defines a mechanism for combining agents e.g. product machine Defines a mechanism for encapsulating agents Hide local signal names from the interface October 21, 2002 ee249 8
9 An Abstract Algebraic Model Concurrency Algebra = (D,, proj, rename) Domain of agent (carrier set) Operation of parallel composition on agents Operation of projection on agents encapsulation, scoping Operation of renaming on agents instantiation Note: we don t say what the carrier and the operations are, just that they must exist! But we must provide some properties that we want satisfied October 21, 2002 ee249 9
10 Concurrency Algebra Let W be a set of signal names A signature γ = ( I, O ) is a pair where I and O are subsets of W We call the set A = I Ο the alphabet of a signature γ Each agent in D has an associated signature γ October 21, 2002 ee249 10
11 Definition of the functions If O and O are disjoint, then E E is defined and the signature is ( ( I I ) ( O O ), O O ) If I B A, then proj( B )( E ) is defined and its signature is ( I, O B ) If r is a renaming function with domain A, then rename( r )( E ) is defined and its signature is ( r( I ), r( O ) ) (where r is naturally extended to sets) October 21, 2002 ee249 11
12 Axioms C1. Associativity of composition ( E E ) E = E ( E E ) C2. Commutativity of composition E E = E E C3. Application of renaming functions rename( r )( rename( r )( E ) ) = rename( r r )( E ) October 21, 2002 ee249 12
13 Axioms C4. Renaming and parallel composition commute rename( r )( E E ) = rename( r A )( E ) rename( r A )( E ) C5. Identity of renaming rename( id A )( E ) = E C6. Application of projections proj( B )( proj( B )( E ) ) = proj( B )( E ) October 21, 2002 ee249 13
14 Axioms C7. Identity of projection proj( A )( E ) = E C8. Projection and parallel composition commute proj( B )( E E ) = proj( B A )( E ) proj( B A )( E ) C9. Projection and renaming commute proj( r( B ) )( rename( r )( E ) ) = rename( r B )( proj( B )( E ) ) October 21, 2002 ee249 14
15 Algebra of Agents Model of agents Concurrency Algebra A Concurrency Algebra Set of agents Parallel composition of agents Projection of agents Renaming of agents October 21, 2002 ee249 15
16 Outline Introduction Concurrency Algebras Trace Algebras Trace Structure Algebras Conservative Approximations Homomorphisms October 21, 2002 ee249 16
17 Objective of Trace Theory Start from models of individual behaviors Construct models of agents that form a concurrency algebra Construct relationships between models of agents from relationships between individual behaviors Again we use an algebra to define a model of individual behaviors October 21, 2002 ee249 17
18 Trace and Trace Structure Algebras Model of individual behaviors Model of agents Trace algebra C Trace Algebra Set of traces Projection of traces Renaming of traces Concurrency Algebra A Concurrency Algebra Set of agents Parallel composition of agents Projection of agents Renaming of agents October 21, 2002 ee249 18
19 What Is a Trace? An element of a set of objects with the following two operations projection renaming Satisfying a set of axioms You must see a pattern here Don t define the objects, define their properties! A trace algebra is a mathematical structure (a set plus the operations) that satisfies the axioms Examples of traces? October 21, 2002 ee249 19
20 Trace Algebra A trace algebra C c over W is a triple ( B C, proj, rename ) where For every A W, B C ( A ) is the set of traces over A Also use B C to denote all traces proj( B ) and rename( r ) are partial functions from B C to B C October 21, 2002 ee249 20
21 Axioms of Trace Algebra T1. Definition of projection proj( B )( x ) is defined if there is A such that x B C ( A ) and B A. When defined proj( B )( x ) is an element of B C ( B ) T2. Application of projection proj( B )( proj( B )( x ) ) = proj( B )( x ) T3. Identity of projection If x B C ( A ) then proj( A )( x ) = x October 21, 2002 ee249 21
22 Axioms of Trace Algebra T5. Definition of renaming rename( r )( x ) is defined iff x B C ( dom( r ) ). When define, rename( R )( x ) is an element of B C ( codom( r ) ) T6. Application of renaming rename( r )( rename( r )( x ) ) = rename( r r )( x ) T7. Identity of renaming If x B C ( A ) then rename( id A )( x ) = x October 21, 2002 ee249 22
23 Axioms of Trace Algebra T9. Projection and renaming commute proj( r( B ) )( rename( r )( x ) ) = rename( r B )( proj( B )( x ) ) T4. Diamond property Let x B C ( A ) and x B C ( A ) be such that proj( A A )( x ) = proj( A A )( x ). For all A A A there is x B C ( A ) such that x = proj( A )( x ) and x = proj( A )( x ) October 21, 2002 ee249 23
24 Examples of trace algebras C I = ( B CI, proj I, rename I ) For every A, the set of traces B CI ( A ) of traces over A is A (finite and infinite sequences over A) proj I ( B )( x ) is the sequence formed by removing every symbol a not in B (the sequence shrinks) rename I ( r )( x ) is the sequence formed by applying r to each element of the sequence October 21, 2002 ee249 24
25 Examples of Trace Algebras The singleton { x o } Here all behaviors are the same. It conveys no information B C ( A ) = 2 A proj( B )( x ) = x B rename( r )( x ) is the natural extension to sets Abstracts time and retains only occurrence information October 21, 2002 ee249 25
26 Examples of Trace Algebras B C ( A ) = ( 2 A ) ω proj( B )( x ) is intersection with B Events carry a time stamp (Tagged Signal Model) B C ( A ) = { x( t ) x : R + 2 A } Real-valued time stamps Projection is the restriction of the image October 21, 2002 ee249 26
27 Examples of Trace Algebras Interpret the set of signals A as a set of state variables A state is an assignment σ : A V where V is the set of possible values of the state variables B C ( A ) = ( A V ) ω Sequence of assignment functions Projection is a restriction of the domain of the assignment function October 21, 2002 ee249 27
28 Outline Introduction Concurrency Algebras Trace Algebras Trace Structure Algebras Conservative Approximations Homomorphisms October 21, 2002 ee249 28
29 Trace and Trace Structure Algebras Model of individual behaviors A trace structure contains a set of traces Model of agents Trace algebra C Trace Algebra Set of traces Projection of traces Renaming of traces Trace structure algebra A Concurrency Algebra Set of agents Parallel composition of agents Projection of agents Renaming of agents October 21, 2002 ee249 29
30 Constructing agents We want to go from individual behaviors to models of agents We want the model of agents to be a concurrency algebra Set of agents, with parallel composition, projection and renaming satisfying the axioms Naturally if a trace is an individual behavior, and agent is a set of those behaviors October 21, 2002 ee249 30
31 Trace Structures A trace structure (or agent) over a trace algebra C is the set of ordered pairs ( γ, P ) where γ is a signature over W (reminder: γ = ( I, O ) ) A is the alphabet of γ P is a subset of B C ( A ) In other words, P is a set of traces over the alphabet A Each trace in P represents a possible behavior of the agent October 21, 2002 ee249 31
32 Note:, proj An Algebra TS and rename of Trace TS denote operations on trace structures, Structures which are derived from the corresponding operations on Let individual Note: the Τ be a traces operations set of proj trace T and on individual structures rename (agents) T traces proj T and rename T are naturally extended to sets of A Trace tracesstructure Algebra is the 4-tuple ( Τ,, proj TS, rename TS ) where For T = T T : γ = (( I I ) (O O ), O O ) P = { x B C ( A ) : proj T ( A )( x ) P and proj T ( A )( x ) P } proj TS ( B )( T ) = ( ( I, O B ), proj T ( B )( P ) ) rename TS ( r )( T ) = ( ( r( I ), r( O ) ), rename T ( r )( P ) ) October 21, 2002 ee249 32
33 Example B C ( A ) = A T = ( ( { a, b }, ), { abab } ) T = ( ( { b, c }, ), { bcb } ) T = T T P = { x B C ( A ) : proj( A )( x ) P and proj( A )( x ) P } P = { abacb, abcab } Note: P is maximal such that its projections correspond to the agents that are being composed October 21, 2002 ee249 33
34 Example B C ( A ) = ( 2 A ) ω T = ( ( { a, b }, ), { <{ a, b }, { a }, { b } > } ) T = ( ( { b, c }, ), { < { b }, { c }, { b } > } ) T = T T P = { < { a, b }, { a, c }, { b } > } Note that under this trace algebra (with time stamps) we don t get non-determinism due to interleaving semantics October 21, 2002 ee249 34
35 Example B C ( A ) = ( 2 A ) ω (also = N 2 A ) T = ( ( { a, b }, ), { < { a }, { a }, { b } > } ) T = ( ( { a, b }, ), { < { b }, { a }, { b } > } ) T = T T Need P B C ( A ) such that proj( A )( P 1 ) = { a } and proj( A )( P 1 ) = { b } Obviously this isn t possible when projection is defined as set intersection In other words, T and T are incompatible October 21, 2002 ee249 35
36 Theorem If C is a Trace Algebra, then the Trace Structure Algebra derived from C is a Concurrency Algebra I.e. the operations of parallel composition, projection and renaming satisfy the axioms of concurrency algebra Note: the result is independent of the particular Trace Algebra (models of behaviors) that we start from October 21, 2002 ee249 36
37 Outline Introduction Concurrency Algebras Trace Algebras Trace Structure Algebras Conservative Approximations Homomorphisms October 21, 2002 ee249 37
38 Trace and Trace Structure Algebras Trace algebra C Trace structure algebra A Abstract Domain Ψ u Ψ l Ψ inv Trace algebra C Trace structure algebra A Detailed Domain Goal: Study the problem of heterogeneous interaction Formalize the concepts of abstraction and refinement October 21, 2002 ee249 38
39 Relationships between models Each semantic domain has a refinement order Based on trace containment T 1 T 2 means T 1 is a refinement of T 2 Guiding intuition: T 1 T 2 means T 1 can be substituted for T 2 Abstraction mapping If a function H between agent domains is monotonic, detailed implies abstract: If T 1 T 2 then H(T 1 ) H(T 2 ) Analogy for real numbers r and s: If r s then r s Conservative approximations A pair of functions Ψ = (Ψ l, Ψ u )is a conservative approximation if Ψ u (T 1 ) Ψ l (T 2 ) implies T 1 T 2 Analogy: r s implies r s Abstract implies detailed October 21, 2002 ee249 39
40 Conservative Approximations if Ψ u ( T impl ) Ψ l ( T spec ) then T impl T spec Intuitively Ψ u ( T impl ) is an upper bound of the implementation (has more behaviors) Intuitively Ψ l ( T spec ) is a lower bound of the specification (has less behaviors) Hence we are being conservative False negatives are possible. False positives are ruled out The verification at the more abstract level should be easier October 21, 2002 ee249 40
41 Outline Introduction Concurrency Algebras Trace Algebras Trace Structure Algebras Conservative Approximations Homomorphisms October 21, 2002 ee249 41
42 Constructing Approximations How do we build a conservative approximation? As before, we prefer to work at the level of the individual behavior October 21, 2002 ee249 42
43 Trace and Trace Structure Algebras Trace algebra C Trace structure algebra A Abstract Domain Homomorphism h Derive Ψ u Ψ l Ψ inv Trace algebra C Trace structure algebra A Detailed Domain Let T spec and T impl be trace structures in A. Then if Ψ u ( T impl ) Ψ l ( T spec ) then T impl T spec October 21, 2002 ee249 43
44 Homomorphisms on Traces Let C and C be two Trace Algebras A homomorphism h is a function from the traces in C (B C ) to the traces in C (B C ) such that h commutes with the operations of projection and renaming h( proj( B )( x ) ) = proj ( B )( h( x ) ) h( rename( r )( x ) ) = rename ( r )( h( x ) ) The homomorphism h is in general many-toone Hence C is in general more abstract than C October 21, 2002 ee249 44
45 Commutative Diagram h( x ) apply proj x apply h apply h apply proj x proj( B )( x ) x = h( proj( B )( x ) ) = proj ( B )( h( x ) ) October 21, 2002 ee249 45
46 Building the Upper Bound Let P be a set of traces, and consider the natural extension to sets h( P ) of h Clearly P h -1 ( h( P ) ) Because h is many-to-one This indeed is an upper bound! Equality holds if h is on-to-one Hence define Ψ u ( T ) = ( γ, h( P ) ) October 21, 2002 ee249 46
47 Building the Upper Bound h -1 ( h( P ) ) h(p) P October 21, 2002 ee249 47
48 Building the Lower Bound We want P h -1 ( lb of P ) If x is not in P, then h( x ) should not be in the lower bound of P Hence define Ψ l ( T ) = h( P ) h( B C ( A ) P ) There is a tighter lower bound October 21, 2002 ee249 48
49 Building the Lower Bound h( P ) - h( B C (A) P ) h -1 ( h( P ) - h( B C (A) P ) ) h( B C (A) P ) B C ( A ) - P October 21, 2002 ee249 49
50 Theorem The pair of functions just defined forms a conservative approximation if Ψ u ( T impl ) Ψ l ( T spec ) then T impl T spec This is an example of a conservative approximation induced by a homomorphism October 21, 2002 ee249 50
51 Examples of homomorphisms From B C ( A ) = A to B C ( A ) = 2 A h( x ) = { a a appears in x at least once } Prove that this is a homomorphism Must show it commutes with projection and renaming What do the upper and lower bounds look like? October 21, 2002 ee249 51
52 Outline Introduction Concurrency Algebras Trace Algebras Trace Structure Algebras Conservative Approximations Homomorphisms Conservative Approximations: inverses October 21, 2002 ee249 52
53 Approximations: Inverses Apply Ψ u Apply Ψ l Consider T such that Ψ u (T) = Ψ l (T) = T Ψ u Ψ l October 21, 2002 ee249 53
54 Approximations: Inverses Apply Ψ u Apply Ψ l Consider T such that Ψ u (T) = Ψ l (T) = T Then Ψ inv (T ) = T Ψ u Ψ l October 21, 2002 ee249 54
55 Approximations: Inverses Apply Ψ u Apply Ψ l Consider T such that Ψ u (T) = Ψ l (T) = T Then Ψ inv (T ) = T Ψ inv Can be used to embed high-level model in low level October 21, 2002 ee249 55
56 Combining MoCs T 1 T 2 Want to compose T 1 and T 2 from different trace structure algebras Ψ inv T 1 T 2 Construct a third, more detailed trace algebra, with homomorphisms to the other two Construct a third trace structure algebra Construct cons. approximations and their inverses Map T 1 and T 2 to T 1 and T 2 in the third trace structure algebra Compose T 1 and T 2 October 21, 2002 ee249 56
57 Combining MoCs T 1 T 2 Want to compose T 1 and T 2 from different trace structure algebras T 1 T 2 T Compose T 1 and T 2 Get T Project back to T 1 and T 2 Map back in the abstract domain Find restrictions due to the interaction at the higher level (constraint application) Ψ inv Find greatest possible T 1 and T 2 that still have the same interaction (don t cares, synthesis) October 21, 2002 ee249 57
58 Key Insight Using this process you can Find restrictions due to the interaction at the higher level (constraint application) Find greatest possible T 1 and T 2 that still have the same interaction (don t cares, synthesis) October 21, 2002 ee249 58
59 A Sample of Trace Algebras A* ( (A) - { }) * pomsets(a) (A V)* (sf: stutter free) (A V)* ( (A))* ((A V V) (A r E )) tomsets((a V --> V) (A r E )) (sf) (A V) 2 (A V) (A) (A*) A {0, 1, X} ((A V --> V) ( (A r C ) - { }))* (sf) (A x V)* Interleaving semantics Explicit simultaneity Partial order models State based asynchronous time State based synchronous time Event based synchronous time Metric time Non-metric time Pre-post State based discrete time Event based discrete time Discrete time with interleaving Combinational GALS Interleaving with values October 21, 2002 ee249 59
Algebraic Trace Theory
Algebraic Trace Theory EE249 Presented by Roberto Passerone Material from: Jerry R. Burch, Trace Theory for Automatic Verification of Real-Time Concurrent Systems, PhD thesis, CMU, August 1992 October
More informationEE249 - Fall 2012 Lecture 18: Overview of Concrete Contract Theories. Alberto Sangiovanni-Vincentelli Pierluigi Nuzzo
EE249 - Fall 2012 Lecture 18: Overview of Concrete Contract Theories 1 Alberto Sangiovanni-Vincentelli Pierluigi Nuzzo Outline: Contracts and compositional methods for system design Where and why using
More informationCausality Interfaces and Compositional Causality Analysis
Causality Interfaces and Compositional Causality Analysis Edward A. Lee Haiyang Zheng Ye Zhou {eal,hyzheng,zhouye}@eecs.berkeley.edu Center for Hybrid and Embedded Software Systems (CHESS) Department of
More informationACLT: Algebra, Categories, Logic in Topology - Grothendieck's generalized topological spaces (toposes)
ACLT: Algebra, Categories, Logic in Topology - Grothendieck's generalized topological spaces (toposes) Steve Vickers CS Theory Group Birmingham 2. Theories and models Categorical approach to many-sorted
More informationEquational Logic. Chapter Syntax Terms and Term Algebras
Chapter 2 Equational Logic 2.1 Syntax 2.1.1 Terms and Term Algebras The natural logic of algebra is equational logic, whose propositions are universally quantified identities between terms built up from
More informationcse303 ELEMENTS OF THE THEORY OF COMPUTATION Professor Anita Wasilewska
cse303 ELEMENTS OF THE THEORY OF COMPUTATION Professor Anita Wasilewska LECTURE 1 Course Web Page www3.cs.stonybrook.edu/ cse303 The webpage contains: lectures notes slides; very detailed solutions to
More informationFormal Models of Timed Musical Processes Doctoral Defense
Formal Models of Timed Musical Processes Doctoral Defense Gerardo M. Sarria M. Advisor: Camilo Rueda Co-Advisor: Juan Francisco Diaz Universidad del Valle AVISPA Research Group September 22, 2008 Motivation
More informationA Brief Introduction to Model Checking
A Brief Introduction to Model Checking Jan. 18, LIX Page 1 Model Checking A technique for verifying finite state concurrent systems; a benefit on this restriction: largely automatic; a problem to fight:
More informationA Brief History of Shared memory C M U
A Brief History of Shared memory S t e p h e n B r o o k e s C M U 1 Outline Revisionist history Rational reconstruction of early models Evolution of recent models A unifying framework Fault-detecting
More informationSemantic Foundations for Heterogeneous Systems. Roberto Passerone. Laurea (Politecnico di Torino) 1994 M. S. (University of California, Berkeley) 1997
Semantic Foundations for Heterogeneous Systems by Roberto Passerone Laurea (Politecnico di Torino) 1994 M. S. (University of California, Berkeley) 1997 A dissertation submitted in partial satisfaction
More informationICS141: Discrete Mathematics for Computer Science I
ICS141: Discrete Mathematics for Computer Science I Dept. Information & Computer Sci., Jan Stelovsky based on slides by Dr. Baek and Dr. Still Originals by Dr. M. P. Frank and Dr. J.L. Gross Provided by
More informationReasoning Under Uncertainty: Introduction to Probability
Reasoning Under Uncertainty: Introduction to Probability CPSC 322 Lecture 23 March 12, 2007 Textbook 9 Reasoning Under Uncertainty: Introduction to Probability CPSC 322 Lecture 23, Slide 1 Lecture Overview
More informationSanjit A. Seshia EECS, UC Berkeley
EECS 219C: Computer-Aided Verification Explicit-State Model Checking: Additional Material Sanjit A. Seshia EECS, UC Berkeley Acknowledgments: G. Holzmann Checking if M satisfies : Steps 1. Compute Buchi
More informationAbstractions and Decision Procedures for Effective Software Model Checking
Abstractions and Decision Procedures for Effective Software Model Checking Prof. Natasha Sharygina The University of Lugano, Carnegie Mellon University Microsoft Summer School, Moscow, July 2011 Lecture
More informationOutline. We will now investigate the structure of this important set.
The Reals Outline As we have seen, the set of real numbers, R, has cardinality c. This doesn't tell us very much about the reals, since there are many sets with this cardinality and cardinality doesn't
More informationPartial model checking via abstract interpretation
Partial model checking via abstract interpretation N. De Francesco, G. Lettieri, L. Martini, G. Vaglini Università di Pisa, Dipartimento di Ingegneria dell Informazione, sez. Informatica, Via Diotisalvi
More informationThe State Explosion Problem
The State Explosion Problem Martin Kot August 16, 2003 1 Introduction One from main approaches to checking correctness of a concurrent system are state space methods. They are suitable for automatic analysis
More informationLogic Model Checking
Logic Model Checking Lecture Notes 10:18 Caltech 101b.2 January-March 2004 Course Text: The Spin Model Checker: Primer and Reference Manual Addison-Wesley 2003, ISBN 0-321-22862-6, 608 pgs. the assignment
More informationLecture Notes: Axiomatic Semantics and Hoare-style Verification
Lecture Notes: Axiomatic Semantics and Hoare-style Verification 17-355/17-665/17-819O: Program Analysis (Spring 2018) Claire Le Goues and Jonathan Aldrich clegoues@cs.cmu.edu, aldrich@cs.cmu.edu It has
More informationLogic Synthesis and Verification
Logic Synthesis and Verification Boolean Algebra Jie-Hong Roland Jiang 江介宏 Department of Electrical Engineering National Taiwan University Fall 2014 1 2 Boolean Algebra Reading F. M. Brown. Boolean Reasoning:
More informationGALOIS THEORY : LECTURE 4 LEO GOLDMAKHER
GALOIS THEORY : LECTURE 4 LEO GOLDMAKHER 1. A BRIEF REVIEW OF S n The rest of the lecture relies on familiarity with the symmetric group of n elements, denoted S n (this is the group of all permutations
More informationUniversal Algebra for Logics
Universal Algebra for Logics Joanna GRYGIEL University of Czestochowa Poland j.grygiel@ajd.czest.pl 2005 These notes form Lecture Notes of a short course which I will give at 1st School on Universal Logic
More informationPredicate Logic. Xinyu Feng 11/20/2013. University of Science and Technology of China (USTC)
University of Science and Technology of China (USTC) 11/20/2013 Overview Predicate logic over integer expressions: a language of logical assertions, for example x. x + 0 = x Why discuss predicate logic?
More informationCSC 7101: Programming Language Structures 1. Axiomatic Semantics. Stansifer Ch 2.4, Ch. 9 Winskel Ch.6 Slonneger and Kurtz Ch. 11.
Axiomatic Semantics Stansifer Ch 2.4, Ch. 9 Winskel Ch.6 Slonneger and Kurtz Ch. 11 1 Overview We ll develop proof rules, such as: { I b } S { I } { I } while b do S end { I b } That allow us to verify
More informationAlgebras with finite descriptions
Algebras with finite descriptions André Nies The University of Auckland July 19, 2005 Part 1: FA-presentability A countable structure in a finite signature is finite-automaton presentable (or automatic)
More informationAutomata-Theoretic Model Checking of Reactive Systems
Automata-Theoretic Model Checking of Reactive Systems Radu Iosif Verimag/CNRS (Grenoble, France) Thanks to Tom Henzinger (IST, Austria), Barbara Jobstmann (CNRS, Grenoble) and Doron Peled (Bar-Ilan University,
More informationa (b + c) = a b + a c
Chapter 1 Vector spaces In the Linear Algebra I module, we encountered two kinds of vector space, namely real and complex. The real numbers and the complex numbers are both examples of an algebraic structure
More informationMathematics Review for Business PhD Students Lecture Notes
Mathematics Review for Business PhD Students Lecture Notes Anthony M. Marino Department of Finance and Business Economics Marshall School of Business University of Southern California Los Angeles, CA 90089-0804
More informationUnifying Theories of Programming
1&2 Unifying Theories of Programming Unifying Theories of Programming 3&4 Theories Unifying Theories of Programming designs predicates relations reactive CSP processes Jim Woodcock University of York May
More informationOne-sided shift spaces over infinite alphabets
Reasoning Mind West Coast Operator Algebra Seminar October 26, 2013 Joint work with William Ott and Mark Tomforde Classical Construction Infinite Issues New Infinite Construction Shift Spaces Classical
More informationCompositional Synthesis with Parametric Reactive Controllers
Compositional Synthesis with Parametric Reactive Controllers Rajeev Alur University of Pennsylvania alur@seas.upenn.edu Salar Moarref University of Pennsylvania moarref@seas.upenn.edu Ufuk Topcu University
More informationFREE PRODUCTS AND BRITTON S LEMMA
FREE PRODUCTS AND BRITTON S LEMMA Dan Lidral-Porter 1. Free Products I determined that the best jumping off point was to start with free products. Free products are an extension of the notion of free groups.
More information1. Propositional Calculus
1. Propositional Calculus Some notes for Math 601, Fall 2010 based on Elliott Mendelson, Introduction to Mathematical Logic, Fifth edition, 2010, Chapman & Hall. 2. Syntax ( grammar ). 1.1, p. 1. Given:
More informationAn Institution for Event-B
An Institution for Event-B Marie Farrell, Rosemary Monahan, and James F. Power Dept. of Computer Science, Maynooth University, Co. Kildare, Ireland Abstract. This paper presents a formalisation of the
More informationTowards a Denotational Semantics for Discrete-Event Systems
Towards a Denotational Semantics for Discrete-Event Systems Eleftherios Matsikoudis University of California at Berkeley Berkeley, CA, 94720, USA ematsi@eecs. berkeley.edu Abstract This work focuses on
More informationOn the Executability of Interactive Computation. June 23, 2016 Where innovation starts
On the Executability of Interactive Computation Bas Luttik Fei Yang June 23, 2016 Where innovation starts Outline 2/37 From Computation to Interactive Computation Executability - an Integration of Computability
More informationA Denotational Semantic Theory of Concurrent Systems
A Denotational Semantic Theory of Concurrent Systems Jayadev Misra Dept. of Computer Science, Univ. of Texas, Austin, 78712, USA Abstract. This paper proposes a general denotational semantic theory suitable
More informationRelational Interfaces and Refinement Calculus for Compositional System Reasoning
Relational Interfaces and Refinement Calculus for Compositional System Reasoning Viorel Preoteasa Joint work with Stavros Tripakis and Iulia Dragomir 1 Overview Motivation General refinement Relational
More informationOrbits of D-maximal sets in E
Orbits of in E Peter M Gerdes April 19, 2012 * Joint work with Peter Cholak and Karen Lange Peter M Gerdes Orbits of in E Outline 1 2 3 Peter M Gerdes Orbits of in E Notation ω is the natural numbers X
More informationStat 451: Solutions to Assignment #1
Stat 451: Solutions to Assignment #1 2.1) By definition, 2 Ω is the set of all subsets of Ω. Therefore, to show that 2 Ω is a σ-algebra we must show that the conditions of the definition σ-algebra are
More informationLinear equations in linear algebra
Linear equations in linear algebra Samy Tindel Purdue University Differential equations and linear algebra - MA 262 Taken from Differential equations and linear algebra Pearson Collections Samy T. Linear
More informationReal-Time Systems. Lecture 15: The Universality Problem for TBA Dr. Bernd Westphal. Albert-Ludwigs-Universität Freiburg, Germany
Real-Time Systems Lecture 15: The Universality Problem for TBA 2013-06-26 15 2013-06-26 main Dr. Bernd Westphal Albert-Ludwigs-Universität Freiburg, Germany Contents & Goals Last Lecture: Extended Timed
More informationDES. 4. Petri Nets. Introduction. Different Classes of Petri Net. Petri net properties. Analysis of Petri net models
4. Petri Nets Introduction Different Classes of Petri Net Petri net properties Analysis of Petri net models 1 Petri Nets C.A Petri, TU Darmstadt, 1962 A mathematical and graphical modeling method. Describe
More informationCOMP2111 Glossary. Kai Engelhardt. Contents. 1 Symbols. 1 Symbols 1. 2 Hoare Logic 3. 3 Refinement Calculus 5. rational numbers Q, real numbers R.
COMP2111 Glossary Kai Engelhardt Revision: 1.3, May 18, 2018 Contents 1 Symbols 1 2 Hoare Logic 3 3 Refinement Calculus 5 1 Symbols Booleans B = {false, true}, natural numbers N = {0, 1, 2,...}, integers
More informationWith Question/Answer Animations. Chapter 2
With Question/Answer Animations Chapter 2 Chapter Summary Sets The Language of Sets Set Operations Set Identities Functions Types of Functions Operations on Functions Sequences and Summations Types of
More informationTrace-based Process Algebras for Real-Time Probabilistic Systems
Trace-based Process Algebras for Real-Time Probabilistic Systems Stefano Cattani A thesis submitted to The University of Birmingham for the degree of Doctor of Philosophy School of Computer Science The
More informationFormal Methods for Java
Formal Methods for Java Lecture 12: Soundness of Sequent Calculus Jochen Hoenicke Software Engineering Albert-Ludwigs-University Freiburg June 12, 2017 Jochen Hoenicke (Software Engineering) Formal Methods
More informationA Framework for. Security Analysis. with Team Automata
A Framework for Security Analysis with Team Automata Marinella Petrocchi Istituto di Informatica e Telematica National Research Council IIT-CNR Pisa, Italy Tuesday 8 June 2004 DIMACS with Maurice ter Beek
More informationPredicate Logic: Sematics Part 1
Predicate Logic: Sematics Part 1 CS402, Spring 2018 Shin Yoo Predicate Calculus Propositional logic is also called sentential logic, i.e. a logical system that deals with whole sentences connected with
More informationSome Remarks on Alternating Temporal Epistemic Logic
Some Remarks on Alternating Temporal Epistemic Logic Corrected version: July 2003 Wojciech Jamroga Parlevink Group, University of Twente, Netherlands Institute of Mathematics, University of Gdansk, Poland
More informationAdvanced Automata Theory 9 Automatic Structures in General
Advanced Automata Theory 9 Automatic Structures in General Frank Stephan Department of Computer Science Department of Mathematics National University of Singapore fstephan@comp.nus.edu.sg Advanced Automata
More informationOperational Semantics
Operational Semantics Semantics and applications to verification Xavier Rival École Normale Supérieure Xavier Rival Operational Semantics 1 / 50 Program of this first lecture Operational semantics Mathematical
More informationA Domain View of Timed Behaviors
A Domain View of Timed Behaviors Roman Dubtsov 1, Elena Oshevskaya 2, and Irina Virbitskaite 2 1 Institute of Informatics System SB RAS, 6, Acad. Lavrentiev av., 630090, Novosibirsk, Russia; 2 Institute
More informationTheorem 4.18 ( Critical Pair Theorem ) A TRS R is locally confluent if and only if all its critical pairs are joinable.
4.4 Critical Pairs Showing local confluence (Sketch): Problem: If t 1 E t 0 E t 2, does there exist a term s such that t 1 E s E t 2? If the two rewrite steps happen in different subtrees (disjoint redexes):
More information3515ICT: Theory of Computation. Regular languages
3515ICT: Theory of Computation Regular languages Notation and concepts concerning alphabets, strings and languages, and identification of languages with problems (H, 1.5). Regular expressions (H, 3.1,
More information1. Propositional Calculus
1. Propositional Calculus Some notes for Math 601, Fall 2010 based on Elliott Mendelson, Introduction to Mathematical Logic, Fifth edition, 2010, Chapman & Hall. 2. Syntax ( grammar ). 1.1, p. 1. Given:
More informationΠ 0 1-presentations of algebras
Π 0 1-presentations of algebras Bakhadyr Khoussainov Department of Computer Science, the University of Auckland, New Zealand bmk@cs.auckland.ac.nz Theodore Slaman Department of Mathematics, The University
More informationIntroduction to Kleene Algebras
Introduction to Kleene Algebras Riccardo Pucella Basic Notions Seminar December 1, 2005 Introduction to Kleene Algebras p.1 Idempotent Semirings An idempotent semiring is a structure S = (S, +,, 1, 0)
More informationCharacterizing the Equational Theory
Introduction to Kleene Algebra Lecture 4 CS786 Spring 2004 February 2, 2004 Characterizing the Equational Theory Most of the early work on Kleene algebra was directed toward characterizing the equational
More informationThe Complexity of Computing the Behaviour of Lattice Automata on Infinite Trees
The Complexity of Computing the Behaviour of Lattice Automata on Infinite Trees Karsten Lehmann a, Rafael Peñaloza b a Optimisation Research Group, NICTA Artificial Intelligence Group, Australian National
More informationThe Journal of Logic and Algebraic Programming
The Journal of Logic and Algebraic Programming 78 (2008) 22 51 Contents lists available at ScienceDirect The Journal of Logic and Algebraic Programming journal homepage: www.elsevier.com/locate/jlap Operational
More informationNotes on Monoids and Automata
Notes on Monoids and Automata Uday S. Reddy November 9, 1994 In this article, I define a semantics for Algol programs with Reynolds s syntactic control of interference?;? in terms of comonoids in coherent
More informationBounded Stacks, Bags and Queues
Bounded Stacks, Bags and Queues J.C.M. Baeten 1 and J.A. Bergstra 2,3 1 Department of Mathematics and Computing Science, Eindhoven University of Technology, P.O. Box 513, NL-5600 MB Eindhoven, The Netherlands,
More informationEquational Logic. Chapter 4
Chapter 4 Equational Logic From now on First-order Logic is considered with equality. In this chapter, I investigate properties of a set of unit equations. For a set of unit equations I write E. Full first-order
More informationAutomata-based Verification - III
COMP30172: Advanced Algorithms Automata-based Verification - III Howard Barringer Room KB2.20: email: howard.barringer@manchester.ac.uk March 2009 Third Topic Infinite Word Automata Motivation Büchi Automata
More informationThe non-logical symbols determine a specific F OL language and consists of the following sets. Σ = {Σ n } n<ω
1 Preliminaries In this chapter we first give a summary of the basic notations, terminology and results which will be used in this thesis. The treatment here is reduced to a list of definitions. For the
More informationTESTING is one of the most important parts of the
IEEE TRANSACTIONS 1 Generating Complete Controllable Test Suites for Distributed Testing Robert M. Hierons, Senior Member, IEEE Abstract A test suite is m-complete for finite state machine (FSM) M if it
More informationLecture 9. Model theory. Consistency, independence, completeness, categoricity of axiom systems. Expanded with algebraic view.
V. Borschev and B. Partee, October 17-19, 2006 p. 1 Lecture 9. Model theory. Consistency, independence, completeness, categoricity of axiom systems. Expanded with algebraic view. CONTENTS 0. Syntax and
More informationHoming and Synchronizing Sequences
Homing and Synchronizing Sequences Sven Sandberg Information Technology Department Uppsala University Sweden 1 Outline 1. Motivations 2. Definitions and Examples 3. Algorithms (a) Current State Uncertainty
More informationarxiv: v2 [cs.dc] 18 Feb 2015
Consensus using Asynchronous Failure Detectors Nancy Lynch CSAIL, MIT Srikanth Sastry CSAIL, MIT arxiv:1502.02538v2 [cs.dc] 18 Feb 2015 Abstract The FLP result shows that crash-tolerant consensus is impossible
More informationExtension theorems for homomorphisms
Algebraic Geometry Fall 2009 Extension theorems for homomorphisms In this note, we prove some extension theorems for homomorphisms from rings to algebraically closed fields. The prototype is the following
More informationModel theory, algebraic dynamics and local fields
Model theory, algebraic dynamics and local fields Thomas Scanlon University of California, Berkeley 7 June 2010 Thomas Scanlon (University of California, Berkeley) Model theory, algebraic dynamics and
More informationDecidable Subsets of CCS
Decidable Subsets of CCS based on the paper with the same title by Christensen, Hirshfeld and Moller from 1994 Sven Dziadek Abstract Process algebra is a very interesting framework for describing and analyzing
More informationOne Year Later. Iliano Cervesato. ITT Industries, NRL Washington, DC. MSR 3.0:
MSR 3.0: The Logical Meeting Point of Multiset Rewriting and Process Algebra MSR 3: Iliano Cervesato iliano@itd.nrl.navy.mil One Year Later ITT Industries, inc @ NRL Washington, DC http://www.cs.stanford.edu/~iliano
More informationBasic Problems in Multi-View Modeling
Basic Problems in Multi-View Modeling Jan Reineke Stavros Tripakis Electrical Engineering and Computer Sciences University of California at Berkeley Technical Report No. UCB/EECS-2014-4 http://www.eecs.berkeley.edu/pubs/techrpts/2014/eecs-2014-4.html
More informationAxiomatic Semantics. Stansifer Ch 2.4, Ch. 9 Winskel Ch.6 Slonneger and Kurtz Ch. 11 CSE
Axiomatic Semantics Stansifer Ch 2.4, Ch. 9 Winskel Ch.6 Slonneger and Kurtz Ch. 11 CSE 6341 1 Outline Introduction What are axiomatic semantics? First-order logic & assertions about states Results (triples)
More informationTutorial on Axiomatic Set Theory. Javier R. Movellan
Tutorial on Axiomatic Set Theory Javier R. Movellan Intuitively we think of sets as collections of elements. The crucial part of this intuitive concept is that we are willing to treat sets as entities
More informationRegular expressions. Regular expressions. Regular expressions. Regular expressions. Remark (FAs with initial set recognize the regular languages)
Definition (Finite automata with set of initial states) A finite automata with set of initial states, or FA with initial set for short, is a 5-tupel (Q, Σ, δ, S, F ) where Q, Σ, δ, and F are defined as
More informationBasics of Relation Algebra. Jules Desharnais Département d informatique et de génie logiciel Université Laval, Québec, Canada
Basics of Relation Algebra Jules Desharnais Département d informatique et de génie logiciel Université Laval, Québec, Canada Plan 1. Relations on a set 2. Relation algebra 3. Heterogeneous relation algebra
More informationModal Logics. Most applications of modal logic require a refined version of basic modal logic.
Modal Logics Most applications of modal logic require a refined version of basic modal logic. Definition. A set L of formulas of basic modal logic is called a (normal) modal logic if the following closure
More informationAutomata Theory. Lecture on Discussion Course of CS120. Runzhe SJTU ACM CLASS
Automata Theory Lecture on Discussion Course of CS2 This Lecture is about Mathematical Models of Computation. Why Should I Care? - Ways of thinking. - Theory can drive practice. - Don t be an Instrumentalist.
More informationComplete Partial Orders, PCF, and Control
Complete Partial Orders, PCF, and Control Andrew R. Plummer TIE Report Draft January 2010 Abstract We develop the theory of directed complete partial orders and complete partial orders. We review the syntax
More informationPredicate Logic. Xinyu Feng 09/26/2011. University of Science and Technology of China (USTC)
University of Science and Technology of China (USTC) 09/26/2011 Overview Predicate logic over integer expressions: a language of logical assertions, for example x. x + 0 = x Why discuss predicate logic?
More informationFirst-Order Logic. 1 Syntax. Domain of Discourse. FO Vocabulary. Terms
First-Order Logic 1 Syntax Domain of Discourse The domain of discourse for first order logic is FO structures or models. A FO structure contains Relations Functions Constants (functions of arity 0) FO
More informationAn Algebraic Theory of Interface Automata
An Algebraic Theory of Interface Automata Chris Chilton a, Bengt Jonsson b, Marta Kwiatkowska a a Department of Computer Science, University of Oxford, UK b Department of Information Technology, Uppsala
More informationWojciech Penczek. Polish Academy of Sciences, Warsaw, Poland. and. Institute of Informatics, Siedlce, Poland.
A local approach to modal logic for multi-agent systems? Wojciech Penczek 1 Institute of Computer Science Polish Academy of Sciences, Warsaw, Poland and 2 Akademia Podlaska Institute of Informatics, Siedlce,
More informationAdvances in the theory of fixed points in many-valued logics
Advances in the theory of fixed points in many-valued logics Department of Mathematics and Computer Science. Università degli Studi di Salerno www.logica.dmi.unisa.it/lucaspada 8 th International Tbilisi
More informationThe algorithmic analysis of hybrid system
The algorithmic analysis of hybrid system Authors: R.Alur, C. Courcoubetis etc. Course teacher: Prof. Ugo Buy Xin Li, Huiyong Xiao Nov. 13, 2002 Summary What s a hybrid system? Definition of Hybrid Automaton
More informationAN INTRODUCTION TO SEPARATION LOGIC. 2. Assertions
AN INTRODUCTION TO SEPARATION LOGIC 2. Assertions John C. Reynolds Carnegie Mellon University January 7, 2011 c 2011 John C. Reynolds Pure Assertions An assertion p is pure iff, for all stores s and all
More informationLecture 4. Algebra. Section 1:. Signature, algebra in a signature. Isomorphisms, homomorphisms, congruences and quotient algebras.
V. Borschev and B. Partee, September 18, 2001 p. 1 Lecture 4. Algebra. Section 1:. Signature, algebra in a signature. Isomorphisms, homomorphisms, congruences and quotient algebras. CONTENTS 0. Why algebra?...1
More informationLecture Notes: Selected Topics in Discrete Structures. Ulf Nilsson
Lecture Notes: Selected Topics in Discrete Structures Ulf Nilsson Dept of Computer and Information Science Linköping University 581 83 Linköping, Sweden ulfni@ida.liu.se 2004-03-09 Contents Chapter 1.
More informationTheoretical Foundations of the UML
Theoretical Foundations of the UML Lecture 17+18: A Logic for MSCs Joost-Pieter Katoen Lehrstuhl für Informatik 2 Software Modeling and Verification Group moves.rwth-aachen.de/teaching/ws-1718/fuml/ 5.
More informationRestricted truth predicates in first-order logic
Restricted truth predicates in first-order logic Thomas Bolander 1 Introduction It is well-known that there exist consistent first-order theories that become inconsistent when we add Tarski s schema T.
More informationProbability Measures in Gödel Logic
Probability Measures in Gödel Logic Diego Valota Department of Computer Science University of Milan valota@di.unimi.it Joint work with Stefano Aguzzoli (UNIMI), Brunella Gerla and Matteo Bianchi (UNINSUBRIA)
More informationAxiomatic Semantics. Lecture 9 CS 565 2/12/08
Axiomatic Semantics Lecture 9 CS 565 2/12/08 Axiomatic Semantics Operational semantics describes the meaning of programs in terms of the execution steps taken by an abstract machine Denotational semantics
More informationAlternating-Time Temporal Logic
Alternating-Time Temporal Logic R.Alur, T.Henzinger, O.Kupferman Rafael H. Bordini School of Informatics PUCRS R.Bordini@pucrs.br Logic Club 5th of September, 2013 ATL All the material in this presentation
More informationFormal Methods for Java
Formal Methods for Java Lecture 20: Sequent Calculus Jochen Hoenicke Software Engineering Albert-Ludwigs-University Freiburg January 15, 2013 Jochen Hoenicke (Software Engineering) Formal Methods for Java
More informationEECS 144/244: Fundamental Algorithms for System Modeling, Analysis, and Optimization
EECS 144/244: Fundamental Algorithms for System Modeling, Analysis, and Optimization Discrete Systems Lecture: Automata, State machines, Circuits Stavros Tripakis University of California, Berkeley Stavros
More informationLecture 2: Axiomatic semantics
Chair of Software Engineering Trusted Components Prof. Dr. Bertrand Meyer Lecture 2: Axiomatic semantics Reading assignment for next week Ariane paper and response (see course page) Axiomatic semantics
More information18.175: Lecture 2 Extension theorems, random variables, distributions
18.175: Lecture 2 Extension theorems, random variables, distributions Scott Sheffield MIT Outline Extension theorems Characterizing measures on R d Random variables Outline Extension theorems Characterizing
More information