Qualitätssicherung von Software (SWQS)
|
|
- Stephanie Jacobs
- 6 years ago
- Views:
Transcription
1 Qualitätssicherung von Software (SWQS) Prof. Dr. Holger Schlingloff Humboldt-Universität zu Berlin und Fraunhofer FOKUS : Modellprüfung II - BDDs
2 Folie 2 Existenzgründer gesucht!
3 Folie 3 Fragen zur Wiederholung Unterschied Verifikation Validierung? Wie kann man Sudoku aussagenlogisch beschreiben? Wie ist die Komplexität des Erfüllbarkeitsproblems? Was versteht man unter Modellprüfung? Unterschied Sudoku Schiebepuzzle?
4 Binary Encoding of Domains Any variable on a finite domain D can be replaced by log(d) binary variables similar to encoding of data types by compilers e.g. var v: {0..15} can be replaced by var v1,v2,v3,v4: boolean (0=0000, 1= 0001, 2=0010, 3=0011,..., 15=1111) State space still in the order of original domain! e.g. three int8-variables can have 2 24 =10 8 states e.g. buffer of length 10 with 10-bit values states Representation of large sets of states? H. Schlingloff, Software-Qualitätssicherung Folie 4
5 Folie 5 Representation of Sets
6 Folie 6 Truth table and tree form formula Reduction: Replace Ite (v,ψ,ψ) by ψ
7 Folie 7 Abbreviations Introduce abbreviations maximally abbreviated for any given order of variables the maximal abbreviated form is uniquely determined!
8 Folie 8 Binary Decision Trees (BDTs) Binary decision tree Elimination of isomorphic subtrees (abbreviations)
9 Folie 9 Binary Decision Diagrams (BDDs) Elimination of redundant nodes (redundant subformulas) Ite (v,ψ,ψ) by ψ formula: ((V1 V2) V4)
10 Folie 10 Calculation of BDDs
11 Folie 11 Boolean operations on BDDs
12 Satisfiability This procedure can be applied for arbitrary boolean connectives (or, and, not) BDD( ) is the constant node p = (p ), (p q) = ( p q) etc. direct algorithms for, possible this amounts to set union, intersection, and complement with respect to the base set Formula φ is satisfiable iff BDD(φ) any path through the BDD to T defines a model H. Schlingloff, Software-Qualitätssicherung Folie 12
13 Binary Encoding of Relations A relation is a subset of the product of two sets Thus, a relation is nothing but a set Example: var v: {0..3}, w:{0..7}; var v0, v1, w0, w1, w2: boolean; divides -Relation: v divides w iff v=1, or v=2 and w even, or v=3 and w in {0,3,6} boolean formula: H. Schlingloff, Software-Qualitätssicherung Folie 13
14 Folie 14 The Influence of Variable Ordering
15 Boolean Quantification Substitution by constants is trivial Boolean quantification:! This works for arbitrary finite domains! H. Schlingloff, Software-Qualitätssicherung Folie 15
16 Bounded Model Checking State s is reachable from s 0 iff it is reachable in 0 steps: s=s 0, or it is reachable in 1 step: R(s 0,s), or it is reachable in 2 steps: s 1 (R(s 0,s 1 ) R(s 1,s)), or it is reachable in 3 steps: s 1 s 2 (R(s 0,s 1 ) R(s 1,s 2 ) R(s 2,s)), or..., or it is reachable in n steps, where n is the diameter of the model Idea: Check each of these formulas sequentially H. Schlingloff, Software-Qualitätssicherung Folie 16
17 Transitive Closure Each finite (transition) relation can be represented as a BDD The transitive closure of a relation R is defined recursively by Thus, transitive closure be calculated by an iteration on BDDs H. Schlingloff, Software-Qualitätssicherung Folie 17
18 Reachability State s is reachable iff s 0 R*s, where s 0 S 0 is an initial state and R is the transition relation Reachability is one of the most important properties in verification most safety properties can be reduced to it in a search algorithm, is the goal reachable? Can be arbitrarily hard for infinite state systems undecidable Can be efficiently calculated with BDDs H. Schlingloff, Software-Qualitätssicherung Folie 18
Qualitätssicherung von Software (SWQS)
Qualitätssicherung von Software (SWQS) Prof. Dr. Holger Schlingloff Humboldt-Universität zu Berlin und Fraunhofer FOKUS 15.7.2014: Modellbasierter Test (Jaroslav Svacina) Specification-based Testing Constructing
More informationBehavior models and verification Lecture 6
Behavior models and verification Lecture 6 http://d3s.mff.cuni.cz Jan Kofroň, František Plášil Model checking For a Kripke structure M = (S, I, R, L) over AP and a (state based) temporal logic formula
More informationModel Checking Revision: Model Checking for Infinite Systems Revision: Traffic Light Controller (TLC) Revision: 1.12
Model Checking mc Revision:.2 Model Checking for Infinite Systems mc 2 Revision:.2 check algorithmically temporal / sequential properties fixpoint algorithms with symbolic representations: systems are
More informationAction Language Verifier, Extended
Action Language Verifier, Extended Tuba Yavuz-Kahveci 1, Constantinos Bartzis 2, and Tevfik Bultan 3 1 University of Florida 2 Carnegie Mellon University 3 UC, Santa Barbara 1 Introduction Action Language
More informationLOGIC SYNTHESIS AND VERIFICATION ALGORITHMS. Gary D. Hachtel University of Colorado. Fabio Somenzi University of Colorado.
LOGIC SYNTHESIS AND VERIFICATION ALGORITHMS by Gary D. Hachtel University of Colorado Fabio Somenzi University of Colorado Springer Contents I Introduction 1 1 Introduction 5 1.1 VLSI: Opportunity and
More informationProgram verification. Generalities about software Verification Model Checking. September 20, 2016
Program verification Generalities about software Verification Model Checking Laure Gonnord David Monniaux September 20, 2016 1 / 43 The teaching staff Laure Gonnord, associate professor, LIP laboratory,
More informationOverview. Discrete Event Systems - Verification of Finite Automata. What can finite automata be used for? What can finite automata be used for?
Computer Engineering and Networks Overview Discrete Event Systems - Verification of Finite Automata Lothar Thiele Introduction Binary Decision Diagrams Representation of Boolean Functions Comparing two
More informationMotivation. CS389L: Automated Logical Reasoning. Lecture 5: Binary Decision Diagrams. Historical Context. Binary Decision Trees
Motivation CS389L: Automated Logical Reasoning Lecture 5: Binary Decision Diagrams Işıl Dillig Previous lectures: How to determine satisfiability of propositional formulas Sometimes need to efficiently
More informationSoftware Testing IV. Prof. Dr. Holger Schlingloff. Humboldt-Universität zu Berlin
Software Testing IV Prof. Dr. Holger Schlingloff Humboldt-Universität zu Berlin and Fraunhofer Institute of Computer Architecture and Software Technology FIRST Outline of this Lecture Series 2006/11/24:
More informationChapter 10 Part 1: Reduction
//06 Polynomial-Time Reduction Suppose we could solve Y in polynomial-time. What else could we solve in polynomial time? don't confuse with reduces from Chapter 0 Part : Reduction Reduction. Problem X
More informationSemantical Characterization of unbounded-nondeterministic ASMs
Semantical Characterization of unbounded-nondeterministic ASMs Berlin, 26/27 Feb 2007 Andreas Glausch Humboldt-Universität zu Berlin Department of Computer Science Abstract State Machines (ASMs) state
More informationFormal Verification. Lecture 7: Introduction to Binary Decision Diagrams (BDDs)
Formal Verification Lecture 7: Introduction to Binary Decision Diagrams (BDDs) Jacques Fleuriot jdf@inf.ac.uk Diagrams from Huth & Ryan, 2nd Ed. Recap Previously: CTL and LTL Model Checking algorithms
More informationFoundations of Computer Science Spring Mathematical Preliminaries
Foundations of Computer Science Spring 2017 Equivalence Relation, Recursive Definition, and Mathematical Induction Mathematical Preliminaries Mohammad Ashiqur Rahman Department of Computer Science College
More informationSymbolic Model Checking
Bug Catching 5-398 Symbolic Model Checking Hao Zheng Dept. of Computer Science & Eng. Univ. of South Florida Overview CTL model checking operates on sets. Calculates the fix points over finite state sets.
More informationLecture 2: Symbolic Model Checking With SAT
Lecture 2: Symbolic Model Checking With SAT Edmund M. Clarke, Jr. School of Computer Science Carnegie Mellon University Pittsburgh, PA 15213 (Joint work over several years with: A. Biere, A. Cimatti, Y.
More informationNegations in Refinement Type Systems
Negations in Refinement Type Systems T. Tsukada (U. Tokyo) 14th March 2016 Shonan, JAPAN This Talk About refinement intersection type systems that refute judgements of other type systems. Background Refinement
More informationComputational problems. Lecture 2: Combinatorial search and optimisation problems. Computational problems. Examples. Example
Lecture 2: Combinatorial search and optimisation problems Different types of computational problems Examples of computational problems Relationships between problems Computational properties of different
More informationChapter 8. NP and Computational Intractability. Slides by Kevin Wayne. Copyright 2005 Pearson-Addison Wesley. All rights reserved.
Chapter 8 NP and Computational Intractability Slides by Kevin Wayne. Copyright 2005 Pearson-Addison Wesley. All rights reserved. 1 Algorithm Design Patterns and Anti-Patterns Algorithm design patterns.
More informationBinary Decision Diagrams and Symbolic Model Checking
Binary Decision Diagrams and Symbolic Model Checking Randy Bryant Ed Clarke Ken McMillan Allen Emerson CMU CMU Cadence U Texas http://www.cs.cmu.edu/~bryant Binary Decision Diagrams Restricted Form of
More informationChapter 8: Data Abstractions
Chapter 8: Data Abstractions Computer Science: An Overview Tenth Edition by J. Glenn Brookshear Presentation files modified by Farn Wang Copyright 28 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
More information8.1 Polynomial-Time Reductions
8.1 Polynomial-Time Reductions Classify Problems According to Computational Requirements Q. Which problems will we be able to solve in practice? A working definition. Those with polynomial-time algorithms.
More informationApplication: Programming Language Semantics
Chapter 8 Application: Programming Language Semantics Prof. Dr. K. Madlener: Specification and Verification in Higher Order Logic 527 Introduction to Programming Language Semantics Programming Language
More informationALGORITHMS EXAMINATION Department of Computer Science New York University December 17, 2007
ALGORITHMS EXAMINATION Department of Computer Science New York University December 17, 2007 This examination is a three hour exam. All questions carry the same weight. Answer all of the following six questions.
More informationAntisymmetric Relations. Definition A relation R on A is said to be antisymmetric
Antisymmetric Relations Definition A relation R on A is said to be antisymmetric if ( a, b A)(a R b b R a a = b). The picture for this is: Except For Example The relation on R: if a b and b a then a =
More informationModel Checking I Binary Decision Diagrams
/42 Model Checking I Binary Decision Diagrams Edmund M. Clarke, Jr. School of Computer Science Carnegie Mellon University Pittsburgh, PA 523 2/42 Binary Decision Diagrams Ordered binary decision diagrams
More informationSyntax and Type Analysis
Syntax and Type Analysis Lecture Compilers Summer Term 2011 Prof. Dr. Arnd Poetzsch-Heffter Software Technology Group TU Kaiserslautern Prof. Dr. Arnd Poetzsch-Heffter Syntax and Type Analysis 1 Content
More informationCS 267: Automated Verification. Lecture 13: Bounded Model Checking. Instructor: Tevfik Bultan
CS 267: Automated Verification Lecture 13: Bounded Model Checking Instructor: Tevfik Bultan Remember Symbolic Model Checking Represent sets of states and the transition relation as Boolean logic formulas
More information2. Syntax and Type Analysis
Content of Lecture Syntax and Type Analysis Lecture Compilers Summer Term 2011 Prof. Dr. Arnd Poetzsch-Heffter Software Technology Group TU Kaiserslautern Prof. Dr. Arnd Poetzsch-Heffter Syntax and Type
More informationPower Set of a set and Relations
Power Set of a set and Relations 1 Power Set (1) Definition: The power set of a set S, denoted P(S), is the set of all subsets of S. Examples Let A={a,b,c}, P(A)={,{a},{b},{c},{a,b},{b,c},{a,c},{a,b,c}}
More informationRange Restriction for General Formulas
Range Restriction for General Formulas 1 Range Restriction for General Formulas Stefan Brass Martin-Luther-Universität Halle-Wittenberg Germany Range Restriction for General Formulas 2 Motivation Deductive
More informationBinary Decision Diagrams
5-44 Bug Catching: Automated Program Verification and Testing based on slides by SagarChaki 2 Carnegie Mellon University BDDs in a nutshell Typically mean Reduced Ordered (ROBDDs) Canonical representation
More informationModel checking pushdown systems
Model checking pushdown systems R. Ramanujam Institute of Mathematical Sciences, Chennai jam@imsc.res.in Update Meeting, IIT-Guwahati, 4 July 2006 p. 1 Sources of unboundedness Data manipulation: integers,
More informationCS357 Lecture: BDD basics. David Dill
CS357 Lecture: BDD basics David Dill BDDs (Boolean/binary decision diagrams) BDDs are a very successful representation for Boolean functions. A BDD represents a Boolean function on variables x, x 2,...
More informationLecture-12: Closed Sets
and Its Examples Properties of Lecture-12: Dr. Department of Mathematics Lovely Professional University Punjab, India October 18, 2014 Outline Introduction and Its Examples Properties of 1 Introduction
More informationLecture 15: The subspace topology, Closed sets
Lecture 15: The subspace topology, Closed sets 1 The Subspace Topology Definition 1.1. Let (X, T) be a topological space with topology T. subset of X, the collection If Y is a T Y = {Y U U T} is a topology
More informationGraph Theory S 1 I 2 I 1 S 2 I 1 I 2
Graph Theory S I I S S I I S Graphs Definition A graph G is a pair consisting of a vertex set V (G), and an edge set E(G) ( ) V (G). x and y are the endpoints of edge e = {x, y}. They are called adjacent
More informationRelational String Verification Using Multitrack
Relational String Verification Using Multitrack Automata Relational String Analysis Earlier work on string analysis use multiple single-track DFAs during symbolic reachability analysis One DFA per variable
More informationLecture - 8A: Subbasis of Topology
Lecture - 8A: Dr. Department of Mathematics Lovely Professional University Punjab, India October 18, 2014 Outline 1 Introduction 2 3 4 Introduction I As we know that topology generated by a basis B may
More informationLecture Notes on Real-world SMT
15-414: Bug Catching: Automated Program Verification Lecture Notes on Real-world SMT Matt Fredrikson Ruben Martins Carnegie Mellon University Lecture 15 1 Introduction In the previous lecture we studied
More informationCS Bootcamp Boolean Logic Autumn 2015 A B A B T T T T F F F T F F F F T T T T F T F T T F F F
1 Logical Operations 1.1 And The and operator is a binary operator, denoted as, &,, or sometimes by just concatenating symbols, is true only if both parameters are true. A B A B F T F F F F The expression
More informationDatabase Theory VU , SS Codd s Theorem. Reinhard Pichler
Database Theory Database Theory VU 181.140, SS 2011 3. Codd s Theorem Reinhard Pichler Institut für Informationssysteme Arbeitsbereich DBAI Technische Universität Wien 29 March, 2011 Pichler 29 March,
More informationNP-Hardness. We start by defining types of problem, and then move on to defining the polynomial-time reductions.
CS 787: Advanced Algorithms NP-Hardness Instructor: Dieter van Melkebeek We review the concept of polynomial-time reductions, define various classes of problems including NP-complete, and show that 3-SAT
More information! Greed. O(n log n) interval scheduling. ! Divide-and-conquer. O(n log n) FFT. ! Dynamic programming. O(n 2 ) edit distance.
Algorithm Design Patterns and Anti-Patterns Chapter 8 NP and Computational Intractability Algorithm design patterns. Ex.! Greed. O(n log n) interval scheduling.! Divide-and-conquer. O(n log n) FFT.! Dynamic
More informationDiscrete Mathematics Lecture 4. Harper Langston New York University
Discrete Mathematics Lecture 4 Harper Langston New York University Sequences Sequence is a set of (usually infinite number of) ordered elements: a 1, a 2,, a n, Each individual element a k is called a
More informationProgramming with Dependent Types Interactive programs and Coalgebras
Programming with Dependent Types Interactive programs and Coalgebras Anton Setzer Swansea University, Swansea, UK 14 August 2012 1/ 50 A Brief Introduction into ML Type Theory Interactive Programs in Dependent
More information13.1 DECISION ANALYSIS WITH DECISION TREES AND TABLES (CONDITION-ACTION ANALYSIS)
Obligatory Reading Fakultät Informatik, Institut für Software- und Multimediatechnik, Lehrstuhl für Softwaretechnologie alzert, Kapitel über Entscheidungstabellen Ghezzi 6.3 Decision-table based testing
More informationLecture1: Symbolic Model Checking with BDDs. Edmund M. Clarke, Jr. Computer Science Department Carnegie Mellon University Pittsburgh, PA 15213
Lecture: Symbolic Model Checking with BDDs Edmund M Clarke, Jr Computer Science Department Carnegie Mellon University Pittsburgh, PA 523 Temporal Logic Model Checking Specification Language: A propositional
More informationVerifying Liveness Properties of ML Programs
Verifying Liveness Properties of ML Programs M M Lester R P Neatherway C-H L Ong S J Ramsay Department of Computer Science, University of Oxford ACM SIGPLAN Workshop on ML, 2011 09 18 Gokigeny all! Motivation
More information1. Draw the state graphs for the finite automata which accept sets of strings composed of zeros and ones which:
P R O B L E M S Finite Autom ata. Draw the state graphs for the finite automata which accept sets of strings composed of zeros and ones which: a) Are a multiple of three in length. b) End with the string
More informationAshish Sabharwal Computer Science and Engineering University of Washington, Box Seattle, Washington
MODEL CHECKING: TWO DECADES OF NOVEL TECHNIQUES AND TRENDS PHD GENERAL EXAM REPORT Ashish Sabharwal Computer Science and Engineering University of Washington, Box 352350 Seattle, Washington 98195-2350
More informationSolving Boolean Equations with BDDs and Clause Forms. Gert Smolka
Solving Boolean Equations with BDDs and Clause Forms Gert Smolka Abstract Methods for solving Boolean equations BDDs [Bryant 1986] Clause forms [Quine 1959] Efficient data structure and algorithms for
More informationAppendix 1. Description Logic Terminology
Appendix 1 Description Logic Terminology Franz Baader Abstract The purpose of this appendix is to introduce (in a compact manner) the syntax and semantics of the most prominent DLs occurring in this handbook.
More informationLecture Notes on Binary Decision Diagrams
Lecture Notes on Binary Decision Diagrams 15-122: Principles of Imperative Computation William Lovas Notes by Frank Pfenning Lecture 25 April 21, 2011 1 Introduction In this lecture we revisit the important
More informationAppendix 1. Description Logic Terminology
Appendix 1 Description Logic Terminology Franz Baader Abstract The purpose of this appendix is to introduce (in a compact manner) the syntax and semantics of the most prominent DLs occurring in this handbook.
More informationFormal Verification Methods 2: Symbolic Simulation
Formal Verification Methods 2: Symbolic Simulation John Harrison Intel Corporation Marktoberdorf 2003 Thu 3st July 2003 (:25 2:0) 0 Summary Simulation Symbolic and ternary simulation BDDs Quaternary lattice
More information! Greed. O(n log n) interval scheduling. ! Divide-and-conquer. O(n log n) FFT. ! Dynamic programming. O(n 2 ) edit distance.
Algorithm Design Patterns and Anti-Patterns 8. NP and Computational Intractability Algorithm design patterns. Ex.! Greed. O(n log n) interval scheduling.! Divide-and-conquer. O(n log n) FFT.! Dynamic programming.
More informationSpecification-based Testing of Embedded Systems H. Schlingloff, SEFM 2008
SEFM School 2008 Specification-based Testing of Embedded Systems Prof. Dr. Holger Schlingloff Humboldt-Universität zu Berlin and Fraunhofer FIRST, Berlin Lecture 4: Mutations, OCL etc. Course Outline L1:
More informationBinary Decision Diagrams (BDD)
Binary Decision Diagrams (BDD) Contents Motivation for Decision diagrams Binary Decision Diagrams ROBDD Effect of Variable Ordering on BDD size BDD operations Encoding state machines Reachability Analysis
More informationSymbolic Methods. The finite-state case. Martin Fränzle. Carl von Ossietzky Universität FK II, Dpt. Informatik Abt.
Symbolic Methods The finite-state case Part I Martin Fränzle Carl von Ossietzky Universität FK II, Dpt. Informatik Abt. Hybride Systeme 02917: Symbolic Methods p.1/34 What you ll learn How to use and manipulate
More informationResearch Collection. Formal background and algorithms. Other Conference Item. ETH Library. Author(s): Biere, Armin. Publication Date: 2001
Research Collection Other Conference Item Formal background and algorithms Author(s): Biere, Armin Publication Date: 2001 Permanent Link: https://doi.org/10.3929/ethz-a-004239730 Rights / License: In Copyright
More informationSyntax-directed model checking of sequential programs
The Journal of Logic and Algebraic Programming 52 53 (2002) 129 162 THE JOURNAL OF LOGIC AND ALGEBRAIC PROGRAMMING www.elsevier.com/locate/jlap Syntax-directed model checking of sequential programs Karen
More informationBoolean Representations and Combinatorial Equivalence
Chapter 2 Boolean Representations and Combinatorial Equivalence This chapter introduces different representations of Boolean functions. It then discusses the applications of these representations for proving
More informationFinal Course Review. Reading: Chapters 1-9
Final Course Review Reading: Chapters 1-9 1 Objectives Introduce concepts in automata theory and theory of computation Identify different formal language classes and their relationships Design grammars
More informationEpistemic Model Checking with Haskell
Epistemic Model Checking with Haskell Malvin Gattinger 2016-12-01, Peking University Haskell in 10 Minutes Simple Explicit Model Checking Symbolic Model Checking Binary Decision Diagrams More Puzzles Even
More informationIntroduction to Finite Model Theory. Jan Van den Bussche Universiteit Hasselt
Introduction to Finite Model Theory Jan Van den Bussche Universiteit Hasselt 1 Books Finite Model Theory by Ebbinghaus & Flum 1999 Finite Model Theory and Its Applications by Grädel et al. 2007 Elements
More information1KOd17RMoURxjn2 CSE 20 DISCRETE MATH Fall
CSE 20 https://goo.gl/forms/1o 1KOd17RMoURxjn2 DISCRETE MATH Fall 2017 http://cseweb.ucsd.edu/classes/fa17/cse20-ab/ Today's learning goals Explain the steps in a proof by mathematical and/or structural
More informationDiscrete, Continuous, and Hybrid Petri Nets
Discrete, Continuous, and Hybrid Petri Nets Bearbeitet von René David, Hassane Alla 1. Auflage 2004. Buch. XXII, 570 S. Hardcover ISBN 978 3 540 22480 8 Format (B x L): 15,5 x 23,5 cm Gewicht: 2080 g Weitere
More informationSymbolic Trajectory Evaluation - A Survey
Automated Verification Symbolic Trajectory Evaluation - A Survey by Mihaela Gheorghiu Department of Computer Science University of Toronto Instructor: Prof. Marsha Chechik January 3, 24 Motivation Simulation
More informationSemantic Subtyping. Alain Frisch (ENS Paris) Giuseppe Castagna (ENS Paris) Véronique Benzaken (LRI U Paris Sud)
Semantic Subtyping Alain Frisch (ENS Paris) Giuseppe Castagna (ENS Paris) Véronique Benzaken (LRI U Paris Sud) http://www.cduce.org/ Semantic Subtyping - Groupe de travail BD LRI p.1/28 CDuce A functional
More informationCS521 \ Notes for the Final Exam
CS521 \ Notes for final exam 1 Ariel Stolerman Asymptotic Notations: CS521 \ Notes for the Final Exam Notation Definition Limit Big-O ( ) Small-o ( ) Big- ( ) Small- ( ) Big- ( ) Notes: ( ) ( ) ( ) ( )
More informationRewriting. Andreas Rümpel Faculty of Computer Science Technische Universität Dresden Dresden, Germany.
Rewriting Andreas Rümpel Faculty of Computer Science Technische Universität Dresden Dresden, Germany s9843882@inf.tu-dresden.de ABSTRACT This is an overview paper regarding the common technologies of rewriting.
More informationIn class 75min: 2:55-4:10 Thu 9/30.
MATH 4530 Topology. In class 75min: 2:55-4:10 Thu 9/30. Prelim I Solutions Problem 1: Consider the following topological spaces: (1) Z as a subspace of R with the finite complement topology (2) [0, π]
More informationElementary Recursive Function Theory
Chapter 6 Elementary Recursive Function Theory 6.1 Acceptable Indexings In a previous Section, we have exhibited a specific indexing of the partial recursive functions by encoding the RAM programs. Using
More informationSome Interdefinability Results for Syntactic Constraint Classes
Some Interdefinability Results for Syntactic Constraint Classes Thomas Graf tgraf@ucla.edu tgraf.bol.ucla.edu University of California, Los Angeles Mathematics of Language 11 Bielefeld, Germany 1 The Linguistic
More informationFinite Model Generation for Isabelle/HOL Using a SAT Solver
Finite Model Generation for / Using a SAT Solver Tjark Weber webertj@in.tum.de Technische Universität München Winterhütte, März 2004 Finite Model Generation for / p.1/21 is a generic proof assistant: Highly
More informationGraph algorithms based on infinite automata: logical descriptions and usable constructions
Graph algorithms based on infinite automata: logical descriptions and usable constructions Bruno Courcelle (joint work with Irène Durand) Bordeaux-1 University, LaBRI (CNRS laboratory) 1 Overview Algorithmic
More informationAutomata Theory for Reasoning about Actions
Automata Theory for Reasoning about Actions Eugenia Ternovskaia Department of Computer Science, University of Toronto Toronto, ON, Canada, M5S 3G4 eugenia@cs.toronto.edu Abstract In this paper, we show
More informationFully-Implicit Relational Coarsest Partitioning for Faster Bisimulation (As Preparation for Fully-Implicit Lumping)
Fully-Implicit Relational Coarsest Partitioning for Faster Bisimulation (As Preparation for Fully-Implicit Lumping) Malcolm Mumme September 24, 2008 1 Abstract The present work applies interleaved MDD
More informationOn the Verification of Sequential Equivalence
686 IEEE TRANSACTIONS ON COMPUTER-AIDED DESIGN OF INTEGRATED CIRCUITS AND SYSTEMS, VOL 22, NO 6, JUNE 2003 On the Verification of Sequential Equivalence Jie-Hong R Jiang and Robert K Brayton, Fellow, IEEE
More informationSymbolic Synthesis of Knowledge-based Program Implementations with Synchronous Semantics
Symbolic Synthesis of Knowledge-based Program Implementations with Synchronous Semantics X. Huang xiaoweih@cse.unsw.edu.au R. van der Meyden meyden@cse.unsw.edu.au ABSTRACT This paper deals with the automated
More informationLimitations of Algorithmic Solvability In this Chapter we investigate the power of algorithms to solve problems Some can be solved algorithmically and
Computer Language Theory Chapter 4: Decidability 1 Limitations of Algorithmic Solvability In this Chapter we investigate the power of algorithms to solve problems Some can be solved algorithmically and
More informationComputability Theory
CS:4330 Theory of Computation Spring 2018 Computability Theory Other NP-Complete Problems Haniel Barbosa Readings for this lecture Chapter 7 of [Sipser 1996], 3rd edition. Sections 7.4 and 7.5. The 3SAT
More informationA set with only one member is called a SINGLETON. A set with no members is called the EMPTY SET or 2 N
Mathematical Preliminaries Read pages 529-540 1. Set Theory 1.1 What is a set? A set is a collection of entities of any kind. It can be finite or infinite. A = {a, b, c} N = {1, 2, 3, } An entity is an
More informationEfficiently Reasoning about Programs
Efficiently Reasoning about Programs Neil Immerman College of Computer and Information Sciences University of Massachusetts, Amherst Amherst, MA, USA people.cs.umass.edu/ immerman co-r.e. complete Halt
More informationDecision Procedures in the Theory of Bit-Vectors
Decision Procedures in the Theory of Bit-Vectors Sukanya Basu Guided by: Prof. Supratik Chakraborty Department of Computer Science and Engineering, Indian Institute of Technology, Bombay May 1, 2010 Sukanya
More informationABC basics (compilation from different articles)
1. AIG construction 2. AIG optimization 3. Technology mapping ABC basics (compilation from different articles) 1. BACKGROUND An And-Inverter Graph (AIG) is a directed acyclic graph (DAG), in which a node
More informationImplementation of Lexical Analysis
Implementation of Lexical Analysis Outline Specifying lexical structure using regular expressions Finite automata Deterministic Finite Automata (DFAs) Non-deterministic Finite Automata (NFAs) Implementation
More informationUnification in Maude. Steven Eker
Unification in Maude Steven Eker 1 Unification Unification is essentially solving equations in an abstract setting. Given a signature Σ, variables X and terms t 1, t 2 T (Σ) we want to find substitutions
More informationComputability Summary
Computability Summary Recursive Languages The following are all equivalent: A language B is recursive iff B = L(M) for some total TM M. A language B is (Turing) computable iff some total TM M computes
More informationCS152: Programming Languages. Lecture 2 Syntax. Dan Grossman Spring 2011
CS152: Programming Languages Lecture 2 Syntax Dan Grossman Spring 2011 Finally, some formal PL content For our first formal language, let s leave out functions, objects, records, threads, exceptions,...
More informationGenerating Efficient Test Oracles from Specifications
FB4: Informatik Generating Efficient Test Oracles from Specifications Studienarbeit im Studiengang Informatik vorgelegt von Markus Bender Betreuer: Prof. Dr. Bernhard Beckert, Universität Koblenz, Prof.
More informationDISCRETE MATHEMATICS
DISCRETE MATHEMATICS WITH APPLICATIONS THIRD EDITION SUSANNA S. EPP DePaul University THOIVISON * BROOKS/COLE Australia Canada Mexico Singapore Spain United Kingdom United States CONTENTS Chapter 1 The
More informationSAT-CNF Is N P-complete
SAT-CNF Is N P-complete Rod Howell Kansas State University November 9, 2000 The purpose of this paper is to give a detailed presentation of an N P- completeness proof using the definition of N P given
More informationTree Decompositions Why Matroids are Useful
Petr Hliněný, W. Graph Decompositions, Vienna, 2004 Tree Decompositions Why Matroids are Useful Petr Hliněný Tree Decompositions Why Matroids are Useful Department of Computer Science FEI, VŠB Technical
More informationLTCS Report. Concept Descriptions with Set Constraints and Cardinality Constraints. Franz Baader. LTCS-Report 17-02
Technische Universität Dresden Institute for Theoretical Computer Science Chair for Automata Theory LTCS Report Concept Descriptions with Set Constraints and Cardinality Constraints Franz Baader LTCS-Report
More informationModel Finder. Lawrence Chung 1
Model Finder Lawrence Chung 1 Comparison with Model Checking Model Checking Model (System Requirements) STM Specification (System Property) Temporal Logic Model Checker M φ Model Finder (http://alloy.mit.edu/tutorial3/alloy-tutorial.html)
More informationMore on Polynomial Time and Space
CpSc 8390 Goddard Fall15 More on Polynomial Time and Space 20.1 The Original NP-Completeness Proof A configuration/snapshot of a machine is a representation of its current state (what info would be needed
More informationBinary Decision Diagrams
Logic and roof Hilary 2016 James Worrell Binary Decision Diagrams A propositional formula is determined up to logical equivalence by its truth table. If the formula has n variables then its truth table
More informationSet Manipulation with Boolean Functional Vectors for Symbolic Reachability Analysis
Set Manipulation with Boolean Functional Vectors for Symbolic Reachability Analysis Amit Goel Department of ECE, Carnegie Mellon University, PA. 15213. USA. agoel@ece.cmu.edu Randal E. Bryant Computer
More informationNP and computational intractability. Kleinberg and Tardos, chapter 8
NP and computational intractability Kleinberg and Tardos, chapter 8 1 Major Transition So far we have studied certain algorithmic patterns Greedy, Divide and conquer, Dynamic programming to develop efficient
More information