Computation Engineering Applied Automata Theory and Logic. Ganesh Gopalakrishnan University of Utah. ^J Springer

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1 Computation Engineering Applied Automata Theory and Logic Ganesh Gopalakrishnan University of Utah ^J Springer

2 Foreword Preface XXV XXVII 1 Introduction 1 Computation Science and Computation Engineering 1 What is 'Computation?' 2 A Minimalist Approach 3 How to Measure the Power of Computers? 5 Complexity Theory 5 Automata Theory and Computing 6 Why "Mix-up" Automata and Mathematical Logic? 7 Why Verify? Aren't Computers "Mostly Okay?" 7 Verifying Computing Systems Using Automaton Models 9 Automaton/Logic Connection ; 10 Avoid Attempting the Impossible 11 Solving One Implies Solving All 11 Automata Theory Demands a Lot From You! 12 A Brief Foray Into History 12 Disappearing Formal Methods. 13 Exercises 14 2 Mathematical Preliminaries Numbers Boolean Concepts, Propositions, and Quantifiers Sets Defining sets Avoid contradictions 17

3 VIII Ensuring uniqueness of definitions Cartesian Product and Powerset Powersets and characteristic sequences Functions and Signature The A Notation * Total, Partial, 1-1, and Onto Functions Computable Functions Algorithm versus Procedure Relations Functions as Relations More A syntax 30 Exercises 31 Cardinalities and Diagonalization Cardinality Basics r Countable sets Cardinal numbers Cardinality "trap" The Diagonalization Method Simplify the set in question Avoid dual representations for numbers Claiming a bijection, and refuting it 'Fixing' the proof a little bit Cardinality of 2 Nat and Nat -> Bool The Schroder-Bernstein Theorem Application: cardinality of all C Programs Application: functions in Nat > Bool Proof of the Schroder-Bernstein Theorem. 46 Exercises 50 4 Binary Relations Binary Relation Basics Types of binary relations Preorder (reflexive plus transitive) Partial order (preorder plus antisymmetric) Total order, and related notions Equivalence (Preorder plus Symmetry) Intersecting a preorder and its inverse Identity relation Universal relation 59

4 IX Equivalence class Reflexive and transitive closure The Power Relation between Machines The equivalence relation over machine types Lattice of All Binary Relations over S Equality, Equivalence, and Congruence Congruence relation 64 Exercises 67 5 Mathematical Logic, Induction, Proofs To Prove or Not to Prove! Proof Methods The implication operation 'If,' or 'Definitional Equality' Proof by contradiction Quantification operators V and Generalized DeMorgan's Law Relating V And Inductive definitions of sets and functions Induction Principles Induction over natural numbers Noetherian induction Structural Putting it All Together: the Pigeon-hole Principle 85 Exercises 86 Dealing with Recursion Recursive Definitions Recursion viewed as solving for a function Fixed-point equations The Y operator Illustration of reduction Recursive Definitions as Solutions of Equations The least fixed-point Fixed-points in Automata Theory 101 Exercises 103

5 X "" 7 Strings and Languages Strings The empty string e Length, character at index, and substring of a string Concatenation of strings Languages How many languages are there? Orders for Strings Operations on languages Concatenation and exponentiation Kleene Star, '*' Complementation Reversal Homomorphism Ensuring homomorphisms Inverse homomorphism An Illustration of homomorphisms Prefix-closure 115 Exercises Machines, Languages, DFA Machines The DFA The "power" of DFAs Limitations of DFAs Machine types that accept non-regular languages Drawing DFAs neatly 129 Exercises NFA and Regular Expressions What is Nondeterminism? How nondeterminism affects automaton operations How nondeterminism affects the power of machines Regular Expressions Nondeterministic Finite Automata Nondeterministic Behavior Without e Nondeterministic behavior with e Eclosure (also known as e-closure) Language of an NFA 147

6 XI A detailed example: telephone numbers Tool-assisted study of NFAs, DFAs, and REs 151 Exercises Operations on Regular Machinery NFA to DFA Conversion Operations on Machines Union Intersection Complementation Concatenation Star Reversal Homomorphism Inverse Homomorphism Prefix-closure More Conversions RE to NFA NFA to RE Minimizing DFA Error-correcting DFAs DFA constructed using error strata DFA constructed through regular expressions Ultimate Periodicity and DFAs 179 Exercises The Automaton/Logic Connection, Symbolic Techniques The Automaton/Logic Connection DFA can 'scan' and also 'do logic' Binary Decision Diagrams (BDDs) Basic Operations on BDDs ' Representing state transition systems Forward reachability Fixed-point iteration to compute the least fixed-point An example with multiple fixed-points Playing tic-tac-toe using BDDs 198 Exercises I 200

7 XII 12 The 'Pumping' Lemma Pumping Lemmas for Regular Languages A stronger incomplete Pumping Lemma An adversarial argument Closure Properties Ameliorate Pumping Complete Pumping Lemmas Jaffe's complete Pumping Lemma Stanat and Weiss' complete Pumping Lemma Exercises Context-free Languages The Language of a CFG Consistency, Completeness, and Redundancy More consistency proofs Fixed-points again! Ambiguous Grammars If-then-else ambiguity Ambiguity, inherent ambiguity A Taxonomy of Formal Languages and Machines Non-closure of CFLs under complementation Simplifying CFGs Push-down Automata DPDA versus NPDA Deterministic context-free languages (DCFL) Some Factoids Right- and Left-Linear CFGs Developing CFGs A Pumping Lemma for CFLs 239 Exercises Push-down Automata and Context-free Grammars Push-down Automata Conventions for describing PDAs Acceptance by final state Acceptance by empty stack Conversion of Pi to P 2 ensuring L(P X ) = N(P 2 ) Conversion of P x to P 2 ensuring N(P 1 ) = L{P 2 ) Proving PDAs Correct Using Floyd's Inductive Assertions Direct Conversion of CFGs to PDAs Direct Conversion of PDAs to CFGs 257

8 XIII Name non-terminals to match stack-emptying possibilities Let start symbol S set up all stack-draining options Capture how each PDA transition helps drain the stack Final result from Figure The Chomsky Normal Form Cocke-Kasami-Younger (CKY) parsing algorithm Closure and Decidability Some Important Points Visited Chapter Summary - Lost Venus Probe 267 Exercises Turing Machines Computation: Church/Turing Thesis "Turing machines" according to Turing Formal Definition of a Turing machine Singly- or doubly-infinite tape? Two stacks+control = Turing machine Linear bounded automata Tape vs. random access memory Acceptance, Halting, Rejection "Acceptance" of a TM closely examined Instantaneous descriptions Examples Examples illustrating TM concepts and conventions A DTM for w#w NDTMs Guess and check An NDTM for ww Simulations Multi-tape vs. single-tape Turing machines Nondeterministic Turing machines The Simulation itself 287 Exercises Basic Undecidability Proofs Some Decidable and Undecidable Problems An assortment of decidable problems Assorted undecidable problems 294

9 XIV 16.2 Undecidability Proofs Turing recognizable (or recursively enumerable) sets Recursive (or decidable) languages Acceptance (ATM) is undecidable (important!) Halting (HaltrM) is undecidable (important!) Mapping reductions Undecidable problems are "ATM in disguise" 305 Exercises Advanced Undecidability Proofs Rice's Theorem Failing proof attempt Corrected proof Greibach's Theorem \ The Computation History Method Decidability of LBA acceptance Undecidability of LBA language emptiness Undecidability of PDA language universality Post's correspondence problem (PCP) PCP is undecidable Proof sketch of the undecidability of PCP 317 Exercises Basic Notions in Logic including SAT Axiomatization of Propositional Logic First-order Logic (FOL) and Validity A warm-up exercise Examples of interpretations Validity of first-order logic is undecidable Valid FOL formulas are enumerable Properties of Boolean Formulas Boolean satisfiability: an overview Normal forms Overview of direct DNF to CNF conversion CNF-conversion using gates DIMACS file encoding Unsatisfiable CNF instances CNF, ^-satisfiability, and general CNF CNF satisfiability 341

10 XV Exercises Complexity Theory and NP-Completeness Examples and Overview The traveling salesperson problem P-time deciders, robustness, and 2 vs A note on complexity measurement The robustness of the Turing machine model Going from "2 to 3" changes complexity Formal Definitions NP viewed in terms of verifiers Some problems are outside NP NP-complete and NP-hard NP viewed in terms of deciders An example of an NP decider Minimal input encodings NPC Theorems and proofs NP-Completeness of 3-SAT Practical approaches to show NPC NP-Hard Problems can be Undecidable (Pitfall) Proof that Diophantine Equations are NPH "Certificates" of Diophantine Equations What other complexity measures exist? NP, CoNP, etc 362 Exercises DFA for Presburger Arithmetic Presburger Formulas and DFAs Presburger formulas Encoding conventions Example 1 representing x < Example 2 Vx3y.(x + y) = Conversion algorithm: Presburger formulas to automata Pitfalls to Avoid The restriction of equal bit-vector lengths 379 Exercises 380

11 XVI 21 Model Checking: Basics An Introduction to Model Checking What Are Reactive Computing Systems? Why model checking? Model checking vs. testing Biichi automata, and Verifying Safety and Liveness Example: Dining Philosophers Model (proctype) and property (never) automata. 394 Exercises Model Checking: Temporal Logics Temporal Logics Kripke structures Computations vs. computation trees Temporal formulas are Kripke structure classifiers! LTL vs. CTL through an example LTL syntax LTL semantics CTL syntax CTL semantics 408 Exercises Model Checking: Algorithms Enumerative CTL Model Checking Symbolic Model Checking for CTL EG p through fixed-point iteration.? Calculating EX and AX LFP and GFP for 'Until' LFP for 'Until' GFP for Until Biichi Automata and LTL Model Checking, Comparing expressiveness Operations on Biichi automata Nondeterminism in Biichi automata Enumerative Model Checking for LTL Reducing verification to Biichi automaton emptiness 433 Exercises 436

12 XVII 24 Conclusions 439 Book web site and tool information 443 A.I Web site and address 443 A.2 Software tool usage per chapter 443 A.3 Possible Syllabi 444 BED Solution to the tic-tac-toe problem 447 References 453 Index 461

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