Formal Systems 2 P. H. Schmitt May 2016 Contents Contents.................................1 List of Figures..............................4 1 Introduction6 1.1 Prerequisites............................7 1.2 Exercises..............................9 2 Axiomatic Set Theory 10 2.1 Basics............................... 11 2.2 The Natural Numbers...................... 16 2.3 Recursion............................. 20 2.4 Integers.............................. 23 2.5 Ordinals.............................. 26 2.6 K¨onig'sLemma (Optional).................... 37 2.7 Cardinals (Optional)....................... 41 2.8 Ramsey Theory(Optional).................... 43 2.8.1 Infinite Ramsey Theory................. 43 2.8.2 Finite Ramsey Theory.................. 47 2.9 Peano Arithmetic with Finite Sets (Optional)............................. 49 2.10 Comments............................. 68 2.11 Exercises.............................. 68 1 3 Modal Logic 72 3.1 Syntax and Semantics...................... 74 3.2 Correspondence Theory..................... 82 3.3 The Tree-Property........................ 93 3.4 Second Order Logic........................ 97 3.5 A Tableau Calculus........................ 106 3.6 Description Logic......................... 116 3.7 Knowledge Representation in the Semantic Web....... 135 3.8 Translation into First-Order Logic................ 147 3.8.1 Decidable Fragments of First-Order Logic....... 149 3.9 Exercises.............................. 151 4 Dynamic Logic 158 4.1 Motivating Example....................... 159 4.2 Syntax and Semantics of Regular Dynamic Logic....... 162 4.2.1 Boogie PL......................... 172 4.3 Propositional Dynamic Logic................... 178 4.4 Decidability of Propositional Dynamic Logic.......... 182 4.5 Alternatives in PDL....................... 194 4.6 Axiomatizations of Dynamic Logic............... 200 4.7 Exercises.............................. 205 5 Temporal Logics 208 5.1 B¨uchi Automata......................... 211 5.2 Linear Temporal Logic...................... 221 5.2.1 Expressiveness of Linear Temporal Logic........ 228 5.3 Bounded Model Checking (Optional).............. 230 5.4 Computation Tree Logic..................... 237 5.5 CTL Model Checking....................... 250 5.6 Exercises.............................. 264 2 6 Solutions to Exercises 267 References................................ 305 Index.................................. 311 3 List of Figures 2.1 Axioms of Zermelo-Fraenkel Set Theory............. 36 2.2 Peano Axioms for PAFin..................... 49 2.3 Axioms of finite set theory for PAFin.............. 51 3.1 Example of a Kripke structure.................. 75 3.2 Some properties of frames.................... 83 3.3 Visualization of the C property................. 85 3.4 Examples of non-characterizable frame classes......... 88 3.5 Modal Tableau Rules....................... 108 3.6 Tableau Rules for ...................... 125 ALC 3.7 XML representation of of meta-data for DC20030602..... 137 3.8 Axiomatic Tripels for RFDS................... 139 3.9 Translated Axiomatic Triples for RFDS............. 141 3.10 A simple RDF graph....................... 145 3.11 A simple RDF graph with blank nodes............. 146 3.12 Translation from Modal to First-order Formulas........ 147 4.1 The Towers of Hanoi....................... 159 5.1 The B¨uchi automaton afin ................... 212 N 5.2 The example automaton .................. 231 Adbp 5.3 Cyclic Semantics for LTL formulas............... 234 4 5.4 M for the example automaton and F FGp ...... 235 3 Adbp ≡ 5.5 Mutual Exclusion (first attempt)................ 238 5.6 Mutual Exclusion (second attempt)............... 239 5.7 Transition system for AFAGp .................. 249 5.8 Mutual Exclusion (repeated from Figure 5.6).......... 261 6.1 Counterexample to (2P 2Q) 2(P Q)......... 278 ! ! ! 6.2 Counterexample to 2(P Q) (2P 2Q).......... 278 _ ! _ 6.3 A B¨uchi automaton accepting Kp;q ............... 300 6.4 Transition system for AGEFp .................. 302 5 Chapter 1 Introduction 6 1.1 Prerequisites We assume that the reader has had some previous exposure to propositional and first-order logic. The lecture notes [Schmitt, 2008] will certainly be good enough. But, also any good textbook on the subject [Fitting, 1990, Monk, 1976, Huth & Ryan, 2000] will do. Here we will occupy ourselves with some interesting yet not mainstream results to get back into the mood. The emphasis is on providing exercises to work on. Meagre Vocabularies To get a better insight into first-order logic logicians have taken a great interest in studying the expressive power of various fragments of first-order formulas. Well know examples are the fragments of Horn formulas, existential or universal formulas or quantifier-free formulas. Here will will not restrict the use of the logical operations, but put limitations on the vocabularies. It is for example well know that it is no restricition to use only relation symbols and forget function symbols of arity strictly greater than 0. Definition 1 (Vocabularies) A vocabulary Σ is called 1.a relational vocabulary if it possibly contains constant symbols, but no function symbols of arity greater than 0. 2.a binary vocabulary if it is relational and all relations symbols in Σ are binary. 3.a triple vocabulary if it is relational and contains exactly on relation symbol rel and this is ternary. We also need a way to compare different types of vocabularies with regard to their expressiveness. Definition 2 (Comparing Vocabularies) Let , be two classes of vocabularies. is called as expressive as V1 V2 V2 V1 if for every signature Σ1 in class 1 there is a signature Σ2 in class 2 such that V V 7 1. for every Σ1-formula F1 there is a Σ2-formula F2 and for every Σ2- structure there is a Σ -structure such that M 1 M1 = F = F M1 j 1 , M j 2 2. and for every Σ -structure there is a Σ -structure such that 1 N 2 M M1 'N This definition may at first sound complicated. But, in the end it is just a formal rendering of what you would intuitivly assume. To get some under- standing for this definition try to do excercise 1.2.2. Definition2 describes the ideal situation. In most cases, in particular also for Exercises 1.2.4 and 1.2.5, the transition from to 1 is only possible for structures that are big enough. M M M Order Relations Definition 3 (Order) A structure (M; R) where R is a binary relation on M is called an order if it satisfies the following axioms 1. x y z(R(x; y) R(y; z) R(x; z)) (transitivity) 8 8 8 ^ ! 2. x( R(x; x)) (anti-reflexivity) 8 : Definition 4 (Linear Order) A order relation (M; R) is called a linear order if it satisfies the following axiom 1. x y z(R(x; y) R(y; x) x = y) (linearity) 8 8 8 _ _ 8 1.2 Exercises Exercise 1.2.1 Let F be a formula in first-order logic, that is true in all infinite structures. Then there is a natural number k such that F is even true in all structures with k or more elements. Exercise 1.2.2 Assume that Σ1, Σ2, F1, F2 satisfy the two statements from Definition2 on page7. Then F1 is a Σ1-tautology iff F2 is a Σ2-tautology. Exercise 1.2.3 Assume that Σ1, Σ2, F1, F2 satisfy the two statements from Definition2 on page7. If validity of Σ1-formulas is undecidable then also validity of Σ2-formulas is undecidable. Exercise 1.2.4 Show that the class of triple vocabularies is as expressive as the class of binary vocabularies. Exercise 1.2.5 Show that the class of binary vocabularies is as expressive as the class of unrestricted relational vocabularies. See also Exercise 3.9.23. 9 Chapter 2 Axiomatic Set Theory 10 Many formal specification languages, among them as prime examples Z and B, use set theoretical concepts and notations. This is an appealing choice, because these concepts are easy to understand and accessible without mathe- matical training. Another advantage is the fact, that there is a well developed mathematical theory of sets. In fact, before set theory was perceived as a foundation for specification languages it was considered as a foundation for all of mathematics. A very intriguing idea: once you accept a few axioms of set theory all mathematical results can be derived from them. In this chapter we will convey a first idea of how this works. 2.1 Basics We will use the Zermelo-Fraenkel (ZF for short) axiom system for set theory. In our presentation we follow the textbooks [Takeuti & Zaring, 1971], [Rubin, 1967], and [R.Drake, 1974]. The full set of ZF axioms is given in Figure 2.1 on page 36. The language of ZF set theory is the language for first-order predicate logic with the binary relation symbol as its only non-logical symbol. In the formulation of the axioms the equality2 symbol = is also used. But note, that using axiom A1 and formula t1 = t2 may be equivalently replaced by a formula containing only . More precisely, axiom A1 states only one implication. The reverse implication,2 i.e. x = y z(z x z y) ! 8 2 $ 2 has nothing to do with set theory, it is a simple consequence of the congruence axioms x = y (p(z; x) p(z; y)) ! $ for any binary relation symbol p. Any free variables in the axioms are implicitely universally quantified. Before we go on, we need some notational conventions, otherwise our formulas would soon be unintelligible. We will use for any formula φ(x) the syntactical construct x φ(x) , called a class term. We intuitively think of x φ(x) as the collectionf j ofg all sets f j g 11 a satisfying the formula φ(a). This is only for notational convenience. The new terms can be eliminated as follows: y x φ(x) is replaced by φ(y) x2 fφ(xj) yg is replaced by u(u y f j g 2 9z(z 2 u^ φ(z))) x φ(x) y (y) is replaced by 8u( (2u) $ f j g 2 f j g 9z(z u^ φ(z))) 8 2 $ Note, that using a class term x φ(x) , does by far not imply that x φ(x) is a set.