Universita` degli Studi di Udine Dipartimento di Matematica e Informatica Dottorato di Ricerca in Informatica XXIV Ciclo - Anno Accademico 2011-2012 Ph.D. Thesis Sets as Graphs Candidate: Supervisors: Alexandru Ioan Tomescu Alberto Policriti Eugenio G. Omodeo December 2011 Tesi di dottorato di Alexandru Ioan Tomescu, discussa presso l'Universit`adegli Studi di Udine Author's e-mail: [email protected] Author's address: Dipartimento di Matematica e Informatica Universit`adegli Studi di Udine Via delle Scienze, 206 33100 Udine Italia Tesi di dottorato di Alexandru Ioan Tomescu, discussa presso l'Universit`adegli Studi di Udine To my parents Tesi di dottorato di Alexandru Ioan Tomescu, discussa presso l'Universit`adegli Studi di Udine Tesi di dottorato di Alexandru Ioan Tomescu, discussa presso l'Universit`adegli Studi di Udine Abstract In set theory and formal logic, a set is generally an object containing nothing but other sets as elements. Not only sets enable uniformity in the formalization of the whole of mathematics, but their ease-of-use and conciseness are employed to represent information in some computer languages. Given the intrinsic nesting property of sets, it is natural to represent them as directed graphs: vertices will stand for sets, while the arc relation will mimic the membership relation. This switch of perspective is important: from a computational point of view, this led to many decidability results, while from a logical point of view, this allowed for natural extensions of the concept of set, such as that of hyperset. Interpreting a set as a directed graph gives rise to many combinatorial, structural and computational questions, having as unifying goal that of a transfer of results and techniques across the two areas. Here, we study sets under the spotlight of combinatorial enumeration, canonical encodings by numbers, random generation, digraph immersions as well-quasi-orders. We also tackle the decidability problem for the celebrated Bernays- Sch¨onfinkel-Ramsey class of first-order formulae, over hypersets, motivated by a recent decidability result for standard sets. This thesis is also devoted to an investigation on the underlying structure of sets; ultimately, by studying the undirected graphs underlying sets, which we call set graphs, we study which graphs can be `implicitly' represented by sets. We elucidate the complexity status of the recognition problem for set graphs, we give characterizations in terms of hereditary graph classes, and put forth polynomial algorithms for certain graph classes. The set interpretation of a graph also leads to simpler proofs of two classical results on claw-free graphs. We have taken advantage of their set-theoretic flavor to formalize them with moderate effort in the set-based proof-checker Referee; these formal proofs are presented in full in an Appendix. Tesi di dottorato di Alexandru Ioan Tomescu, discussa presso l'Universit`adegli Studi di Udine Tesi di dottorato di Alexandru Ioan Tomescu, discussa presso l'Universit`adegli Studi di Udine Acknowledgments First of all, I thank Alberto Policriti for his support, guidance, and confidence in me. His supervision, together with that of Eugenio, has been nothing but excellent. Alberto exposed me to a wealth of different topics and gave me the opportunity to freely choose the ones on which to concentrate my research on. His contagious optimism and his ability to see the hidden beauty behind many results were most motivating for me. I equally thank Eugenio Omodeo, especially for his infinite patience and for his great attention to detail. Meeting Eugenio back in 2006 was one of those random events by virtue of which, after a long series of other coincidences, I started my PhD studies in Udine. I will always remember my, quite often, weekly visits to Trieste and Monfalcone. Many thanks go also to Martin Milaniˇc,whom I met at the middle of my PhD studies. Working with Martin opened up a whole new perspective in my research; he was always very kind and available with explanations and guidance. Martin also contributed to a great stay in Friuli, given our relatively close distance. I thank the other persons with whom I was most fortunate to work with in these last three years: Giovanna D'Agostino, Romeo Rizzi, Francesco Vezzi, Cristian Del Fabbro, Oliver Schaudt. I also thank Angelo Montanari and Andrea Sgarro for assisting me in different ways in these last years. I am also happy for having met Agostino Dovier, Carla Piazza, Alberto Marcone, Francesco Fabris, Luca Bortolussi, Alberto Casagrande which made up a very stimulating working environment in Udine and in Trieste. I am also grateful to Domenico Cantone and Martin Charles Golumbic for agreeing to referee this thesis, and for their helpful comments and suggestions. Last, but not least, I thank my other Italian colleagues and friends for a warm and pleasant company during my stay in Udine, among which: le simpaticissime Francesca and Laura, i gemelli Giorgio il buono, Giovanni il cattivo, and Felice il napoletano. Tesi di dottorato di Alexandru Ioan Tomescu, discussa presso l'Universit`adegli Studi di Udine Tesi di dottorato di Alexandru Ioan Tomescu, discussa presso l'Universit`adegli Studi di Udine Contents Introduction 1 1 Basic Concepts 11 1.1 Well-founded sets . 11 1.2 Graphs and digraphs . 12 1.3 Sets as digraphs . 15 1.4 Bisimulation, hyper-extensionality and hypersets . 18 2 Combinatorial Enumeration of Sets 23 2.1 Counting sets as extensional acyclic digraphs . 24 2.1.1 Counting labeled extensional acyclic digraphs . 26 2.1.2 Counting weakly extensional acyclic digraphs by sources, vertices of maximum rank, or arcs . 27 2.2 An Ackermann-like enumeration of hypersets . 32 2.2.1 A new look at the Ackermann order . 33 2.2.2 An Ackermann order on hereditarily finite hypersets . 35 2.3 Random generation of sets . 40 3 Infinite Enumeration of Sets and Well-Quasi-Orders 51 3.1 Well-quasi-orders and digraph immersion . 52 3.2 Slim digraphs and their structure . 54 3.3 Encoding slim digraphs . 57 4 Set Graphs { The Structure Underlying Sets 61 4.1 Basic properties . 62 4.1.1 Necessary or sufficient conditions . 62 4.1.2 Operations preserving set graphs . 64 4.1.3 Unicyclic set graphs . 67 4.1.4 Related graph classes and notions . 68 4.2 Complexity issues . 71 4.2.1 Recognizing set graphs is hard . 71 4.2.2 The complexity of counting extensional orientations . 76 4.2.3 Tractability on graphs of bounded tree-width . 77 4.2.4 The complexity of finding a separating code . 80 4.3 Claw conditions and set graphs . 81 4.3.1 Claw-free graphs . 81 4.3.2 Claw disjoint graphs . 86 4.3.3 K1;r+2-free graphs and r-extensionality . 89 Tesi di dottorato di Alexandru Ioan Tomescu, discussa presso l'Universit`adegli Studi di Udine ii Contents 5 Connected Claw-Free Graphs Mirrored into Transitive Sets 93 5.1 Simpler proofs for two properties of connected claw-free graphs . 94 5.2 Formalizing connected claw-free graphs in a set-based proof-checker . 96 5.2.1 The Referee system in general . 97 5.2.2 The Referee system in action . 99 5.2.3 Specifications of Hamiltonicity proof and of the perfect matching theorem . 106 5.2.4 An outward look . 108 6 Sets Modeling Bernays-Sch¨onfinkel-Ramsey 8∗-formulae 111 6.1 The decidability problem . 112 6.1.1 The computational complexity of deciding satisfiability . 114 6.1.2 A digraph-based satisfiability algorithm for 9∗8-sentences and for 98∗-sentences over hypersets . 115 6.2 Infinite set models . 117 6.2.1 Stating infinity . 121 6.2.2 An apparatus for starting off an infinite well-founded spiral . 122 6.2.3 Infinite well-founded models under ZF, ZF − FA + AFA and ZF − FA124 6.2.4 An apparatus for starting off an infinite non-well-founded spiral . 127 6.2.5 Infinite non-well-founded models under ZF − FA + AFA and ZF − FA131 6.3 Expressiveness of the set theoretic BSR class . 137 6.3.1 A satisfiability-preserving translation from a logical context . 137 6.3.2 Considerations on parsimonious sets and finite representability . 140 6.3.3 Bizarre infinitude . 142 Summary of results . 145 Conclusions 147 A Finiteness: a Proof-Scenario Checked by Referee 151 B Connected Claw-Free Graphs: a Proof-Scenario Checked by Referee 157 B.1 Basic laws on the union-set global operation . 157 B.2 Transitive sets . 160 B.3 Basic laws on the finitude property . 161 B.4 Some combinatorics of the union-set operation . 162 B.5 Claw-free, transitive sets and their pivots . 164 B.6 Hanks, cycles, and Hamiltonian cycles . 170 B.7 Hamiltonicity of squared claw-free sets . 180 B.8 Perfect matchings . 181 B.9 Each claw-free set admits a near-perfect matching . 184 B.10 From membership digraphs to general graphs . 187 Bibliography 189 Tesi di dottorato di Alexandru Ioan Tomescu, discussa presso l'Universit`adegli Studi di Udine Introduction Set theory was initially proposed as a study of infinite sets, its birthplace being a series of papers published between 1874 and 1884 by Georg Cantor. In these, Cantor proved the uncountability of real numbers, introduced cardinal and ordinal numbers and formu- lated the celebrated Continuum Hypothesis. During the so-called \foundational crisis in mathematics", one of the representative contradictions appearing was Bertrand Russell's 1901 paradox concerning the existence of a set of all sets. Actually, it is the naturalness involved in the spontaneous concept of set that, unless properly tamed, leads to Russell's and to other similar antinomies. David Hilbert acknowledged that, on the one hand, set theory had pointed out the necessity to perfect logical theory, and that, on the other hand, set theory itself, once established axiomatically, can lie at the foundations of mathematics.