Algebras of Partial Functions Brett McLean Thesis submitted for the degree of Doctor of Philosophy Department of Computer Science University College London June 2018 I, Brett McLean, confirm that the work presented in this thesis is my own. Where information has been derived from other sources, I confirm that this has been indicated in the thesis. Abstract This thesis collects together four sets of results, produced by investigating modifications, in four distinct directions, of the following. Some set-theoretic operations on partial functions are chosen—composition and intersection are examples—and the class of algebras isomorphic to a collection of partial functions, equipped with those operations, is studied. Typical questions asked are whether the class is axiomatis- able, or indeed finitely axiomatisable, in any fragment of first-order logic, what computational complex- ity classes its equational/quasiequational/first-order theories lie in, and whether it is decidable if a finite algebra is in the class. The first modification to the basic picture asks that the isomorphisms turn any existing suprema into unions and/or infima into intersections, and examines the class so obtained. For composition, intersec- tion, and antidomain together, we show that the suprema and infima conditions are equivalent. We show the resulting class is axiomatisable by a universal-existential-universal sentence, but not axiomatisable by any existential-universal-existential theory. The second contribution concerns what happens when we demand partial functions on some finite base set. The finite representation property is essentially the assertion that this restriction that the base set be finite does not restrict the algebras themselves. For composition, intersection, domain, and range, plus many supersignatures, we prove the finite representation property. It follows that it is decidable whether a finite algebra is a member of the relevant class. The third set of results generalises from unary to ‘multiplace’ functions. For the signatures invest- igated, finite equational or quasiequational axiomatisations are obtained; similarly when the functions are constrained to be injective. The finite representation property follows. The equational theories are shown to be coNP-complete. In the last section we consider operations that may only be partial. For most signatures the relevant class is found to be recursively, but not finitely, axiomatisable. For others, finite axiomatisations are provided. Contents 1 Introduction 11 2 Mathematical background 15 2.1 First-order logic . 15 2.1.1 Syntax . 15 2.1.2 Semantics . 17 2.2 Universal algebra . 19 2.3 Model theory . 21 2.4 Computability theory . 24 2.5 Algebraic logic . 28 2.5.1 Ordered structures . 28 2.5.2 Representation . 30 3 Related work 33 3.1 Algebras of relations . 33 3.1.1 Relation algebras . 33 3.1.2 Representable relation algebras . 34 3.1.3 Subsignatures and supersignatures of the relation algebra signature . 35 3.1.4 The fundamental theorem of relation algebras . 36 3.1.5 Kleene algebras . 37 3.1.6 Higher-order relations . 39 3.1.7 Complete representations . 41 3.2 Algebras of partial functions . 41 3.2.1 Unary functions . 42 3.2.2 The finite representation property . 45 3.2.3 Multiplace functions . 46 3.2.4 Tabular summary . 46 4 Complete representation by partial functions 49 4.1 Introduction . 49 4.2 Representations and complete representations . 50 6 Contents 4.3 Atomicity . 53 4.4 Distributivity . 57 4.5 A representation . 59 4.6 Axiomatising the class . 61 5 The finite representation property 65 5.1 Introduction . 65 5.2 Algebras of partial functions . 66 5.3 Uniqueness of presents and futures . 69 5.4 The finite representation property . 71 5.5 Entirely algebraic constructions . 78 5.6 Failure of the finite representation property . 79 6 Multiplace functions for signatures containing antidomain 81 6.1 Introduction . 81 6.2 Algebras of multiplace functions . 82 6.3 Composition and antidomain . 87 6.4 The representation . 101 6.5 Injective partial functions . 105 6.6 Intersection . 107 6.7 Preferential union . 111 6.8 Fixset . 113 6.9 Equational theories . 115 7 Disjoint-union partial algebras 119 7.1 Introduction . 119 7.2 Disjoint-union partial algebras . 121 7.3 Axiomatisability . 125 7.4 A recursive axiomatisation via games . 127 7.5 Non-axiomatisability . 131 7.6 Signatures including intersection . 142 7.7 Decidability and complexity . 147 8 Conclusion 149 Index 153 Bibliography 157 List of Figures 4.1 An algebra refuting right-distributivity over meets . 59 4.2 Algebra for which θ does not represent range correctly . 63 5.1 The algebra F1. Dashed lines for f, solid lines for g .................... 68 5.2 Left: a non-representation of F1. Right: a representation of F1 .............. 69 5.3 The algebra F2. Dashed lines for f, solid lines for g .................... 80 List of Tables 3.1 Summary of representability results for partial functions . 47 6.1 Summary of representation classes for n-ary functions . 87 Acknowledgement I wish to put on record what an excellent supervisor Robin Hirsch is. His style is to provide as much support as is needed—he is always generous with his time—whilst simultaneously granting complete intellectual freedom. I believe this unpressured atmosphere has been central to making my PhD journey an unexpectedly smooth ride. Though Robin made it clear the selection of topic was entirely my own, his recommendation proved remarkably well chosen, since there are many interesting questions in this area amenable to attack by a novice researcher and they sit well together as a cohesive whole. Robin’s week-to-week direction, ideas and suggestions have also been tremendously effective. Thank you Robin. Chapter 1 Introduction Across mathematics as a whole, functions have a much more pervasive presence than relations, their more general cousins. However, with regard to the metamathematics of reasoning about these entities, the situation has, historically, been reversed. Investigations into the laws obeyed by binary relations began in the latter half of the 19th century, starting with De Morgan [82], followed by Peirce [84], Schroder¨ [94], and Russell [88]. For a brief account of this early period, see [85]. Following four decades of almost total neglect, the subject was revived by Tarski’s 1941 article On the calculus of relations [102], and since then, binary and higher- order relations have been continuously active topics of research. Initially, the purpose of this work was algebraic logic in the strict sense. That is to say, the relations were providing the semantics for logical formulas.1 This is perhaps the explanation as to why, until recently, the corresponding theory of functions was relatively less developed, since the semantics of formulas is not a role so naturally suited to functions. However, increasingly in the history of relations, computer science has become a source of motivation, with binary relations providing the semantics for, in particular, (nondeterministic) computer programs. In this view, the relation relates states of the machine before the program is executed to possible states after it is executed. Despite the ubiquity of functions in mathematics, prior to the turn of this century the only sustained period of activity on reasoning with (possibly partial) functions was the 1960s, when the semigroup theorist Boris Schein and associates were active in this area.2;3 In the last fifteen years however, interest has rekindled, and a regular stream of papers has been appearing, with computer science considerations a prime motivation. The contemporary framework for studying one of these types of entities—either relations or some specialisation thereof—is to first select some operations of interest acting on the entities. For example composition and union are both binary operations we can perform on binary relations; the precise set we choose we call the signature. Then any collection of our entities closed under the chosen operations forms an algebraic structure. The object of study is then the isomorphic closure of the class of all these ‘concrete’ algebras, and we call this class the representation class. 1A particular interest of Tarski’s was for these formulas to be those of the language of set theory [105]. 2Schein’s survey article [91] records the results obtained during this time. 3Later, there was a line of work in the category-theoretic tradition [5, 87, 27], but that is outside the interests of this thesis. 12 Chapter 1. Introduction Aspirations for reasoning about programs by means of binary relations extend beyond reasoning about program behaviour in the abstract, to the possibility of practical real-world applications to auto- mated program verification [7, 50, 28, 6]. However, a recurring theme in work on relations is the discov- ery of their poor logical and computational behaviour. To give some explicit examples of what is meant by this, often the representation class is found not to have a simple (for example finite) axiomatisation in first-order logic [80, 46], its validities in fragments of first-order logic are found to have high compu- tational complexity [35] and, for finite structures, membership of the representation class is found to be undecidable [40]. The uninitiated may be surprised to learn that the observed logical and computational unruliness seems to be an intrinsic feature of relations, rather than stemming from any provision for expressing ‘unbounded iteration’ and the well-known associated complications this presents with regard to comput- ability. Indeed the predisposition towards negative results exists even when only rather simple operations are used [3]. An alternative approach beckons. Though the nondeterministic paradigm certainly has its uses, in practice many programs are actually deterministic. This suggests that algebras of partial functions might provide a useful framework for reasoning about programs, avoiding the use of troublesome relations. Encouragingly, investigations into classes of algebras of partial functions have tended to indicate they are much better behaved than classes of algebras of binary relations.