CATEGORICAL MODELS OF FIRST-ORDER CLASSICAL PROOFS

Transcription

1 CATEGORICAL MODELS OF FIRST-ORDER CLASSICAL PROOFS Submitted by Richard M c Kinley for the degree of Doctor of Philosophy of the University of Bath 2006 COPYRIGHT Attention is drawn to the fact that copyright of this thesis rests with its author. This copy of the thesis has been supplied on condition that anyone who consults it is understood to recognise that its copyright rests with its author and no information derived from it may be published without the prior written consent of the author. This thesis may be made available for consultation within the University library and may be photocopied or lent to other libraries for the purposes of consultation.

2 Abstract This thesis introduces the notion of a classical doctrine: a semantics for proofs in firstorder classical logic derived from the classical categories of Führmann and Pym, using Lawvere s notion of hyperdoctrine. We introduce a hierarchy of classes of model, increasing in the strength of cut-reduction theory they model; the weakest captures cut reduction, and the strongest gives De Morgan duality between quantifiers as an isomorphism. Whereas classical categories admit the elimination of logical cuts as equalities, (and cuts against structural rules as inequalities), classical doctrines admit certain logical cuts as inequalities only. This is a result of the additive character of the quantifier introduction rules, as is illustrated by a concrete model based on families of sets and relations, using an abstract Geometry of Interaction construction. We establish that each class of models is sound and complete with respect to the relevant cut-reduction theory on proof nets based on those of Robinson for propositional classical logic. We show also that classical categories and classical doctrines are not only a class of models for the sequent calculus, but also for deep inference calculi due to Brünnler for classical logic. Of particular interest are the local systems for classical logic, which we show are modelled by categorical models with an additional axiom forcing monoidality of certain functors; these categorical models correspond to multiplicative presentations of the sequent calculus with additional additive features.

3 Acknowledgements There are many without whom this work would languish unfinished. I must begin by thanking David Pym for introducing to me the core ideas in this work, and for the guidance he has provided. In matters both sacred and profane, both integral and peripheral, he has provided invaluable, fruitful and enjoyable discussion and advice. To him I am deeply grateful. I thank Carsten Führmann for too many helpful discussions to remember, and his for his encouragement. I am very grateful to John Power and Daniel Richardson for agreeing to examine the thesis, and for their comments. I have also benefited from discussions with Kai Brünnler, Matthew Collinson, Alessio Guglielmi, Francois Lamarche, Edmund Robinson and Lutz Strassburger. Thanks to my friends and family, who have continued to cheer this thesis on, despite resolutely refusing to read it. The greatest thanks must go to my wife, Mary. Throughout she has been a constant source of love and encouragement. Diagrams were typeset using Paul Taylor s diagrams and prooftree packages. The author was supported by a University Studentship from the University of Bath. 1

4 Contents 1 Introduction Structure of this thesis Pre-background: established theory Sequent calculi Multiplicative and additive presentations One-sided calculi Calculi for intuitionistic logic Cut-elimination First-order LK Proof nets Categorical logic and categorical preliminaries Interpretation of proofs in a category Indexed categories and hyperdoctrines

5 2.3.3 Monoidal categories and functors Symmetric linearly distributive and -autonomous categories Background: recent work Proof nets for propositional classical logic Classical categories The calculus of structures for classical logic System SKSg A local system First-order logic in the calculus of structures Formalism A and Formalism B Proof nets for first-order classical logic First-order proof nets and quantifier boxes Static properties Cut-elimination in first-order proof nets Other (in)equations Classical doctrines Categorical axioms for quantifiers Classical doctrines

6 5.3 Some properties of classical doctrines Sharp Classical Doctrines Linear functors, duality and MIX A concrete example Modelling classical first-order proofs Interpretation of the sequent calculus in classical doctrines Completeness The term model Permutations with structural rules and sharp doctrines Permutations between quantifiers Categorical semantics and the calculus of structures Providing a semantics for deep inference systems Locality and categorical semantics The first-order medial rules Towards a native notion of proof equality for SKS Classical doctrines with equality and axiomatic theories Nonlogical axioms in sequent calculi Equality and algebraic theories

7 8.3 Theories with relations Conclusions and further work 215 References 220 A Monoidal categories 226 A.1 Adjunctions in monoidal categories A.1.1 Proof of Theorem A.1.2 Proof of Lemma B Generalized quantifiers 235 5

8 Chapter 1 Introduction This thesis is concerned with the denotations of propositions and of proofs in classical logic. This question seems, on the face of things, to be answered very quickly; every first-year mathematics undergraduate is familiar with the notion of a truth value: to every atomic proposition we may assign either the value (true) or (false). The truth values of compound propositions are then determined by truth tables, as functions of the truth values of their constituent atomic propositions. For propositional logic, this is enough; a model of propositional classical logic in this sense is a Boolean algebra. The simplest model of this type is given by the Boolean algebra {, }. We use the notation φ for the object of a Boolean algebra representing the meaning of a proposition φ. This mathematical structure may be seen as a lattice, with φ ψ if φ ψ. Treating a logic as an ordering on propositions scales well to a number of non-classical logics; for example, associated to intuitionistic logic is the notion of Heyting semilattices. While this very simple notion of model has been very successful as a tool, allowing the development of model theory, it is unsatisfying from the perspective of modelling logic itself. Any two logically equivalent propositions are equal, and in particular the denotation of any tautology is. Moreover, the model gives us no information about the structure of proof. For this reason, we will refer to this paradigm for the semantics of logic as provability semantics, contrasted with the subject of this thesis: the semantics of proofs. 6

9 The notion that the meaning of a proposition should be tied to its set of proofs originates with Heyting. To Heyting, a proof of a formula φ ψ should consist of nothing more or less than a proof of φ and a proof of ψ, a proof of φ ψ should be either a proof of φ or a proof of ψ (plus an indication of which we have proved), and a proof of φ ψ should be a function mapping proofs of φ to proofs of ψ. It seems intuitive that under this interpretation classes of logically equivalent propositions are no longer identified: a proof of ψ ψ is not a proof of ψ. What is not immediately obvious is how we might formalize this notion; in order to do so, we must first know when two proofs are identical. The first steps toward this were made by Prawitz, who suggested that, for natural deduction (for intuitionistic logic), two proofs should be considered equal if they have the same normal form under normalization (βη-reduction). A natural setting in which to evaluate proofs is a category: a categorical semantics for set of proofs in a category C consists of 1. Evaluating each proposition φ as an object φ of C, and 2. each proof Φ from φ to ψ as a morphism Φ : φ ψ. The provability semantics mentioned above is retrieved by insisting C be a pre-order. Soon after Prawitz suggested normalization as a source for proof equality, it was noticed by Lambek that this is precisely the equality on proofs given by interpreting proofs as morphisms in a cartesian closed category. Thus, it is often said a categorical model of intuitionistic logic is a cartesian closed category. This is somewhat misleading: there may be several proof theories for each logic. For example, the multipleconclusioned sequent calculus for intuitionistic logic [21] has more proofs than the usual single conclusioned calculus, as we will see below, and at a sufficient level of detail even the correspondence between natural deduction and the sequent calculus is not precise, although it is very close [59]. Thus it is technically incorrect to say that, for example, that cartesian closed categories are a class of models of propositional intuitionistic logic; they are a model of natural deduction for propositional intuitionistic logic. The structure of our calculus imposes constraints on proof-equality; for example, in 7

10 the sequent calculus, we require that the rules of the calculus preserve proof equality (and in general, cut-reducibility). Taking the reductions on a proof calculus as equalities is enough to give a semantics for the single-conclusioned calculus of intuitionistic logic, and for linear logic, but in general it fails; in particular for a multiple-conclusioned sequent calculus with weakening and contraction. The problem is illustrated by the following example, usually attributed to Lafont [28]. Given two proofs Φ 1 and Φ 2 of the same sequent,consider the proof Φ 1 Γ WR Γ φ, Φ 2 Γ WL Γ,φ CUT (1.1) Γ,Γ, CL, CR. Γ Suppose we want our semantics to admit cut-reduction as equality. Then this proof, under the usual set of cut-reductions, reduces to both Φ 1 and Φ 2. Therefore, this naïve approach identifies every proof of a given sequent, so obtain, not a semantics of proofs as desired, but nothing more than provability semantics. It is important to note that, while this example was conceived as a demonstration of the inconsistencies in the proof theory of classical logic, it is equally applicable to the multiple conclusioned calculus for intuitionistic logic, since it makes no use, explicit or implicit, of negation. The collapse arises from unrestricted use of weakening and contraction. One tactic for repairing this lack of a denotational semantics is allow a restricted set of reductions; in the above example, we might allow the left-hand reduction but not the right-hand. This is the approach taken in the λµ calculus [55] for classical logic. The two choices correspond to a call-by-name or a call-by-value evaluation strategy. This gives rise to models of classical proofs, either via fibrations of cartesian closed categories [57] or control and co-control categories [61]. This is at the expense of breaking symmetry. 8

11 Recently, several successful attempts to give a non-trivial semantics to the classical logic have appeared which preserve the symmetry of cut-reduction. Classical categories [23, 22, 24] have morphisms that denote proofs in Gentzen s LK, and an order enrichment modelling the standard theory of cut-reduction in LK. These models are sound and complete with respect to the ordering on proofs. Other notable projects in the semantics of classical proofs include the work of Hyland [38] on abstract interpretation of classical proof, which are similar to a classical category for which and are modelled by the same monoidal product. The difference lies in the omission of one axiom on the structure of the ordering on proofs. The work of Bellin, Hyland, Robinson and Urban [7] is ambitious in considering models of classical logic which reject the functoriality of conjunction and disjunction (corresponding to the rejection of axiom expansion in the proof theory). This leads to a more general notion of model, but one for which there are few natural examples. Dosen and Petric [20], by analyzing the coherence of distributive categories via the category of sets and relations, derive a notion of Boolean category, which is somewhat degenerate in requiring finite products and coproducts. The work of Lamarche and Strassburger [44, 45], does not analyze the symmetric cut reduction system of LK, nor the restricted, asymmetric reduction of the λµ-calculus, but instead a reduction system which rejects both reductions in Lafont s example, introducing (implicitly) a primitive mix law [26], to which a degenerate cut reduces: Φ 1 Γ WR Γ φ, Γ, Γ, Φ 2 Γ WL Γ,φ CUT Φ 1 Γ Φ 2 Γ, Γ, Γ Mix In a classical category the mix rule is denotationally equivalent to the degenerate cut with φ =, or alternatively with φ =. Mix also gives rise to a product on proof spaces: Φ Ψ = Φ Γ Ψ Γ Mix Γ,Γ, CL, CR Γ 9

12 In classical categories this operation is idempotent; Φ Φ = Φ. In the Boolean categories of [45], there are given examples where this is also the case (which are a special case of classical categories) and also models where this equality does not hold; these non-idempotent models are intruiguing but not well understood. This thesis concentrates on the classical categories of Führmann and Pym, and extends their results to first-order LK, in the spirit of Lawvere [49] and Seely [60]. In particular, we will see that the Adjointness in Foundations view of quantification (where quantifiers are adjoints to substitution) can be unified with the naïve view of quantifiers as an infinitary conjunction or disjunction over the domain of quantification. While it does not directly address the other notions of model above, it is worth noting that techniques developed in this thesis should adapt successfully to them. The issue of finding a notion of equality on proofs is separate from, but related to, the issue of finding canonical objects representing those proofs; the essence of the proof. For multiplicative linear logic (without units), proof nets [26] are such canonical objects, but for classical logic (for any logic with weakening) the usual approach to weakening [58, 12] relies on attaching a weakening to a particular node of the net. Since we do not wish to distinguish between proofs which differ only by the attachment of a weakening, we must quotient by a re-wiring relation on nets. The search for a notion of proof net which gives canonical cut-free proofs has led to the development by Hughes of abstract combinatorial proofs [36, 37]. Nevertheless, the non-canonical proof nets are technically useful objects in the study of the sequent calculus; the proof of completeness for classical categories [23] relies on the proof nets of Robinson [58], with a cut-reduction equivalent to that for LK, to significantly simplify the calculations. Our first task will be to develop a similar system for first-order logic. Another area of research which has of late been very active is the search for new deductive systems, with the hope being that the problem of identity of proofs is more easily solved in those systems. These deep inference systems [33, 13, 63] have much in common with categorical logic; in particular, the notions of type A bureacracy and type B bureacracy correspond precisely to the categorical notions of bifunctoriality and naturality [35, 52, 64]. For some of these systems, categorical coherence gives a natural equational (or inequational) theory on proofs, but the most sucessful deep inference systems make use of a notion previously uninvestigated in categorical logic 10

13 the medial rule. Moreover, while coherence correpsonds precisley to cut-reduction in sequent calculi, it is not clear how it relates to notions of reduction in deep inference systems. We take in this thesis some small steps to addressing this problem; we give conditions for the models of Führmann and Pym to capture the medial rule, and show how coherence may connect to reduction in deep inference systems. A large part of this thesis will be concerned with De Morgan duality: in classical logic A B is logically equivalent to ( A B). When using classical logic in a practical setting, it is often a useful short-cut to consider only one half of any De Morgan dual pair, and/or use a one-sided presentation of the logic; this will, for example, half the length of any presentation of the system. However, when considering the system for its own merits, one may appreciate the beauty of the symmetry (which is, of course, hidden in a one-sided approach). Moreover, in categorical logic logical equivalence does not imply isomorphism; for example φ is not ismorphic to φ φ We will refer to the concept of isomorphic De Morgan duality as proof-theoretic duality. We will continue to use a two sided system in this thesis. By using a setting in which duality is not assumed (symmetric linearly distributive categories) rather than one where it is ( -autonomous categories), we can analyze the properties needed of the theory on proofs to derive duality of, for example, existential and universal quantification. Note that this does not stop us from using categorical duality; the structure used to model a connective (for example, a monoidal functor) will be categorically dual to that of its De Morgan dual (in this case, a comonoidal functor), and so we will be able to economize on paper by concluding a proof about the existential quantifier with the words and dually for the universal quantifier, without committing to their being proof-theoretically dual Structure of this thesis Chapter 2: We recall some established results in proof-theory and categorical logic. Chapter 3: We cover some more recent developments: proof nets for classical logic, classical categories, and deep inference systems. 11

14 Chapter 4: We give details of a calculus of proof nets for first-order classical logic with quantifiers, and demonstrate the key properties: sequentialization and cut-elimination. Chapter 5: We give three notions of model of first order classical proofs, in order of increasing strength. The first, classical doctrines, models quantifiers as functors monoidal in one connective. The second, sharp classical doctrines, embodies the well-behavedness of quantifiers with respect to structural rules and has quantifiers (co)monoidal in both connectives. The third, dual classical doctrines, introduces enough extra structure to ensure that existential quantification is De Morgan dual to universal quantification. Chapter 6: We introduce, for each notion of model, a corresponding notion of theory on proofs, giving soundness and completeness result for each class. For sharp doctrines/theories, we observe that ideas from the calculus of structures allow a smoother axiomatization for monoidality in the proof nets, associated to the additive nature of the sequent calculus quantifier rules. Chapter 7: We comment on the interplay between deep inference systems and our categorical models, and observe that the models give a natural notion of proof equality on calculus of structures proofs. Chapter 8: We extend our models to include equality predicates, algebraic and nonalgebraic signatures, and non-logical axioms. We provide models of modal and (multiple conclusioned) intuitionistic propositional logics by viewing them in terms of their first-order Kripke semantics. Chapter 9: We give conclusions and ideas for future work. 12

15 Chapter 2 Pre-background: established theory 2.1 Sequent calculi The sequent calculus was introduced in Gentzen s seminal paper [25] as a setting for studying the properties of formal proof. He gave systems LJ (for intuitionistic logic) and LK (for classical logic), which have since formed the template for numerous calculi for other logics. A key feature of sequent systems is the cut rule, by which two proofs in the system may be composed, and a key property of sequent systems is that judgments that can be derived with cut may also be derived without cut; the procedure for deriving the cut-free proof being known as cut-elimination. The lack of a cut-elimination procedure is considered, in most cases, a serious defect of a sequent calculus. A sequent calculus is a system for deriving judgments of the form Γ, where Γ and are sequences 1 of formulae, with formulae given by (a sub-grammar of) the grammar A := a A A A A A ya ya, 1 In certain contexts Γ and are taken to be multisets or sets, making weakening and contraction implicit; in the context of semantics of proofs this is fatal, since we must keep track of where weakenings are made. 13

16 AX ϕ ϕ Γ, ϕ, ψ L Γ, ϕ ψ Γ ϕ, Γ ψ, R Γ, Γ ϕ ψ,, Γ ϕ, ψ, R Γ ϕ ψ, Γ, ϕ Γ, ψ L Γ, Γ, ϕ ψ, R L Γ, ϕ, ψ, Γ EL Γ, ψ, ϕ, Γ Γ, ϕ, ψ, ER Γ, ψ, ϕ, Γ ϕ, Γ, ϕ CUT Γ, Γ, Table 2.1: LB: a sequent calculus for the core system and the intended reading is that the conjunction of the formulae in Γ entails the disjunction of the formulae in. The simplest logic we will consider in this thesis has the sequent calculus given in Table 2.1. We will call this the core system, and the calculus LB. The rules of the core system will be a subset every other system we consider in this thesis. Using the rule L rule as an example, we give some terminology: Γ, ϕ Γ, ψ L. Γ, Γ, ϕ ψ, We call the formula introduced by the rule (in this case ϕ ψ) the principal formula. 14

17 Γ, ϕ L Γ ϕ, Γ ϕ, R Γ, ϕ Table 2.2: Optional rules for negation The formulae above the line which make up the principal formula are the active formulae, and the other formulae are the context. An alternative presentation for LB (the efficacy of which was demonstrated in [18] and then [23]) is to replace the rules R and L by constants: φ, ψ φ ψ K, φ ψ φ, ψ K, and simulate R and L by cutting against these constants. We can extend LB by adding more constants, or alternatively more rules. For example, Table 2.1 gives rules for negation, equivalent to adding the constants K R, φ, φ K L, φ, φ giving multiplicative linear logic (MLL). Table 3.4 gives structural rules, equivalent to adding the constants K CL, φ φ, φ K CR, φ, φ φ K WR, φ K WL. φ We may use these constants to define a calculus for classical logic, which we will call LK: 15

18 Γ, ϕ, ϕ CL Γ, ϕ Γ WL Γ, ϕ Γ, ϕ, ϕ CR Γ, ϕ Γ WR Γ, ϕ Table 2.3: Optional rules for weakening, contraction Definition The system LK is given by LB, plus the constants (alternatively, rules) for contraction, weakening, and negation Multiplicative and additive presentations The definition given by Gentzen in his original paper for LK differs from that given above; not only in the (somewhat technical) use of constants, but more substantially. Gentzen s original formulation for the rule introducing conjunction on the left, for instance, is the pair of rules Γ, ψ La, Γ, ϕ ψ Γ, ϕ La, Γ, ϕ ψ while the formulation of the rule introducing conjunction on the right is Γ ϕ, Γ ϕ ψ, Γ ψ, Ra (These formulations are known as additive rules, and the previous presentation multiplicative rules.) In a system with with weakening and contraction, L and L a are inter-derivable, as are R and R a. In the absence of these structural rules, however, the connectives that these two families of rules define are very different. These logics without structural rules are known as linear logics [26], which might contain both multiplicative and additive conjunction and disjunction. Of principal importance to us will be the logic MLL, given by adding negation to LB. It has a very well behaved proof 16

19 Ax GS1 ϕ, ϕ,ϕ,, ϕ CUTGS1p Table 2.4: System GS1p: Axiom and CUT theory and model theory, which we will discuss below One-sided calculi Given the duality of the classical sequent calculus, it is common to consider Gentzen Schütte style systems, where sequents have no antecedant, only a conclusion. For example, the system GS1p is given by removing the axiom, cut and left rules of LK, and adding the rules in table 2.4 It will be useful for later chapters to see how a proof in GS1p may be translated into LK: Lemma Any proof Φ in GS1p can be transformed into a proof in LK. Proof. Suppose we have a GS1p proof Φ. Since GS1p and LK differ only on cut and axiom, we transform only instance of these rules. We replace each occurrence of an axiom ϕ,ϕ Ax GS1p with Ax LK. ϕ ϕ R ϕ,ϕ (2.1) The case of cut is little more complicated. There are three obvious ways of defining the GS1p cut rule in LK: 17

20 ϕ, Ax LK ϕ ϕ L ϕ, ϕ,ϕ CUTLK ϕ CUT LK,, (2.2) ϕ, L ϕ,, ϕ (2.3) CUTLK,, and ϕ, ϕ, R ( ϕ ϕ),,, Ax LK ϕ ϕ L ϕ,ϕ L ϕ ϕ CUTLK. (2.4) These definitions are coherent, in the sense that they are identified in a semantics that validates logical cut as equality (in particular the semantics of Führmann and Pym [23]). This is clear from eliminating the logical cut in (2.4), and representing negation as cut against a constant Calculi for intuitionistic logic In addition to LK for classical logic, Gentzen [25] introduced a sequent calculus LJ for intuitionistic logic. The calculus is a restriction of the additive calculus, minus negation, in which the right hand of the sequent consists of at most one formula. We give the calculus in Table 2.5. It is not, however, necessary to restrict all the rules of the sequent calculus in this way. Dummett [21] gives a calculus for intuitionistic logic for which, in the propositional fragment, only negation and implication are restricted. The calculus is obtained by adding the rules in Table 2.6 to LB, plus structural rules. 18

21 AX ϕ ϕ Γ, ψ i L Γ, ψ 1 ψ 2 Γ ϕ Γ ψ R Γ ϕ ψ Γ ψ i Γ ψ 1 ψ 2 R Γ, ϕ Γ, ψ L Γ, ϕ ψ Γ, φ ψ L Γ φ ψ Γ ϕ Γ, ψ R Γ, ϕ ψ Γ, ϕ, ϕ CL Γ, ϕ Γ WL Γ, ϕ Γ WR Γ ϕ Γ ϕ, Γ, ϕ CUT Γ, Γ (Where is either empty, or contains at most one formula.) Table 2.5: LJ: a sequent calculus for intuitionistic logic 19

22 Γ, φ ψ R Γ φ ψ Γ ϕ, Γ, ψ L Γ, Γ ϕ ψ, Γ, φ R Γ φ Γ ϕ, L Γ, ϕ Table 2.6: Negation and implication in Dummett s calculus for intuitionistic logic. This calculus is sound and complete for intuitionistic logic (with respect to, say, the usual Kripke semantics) but its proof theory is very different to that of the singleconclusioned system, as we noted in the introduction Cut-elimination Conceptually, the proof of cut-elimination for a sequent system is very simple; the underlying notion is the logical cut. A logical cut applies in the following circumstance, where the cut formula is introduced, in both premises of the cut, by the introduction rule for its main connective: L R Γ, φ Γ φ,. CUT Γ, Γ, The reduct of a logical cut has a cut or cuts between the principal formulae of the cut formula. For example, the logical cut for reduces Γ, φ Γ, ψ Γ φ, ψ, L R Γ, Γ φ ψ, Γ φ ψ, (2.5) CUT Γ, Γ, Γ,, 20

23 to Γ, φ Γ φ, ψ, Γ, ψ CUT Γ, Γ ψ,, (2.6) CUT. Γ, Γ, Γ,, Assuming the top cut is also a logical cut, the procedure may continue by induction on the structure of the cut formula. Of course, we may not assume that all cuts are logical. In the lower cut of 2.6 the cut formula is a result of another application of cut. The proof of cut-elimination therefore requires a number of permutation laws or commuting conversions. It is for this reason that a calculus of proof nets (see Section 2.2) is essential in giving short proofs of meta-theorems such as completeness; proof nets allow cuts to appear in parallel. The final obstacle to a simple proof of cut-elimination lies in the structural rules. Because of these, the cut-formula might not be the consequence of a logical rule, even after the application of commuting conversions. Consider a proof of the following shape: Γ WR Γ, φ Γ φ,. CUT Γ, Γ, (Here φ does not result from any logical rule). In eliminating the cut, we must keep the context Γ and. This is achieved by successive weakenings: Γ WL, WR. Γ, Γ, We have deleted a proof of arbitrary size. Now consider a proof of the shape Γ φ, φ, CR Γ, φ Γ φ,. CUT Γ, Γ, If we push the cut above the contraction, we copy the contexts Γ and, since we must make two copies of the proof of Γ, φ. These copies must be removed by 21

24 contraction: Γ, φ Γ, φ Γ φ, φ, CUT Γ, Γ φ,, CUT Γ, Γ, Γ,, CL, CR. Γ, Γ,, By contrast with the logical cuts, the elimination of these cuts against structural rules removes or copies a subproof of arbitrary size. An elimination of this kind is referred to as non-local, while a reduction which removes only a bounded subproof is called local. The multiplicative system with the cut rule we have seen so far behaves much better with respect to the required permutation rules than the additive system with the same rule, in the sense that if a permutation of rule order is possible, it is possible without any extra weakenings and contractions. By contrast, to permute the cut above the logical rule in the proof Γ, φ θ, Γ, ψ θ, L a Γ, θ Γ, φ ψ θ,. CUT, Γ, Γ, φ ψ, we must both duplicate one proof: Γ, φ θ, Γ, θ Γ, ψ θ, Γ, θ CUT CUT Γ, Γ, φ, Γ, Γ, φ, R a, Γ, Γ, φ ψ, so the permutation is no longer local. Indeed, if one translates the R a rule into the multiplicative calclulus, the connection to the non-local rule for cut against contraction becomes clear. The purely multiplicative system is conceptually clean, in the sense that we can simulate the rules L and R by cut against a constant, without additional weakening or contraction. With this observation, we can derive the well-behavedness of the other connectives from the fact that cuts can permute past each other. By contrast, to simulate the additive rule R a with a multiplicative cut, we need con- 22

25 traction: Γ ψ, Γ φ, φ, ψ φ ψ CUT Γ, ψ φ ψ, CUT Γ, Γ φ ψ,, CL, CR Γ φ ψ,. Thus, there is no easy way of relating the properties of R a to cut in the same way we can with R. The additive cut rule, Γ, φ Γ φ, CUTa Γ fits more naturally with the additive rules. We can simulate R a with weakening (and without contraction). Permuting an additive cut above another additive cut is a nonlocal process. This presents us with a question. The additive formulation of classical logic, with an additive cut, is a widely used system with a perfectly good theory of cut-reduction, but one with different properties to those of the multiplicative system. How compatible are the two systems; i.e., does the system of proof inequality given by Führmann and Pym collapse in the presence of additive cut-reduction? In this thesis we will begin to answer that question, by showing that the additition of reductions from the additive theory yields models of local proof systems in the calculus of structures for classical logic [15, 13] which do not collapse. We do not give a detailed account of the admissibility of cut in a sequent system; for accounts of the proof for various different formulations of classical logic, see Gentzen s original paper [25], and also [66] and [54] First-order LK Given a first-order language, we now give the formulation of first-order classical proofs over that language. To make the transition from the syntax to the semantics easier, we 23

26 introduce a slight modification to the language of formulae; we add notation to make the dummying of variables in a formula explicit, and we restrict the propositional connectives to act only on formulae over the same variables (including dummied variables). Note that this change is not intended to give any extra proofs, but merely to make some information implicit in the usual grammar explicit, and to make the structure of formulae more suited to the treatment we give in Chapter 6. A first-order language L = (X, (V X ), A, F) consists of a nonempty set X of variable sorts, an infinite set V X of variable symbols for each sort X in X, a set A of sorted atomic formulae, and a set F of sorted function symbols. As is standard (see [60]), we introduce the notion of type as a finite sequence of sorts. (This will allow us to think of formulae and terms as being unary, which will unify the standard quantifiers we give now with the generalised quantifiers given in chapter 8). We may now replace the notion of a sorted formula/function symbol with that of a typed formula/function symbol; to each atomic formula a and function symbol f we associate a type for the sorts of the arguments, and in addition for a function symbol we associate a sort for the result of the function symbol. We introduce the type 1 (which one may think of as the empty sequence) as the type of nullary formulae (sentences) and nullary terms (constants). Each instance of an atomic formula and function symbol has a tuple/tuples of variable symbols associated to it, so that from an atomic a with type X, X, we may form instances a(x, y), a(y, x) etc., where x, y are variables from V X. We will denote by T the set of terms, by which we mean the set formed by closing F under substitution. We will assume that each instance of a term has a tuple of properly typed variables associated to its domain and codomain. It is clear, then, how typing extends to instances of terms, but it is not so clear how it extends to the usual grammar of formula instances. A substitution instance of an atom is given by the grammar A := a( x) [t/y]a, where a has type X 1, X 2,..., X n and x is an n-tuple of variable names such that x i has type X i, and where y is a variable name of type Y and t is a term of type Y. Formula instances are now given by the grammar B := b (B B) (B B) B y.b y.b, 24

27 where b is a substitution instance of an atom. Note that substitution is defined here on atoms. It is trivial to extend the definition of substitution to formulae, only being careful to note that some alpha conversion implicitly occurs when a term with free variable x is applied to a formula which has x bound. For example, if we wish to substitute a term t with domain x and codomain y into the formula x.a(x, y), then we must first rename the bound x to, say, x, to avoid capturing the free x in t. If φ and ψ are formulae of type X, what is the type of (φ φ)? If we are able to give an answer, it is only for a particular instance of (φ φ) ( the formula (φ(x) φ(x)) has type X, and the formula (φ(x) φ(y)) has type X, X). Yet, when considered on their own, clearly φ(x) and φ(y) should have the same interpretation. That is, if I have one proof containing φ(x) and no other variables than x, and another containing φ(y) and y, If I can interpret φ(x) as a particular function from objects to truth values (in the standard Tarskian semantics) then I can interpret φ(y) as the same function. In each other s presence, one must interpret each by a different function, and by default we must dummy the argument of each onto the other (so both φ(x) and ψ(y) are interpreted as binary functions, but each actually depends on only one of its inputs. To ensure that the meaning of a formula is independent of context, we must annotate a formula with information about the variables present in context on which it does not depend. Following [60], we assume that the function symbols of our language include binary projections π X,Y : X, Y X for each pair X and Y of sorts, and nullary projections π X : X 1 for each sort X. Given a formula φ over a type X, W we will use the abbreviation [y]φ for the substitution [π X,Y /x]φ (a formula over X, Y, W ), where the variable associated to the sort Y does not appear associated to the type X, W. (If φ is over 1, we will use [y]φ as shorthand for π Y φ.) Similarly, given Γ = φ 1, φ 2,...φ n, where each φ i does not depend on x, we define [x]γ = [x]φ 1, [x]φ 2,...[x]φ n. By contrast, a formula φ that genuinely does depend on x we will refer to as being properly over x. Remark The requirement that φ not be over y before we may form Y would be redundant if terms have types rather than sorts as codomains. Then we way form projections π X,Y X where X is a type, and define the abbreviation [y]φ to be [π X,Y / x]φ, where X is the type of φ, and the variable associated to X is different to those appearing in Y. Here Y cannot possible have associated to it the same variable as any of the sorts in X. 25

28 Alternatively, one may define [x]φ to be the result of making, for each sort Z in the type X of φ, the substitution [π X,Z /z], where z is the variable associated to Z. Again, we must specify that the variable associated to X is not the same as that associated to Z (so, in this case, the projection really is a projection from X, X to X, and not just the identity term on X. (Clearly the order of these substitutions is irrelevant, so we may think of the occuring simultaneously). Once again, if φ contains the variable x then we will be unable to carry out the substitution necessary on x, and so if φ is over a type containing x, the expression [x]φ is not well-formed. With this notation in place, we may restrict conjunction and disjunction to acting fibrewise ; that is, we may form the conjunction and disjunction only of formulae over the same type. We recover the conjunction x = 1 y = 1, for example, as [y]x = 1 [x]y = 1. the operators and to act only on formulae of the same type. A typed substitution instance of an atom is given by the grammars below: A 1 := a [s/y]a y:y, where a is an atom of type 1 and s is a term with codomain Y and domain 1, with the y the variable associated to the domain of s; A x:x := a( x) [t/y]a z,y, v:z,y,v, where x is a tuple of variables of type X, t is a term with domain variables w of type W and codomain variable y of sort Y, z is a tuple of variables of type Z, the type X equals Z, W, V., and x = z, w, v. That is, a typed substitution instance over x : X of an atom is either an atom of type X with free variables x, or the result of substituting a term into a typed substitution instance of a different type which has type X and variables x. Grammars for typed formula instances over 1 (sentences) and over X (formulae with free variables x of type X), are given below: B 1 := b 1 (B 1 B 1 ) (B 1 B 1 ) B 1 z.b z:z z.b z:z, where b 1 is a typed substitution instance over 1 of an atom and z is a variable of type 26

29 Z; B x:x := b x:x (B x:x B x:x ) (B x:x B x:x ) B x:x z.b w,zȳ:w,z,y z.b w,zȳ:w,z,y. where b x:x is a typed substitution instance over x : X of an atom, and z is a variable name of sort Z, w is a sequence of varibles of type W, ȳ is a sequence of variables of type Y, and x = w, ȳ. As above, we extend the definition of substitution from typed instances of atoms to typed instances of formulae by induction on the structure of the formula (once again taking care to apply any necessary alpha conversion where necessary). Since the notation [x] is shorthand for a particular substitution, it is defined on typed formula instances. The only change this notation makes to the propositional fragment of LK is that we may consider each rule therein as acting fibrewise; each formula in such a rule is over the same type. The usual formulation of LK for first-order logic is given in Table 2.7. The Table 2.8 shows how these rules may be adapted to our modified grammar of formulae. The notation allows us to state the side conditions for L and R within the body of the rules; recall from abaove that the definition of [x]φ requires that x not appear as the variable associated to any sort in the type of φ. This is clearly no great innovation, but it does make the adjunction present more obvious. These rules take a proof over one type (perhaps X, Y ) to another (in this case, Y ). The rules R and L, like the propositional rules, act fibrewise. To maintain the fibrewise nature of the rules we must decorate the active formula with copies of some of the [y i ], since it no longer depends on the variable quantified over. To ensure the well-behavedness of substitution in our proofs, we require an eigenvariable condition. To each instance of L and R we associate a unique variable ( eigenvaraiable) and we require that this variable only appear above the relevant quantifier in the proof. Clearly we may alpha-rename any proof which does not satisfy this property so that the renamed proof does. Given this condition, the following result is standard: Lemma The sequent Γ is provable in LK iff the sequent [t/x]γ [t/x] is provable. 27

30 Γ, φ xl Γ, x.φ Γ [t/x]φ, xr Γ x.φ, Γ, [t/x]φ xl Γ, x.φ Γ φ, xr Γ x.φ, Where for L and R, x is not free in Γ and in. Table 2.7: System LK: quantifiers [x]γ, φ [x] xl Γ, x.φ Γ [t/x]φ, xr Γ [y i1 ][y i2 ]...[y im ] x.[x /x]φ, Γ, [t/x]φ xl Γ, [y i1 ][y i2 ]...[y im ] x.φ [x]γ φ, [x] xr Γ x.φ, Where φ is over a type containing x of sort X, t is term with domain variables y 1, y 2,..., y n of type Y = Y 1, Y 2,...Y n and codomain variable x, x is a variable name of type X, and y i1, y i2...y im is the subsequence of (y i ) not free in x.φ,. Table 2.8: System LK: quantifiers with typing 28

31 Proof. If x is not free in Γ, the two sequents are clearly the same. Otherwise, x is not an eigenvariable of any quantifier, and so any proof of Γ will remain a proof if we replace x with t. 2.2 Proof nets Proof nets were introduced as a syntax for linear logic by Girard [27], initially for the fragment MLL of linear logic; that is, MLL without units. In many ways a proof net is a sequent proof without the sequentiality, as applications of rules which in the sequent calculus may be permuted exists in parallel in a proof net. A proof net is, informally, a bipartite directed graph, with one sort of nodes labeled with propositions and the other with names of sequent calculus rules, such that locally the graph has structure as given in Table 2.9. However, not all such graphs will represent proofs in the sequent calculus. We will refer to the class of all these graphs (whether or not they represent proofs) proof structures. Remark It is customary to distinguish between the conjunction and disjunction of MLL and those of classical/intuitionistic logic; here we follow the custom of [18] and use for the multiplicative linear conjunction ( tensor ) and for the multiplicative linear disjunction ( par ). For example, the following is a proof net demonstrating an entailment that is very important in the categorical semantics of classical proofs: A (B C) (A B) C. Ax Ax Ax B : L C : L A : R B : R L R A : L B C : L A B : R C : R L A (B C) : L R (A B) C We can, however, construct more structures than there are proofs; specifically, we can form graphs such as 29

32 Ax φ : L φ : R φ : L ψ : L φ : R ψ : R L R φ ψ : L φ ψ : R φ : L ψ : L φ : R ψ : R L R φ ψ : L φ ψ : R φ : L φ : R L R φ : L φ : R φ : L φ : R Cut Table 2.9: Proof nets for MLL 30

33 Ax Ax A : L B : L A : R B : R L R A B A B which conform to the given specification, but represent unsound proofs. To eliminate such structures, we need to introduce a correctness criterion. Such a criterion for MLLnets is given in [26], and refined by Danos and Regnier [19]. Definition A (Danos-Regnier) switching σ is the choice of one of the hypotheses for each node of the following forms: L, R, CL, CR. We shall say that the remaining nodes are unswitched. Definition Let S be a proof structure and σ a switching on it. Then the (Danos- Regnier) graph of σ, DR(σ, S), is the following undirected graph: Its vertices are the propositional vertices of S; Its edges join conclusions of rule nodes to hypotheses as follows. If the rule node is unswitched, then each conclusion is joined to each hypothesis. If the rule node is switched, then the conclusion is joined only to the hypothesis chosen by σ. The exceptions are axioms and cut, where the two formul are joined. Definition A proof structure S is a proof-net if, for every switching σ, DR(σ, S) is directed and acyclic. Clearly, from any sequent proof we can inductively generate a proof net: The converse is a non-trivial result, known as sequentialization: Theorem Each correct proof net S is the image of some sequent proof. For example, the net 2.2 sequentializes to A A B B R A, B A B C C L A, B C A B, C R A, B C (A B) C L A (B C) (A B) C 31

34 (but also to three other sequent proofs which differ on the ordering of L and R and of R and L.) Note that the representation for proof nets we have given is somewhat wasteful; we could drop the propositional nodes, and arrive at nets as follows L R L R as used in [16]. This makes for nets with which calculations may more easily be made. Alternatively, we can drop the rule nodes and arrive at nets of the form B : L C : L A : R B : R A : L B C : L A B : R C : R A (B C) : L (A B) C These nets are less cluttered, but retain the character of the sequent calculus, and it is this style we will use. Cut-reduction for MLL proof nets is particularly simple. Begin by expanding all logical cuts: φ : L ψ : L φ : R ψ : R φ : L ψ : L φ : R ψ : R φ ψ : L φ ψ : R and then all that remains are cuts of the form 32

35 φ : L φ : R φ : L φ : R... φ : R φ : L φ : R which can be reduced to axiom links. This procedure is deterministic and terminating, and we acquire a unique normal form for each net. If we limit axioms to occurring only on atoms, the normal form will be a linking of atoms on the left to those on the right, and a series of introductions of conjunctions and disjunctions. Extending proof nets for MLL to logics with weakening and contraction has proven difficult. While MLL proof nets are simple and act as canonical objects for proofs, proofs involving contraction or weakening involve a certain amount of bureaucracy, associated with the order in which contractions occur and the the location to which a weakening is attached. For example, if we add units to MLL, we need a representation of the rule Γ Γ, In general, weakenings (of which this is a special case) must be anchored to some other node of the net M φ : X φ : X ψ : L since unanchored nets implicitly admit mix (and mix is not sound in MLL). These links need to be rewired, as otherwise they obstruct the flow of cut reduction; the net φ : R φ : L φ : R ψ : L 33

36 cannot be reduced without moving the weakening. These rewirings are necessary, so we have lost uniqueness of normal forms. (However, see [36] for a solution to this problem.) As proof nets diverge further from those for MLL, the approaches separate into two distinct families: 1. Syntactic proof nets, where the goal is to provide a representation of proofs as close to the sequentialized proofs as possible, while still eliminating the bureaucracy associated with ordering of rules; and 2. Semantic proof nets, where the aim is to give objects as close to canonical objects as possible, at the expense (perhaps) of giving intelligible proofs. Our intention is to use proof-nets to eliminate commuting conversions in our calculations, rather than as canonical objects of a category. In fact, we will want to model some commuting conversions non-trivially in our categories. The nets we use will be unapologetically those of the first type. 2.3 Categorical logic and categorical preliminaries We give some preliminaries specific to the semantics of proofs; for more a more general introduction to category theory, and further details, see [50]. The field of categorical proof theory, in which one considers the propositions of a logic as objects and the proofs as morphisms in a category began with Lambek [46]. It is most richly developed for intuitionistic natural deduction, where a model is given by a bicartesian closed category; conjunction is modeled by categorical product, disjunction by coproduct, and implication by exponentiation (for details see [47]). We give details now of some extensions to this basic idea. 34

37 2.3.1 Interpretation of proofs in a category Given a set P of proofs derived in a given calculus, let a theory T on those terms be a relation on proofs of the same entailment which is reflexive and transitive. The relation given by Prawitz on natural deduction proofs (two proofs are related if they have the same normal form) is an example of such a theory. We recall in this section what it means to model P or T, for a model to be sound, for a class of models to be complete, and for a model to be initial. Remark We assume, for this discussion, that the calculus in question is propositional, leaving the discussion of first-order soundness etc. for the specific case of classical doctrines. Definition Given a category C and a set P of proofs, an interpretation of P in C is given by: 1. A function C : φ φ, mapping propositions to objects of C; and 2. A function C : (Φ : φ ψ) Φ : φ ψ, mapping proofs from P to morphisms of C - that is, it is a functor from P to C. For the interpretation C to be a model of T, all the relations holding in T should hold of their images under C. Of course, if C is an ordinary category, the only relationship that can hold between morphisms is equality, and so we may only model equivalence relations. Definition An interpretation C is sound with respect to an equivalence relation T if, for every equivalence Φ Ψ in T, C Φ = C Ψ. We use the term completeness, in this context, as a property of a class of models. Definition A class M of models is complete with respect to a theory T if we have the following: if C Φ = C Ψ holds of every model in M, then Φ Ψ is in T. 35