A New Category for Semantics

Transcription

1 A New Category for Semantics Andrej Bauer and Dana Scott June 2001 Domain theory for denotational semantics is over thirty years old. There are many variations on the idea and many interesting constructs that have been proposed by many people for realizing a wide variety of types as domains. Generally, the effort has been to create categories of domains that are cartesian closed (that is, have products and function spaces interpreting typed - calculus) and permit solutions to domain equations (that is, interpret recursive domain definitions and perhaps untyped -calculus). What has been missing is a simple connection between domains and the usual set-theoretical structures of mathematics as well as a comprehensive logic to reason about domains and the functions to be defined upon them. In December of 1996, Scott realized that the very old idea of partial equivalence relations on types could be applied to produce a large and rich category containing many specific categories of domains and allowing a suitable general logic. The category is called Equ, the category of equilogical spaces. The simplest definition is the category of Ì ¼ -spaces and total equivalence relations with continuous maps that are equivariant (meaning, preserving the equivalence relations). An equivalent definition uses algebraic (or continuous) lattices and partial equivalence relations, together with continuous equivariant maps. This category is not only cartesian closed, but it is locally cartesian closed (that is, it has dependent sums and products). Moreover, it contains as a full subcategory all ¼ -spaces (and therefore the category of sets and the category of domains). The logic for this category is intuitionistic and can be explained by a form of the realizability interpretation. The project now is to use this idea as a unifying platform for semantics and reasoning. Our small group of faculty and students at Carnegie Mellon, namely Steven Awodey, Andrej Bauer, Lars Birkedal, and Jesse Hughes, has begun work on this program. A selection of our theses and papers is listed in the bibliography. 1

2 Note: Equilogical Spaces We build on the known cartesian closed category CLat of continuous lattices and continuous maps. Types: Pairs µ, where is a continuous lattice and is a partial equivalence relation on. We set Ü Ü Ü. A type is also called an equilogical space. Equivalence Classes: Ü Ü ¼ Ü Ü ¼, and Ü Ü ¾, which we regard as a topological space with the quotient topology inherited from the subspace of. Theorem: Each continuous lattice can be regarded as an equilogical space where and each Ü Ü. Some Flat Lattices: We set Ä Ò ¼ ½ Ò ½ and Ľ ¼ ½ Ò. The integer types are ¼ Ä ¼ µ and ½ Ä ½ ¼ ¼µ µ Ò Ä Ò µ Ò µ and Æ Ä½ µ µ Any set can similarly be made into a type via a flat lattice. 2

3 Defining a Category Exponentials: µ, where ¼ iff Ü Ü ¼ always implies ܵ ¼ ܼ µ. Mappings: iff ¾. We also write dom and cod. Composition: Æ Æ, provided we have and. Mappings of quotients: If, then, where Ü µ ܵ, where and Ü ¾. Products: µ, where Ü Ýµ Ü ¼ Ý ¼ µ iff Ü Ü ¼ and Ý Ý ¼. Coproducts: Ä ¾ µ, where Ü Ýµ ¼ Ü ¼ Ý ¼ µ iff either ¼ ¼ and Ü Ü ¼, or ¼ ½ and Ý Ý ¼. Theorem: Equ is a (concrete) cartesian closed category with coproducts. 3

4 Categorical Embeddings Theorem: CLat is a full and faithful subcategory of Top ¼, the category of T ¼ -spaces and continuous mappings, and the latter category is fully and faithfully embedded in Equ. The embeddings preserve products throughout and coproducts from Top ¼ to Equ. They also preserve all exponents from CLat and those from Top ¼ that exist. Proof: Continuous lattices can be defined as T ¼ -spaces which are injective with respect to subspace embeddings; hence, they form a full subcategory of Top ¼. The second half of the theorem is proved by remarking that every T ¼ -space is embeddable as a subspace of an algebraic lattice, namely the powerset of the set of open subsets of the given space. The embedding is defined by mapping each point to its neighborhood filter. It needs to be checked that continuous functions between spaces can always be extended to continuous functions between the powersets. The equilogical space then consists of the powerset lattice and the identity relation on the subspace. 4

5 Representation via Quotients Theorem: is a faithful but non-full functor from Equ to Top; however, there is a full and faithful functor Q from Top to Equ such that QÌ is naturally homeomorphic to Ì in Top. Thus, every topological space is a quotient of a T ¼ -space in a functorial way. Proof: An earlier, but weaker, version of this argument was given by Jens Blanck in the paper Domain representations of topological spaces. The improved result was provided to us by Y. Ersov (private communication). Subsequently, after a lecture, Peter Johnstone suggested a simpler topology. Inasmuch as equivariant mappings are regarded as equivalent (i.e., equal in the category) if they do equivalent things to equivalent arguments, then the corresponding mappings on the quotients are the same, and conversely. This is why the functor is faithful. To see it is not full, consider the equilogical space Ä ½ µ either ¼ or ¼ and ¼ µ Clearly we have ¼, and the quotient topology on this set is the trivial indiscrete topology. On such a twoelement space, there are four continuous self-maps; however, in Equ there are only three maps, because on the underlying lattice all functions have to be monotone. 5

6 Representation via Quotients (continued) Next, consider the type where Ä ½ Ä ½, and where we have ¼µ ¼ µ ¼ µ ¼µ. Then is again the indiscrete two-element space, but all four functions are possible in Equ. Thus, different representations of spaces as quotients give us different results about the mappings between them. Next, suppose Ì is a given topological space with Op Ì µ as its collection of open subsets. We next form the set Ì and give it the following topology: Í Ë ½ Ò ¼ Í Ò Ò ¾ Op Ì µ if, and only if, for all Ò ¾, we have Í Ò int Í Ò ½ µ, where int is the interior operation for Ì. Note that we will have ½ Í Ò ¼ Í Ò ¾ Op Ì µ It is easily checked that this defines a topology on Ì. Suppose two points Ü µ and Ý Ñµ are distinct. We can assume that Ñ. Define a set Í Ë ½ Ò ¼ Í Ò Ò where Í Ò Ý Ì if Ò Ñ if Ò Ñ if Ò Ñ Then Í is an open neighborhood of Ý which does not contain Ü. So, the space Ì is T ¼. 6

7 Representation via Quotients (continued) We then put an equivalence relation on Ì by making each Ü Òµ Ò ¾ an equivalence class. By embedding Ì into a powerset lattice as a subspace, we have an equilogical space QÌ by using the equivalence classes indicated. Every continuous mapping Ì Ì ¼ obviously gives a mapping Ì Ì ¼, defined by µ Ü Òµµ ܵ Òµ, which is continuous and equivariant. As mappings between subspaces can be continuously extended to mappings between the lattices, we thus have a way to define Q QÌ QÌ ¼ in Equ. In this way Q becomes a functor. By direct construction, the topology on QÌ exactly replicates the topology on Ì. Moreover, the functions Q provide exact copies of all expected continuous functions. Note: If Ì is a T ¼ -space regarded as an equilogical space, then the equilogical space QÌ is not generally isomorphic to the given Ì. There is an obvious epi-mono from QÌ to Ì, but it is not an isomorphism because of not finding an inverse function. Note: The category Equ could be presented as (total) equivalence relations on T ¼ -spaces. The representation theorem just presented might make it plausible that we would obtain an equivalent category by using equivalence relations on all topological spaces. This is not true, however, and in fact the latter category is much larger than Equ, in the sense of allowing constructions not available in Equ. Note: The functor does not in general preserve products, but it does preserve coproducts. 7

8 Predicates on Types General Predicates: È È È µ is a predicate on type iff È is a continuous lattice, and È È. Stable Predicates: These are predicates of the form È È µ, where is the one-element lattice. Note: These definitions could be expanded to predicates of several arguments. This is not necessary, however, in view of the underlying type structure which has products. Subtypes: Given a predicate È on type, we define Ë Ü : È Üµ È Ë µ, where Ô Üµ Ë Ô ¼ Ü ¼ µ iff Ü Ü ¼ and further Ô È Ü and Ô ¼ È Ü ¼. Note: In general, the subtypes of a type form a completely unbounded totality; however, the stable subtypes of a type correspond to the powerset of. In the category Equ the subtypes correspond (up to isomorphism) to monomorphisms and the stable subtypes to the regular monos. 8

9 Partial Mappings Stable Partial Mappings: provided there is a stable subtype Ë of such that Ë. We write dom, cod and def Ë. Note: Strictly speaking mappings are to be understood as triples (giving domains and codomains explicitly), and partial mappings have to be quadruples (with the stable subdomain given as well). A fully rigouous notation is too heavy for this outline. Invoking the Principle of Tolerance, we also regard (total) mappings as partial mappings, and a type as being a stable subtype of itself. Note: In the case of partial mappings, we also have to agree that Ü µ in the case that Ü ¾ def. Inverse Images: Given a stable partial mapping with def Ë and a predicate É on type, we define ½ É as the predicate É ½ É µ, where for Ü ¾ we have Õ ½ É Ü iff ܵ ¾ Ë and Õ É Üµ Ë µ. 9

10 Valuations and Terms Variables: A denumerable set Var. We write Ù Ú Û Ù ¼ Ú ¼ to range over Var. Valuations: A function «from variables to pairs, so that for a variable Ù we have «Ùµ Ü µ, where is a type. We sometimes write as shorthand «Ùµ ¼ and «Ùµ ½ Ü. Note: In specifying «Ùµ above, we allow the possibility that Ü Ü may not hold. We find then that Ü ¾, and say the value of the variable is undefined. Terms: Built up from variables by using these modes of composition:, µ, µ, and Ù :, where, first, is a constant of the form Ü µ for some type and some element Ü ¾, next is a stable partial function, and are previously obtained terms, and Ù is a variable and is a type. Note: For simplicity, we allow constants and types to occur in terms and formulae and, thus, do not distinguish between use and mention. 10

11 Logical Formulae Atomic Formulae: Expressions of the form È µ and, where and are terms. Note: We abbreviate as E for the existence predicate. General Formulae: Built up from atomic formulae by using these modes of composition: ³, ³, ³, ³, Ù : ³, and Ù : ³, where ³ and are previously obtained formulae, Ù is a variable, and is a type. Types of terms: Relative to a valuation «, define Ù ««Ùµ ¼, «¼, µ «, provided, µ «, provided «and «, and Ù : «, where «µ Ùµ. Note: There may be a small difficulty here, inasmuch as the types and do not always uniquely determine. Note further that the notation «Üµ Ùµ indicates that the valuation «has had the type and value of the variable Ù changed as shown. 11

12 Realizability Semantics Potential Realizers of Formulae: valuation «, we define Relative to a given È µ È and, ³ ³ and ³ Ä ¾ ³, ³ ³ and ³, Ù : ³ ³ and Ù : ³ ³. Values of Terms: Relative to a given valuation «, we define Ù ««Ùµ ½, and «½, µ ««µ, provided and «, µ «««µ, provided «and «, and Ù : «¾ ܵ ¾ «Ü µ Ùµ whenever Ü ¾, provided «µ Ùµ. Note: The values of applications above have to be taken as (undefined) in case the provisos fail. The value of a -term has to be a total function unless it is undefined. 12

13 Realizability Semantics (Continued) Realizing Formulae: «, we define Relative to a given valuation Ô È µ «iff Ô È «and È is a predicate of type «, Ô «iff Ô and ««and ««¾ «, Ô ³ «iff Ô ¾ ³ and Ô ¼ ³ «and Ô ½ «, Ô ³ «iff Ô ¾ ³ and either Ô ¼ ¼ and Ô ½ ³ «or Ô ¼ ½ and Ô ¾ «, Ô ³ «iff Ô ¾ ³ and, for all Õ ¾ ³, if Õ ³ «, then Ô Õµ «, Ô ³ «iff Ô and no Õ ³ «, Ô Ù : ³ «iff Ô ¾ Ù : ³ and whenever Ü ¾, then Ô Üµ ³ «Ü µ Ùµ, Ô Ù : ³ «iff Ô ¾ Ù : ³ and Ô ¼ ¾ and Ô ½ ³ «Ô ¼ µ Ùµ. 13

14 Principles of Abstraction Predicate Abstraction: Given a valuation «and a type, there is a unique predicate È on type with È ³ such that the formula Ù : È Ùµ ³ is realizable relative to «. Note: We can write Ù : ³ for the subtype determined by the predicate È. Function Abstraction (I): Given a valuation «and types and, if the formulae Ù : Ú : ³ Ú : ³ and Ù : Ú ¼ : Ú : Ú Ú ¼ ³ Ú : ³ Ú Ú ¼ are realizable relative to «, then there is a stable partial map such that the formula Ù : Ú : Ú Ùµ ³ is also realizable relative to «. Function Abstraction (II): Given a valuation «, a type, and a term of type, there is a stable partial map so that the formula Ù : Ùµ is realizable relative to «. 14

15 Principles of Extensionality Lambda Abstraction: Given types and and a total function, the formula Ú : Ù : Ùµµ Úµ Úµ is realizable (relative to all valuations). Lambda Extensionality: and, the formula Given a type and terms Ù : Ù : Ù : is realizable (relative to all valuations). Function Extensionality: Given types and and three distinct variables, the formula Ú Ú ¼ : Ù : Ú Ùµ Ú ¼ Ùµ Ú Ú ¼ is realizable (relative to all valuations). 15

16 Principles of Logic Theorem: All the usual axioms and theorems of typed firstorder intuitionistic logic and typed lambda calculus are realizable, including the general Principle of Stability of Identity: Ù Ú : Ù Ú Ù Ú Moreover, all types are either empty or not empty, and so the sentence is realizable. Ù : Ù Ù Ù : Ù Ù Note: The exponential types provide that part of higher-order logic appropriate to the underlying cartesian closed category of types. Note: The Law of the Excluded Middle in the form Ù Ú : Ù Ú Ù Ú already fails to be realizable for the type Ä ¼, even though this type only has two elements and. It can be shown in general that Excluded Middle holds only for types isomorphic in Equ to sets (i.e., discrete topological spaces). Theorem: For the type Ä ¼, the following formula is realizable: Ù : Ä ¼ Ù Ù Note: A similar formula would hold for those types, where has exactly two elements. Not all such types are mutually isomorphic, however. 16

17 Double Negation and Stability Theorem: The stable subtypes of a type form a Boolean algebra where, for given stable predicates È and É, the Boolean operations correspond to: Ù : È Ùµ É Ùµ Ù : È Ùµ É Ùµ Ù : È Ùµ Note: In terms of elements, these types correspond to the intersections, unions and complements of subsets of. Theorem: Families of stable predicates on a type indexed by a type Á can be represented by stable predicates È on a type Á. The intersection and union of such families of stable predicates correspond to subtypes: Ù : : Á È Ùµµ Ù : : Á È Ùµµ where µ Á Á µ is the cannonical pairing function. 17

18 The Principle of Choice Definition: We say that choice holds from type to type if, and only if, relative to all valuations «and all formulae ³, the formula Ù : Ú : ³ Û : Ù : Ú : Ú Û Ùµ ³ is realizable relative to «, provided that Ù, Ú and Û are distinct variables and Û is not free in ³. We say choice holds for if it holds from to all types. Theorem: Choice holds for a type if, and only if, it is isomorphic to a Ì ¼ -topological space (i.e., a type µ, where the partial equivalence relation has only singleton equivalences classes in, which is then homeomorphic to ). Note: Isomorphism for types and in Equ can be expressed in the internal logic by the realizability of the formula: Û : Û ¼ : Ù : Ú : Ú Û Ùµ Ù Û ¼ Úµ provided the four variables are distinct. Isomorphic types have the same (properly formulated) logical properties. 18

19 Characterizing Mappings Theorem: A morphism is a monomorphism if, and only if, is injective if, and only if, the following sentence is realized: Ù Ù ¼ : Ùµ Ù ¼ µ Ù Ù ¼ Theorem: A morphism is a regular mono if, and only if, is a subspace embedding if, and only if, the following sentence is realized: Ù Ù ¼ : Ùµ Ù ¼ µ Ù Ù ¼ Ú : Ù : Ú Ùµ Ù : Ú Ùµ Theorem: A morphism is an epimorphism if, and only if, is surjective if, and only if, the following sentence is realized: Ú : Ù : Ú Ùµ Theorem: A morphism is regular epi if, and only if, is a quotient map if, and only if, the following sentence is realized: Ú : Ù : Ú Ùµ Theorem: Up to isomorphism, every morphism can be factored as Æ Æ Õ with Õ ¼ ¼ and with Õ regular epi, epi-mono, and regular mono. 19

20 Internalizing Topology Theorem: Among topological spaces, Ä ¼ is characterized up to isomorphism by the realizability of these formulae: Ù : Ù Ù Ù : Ù Ù and Ù Ú : Ù Ú Û : Û ¼ : Û Û ¼ Ù Û Û ¼ Ú Problem: The realizable first-order sentences with quantification only over are decidable, since they each reduce to a property of a finite type. But is there a nice axiomatization? A similar question could be asked about higher-order properties using types generated from using products, sums, and exponents. Or about other types made from finite lattices. Theorem: Given a type, the open subsets of the quotient space are in a one-one correspondence with the elements of and with stable subtypes of the form Ù : Ùµ, where holds. Note: The closed subsets correspond to subtypes Ù : Ùµ Corollary: Given types and and a mapping, the inverse image by of any open subset of is again an open subset of. 20

21 Proving Topological Properties Theorem: The subtype Û : Û µ is open in. Corollary: Given a type, the intersection of two open subsets of is again open. Theorem: Given a type Á, the subtype Ù : Á : Á Ù µ is open in Á. Corollary: Given a type, the union of any family of open subsets of is again open. Corollary: The topology of the quotient space of a type has a countable basis if, and only if, the following is realizable: Û : µ Æ Ù Ú : Ò : Æ Û Òµ Ù Û Òµ Ú Ù Ú where Ù Ú abbrevates Ø : ٠ص Ú Øµ 21

22 Compactness Definition: A stable subtype à of a type is compact in if, and only if, the subtype is open. Û : Ù : Ã Û Ùµ Note: We simplified the writing of the subtype by not invoking the necessary injection mapping in Ã. The possibility of such a definition was brought to our attention by M. Escardó. Proposition: Because the intersection of two open sets is open, it follows that the union of two compact subtypes is compact. Proposition: Because the inverse image of an open set is open, it follows that a closed subtype of a compact subtype is compact. Proposition: Given and a stable compact subtype à of, the stable image subtype is compact in. Ú : Ù : Ã Ú Ùµ Theorem (M. Escardó): The stable compact subtypes of a type represent subsets of the topological space which are compact in the usual sense of open coverings. In case is topological, the converse holds. 22

23 Separation, Products and Coproducts Definition: A type satisfies Hausdorff separation if, and only if, the subtype Û : Û ¼ Û ½ is closed in, where ¼ and ½ are the two canonical projection mappings. Corollary: If, and satisfies Hausdorff separation, then Û : Û ½ Û ¼ µ is closed in. Theorem: If a type A satisfies Hausdorff separation, then every stable compact subtype of is closed. Theorem: The product and coproduct of two types satisfying Hausdorff separation also have that property. Theorem: If à and Ä are stable compact subtypes of types and, respectively, then Ã Ä is a stable compact subtype of. Theorem: If à and Ä are stable compact subtypes of types and, respectively, then Ã Ä is a stable compact subtype of. 23

24 Fixed Points Definition: A type has the (strong) fixed-point property (fpp) if, and only if, the sentence Û : Ù : Ù Û Ùµ is realizable. Definition: A type has the stable fixed-point property (sfpp) if, and only if, the sentence Û : Ù : Ù Û Ùµ is realizable. Note: If a type has the sfpp and if, then the continuous function has a fixed point in. Definition: A type has the general fixed-point property (gfpp) if, and only if, for any type Á, every power Á has the sfpp. Note: If a type has the gfpp, then so does any power Â. 24

25 Theorem: Some Proofs about Fixed Points If a type has the gfpp, then it has the fpp. Proof: Suppose the type has the gfpp. Let be defined by: Û : Ú : Ú Û Úµµ Let be a mapping in Equ corresponding to a fixed point of. We will then have as realizable the sentence Ú : Úµ Ú Úµµ The desired conclusion now follows. Theorem: If types and have the gfpp, then so does. Proof: It will be sufficient to show that has the fpp, for if and both have the gfpp, then so do Á and Á. But Á Á µ Á So we can then argue that has the gfpp. Let and be fixed-point producing mappings, as in the previous proof. Suppose we are given a mapping. Define Ù : Ù Ú : Ù Úµµ ½ µµ ¼ and Ú : Úµµ ½ µ The pair µ gives the desired fixed point of, and since the formulae work uniformily in, we have proved the fpp for the product. 25

26 Complete Lattices and Fixed Points Note: Every poset can be considered as an equilogical space by using the obvious embedding of a poset into the algebraic lattice of downward closed subsets of the poset. The mappings in Equ between posets then correspond exactly to the monotone functions between the posets. Theorem: Every complete lattice considered as an equilogical space has the gfpp. Proof: Let Ä be a type, where Ä is (isomorphic to) the algebraic lattice of downward closed subsets of Ä considered as a poset. Suppose, as a poset, Ä is a complete lattice. Now, if A is any type, then Ä is also a complete lattice under the pointwise ordering, and all mappings Ä Ä are monotone. By the Tarski Fixed-Point Theorem, all such mappings have fixed points. This is sufficient to conclude that Ä has the sfpp. Note: The argument just given shows that the cartesian closed category of types with the gfpp is larger that the category of continuous lattices, which is an obvious subcategory of Equ. Indeed, a type produces a complete lattice which in general need not be a continuous lattice. Indeed, a T ¼ -space is exponentiable in the category of T ¼ -spaces if, and only if, its lattice of open sets is a continuous lattice. Note: The well known D½-construction shows that there are many types (indeed, continuous lattices) such that. In fact, non-trivial continuous lattices can be found where we have. All such types (lattices or not) have the fpp. 26

27 A -Calculus Proof Note: In the category of T ¼ -spaces, the continuous lattices are distinguished as retracts of every space of which they are a subspace. In particular, the obvious evaluation mapping Ä ¼ Ä ¼ is one half of a retraction pair, if D is a continuous lattice. Moreover, since a retract of a continuous lattice is again a continuous lattice, this one retraction set-up characterizes these lattices. Unfortunately, the corresponding statement in Equ does not characterize the continuous lattices. Theorem: Given in Equ, the evaluation mapping is one half of a retraction pair. Proof: Now define Recall that by definition we have We now calculate Ù : Ú : Ú Ùµ Û : : Û Ù : Ù µµ Ùµµ µ Ùµ Ù ¼ : Ù ¼ µµ Ù µ which establishes what we want. Note: Inasmuch as need not be a continuous lattice, we have a counter-example to a possible characterization. Note, too, that could be replaced by any other type in the theorem. 27

28 The Brouwer Fixed-Point Theorem Definition: There are two principal models of the real numbers in Equ, namely the topological space of reals considered as a type Ê Ø, and the Cauchy sequences of rational numbers considered as an equilogical space Ê using the usual equivalence relation. There is an epi-mono from Ê to Ê Ø, but the spaces are not isomorphic (because there is no continuous way to choose a Cauchy sequence for each real). The unit interval in each model is denoted by ¼ ½ Ø and ¼ ½. Theorem: The unit intervals ¼ ½ Ø and ¼ ½ both have the sfpp but not the stronger fpp; hence, they cannot have the gfpp. Note: The positive part of the proof is by the Brouwer Theorem, and the negative part (as pointed out to us by Peter Freyd) comes easily from examples from Catastrophe Theory. (Again, a continuous choice of fixed points is impossible.) Note: The usual algebra of reals is available in both models, and the ordering relation is stable. Linearity of ordering only holds in a weaker form for Ê : Ù Ú Û : Ê Ù Ú Ù Û Û Ú For Ê Ø a double negation must be inserted before the implication. So, the two models do not satisfy the same first-order formulae, and indeed Ê is the more satisfactory model, giving us the basic principles of a Bishop-style analysis but with a stable apartness relation. 28

29 The Finite Arithmetic Types Theorem: Induction over the integers is realizable as: Ò : Æ È Òµ È Ò ½µ and as: È ¼µ Ò : Æ È Òµ Ò : Æ È Òµ È Òµ Ò : Æ È Òµ Ò : Æ È Òµ Ñ : Æ Ñ Ò È Ñµ Theorem: The finite types built from the integer types Ò and Æ by means of,, and give us just the Kleene-Kreisel countable functionals. Theorem: The realizable first-order formulae of arithmetic, involving any integer functions and quantification only over Æ, are just the classically valid formulae; however, higher-order formulae need not be classical, as the formulae Ù : Æ Æ Ò : Æ Ù Òµ ¼ Ò : Æ Ù Òµ ¼ and Ù : Æ Æ Ò : Æ Ù Òµ ¼ Ò : Æ Ù Òµ ¼ are realizable. Theorem: The Axiom of Choice is realizable from one finite type to another. 29

30 BIBLIOGRAPHY The papers are listed in reverse chronological order and can be downloaded via S. Awodey, A. Bauer. Propositions as [Types]. Preprint, Institut Mittag-Leffler, Sweden. June S. Awodey, A. Bauer. Sheaf Toposes for Realizability. Preprint CMU-PHIL-117. April A. Bauer, L. Birkedal, D.S. Scott. Equilogical Spaces. September Revised February To appear in Theoretical Computer Science. A. Bauer The Realizability Approach to Computable Analysis and Topology. Ph.D. Thesis. September S. Awodey, J. Hughes. The Coalgebraic Dual of Birkhoff s Variety Theorem. Preprint CMU. October L. Birkedal, J. van Oosten. Relative and Modified Relative Realizability. Preprint 1146, Department of Mathematics, Universiteit Utrecht. March A. Bauer, L. Birkedal. Continuous Functionals of Dependent Types and Equilogical Spaces. Proceedings of Computer Science Logic Conference S. Awodey, L. Birkedal, D.S. Scott. Local Realizability Toposes and a Modal Logic for Computability. January To appear in Mathematical Structructures in Compter Scicience. 30

31 BIBLIOGRAPHY (continued) L. Birkedal. A General Notion of Realizability. Proceedings of LICS December L. Birkedal. Developing Theories of Types and Computability via Realizability. PhD-thesis. Electronic Notes in Theoretical Computer Science, 34, December Available at S. Awodey and L. Birkedal. Elementary Axioms for Local Maps of Toposes. Technical Report No. CMU-PHIL103. November To appear in Journal of Pure and Applied Algebra. S. Awodey. Topological Representation of the Lambda Calculus. September Mathematical Structures in Computer Science, vol. 10 (2000), pp L. Birkedal, A. Carboni, G. Rosolini, and D.S. Scott. Type Theory via Exact Categories. LICS July D.S. Scott. A New Category?: Domains, Spaces and Equivalence Relations. Unpublished Manuscript. December