Th. Streicher. Correctness and Completeness of a Categorical Semantics of the Calculus of Constructions. PhD thesis, Univ. Passau, 1989. Appeared as technical report MIP - 8913.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....is partial. Partiality is introduced in the interpretation of application. But the interpretation of a derivable judgement will be defined and true. The method with a partial interpretation function has also been used by Streicher for a categorical interpretation of the calculus of constructions [15]. Interpretation of set expressions: Pix : A 0 :A 1 [x] ae = x : This is defined iff [ A 0 ] ae is defined and [ A 1 [x] ae x is defined whenever u 2 [ A 0 ] ae. Interpretation of element expressions: x] ae = ae(x) This is always defined. x : A:a[x] ae = fhu; a[x] ae x ....

....4 This is defined iff [ a 1 ] ae and [ a 0 ] ae are defined, and [ a 1 ] ae is a function the domain of which contains [ a 0 ] ae. Observe that it is possible to interpret polymorphic application in set theory. This is not the case for interpretations of type theory in general, compare Streicher [15]. Interpretation of context expressions: ffl] f;g: This is always defined. Gamma; x : A] fae x jae 2 [ Gamma] u 2 [ A] aeg: This is defined iff Gamma is defined and [ A] ae is defined whenever ae 2 [ Gamma] Interpretation of judgement expressions: Gamma context ] iff ....

T. Streicher. Correctness and Completeness of a Categorical Semantics of the Calculus of Constructions. PhD thesis, Fakultat fur Mathematik und Informatik, Universitat Passau, 1988.

....dependent type theories. Some of the earliest work in this area was undertaken by Cartmell [Car86] with additional work by Taylor [Tay86] Here we shall give a presentation of categories with attributes which is based on on Pitts account in [Pit95] Further useful information can be found in [Str89], HP89] CGW89] and [Ben85] Categories with Attributes Definition 9.1.1 A category with attributes is specified by a category C with terminal object (called the base category) which is equipped with the following structure: ffl For each object X in C, a collection of fibrations over X, ....

T. Streicher. Correctness and Completeness of a Categorical Semantics of the Calculus of Constructions. PhD thesis, Universitat Passau, 1989. Reference MIP - 8913.

....solves this coherence problem, but only for the pure system of typed combinators without rec: Proposition 15 If a; a 2 T(A) are such that strip a = strip a ; then a convA a Proof: We don t know any direct proof of this proposition. The following indirect argument is due to Streicher [29]. First, we prove unicity of a typed decoration of a raw term t ffl A for t in normal form. The proposition results then from the normalisation theorem and the fact that strip preserves conversion. This proposition does not hold for the system with rec: Indeed, it is then possible to exhibit raw ....

T. Streicher. Correctness and Completeness of a Categorical Semantics of the Calculus of Constructions. PhD thesis, Fakultat fur Mathematik und Informatik, Universitat Passau, 1988. 17

....has the same logical structure as the M:A fragment but needs some extra structure. To interpret the kind Type, for instance, we must require the existence of a chosen object which obeys some equations regarding substitution and quantification. A treatment of the intuitionistic case is in Streicher [21]. A Kripke resource model is a Kripke resource structure that has enough points to interpret not only the constants of S but also the lL calculus terms defined over S and a given context G. Formally, a Kripke resource model is made up of five components: a Kripke resource structure that has ....

....the category of families of sets can just be considered as a presheaf Fam: Ctx ; Set] rather than as an indexed category; we will adopt this view in the sequel. We can explicate the structure of Ctx by describing Fam as a contextual category [4] The following definition is from Streicher [21]: Definition 8. The contextual category Fam, along with its denotation DEN:Fam Set and length, is described as follows: 1) 1 is the unique context of length 0 and DEN(1) f 0g; 2) If D is a context of length n and A:DEN(D) Set is a family of sets indexed by elements of DEN(D) then D ....

T Streicher. Correctness and Completeness of a Categorical Semantics of the Calculus of Constructions. PhD thesis, Universitat Passau, 1988.

....the existence of a chosen object, call it in each bre. The object must obey several equations: it must be preserved on the nose by any f and must behave well under quanti cation. Details of the treatment of the A:K fragment in the case of contextual categories are in Streicher s thesis [36]. The analogous development in our setting is similar and we omit the details. De nition 18 Let be a calculus signature. A Kripke resource model is a 5 tuple hfJ r : W ; C op ; Cat] j r 2 Rg; J K; join; share; j= i where fJ r : W ; C op ; Cat] j r 2 Rg is a Kripke resource ....

....plays two other roles too. Firstly, there is dependent typing partiality to bootstrap the de nition. And, secondly, there is Kripke semantic partiality of information, in which the further up the world structure one goes, the more objects have de ned interpretations. We refer to Streicher [36], Pym [31] and Mitchell and Moggi [21] for some comments regarding these matters. The following lemma follows easily from the de nition. Lemma 19 join and share are functors. Proof We need to show that both join and share preserve identities and composition. We omit the details. We now ....

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T Streicher. Correctness and Completeness of a Categorical Semantics of the Calculus of Constructions. PhD thesis, Universitat Passau, 1988.

....The first is the term model, constructed out of the syntax of the type theory. The main purpose behind its construction is to show a Henkin style completeness theorem. The second model, a set theoretic one built in the spirit of Cartmell and Streicher s contextual category of families of sets [Car86, Str88], is much more interesting. The model constructs two kinds of indexed products in Set C op , where C is a small monoidal category. The first is the usual cartesian product, as described in Cartmell for instance. The second is a restriction of Day s tensor product [Day70] The Kripke resource ....

....the existence of a chosen object, call it W, in each fibre. The object W must obey several equations: it must be preserved on the nose by any f and must behave well under quantification. Details of the treatment of the A:K fragment in the case of contextual categories are in Streicher s thesis [Str88]. The analogous development in our setting is similar and we omit the details. Definition 53 Let S be a lL calculus signature. A Kripke resource S lL model is a 5 tuple hfJ r : W ; C op ; Cat] j r 2 Rg;J GammaK; join;share; j= S i where fJ r : W ; C op ; Cat] j r 2 Rg is a Kripke resource ....

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T Streicher. Correctness and Completeness of a Categorical Semantics of the Calculus of Constructions. PhD thesis, Universitat Passau, 1988.

....techniques seem better suited to generalizations concerned with weaker type theories [86] For instance, at a logical level, the base and bres deal, respectively, with terms and propositions. Nevertheless, these ideas owe much to work of Cartmell [28] Pitts [138] Seely [160] Streicher [167] and others. We take a Kripke prestructure to be a functor J : W; D op ; V] where W is a small category (of worlds) D op = W D op W , where W ranges over the objects of W, and each DW (the base at W ) is small; V, a subcategory of the category of small categories, Cat, is a ....

T.Streicher. Correctness and completeness of a categorical semantics of the calculus of constructions. PhD thesis, Universitat Passau, 1988.

....to the Martin Lof style presentation. Whereas this equivalence is easy to prove in the case of the simply typed calculus (and hence it is not really necessary to differentiate between the two approaches in this case) the difference becomes crucial as soon as we add, for example, dependent types [15] 1 . This difference becomes crucial again when we consider calculi with explicit substitutions. This paper presents calculi for both approaches 1 The equivalence proofs can still be done [8] but some of the required properties of the type theories, like confluence and subject reduction, are ....

Thomas Streicher. Correctness and Completeness of a Categorical Semantics of the Calculus of Constructions. PhD thesis, Universitat Passau, June 1989.

.... intuitionistic setting are obtainable from the small complete internal poset Omega , whose object of objects is the subobject classifier [49, Ch.3] Subsequent research in categorical logic has considered the richer type systems above, with fibrations emerging as the central unifying concept [102,6,45, 26,105,84,48]. We shall return to these issues in Chapter 5 below. Type theoretically, the notion of subset , or subset type has been much less clearly defined. Probably the most closely argued and theoretically satisfying has been the work of the Goteborg group [79] One of the troubling (and desirable ) ....

T.Streicher, Correctness and completeness of a categorical semantics of Constructions, thesis, Passau, 1989.

....CC has been studied by a number of authors from a categorical point of view, e.g. see [HP89] in [Ehr89] the notion of a dictos is introduced, and in [Jac91] the more general notion of a CC category is used. A very natural semantics based on the concept of Realizability is the set semantics. In [Str89] a mild generalization (D sets) is investigated in great detail and used to show some independence results. 8 It should be noted that there are two different ways to present a type theory: a presentation based on conversion as in the presentation of PTS or a presentation where equality is a ....

.... compare this to the more conventional construction presented in [Wer92] 1.5. The use of categories The machinery of Category Theory has been used and proven useful for the semantic investigation of Type Theories. e.g. this has been extensively studied in the work of Thomas Streicher [Str89] 12 and Bart Jacobs [Jac91] Moreover in [Rit92] the categorical semantics of CC is used directly as a starting point for an implementation. It has been noted 13 that the naive use of categorical notions does not necessarily produce a sound interpretation of the syntax. Often we have to ....

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Thomas Streicher. Correctness and Completeness of a Categorical Semantics of the Calculus of Constructions. PhD thesis, Universitat Passau, Passau, West Germany, June 1989.

....the Martin Lof style presentation. Whereas this equivalence is easy to prove in the case of the simply typed calculus (and hence it is not really necessary to differentiate between the two approaches in this case) the difference becomes crucial as soon as we add, for example, dependent types [Str89] 1 . This difference becomes crucial again when we consider calculi with explicit substitutions. This paper presents calculi for both approaches and shows their equivalence (see section 3) This is because we want to connect the implementation, which is based on the second approach, with the ....

Thomas Streicher. Correctness and Completeness of a Categorical Semantics of the Calculus of Constructions. PhD thesis, Universitat Passau, June 1989.

....for the ontologies of greatest interest in computer science, e.g. dependent type theories, Constructions, linear and modal logics. Both the syntax and semantical analysis of new programming languages is being shaped by these semantic frameworks. The reader is urged to consult the work in e.g. [31, 97, 96, 2, 75, 51, 49, 48, 107, 108, 71, 83, 106, 88] and others cited in the appendix, for further details. The author would like to thank Anil Nerode for introducing him to the realizability interpretation of IZF, and encouraging further study of the field, and Peter Freyd for many insights about the effective topos and PERs. Many thanks also to ....

Streicher, T. [1989], Correctness and Completeness of a Categorical Semantics of the Calculus of Constructions, Ph. D. dissertation,Univ. Passau.

....of a chosen object, call it Omega Gamma in each fibre. The object Omega must obey several equations: it must be preserved on the nose by any f and must behave well under quantification. Details of the treatment of the A:K fragment in the case of contextual categories are in Streicher s thesis [32]. The analogous development in our setting is similar and we omit the details. Definition 17 Let Sigma be a calculus signature. A Kripke resource Sigma model is a 5 tuple hfJ r : W ; C op ; Cat] j r 2 Rg; J GammaK; join; share; j= Sigma i where fJ r : W ; C op ; Cat] j r 2 Rg is a ....

....plays two other roles too. Firstly, there is dependent typing partiality to bootstrap the definition. And, secondly, there is Kripke semantic partiality of information, in which the further up the world structure one goes, the more objects have defined interpretations. We refer to Streicher [32], Pym [28] and Mitchell and Moggi [18] for some comments regarding these matters. The following lemma follows easily from the definition. Lemma 18 join and share are functors. Proof We need to show that both join and share preserve identities and composition. We omit the details. We now ....

[Article contains additional citation context not shown here]

T Streicher. Correctness and Completeness of a Categorical Semantics of the Calculus of Constructions. PhD thesis, Universitat Passau, 1988.

....to the Martin Lof style presentation. Whereas this equivalence is easy to prove in the case of the simply typed calculus (and hence it is not really necessary to differentiate between the two approaches in this case) the difference becomes crucial as soon as we add, for example, dependent types [14] 1 . This difference becomes crucial again when we consider calculi with explicit substitutions. This paper presents calculi for both approaches and shows their equivalence (see section 3) This is because we want to connect the implementation, which is based on the second approach, with the ....

Thomas Streicher. Correctness and Completeness of a Categorical Semantics of the Calculus of Constructions. PhD thesis, Universitat Passau, June 1989.

....this coherence problem, but only for the pure system of typed combinators without rec: Proposition 15 If a; a 0 2 T(A) are such that strip a = strip a 0 ; then a convA a 0 : Proof: We don t know any direct proof of this proposition. The following indirect argument is due to Streicher [29]. First, we prove unicity of a typed decoration of a raw term t ffl A for t in normal form. The proposition results then from the normalisation theorem and the fact that strip preserves conversion. 2 This proposition does not hold for the system with rec: Indeed, it is then possible to exhibit ....

T. Streicher. Correctness and Completeness of a Categorical Semantics of the Calculus of Constructions. PhD thesis, Fakultat fur Mathematik und Informatik, Universitat Passau, 1988.

....techniques seem better suited to generalizations concerned with weaker type theories [86] For instance, at a logical level, the base and fibres deal, respectively, with terms and propositions. Nevertheless, these ideas owe much to work of Cartmell [28] Pitts [138] Seely [160] Streicher [167] and others. We take a Kripke prestructure to be a functor J : W; D op ; V] where W is a small category (of worlds) D op = W D op W , where W ranges over the objects of W, and each DW (the base at W ) is small; V, a subcategory of the category of small categories, Cat, is a ....

T.Streicher. Correctness and completeness of a categorical semantics of the calculus of constructions. PhD thesis, Universitat Passau, 1988.

....gives up the principle of identifying all types with propositions. The former idea was adopted by Martin Lof (see e.g. ML84, NPS90] who developed predicative type theory, the latter is represented by the various impredicative theories like the most prominent Calculus of Constructions (see e.g. [Coq85, CH88, Str89]) and its derivatives. Since impredicative universes are good for modeling polymorphism we have chosen an impredicative system, namely the Extended Calculus of Constructions (ECC) Luo90, Luo94] for our purposes. ECC is an extension of the Calculus of Constructions with predicative universes ....

....indexed products. 12 Chapter 1. Introduction A realizability model for # cpo s To show that the theory we presented is consistent, we have to provide a model for ECC # and the axioms. This is the content of Chapter 8. A PER model for ECC was already outlined in [Luo90, Luo94] whereas [Str89, Str91] described a categorical model of the Calculus of Constructions and proved its correctness. The problem with models for dependently typed calculi is that well formedness cannot be separated from validity of sequents. Thus it is quite complicated to give a correct interpretation function. Luo ....

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T. Streicher. Correctness and Completeness of a Categorical Semantics of the Calculus of Constructions. PhD thesis, Universitat Passau, 1989. Available as report MIP-8913, University of Passau.

....of this, namely the categorical semantics of dependent products in Section 6.5. Several different, but interconnected, categorical structures have been proposed for interpreting the basic framework of dependent types by Seely [ 1984 ] Cartmell [ 1986 ] Taylor [ 1986 ] Ehrhard [ 1988 ] Streicher [ 1989, 1991 ] Hyland and Pitts [ 1989 ] Obtu lowicz [ 1989 ] Curien [ 1989 ] and Jacobs [ 1991 ] This reflects the fact that the categorical interpretation of dependent types is undoubtedly more complicated than the other varieties of categorical logic explained in this chapter. This is due ....

Th. Streicher. Correctness and Completeness of a Categorical Semantics of the Calculus of Constructions. PhD thesis, Univ. Passau, 1989. Tech. Report MIP-8913.

....has the same logical structure as the M :Afragment but needs some extra structure. To interpret the kind Type, for instance, we must require the existence of a chosen object which obeys some equations regarding substitution and quantification. A treatment of the intuitionistic case is in Streicher [20]. A Kripke resource model is a Kripke resource structure that has enough points to interpret not only the constants of Sigma but also the calculus terms defined over Sigma and a given context Gamma. Formally, a Kripke resource model is made up of five components: a Kripke resource structure ....

....the category of families of sets can just be considered as a presheaf Fam: Ctx op ; Set] rather than as an indexed category; we will adopt this view in the sequel. We can explicate the structure of Ctx by describing Fam as a contextual category [4] The following definition is from Streicher [20]. Definition 15 The contextual category Fam, together with its length and denotation DEN:Fam Set, is described as follows: 1. 1 is the unique context of length 0 and DEN(1) f;g 2. If D is a context of length n and A:DEN(D) Set is a family of sets indexed by elements of DEN(D) then D ....

T Streicher. Correctness and Completeness of a Categorical Semantics of the Calculus of Constructions. PhD thesis, Universitat Passau, 1988.

....as discussed in [Pfenning 1991] are complete. A semantical reason for strengthening the equality is that in almost all models of type systems the j rule is valid. Further CR fij is essential if we relate the syntax of PTSs to the semantics: The systems that are really interpreted in e.g. [Streicher 1989] and [Jacobs 1991] are not the ones as formulated above (with equality on the pseudoterms as a side condition) but with an equality judgement inside the context. If i is a PTS, write i = for the associated semantical version of the system (with equality judgement) replacing by = The ....

Th. Streicher, Correctness and completeness of a categorical semantics of the Calculus of Constructions. Ph. D. thesis, University of Passau, Germany.

....is partial. Partiality is introduced in the interpretation of application. But the interpretation of a derivable judgement will be defined and true. The method with a partial interpretation function has also been used by Streicher for a categorical interpretation of the calculus of constructions [15]. Interpretation of set expressions: Pix : A 0 :A 1 [x] ae = Y u2[ A 0 ] ae [ A 1 [x] ae u x : This is defined iff [ A 0 ] ae is defined and [ A 1 [x] ae u x is defined whenever u 2 [ A 0 ] ae. Interpretation of element expressions: x] ae = ae(x) This is always defined. x : ....

....This is defined iff [ a 1 ] ae and [ a 0 ] ae are defined, and [ a 1 ] ae is a function the domain of which contains [ a 0 ] ae. Observe that it is possible to interpret polymorphic application in set theory. This is not the case for interpretations of type theory in general, compare Streicher [15]. Interpretation of context expressions: ffl] f;g: This is always defined. Gamma; x : A] fae u x jae 2 [ Gamma] u 2 [ A] aeg: This is defined iff Gamma is defined and [ A] ae is defined whenever ae 2 [ Gamma] Interpretation of judgement expressions: Gamma context ] iff ....

T. Streicher. Correctness and Completeness of a Categorical Semantics of the Calculus of Constructions. PhD thesis, Fakultat fur Mathematik und Informatik, Universitat Passau, 1988.

....Several presentations are used in the literature, each of which serves a specific purpose. For example, the labelled syntax 1 These proofs are concerned with a different syntax but may be adapted to that of this paper. we consider is best suited to give a semantics of type systems 2 (see [2, 19, 22]) while the standard syntax is best suited for proof checking (see [11, 18] Our work establishes the equivalence between the two presentations for a large class of systems. Contents of the paper and prerequisites In Section 2, we introduce the standard and labelled syntaxes of algebraic type ....

Th. Streicher. Correctness and Completeness of a Categorical Semantics of the Calculus of Constructions. PhD thesis, Univ. Passau, 1989. Appeared as technical report MIP - 8913.

....Secondly, we introduce a notion of canonical derivation in order to give a good inductive definition of the interpretation. As pointed out by Coquand [Coq89] it is a nice meta theoretic property for a calculus that a judgement have at most one derivation (up to conversion) The work by Streicher [Str88] shows that this property is very helpful, and sometimes necessary, to define a semantics by induction on derivations. When a calculus lacks such a property, a definition of semantics may assign different denotations to the same judgement. Streicher [Str88] gives a way of solving this problem, ....

....(up to conversion) The work by Streicher [Str88] shows that this property is very helpful, and sometimes necessary, to define a semantics by induction on derivations. When a calculus lacks such a property, a definition of semantics may assign different denotations to the same judgement. Streicher [Str88] gives a way of solving this problem, first defining denotations by induction on the structure of pre judgements (instead of on derivations) and then proving that the definition gives a unique denotation to every derivable judgement. It is obvious that our presentation of ECC does not have the ....

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T. Streicher, Correctness and Completeness of a Categorical Semantics of the Calculus of Constructions, PhD Dissertation, Passau.

....can not be understood as arbitrary sets. The intuitionistic set theoretic model for polymorphic calculus can be extended to CC; this was considered by many people including [HPit87] Ehr88] Luo88a] who extended the model of partial equivalence relations for second order calculus to CC, and [Str88] who considered model construction for CC in Cartmell s framework of contextual categories [Car78,86] CC is a very strong functional system. As Girard pointed out, any further attempt to extend the calculus must be very cautious [Gir86] Adding another impredicative level to the calculus would ....

.... [BLam84] and [Mac86] The idea of lifting propositions (in the impredicative universes of Constructions) as higher level types, in order to use Sigma types to express abstract structures and mathematical theories, was investigated in [Luo88a] Luo89a,b] Recently, Coquand [Coq89] and Streicher [Str88] considered using an explicit lifting operator to lift propositions, and view the calculus with type inclusions as 9 In the presentation of GCC in [Coq86a] page 235) the rules stating Type j : Type j 1 were inadvertently missing. INTRODUCTION 13 an abbreviation [Coq89] There are several ....

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T. Streicher, Correctness and Completeness of a Categorical Semantics of the Calculus of Constructions, PhD Dissertation, Passau.

....derivation for any typing judgement. Therefore, in such calculi, giving meaning to derivations is the same as giving meaning to derivable judgements. But for other calculi, such as Martin Lof s Intuitionistic Type Theory (ITT) Mar84] see [Sal88] and the Calculus of Constructions [CH88] see [Str88]) and of immediate concern to us Cardelli and Wegner s Fun, this is not so, and one must prove that derivations yielding the same judgement are given the same meaning. This idea has also appeared in the context of category theory and our use of the term coherence is partially inspired by ....

T. Streicher. Correctness and completeness of a categorical semantics of the Calculus of Constructions. PhD thesis, Passau University, 1988.

....that fij reduction is normalizing and in both cases the proof is given for a large collection of Pure Type Systems. Strong normalization of fij reduction for CC with fij conversion can be proved by adapting the proof for the fi case in [Geuvers and Nederhof 1991] In [Coquand 1990] and in [Streicher 1988] the syntax is built up more explicitly using a T operator. This is done for semantical reasons (the latter therefore discusses even more explicit versions of the calculus) to be better able to describe the interpretation of the syntax in the model. The idea is to view Prop as a special base type ....

T. Streicher, Correctness and completeness of a categorical semantics of the calculus of constructions, Ph.D. Thesis, Passau University, Germany.

....important, application is to contribute to a better understanding of the various presentations of type systems. Several presentations are used in the literature, each of which serves a specific purpose. For example, the labelled syntax we consider is best suited to give a semantics of type systems[2, 25, 27] while the standard syntax is best suited for programming and proof checking. Our work establishes the equivalence between the two presentations for a large class of systems. Contents of the paper In Section 2, we introduce the standard and labelled syntaxes of algebraic type systems. The Subject ....

Th. Streicher. Correctness and Completeness of a Categorical Semantics of the Calculus of Constructions. PhD thesis, University of Passau, 1989. Appeared as technical report MIP - 8913.

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Th. Streicher. Correctness and Completeness of a Categorical Semantics of the Calculus of Constructions. PhD thesis, Univ. Passau, 1989. Appeared as technical report MIP - 8913.

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Thomas Streicher. Correctness and Completeness of a Categorical Semantics of the Calculus of Constructions. PhD thesis, Universitat Passau, 1988.

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