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15
Syntax and Semantics of Dependent Types
 Semantics and Logics of Computation
, 1997
"... ion is written as [x: oe]M instead of x: oe:M and application is written M(N) instead of App [x:oe] (M; N ). 1 Iterated abstractions and applications are written [x 1 : oe 1 ; : : : ; x n : oe n ]M and M(N 1 ; : : : ; N n ), respectively. The lacking type information can be inferred. The universe ..."
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Cited by 40 (4 self)
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ion is written as [x: oe]M instead of x: oe:M and application is written M(N) instead of App [x:oe] (M; N ). 1 Iterated abstractions and applications are written [x 1 : oe 1 ; : : : ; x n : oe n ]M and M(N 1 ; : : : ; N n ), respectively. The lacking type information can be inferred. The universe is written Set instead of U . The Eloperator is omitted. For example the \Pitype is described by the following constant and equality declarations (understood in every valid context): ` \Pi : (oe: Set; : (oe)Set)Set ` App : (oe: Set; : (oe)Set; m: \Pi(oe; ); n: oe) (m) ` : (oe: Set; : (oe)Set; m: (x: oe) (x))\Pi(oe; ) oe: Set; : (oe)Set; m: (x: oe) (x); n: oe ` App(oe; ; (oe; ; m); n) = m(n) Notice, how terms with free variables are represented as framework abstractions (in the type of ) and how substitution is represented as framework application (in the type of App and in the equation). In this way the burden of dealing correctly with variables, substitution, and binding is s...
The Groupoid Interpretation of Type Theory
 In Venice Festschrift
, 1996
"... ion and application Suppose that M 2 Tm(B). We define its abstraction A;B (M) 2 Tm(\Pi LF (A; B)) on objects by A;B (M)(fl)(a) = M(fl; a) A;B (M)(fl)(q) = M(id fl ; q) If p : fl ! fl 0 then we need a natural transformation A;B (M)(p) : p \Delta A;B (M)(fl) ! A;B (M)(fl 0 ) At object a ..."
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ion and application Suppose that M 2 Tm(B). We define its abstraction A;B (M) 2 Tm(\Pi LF (A; B)) on objects by A;B (M)(fl)(a) = M(fl; a) A;B (M)(fl)(q) = M(id fl ; q) If p : fl ! fl 0 then we need a natural transformation A;B (M)(p) : p \Delta A;B (M)(fl) ! A;B (M)(fl 0 ) At object a 2 A(fl 0 ) it is given by M(p; id a ). Conversely, if M 2 Tm(\Pi(A; B)) we define a dependent object \Gamma1 A;B 2 Tm(B). Its object part is given by \Gamma1 A;B (M)(fl; a) = M(fl)(a) For the morphism part assume p : fl ! fl 0 and q : p \Delta a ! a 0 . We define \Gamma1 A;B (M)(p; q) = M(fl 0 )(q) ffi (id fl 0 ; q) \Delta M(p) p \Delta a We claim that \Gamma1 A;B (M)(p; q) : (p; q) \Delta \Gamma1 A;B (M)(fl; a) ! \Gamma1 A;B (M)(fl 0 ; a 0 ) as required. To see this, first note that M(fl 0 )(q) : (id fl 0 ; q) \Delta M 0 (fl 0 )(p \Delta a) !M 0 (fl 0 )(a 0 ) because q : p \Delta a ! a 0 . On the other hand M(p) : p \Delta M(fl) !M(fl 0 )...
The Gentle Art of Levitation
"... We present a closed dependent type theory whose inductive types are given not by a scheme for generative declarations, but by encoding in a universe. Each inductive datatype arises by interpreting its description—a firstclass value in a datatype of descriptions. Moreover, the latter itself has a de ..."
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We present a closed dependent type theory whose inductive types are given not by a scheme for generative declarations, but by encoding in a universe. Each inductive datatype arises by interpreting its description—a firstclass value in a datatype of descriptions. Moreover, the latter itself has a description. Datatypegeneric programming thus becomes ordinary programming. We show some of the resulting generic operations and deploy them in particular, useful ways on the datatype of datatype descriptions itself. Surprisingly this apparently selfsupporting setup is achievable without paradox or infinite regress. 1.
Recursion over Objects of Functional Type
, 1999
"... This paper presents an extension of the simpletyped lambdacalculus allowing iteration and case... ..."
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Cited by 14 (0 self)
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This paper presents an extension of the simpletyped lambdacalculus allowing iteration and case...
A Modal Lambda Calculus with Iteration and Case Constructs
 TYPES FOR PROOFS AND PROGRAMS: INTERNATIONAL WORKSHOP, TYPES ’98, KLOSTER IRSEE
, 1997
"... An extension of the simplytyped lambdacalculus allowing iteration and case reasoning over terms defined by means of higher order abstract syntax has recently been introduced by Joëlle Despeyroux, Frank Pfenning and Carsten Schürmann. This thorny mixing is achieved thanks to the help of the operato ..."
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Cited by 8 (1 self)
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An extension of the simplytyped lambdacalculus allowing iteration and case reasoning over terms defined by means of higher order abstract syntax has recently been introduced by Joëlle Despeyroux, Frank Pfenning and Carsten Schürmann. This thorny mixing is achieved thanks to the help of the operator ` ' of modal logic IS4. Here we give a new presentation of their system, with reduction rules, instead of evaluation judgments, that compute the canonical forms of terms. Our presentation is based on a modal lambdacalculus that is better from the user's point of view, is more concise and we do not impose a particular strategy of reduction during the computation. Our system enjoys the decidability of typability, soundness of typed reduction with respect to typing rules, the ChurchRosser and strong normalization properties. Finally it is a conservative extension of the simplytyped lambdacalculus.
Specifying Interactions With Dependent Types
 In Workshop on subtyping and dependent types in programming
, 2000
"... this paper we consider how to express specications of interactions in dependent type theory. The results so far are modest, though we hope we have identied some key structures for describing contracts between independent agents, and shown how to dene them in a dependently typed framework. These are ..."
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this paper we consider how to express specications of interactions in dependent type theory. The results so far are modest, though we hope we have identied some key structures for describing contracts between independent agents, and shown how to dene them in a dependently typed framework. These are called below transition systems (2.2) and interaction systems (2.3). Both are coalgebras; transition systems for a functor Fam , and interaction systems for its composite with itself, Fam (Fam ). These structures seems to have interesting connections with predicate transformer semantics for imperative programs, as initiated by Dijkstra, and also with the renement calculus of Back and von Wright as described in their book [2]. We restrict attention to situations in which the system and its environment communicate by exchanging messages in strict alternation, as with the moves in a two player game.
Ordinals and Interactive Programs
, 2000
"... The work reported in this thesis arises from the old idea, going back to the origins of constructive logic, that a proof is fundamentally a kind of program. If proofs can be ..."
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Cited by 5 (2 self)
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The work reported in this thesis arises from the old idea, going back to the origins of constructive logic, that a proof is fundamentally a kind of program. If proofs can be
State Dependent IOMonads in Type Theory
, 2004
"... We introduce the notion of state dependent interactive programs for MartinLof Type Theory. These programs are elements of coalgebras of an endofunctor on the presheaf Set. We prove the existence of final coalgebras for these functors. This shows as well the consistency of type theory plus rules e ..."
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We introduce the notion of state dependent interactive programs for MartinLof Type Theory. These programs are elements of coalgebras of an endofunctor on the presheaf Set. We prove the existence of final coalgebras for these functors. This shows as well the consistency of type theory plus rules expressing the existence of weakly final coalgebras for these functors, which represents the type of interactive programs. We define in this type theory the bisimulation relation, and give some simple examples for interactive programs. A generalised monad operation is defined by corecursion on interactive programs with return value, and a generalised version of the monad laws for this operation is proved. All results have been verified in the theorem prover Agda which is based on intensional type theory.
Reductionfree normalisation for system F
, 1996
"... We present a semantical proof of existence of normal forms for system F including jequality. A reductionfree normalisation function can be obtained from this. The proof uses the method of glueing (a variant of) the term model along the global sections functor, carried out in the internal language ..."
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We present a semantical proof of existence of normal forms for system F including jequality. A reductionfree normalisation function can be obtained from this. The proof uses the method of glueing (a variant of) the term model along the global sections functor, carried out in the internal language of a category of presheaves. As a byproduct we obtain an semantical explanation of higherorder abstract syntax. The paper extends a previous one (Altenkirch, Hofmann, and Streicher 1996) in which a combinatory version of system F has been treated. 1 Introduction In this paper we give a semantical proof of reductionfree normalisation for F fij , a version of Girard's system F with full fijequality for both kinds of abstraction. This generalises the semantical normalisation algorithms for simplytyped systems (Berger and Schwichtenberg 1991; Coquand and Dybjer 1996; Altenkirch, Hofmann, and Streicher 1995) to polymorphism. As in those approaches we do not prove strong normalisation but co...
A Correspondence between MartinLöf Type Theory, the Ramified Theory of Types and Pure Type Systems
 Journal of Logic, Language and Information
, 2001
"... In Russell's Ramified Theory of Types rtt, two hierarchical concepts dominate: orders and types. The use of orders has as a consequence that the logic part of rtt is predicative. The concept of order however, is almost dead since Ramsey eliminated it from rtt. This is why we find Church's simple the ..."
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Cited by 3 (1 self)
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In Russell's Ramified Theory of Types rtt, two hierarchical concepts dominate: orders and types. The use of orders has as a consequence that the logic part of rtt is predicative. The concept of order however, is almost dead since Ramsey eliminated it from rtt. This is why we find Church's simple theory of types (which uses the type concept without the order one) at the bottom of the Barendregt Cube rather than rtt. Despite the disappearance of orders which have a strong correlation with predicativity, predicative logic still plays an influential role in Computer Science. An important example is the proof checker Nuprl, which is based on MartinLöf's Type Theory which uses type universes. Those type universes, and also degrees of expressions in Automath, are closely related to orders. In this paper, we show that orders have not disappeared from modern logic and computer science, rather, orders play a crucial role in understanding the hierarchy of modern systems. In order to achieve our goal, we concentrate on a subsystem of Nuprl. The novelty of our paper lies in: 1) a modest revival of Russell's orders, 1 2) the placing of the historical system rtt underlying the famous Principia Mathematica in a context with a modern system of computer mathematics (Nuprl) and modern type theories (MartinLöf's type theory and PTSs), and 3) the presentation of a complex type system (Nuprl) as a simple and compact PTS.