Amy Felty. Encoding dependent types in an intuitionistic logic. In Gerard Huet and Gordon D. Plotkin, editors, Logical Frameworks, pages 214-251. Cambridge University Press, 1991.  Home/Search   Document Details and Download   Summary   Related Articles   Check

....is worth recalling that the substitution and identity principles are are needed to ensure that the type theory makes sense, since the syntatic restrictions inherent in CLF make it impossible to generate proofs of A A or to compose proofs of A B and B C in any other way. With one exception [Fel91], prior presentations of LF and LLF have been based on a syntax in which not every term is canonical. A diculty is that equality cannot then be axiomatized in a manifestly decidable, syntax directed way. In their original presentation of LF, Harper et al. de ne equality in terms of ....

....into canonical form is incremental and can be aborted as soon as it is evident that the canonical forms of the two terms being compared will not be the same, an important concern for ecient implementation. Felty has described a canonical LF in which only canonical forms are well typed [Fel91]. This o ers a number of advantages over other approaches: equality itself need not be axiomatized at all, because terms are equal just when they are identical (up to equivalence) And the representation methodology has an attractive simplicity: one establishes a compositional bijection between ....

Amy Felty. Encoding dependent types in an intuitionistic logic. In Gerard Huet and Gordon D. Plotkin, editors, Logical Frameworks, pages 214-251. Cambridge University Press, 1991.

....with CLF, but is not a type theory and does not identify the logical connectives inherited from lax logic and linear logic as we do here. The method of de ning a type theory by a typed operational semantics goes back to the Automath languages [dB93] and has been applied to LF by Felty [Fel91]. Our canonical formulation signi cantly extends and streamlines the ideas behind Felty s canonical LF and its extension to LLF [CP98] the need for con uence and normalization results is eliminated. 5 Conclusion In this paper, we have presented the basic design of a logical framework that ....

Amy Felty. Encoding dependent types in an intuitionistic logic. In Gerard Huet and Gordon D. Plotkin, editors, Logical Frameworks, pages 214-251. Cambridge University Press, 1991.

....user declarations of appropriate constructors. Polymorphic type variables are avoided in the COL because, as will be seen, the translation between meta level and object level lambda terms is by induction on the structure of types. Type variables complicate this induction. However, previous work [6, 9] do show that it should be possible to extend COL to a limited, dependently typed language, namely LF. However, simple types already suce for many languages to interoperate and it is unclear if the dependent type enrichment will be meaningful in practice. Furthermore, it is possible to embed LF ....

....Tm at COL type A c . The parameter l represents an embedding level, which is needed to calculate de Bruijn indices. Quanti ers of the (intuitionistic) higher order meta logic have di erent meanings depending if they are on the left or right hand sides of implications ( Following the style of [6], the relation is labeled positive ( if it appears on the left of an even number of implications and negative ( if appears on the left of an odd number of implications. In a logic programming context, positive clauses correspond to de nite clauses while negative ones correspond to goals. We ....

A. Felty. Encoding dependent types in an intuitionistic logic. In G. Huet and G. Plotkin, editors, Logical Frameworks. Cambridge University Press, 1991.

....worth recalling that the substitution and identity principles are are needed to ensure that the type theory makes sense, since the syntatic restrictions inherent in CLF make it impossible to generate proofs of A A or to compose proofs of A B and B C in any other way. With one exception [Fel91], prior presentations of LF and LLF have been based on a syntax in which not every term is canonical. A di#culty is that equality cannot then be axiomatized in a manifestly decidable, syntax directed way. In their original presentation of LF, Harper et al. define equality in terms of ....

....into canonical form is incremental and can be aborted as soon as it is evident that the canonical forms of the two terms being compared will not be the same, an important concern for e#cient implementation. Felty has described a canonical LF in which only canonical forms are well typed [Fel91]. This o#ers a number of advantages over other approaches: equality itself need not be axiomatized at all, because terms are equal just when they are identical (up to # equivalence) And the representation methodology has an attractive simplicity: one establishes a compositional bijection between ....

Amy Felty. Encoding dependent types in an intuitionistic logic. In Gerard Huet and Gordon D. Plotkin, editors, Logical Frameworks, pages 214--251. Cambridge University Press, 1991.

....cant problems. The original presentation of LF by Harper, Honsell and Plotkin [5] hereafter HHP ) avoided the diculties of reduction by using conversion as de nitional equality even though this destroyed the property that every term is equal to some canonical form. Felty s Canonical LF [4] is a version of LF where all well typed objects and families are in canonical form, avoiding all issues of de nitional equality. Felty showed that Canonical LF is essentially the same as full LF if typing derivations are restricted to pre canonical terms (those whose normal forms are ....

Amy Felty. Encoding dependent types in intuitionistic logic. In Gerard Huet and Gordon D. Plotkin, editors, Logical Frameworks, pages 214-251. Cambridge University Press, 1991.

....with CLF, but is not a type theory and does not identify the logical connectives inherited from lax logic and linear logic as we do here. The method of defining a type theory by a typed operational semantics goes back to the Automath languages [dB93] and has been applied to LF by Felty [Fel91]. Our canonical formulation significantly extends and streamlines the ideas behind Felty s canonical LF and its extension to LLF [CP98] the need for confluence and # normalization results is eliminated. 5 Conclusion In this paper, we have presented the basic design of a logical framework that ....

Amy Felty. Encoding dependent types in an intuitionistic logic. In Gerard Huet and Gordon D. Plotkin, editors, Logical Frameworks, pages 214--251. Cambridge University Press, 1991.

....signi cant problems. The original presentation of LF by Harper, Honsell and Plotkin [5] hereafter HHP ) avoided the diculties of reduction by using conversion as de nitional equality even though this destroyed the property that every term is equal to some canonical form. Felty s Canonical LF [4] is a version of LF where all well typed objects and families are in canonical form, avoiding all issues of de nitional equality. Felty showed that Canonical LF is essentially the same as full LF if typing derivations are restricted to pre canonical terms (those whose normal forms are canonical) ....

Amy Felty. Encoding dependent types in intuitionistic logic. In Gerard Huet and Gordon D. Plotkin, editors, Logical Frameworks, pages 214-251. Cambridge University Press, 1991. 41

....to be fully realized. Only valid proofs can be typed (by the formulas they prove) For example (replacing all and imp with 8 and ) all i x: p x ) p x) x: imp i (p x) p x) y:y) has type true 8x(p x ) p x) Prolog can be used to implement dependent type checking for this signature. In [5] it was shown that (pre canonical) LF specifications have natural encodings in Prolog via the formulas as types principle. Essentially, dependent type expressions are translated into terms (i.e. meta level data structures) of simple types in Prolog. The relevant LF type inferencing rules are ....

.... inferencing rules are given here for reference (x can not appear free in N or A) 4 Gamma M : Pix : A:B Gamma N : A Gamma MN : N=x]B Gamma; x : A M : B Gamma x : A:M : Pix : A:B Prolog implementation of dependent type checking for proof terms of first order logic (from [5]) typing (impi A B M) true (imp A B) typing A bool, typing B bool, pi x (typing x (true A) typing (M x) true B) typing (impe A B P Q) true B) typing A bool, typing B bool, typing P (true (imp A B) typing Q (true A) typing (alle B T P) true (B T) typing T i, ....

[Article contains additional citation context not shown here]

A Felty. Encoding Dependent Types in an Intuitionistic Logic. In G. Huet and G. Plotkin editors, Logical Frameworks, pages 214-252. Cambridge University Press, 1991.

....valid proofs can be typed (by the formulas they prove) For example (replacing all and imp with 8 and ) all i x: p x ) p x) x: imp i (p x) p x) y:y) has type (true 8x(p x ) p x) 3.1.2 Logic programming and LF Prolog can be used to implement dependent type checking for this signature. In [5] it was shown that (canonical) LF specifications have natural encodings in Prolog via the formulas as types principle. Essentially, dependent type expressions are translated into terms (i.e. meta level data structures) of simple types in Prolog (we will formalize this encoding of LF expressions ....

....dependent type expressions are translated into terms (i.e. meta level data structures) of simple types in Prolog (we will formalize this encoding of LF expressions to simply typed terms in section 3. 2) Prolog implementation of dependent type checking for proof terms of first order logic (from [5]) typing (impi A B M) true (imp A B) typing A bool, typing B bool, pi x (typing x (true A) typing (M x) true B) typing (impe A B P Q) true B) typing A bool, typing B bool, 49 typing P (true (imp A B) typing Q (true A) typing (alle B T P) true (B T) typing T ....

[Article contains additional citation context not shown here]

Amy Felty. Encoding dependent types in an intuitionistic logic. In G'erard Huet and Gordon D. Plotkin, editors, Logical Frameworks. Cambridge University Press, 1991.

....[4] There are two ways of giving LF specifications a computational interpretation. The typeinferencing rules of LF can be seen as a proof search mechanism, and can thus be given an operational meaning directly. In [10] Pfenning defined the programming language Elf based on this approach. In [3], however, Felty demonstrated how LF specifications can be translated into formulas of intuitionistic logic. These translated specifications can then be executed as logic programs. My formulation of LF substitutions follows the latter approach. Since terms can appear inside types, substitution on ....

Amy Felty. Encoding dependent types in an intuitionistic logic. In G'erard Huet and Gordon D. Plotkin, editors, Logical Frameworks. Cambridge University Press, 1991.

....typing judgements DTT have increased information content; an inhabitation judgement, for example, can bear the information that a term meets a partial specification. The translation of DTT into HOL uses the more or less familiar idea of sending a dependent type to a predicate. It is also used in [2] and on a more abstract level in section 4.3.5 of [8] the translation is extracted from the interpretation of DTT in a topos. A detailed description of this translation forms the basis for an actual implementation, which is briefly described in section 6. Our practical experience with the ....

.... Gamma a : A where it is understood that all free variables of a occur in Gamma. Finally, one wants judgements about equality of terms. These take the form Gamma a = b : A for terms Gamma a : A and Gamma b : A. As a typical example one can state the following in DTT. n : Nat[2] n 4 = n 5 : Nat[2] Notice how much information is contained in this judgement. Indeed, the facility for compact expression of complex facts is what makes DTT so attractive. The above judgements are the four kinds of judgement distinguished by Martin Lof [11] Products and sums. The most ....

[Article contains additional citation context not shown here]

A. Felty, `Encoding Dependent Types in an Intuitionistic Logic', in Logical Frameworks , edited by G. Huet and G. Plotkin (Cambridge University Press, 1991), pp. 215--251.

....and signature are combined into one concept. We also adopt fij equivalence as definitional equality. This is a common practice despite difficulties involving j reduction and the Subject Reduction property. This paper also takes as technical starting points the works of Felty and Miller [7, 2]. Please consult [4] for a more extensive treatment, with proofs, of the subject matter. Operations such as unification, rewriting and type inferencing require the basic operation of substitution for free variables. With dependently typed expressions, this means simultaneously applying a ....

....in natural deduction and sequent calculi can be implemented in a relatively straightforward manner in such a language. Thus it is natural to formulate, in such a logic programming language, the object language of LF and its type assignment calculus. This was first accomplished by Felty in [2]. The idea of formulating substitutions as logic programming was first proposed by Miller in [7] 2 . Substitutions on simply typed terms respect the types of terms. One way of considering the application of a substitution as in A = B is the type checking of two terms (A and B) in parallel, ....

[Article contains additional citation context not shown here]

Amy Felty. Encoding dependent types in an intuitionistic logic. In G'erard Huet and Gordon D. Plotkin, editors, Logical Frameworks. Cambridge University Press, 1991.

....a direct extension of the current implementation techniques for LF [17] would require carrying types around in order to handle properly objects of type . To our knowledge, this is the first formulation of a type theory that focuses uniquely on long forms. It was inspired by Felty s canonical LF [6]. Traditional presentations of linear logic define the Context splitting=merging s dot Delta = Delta 1 Delta Psi = Psi 0 1 Psi 00 s lin1 ( Psi; x: A) Psi 0 ; x: A) 1 Psi 00 Psi = Psi 0 1 Psi 00 s int ( Psi; x : A) Psi 0 ; x : A) 1 ( Psi 00 ; x : A) Psi = ....

A. Felty. Encoding dependent types in an intuitionistic logic. In G. Huet and G. D. Plotkin, editors, Logical Frameworks, pages 214--251. Cambridge University Press, 1991.

....thesis, 32] copy clauses were extended to the LF dependent type system. The theorems and proofs of Chapter 2 were extended to dependent types. Copy clauses became four place predicates mutually defined on three levels (terms, types and kinds) This work was built upon similar efforts by Felty [13]. Formulating a notion of substitution that preserves the relationship between proof (as term) and formula (as type) can clearly be useful. It was hoped that these copy clauses can lead to program transformers and theorem proving by analogy techniques. To incorporate that work into the context of ....

Amy Felty. Encoding dependent types in an intuitionistic logic. In G'erard Huet and Gordon D. Plotkin, editors, Logical Frameworks. Cambridge University Press, 1991.

.... : M N if is a primitive type 8x( x; x : 1 ] Gamma oe [ Mx; Nx : 2 ] if is 1 2 These functions are a (corrected) version of those used by Miller [11] to specify equality and substitution for simply typed terms and are similar to those used by Felty [4] to code a dependent typed calculus in a higher order intuitionistic logic. The remaining rules of Figure 2 are specified in a straightforward manner, by including the following clauses at each primitive type. M P ) P N) oe (M N) M # P ) N # P ) oe (M N) M ....

Amy Felty. Encoding dependent types in an intuitionistic logic. In G'erard Huet and Gordon Plotkin, editors, Logical Frameworks, pages 215--251. Cambridge University Press, 1991.

....they also include much more. As a result of this mismatch, although the main ideas are rather simple, carrying out the full formalization of the correspondence between these two languages is more difficult than one might expect. The encoding presented here is an extension of an encoding in Felty [8] of the Logical Framework (LF) in a slightly less expressive logic than the one used here. Although LF types are more expressive than simple types, they are less expressive than CC types. Correctly handling the polymorphism of CC requires a significant extension over the LF encoding. In our ....

....given an arbitrary derivable assertion, there is not necessarily a canonical assertion with the same basic structure. 1 Because of the restricted form of canonical derivations, we lose some proofs. We can, in a sense, gain them back using a technique similar to that used for LF in Felty [8]. Given an arbitrary derivation of Gamma ff, we can define a function which reads off a series of context lemmas Delta. It is then possible to obtain a canonical derivation of the same basic structure of Gamma; Delta ff. In this paper, we will only consider canonical derivations. By ....

[Article contains additional citation context not shown here]

Amy Felty. Encoding dependent types in an intuitionistic logic. In G'erard Huet and Gordon Plotkin, editors, Logical Frameworks, pages 215-- 251. Cambridge University Press, 1991.

....item a of type , the following function defined by induction on the structure of yields the necessary congruence rule. a : a Gamma a if is a primitive type 8 1 x( x : 1 ] oe [ ax : 2 ] if is 1 2 (This function is similar to the one used by Felty [7] to code a dependent typed calculus in hh and by Miller [20] to specify equality and substitution for simply typed terms. To complete the specification, we introduce the predicate at each primitive type . The following clauses express the reflexive, symmetric, transitive closure ....

Amy Felty. Encoding dependent types in an intuitionistic logic. In G'erard Huet and Gordon Plotkin, editors, Logical Frameworks. Cambridge University Press, 1991.

....with dependent types have also been proposed as specification languages for representing a wide variety of logics. Examples include the AUTOMATH languages [5] type theories developed by Martin Lof [26] the Logical Framework (LF) 21] LF [15] and the Calculus of Constructions [4] In [11], we show that LF signatures can be encoded directly and naturally as formulas in the subset of hohh that does not allow predicate quantification. This encoding demonstrates a close correspondence between the two approaches. In addition, an encoded signature can serve as a set of tactics providing ....

Amy Felty. Encoding dependent types in an intuitionistic logic. In G'erard Huet and Gordon Plotkin, editors, Logical Frameworks, pages 215--251. Cambridge University Press, 1991.

No context found.

Amy Felty. Encoding dependent types in an intuitionistic logic. In Gerard Huet and Gordon D. Plotkin, editors, Logical Frameworks, pages 214-251. Cambridge University Press, 1991.

No context found.

Amy Felty. Encoding dependent types in an intuitionistic logic. In Gerard Huet and Gordon D. Plotkin, editors, Logical Frameworks, pages 214--251. Cambridge University Press, 1991.

No context found.

Amy Felty. Encoding dependent types in an intuitionistic logic. In Gerard Huet and Gordon D. Plotkin, editors, Logical Frameworks, pages 214--251. Cambridge University Press, 1991.

Online articles have much greater impact   More about CiteSeer.IST   Add search form to your site   Submit documents   Feedback  

CiteSeer.IST - Copyright Penn State and NEC