Equivalences between Pure Type Systems and Systems of Illative Combinatory Logic

Bunder, Martin W.; Dekkers, W.J.M.

doi:10.1305/ndjfl/1117755149

Cited by 5 publications

(6 citation statements)

References 14 publications

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“…The translation [ ] of Bunder and Dekkers [3] translates the pseudoterms and statements of PTSs into terms of illative combinatory logic (ICL) as follows: ∈ F V (XY )) where G = λxyz.Ξx(Syz) (S is the combinator equivalent to λxyz.xz(yz)). Terms in ICL can be represented without any free variables at all using the combinators S and K (equivalent to λxy.x).…”

Section: Definition 11 (Inhabited and Normal Form Inhabited Sorts)mentioning

confidence: 99%

“…If M and N are pseudoterms, M : A is a statement, Γ is a context if it is a sequence of statements; Γ ⊢ M : A is then called a judgement. A PTS has a set of axioms A each of the form c : s where c ∈ C and s ∈ S. Then it has a set R of triples (s 1 , s 2 , s 3 ) ∈ S 3 , which determine under what conditions a term Πx:A.B is in a sort. Most PTSs are known by a "specification" (S, A, R) (as usually C = S).…”

Section: Pure Type Systemsmentioning

confidence: 99%

“…The translation [ ] of Bunder and Dekkers [3] translates the pseudoterms and statements of PTSs into terms of illative combinatory logic (ICL) as follows:…”

Section: Pure Type Systemsmentioning

confidence: 99%

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Are there Hilbert-style Pure Type Systems?

Bunder¹,

Dekkers²

2008

Logical Methods in Computer Science

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Abstract. For many a natural deduction style logic there is a Hilbert-style logic that is equivalent to it in that it has the same theorems (i.e. valid judgements Γ ⊢ P where Γ = ∅). For intuitionistic implicational logic, the axioms of the equivalent Hilbert-style logic can be propositions which are also known as the types of the combinators I, K and S.Natural deduction versions of illative combinatory logics have formulations with axioms that are actual type statements for I, K and S. As pure type systems (PTSs) are, in a sense, equivalent to systems of illative combinatory logic, it might be thought that Hilbert style PTSs (HPTSs) could be based in a similar way. This paper shows that some PTSs have very trivial equivalent HPTSs, with only the axioms as theorems and that for many PTSs no equivalent HPTSs can exist. Most commonly used PTSs belong to these two classes.For some PTSs however, including λ * and the PTS at the basis of the proof assistant Coq, there is a nontrivial equivalent HPTS, with axioms that are type statements for I, K and S.

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Section: Definition 11 (Inhabited and Normal Form Inhabited Sorts)mentioning

confidence: 99%

Section: Pure Type Systemsmentioning

confidence: 99%

See 1 more Smart Citation

Are there Hilbert-style Pure Type Systems?

Bunder¹,

Dekkers²

2008

Logical Methods in Computer Science

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“…A comparison of the definition of a PTS to the definitions of the original versions of the calculus introduced by Coquand shows that the latter is a special case of the former. More recently, Bunder and Dekkers have been studying variants of PTSs for the purpose of comparing them with systems of illative combinatory logic [20] (see also [21] and [22]), and as a result the exact differences between these different formulations now seem more important. The purpose of this paper is to study these different formulations of the calculus of constructions and to compare them.…”

Section: Introductionmentioning

confidence: 99%

Variants of the basic calculus of constructions

Bunder

Seldin

2004

Journal of Applied Logic

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In this paper, a number of different versions of the basic calculus of constructions that have appeared in the literature are compared and the exact relationships between them are determined. The biggest differences between versions are those between the original version of Coquand and the version in early papers on the subject by Seldin. None of these results is very deep, but it seems useful to collect them in one place.

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“…A comparison of the definition of a PTS to the definitions of the original versions of the calculus introduced by Coquand shows that the latter is a special case of the former. More recently, Bunder and Dekkers have been studying variants of PTSs for the purpose of comparing them with systems of illative combinatory logic [3], and as a result the exact differences between these different formulations now seem more important. The purpose of this paper is to study these different formulations of the calculus of constructions and to compare them.…”

mentioning

confidence: 99%

Editorial

Kamareddine¹

2004

Journal of Applied Logic

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EditorialLogics, Types and Rewriting have existed for centuries (cf. [9]). The 20th century however, has given birth to new theories of logic, types and rewriting which played and continue to play a major role in all aspects of computation. The more and more computer technology dominates our lives, the more logic, types and rewriting become fundamental to this technology. In particular, methods of how best to combine, represent and implement calculi of logic, types and rewriting become vital for making these calculi useful in practice. It is already known that the areas of logic, types and rewriting converge (cf.[9]). Heyting [7], Kolmogorov [11], Curry and Feys [5] (improved by Howard [8]), and de Bruijn [10], all observed the "propositions as types" or "proofs as terms" (PAT) correspondence. In PAT, logical operators are embedded in the types of λ-terms rather than in the propositions and λ-terms are viewed as proofs of the propositions represented by their types. Advantages of PAT include the ability to manipulate proofs, easier support for independent proof checking, the possibility of the extraction of computer programs from proofs, and the ability to prove properties of the logic via the termination of the rewriting system. However, many useful computational systems are not based on the PAT principle. This volume contains three papers on variants of logics, types and rewriting as follows:1570-8683/$ -see front matter 

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Equivalences between Pure Type Systems and Systems of Illative Combinatory Logic

Cited by 5 publications

References 14 publications

Are there Hilbert-style Pure Type Systems?

Are there Hilbert-style Pure Type Systems?

Variants of the basic calculus of constructions

Editorial

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