Wil Dekkers

Systems of illative combinatory logic complete for first-order propositional and predicate calculus

Illative combinatory logic consists of the theory of combinators or lambda calculus extended by extra constants (and corresponding axioms and rules) intended to capture inference. The paper considers systems of illative combinatory logic that are sound for first-order propositional and predicate calculus. The interpretation from ordinary logic into the illative systems can be done in two ways: following the propositions-as-types paradigm, in which derivations become combinators or, in a more direct way, in which derivations are not translated. Both translations are closely related in a canonical way. The two direct translations turn out to be complete. The paper fulfills the program of Church [1932], [1933] and Curry [1930] to base logic on a consistent system of A-terms or combinators. Hitherto this program had failed because systems of ICL were either too weak (to provide a sound interpretation) or too strong (sometimes even inconsistent). ?

Completeness of the propositions-as-types interpretation of intuitionistic logic into illative combinatory logic

Illative combinatory logic consists of the theory of combinators or lambda calculus extended by extra constants (and corresponding axioms and rules) intended to capture inference. In a preceding paper, [2], we considered 4 systems of illative combinatory logic that are sound for first order intuitionistic propositional and predicate logic. The interpretation from ordinary logic into the illative systems can be done in two ways: following the propositions-as-types paradigm, in which derivations become combinators, or in a more direct way, in which derivations are not translated. Both translations are closely related in a canonical way. In the cited paper we proved completeness of the two direct translations. In the present paper we prove that also the two indirect translations are complete. These proofs are direct whereas in another version, [3], we proved completeness by showing that the two corresponding illative systems are conservative over the two systems for the direct translations. Moreover we shall prove that one of the systems is also complete for predicate calculus with higher type functions. ?

Completeness of two systems of illative combinatory logic for first-order propositional and predicate calculus

Abstract. Illative combinatory logic consists of the theory of combinators or lambda calculus extended by extra constants (and corresponding axioms and rules) intended to capture inference. The paper considers 4 systems of illative combinatory logic that are sound for first-order propositional and predicate calculus. The interpretation from ordinary logic into the illative systems can be done in two ways: following the propositions-as-types paradigm, in which derivations become combinators, or in a more direct way, in which derivations are not translated. Both translations are closely related in a canonical way. In a preceding paper, Barendregt, Bunder and Dekkers, 1993, we proved completeness of the two direct translations. In the present paper we prove completeness of the two indirect translations by showing that the corresponding illative systems are conservative over the two systems for the direct translations. In another version, DBB (1997), we shall give a more direct completeness proof. These papers fulfill the program of Church and Curry to base logic on a consistent system of

Pure type systems with more liberal rules

\lambda

Inhabitation of types in the simply typed lambda calculus

-terms or combinators. Hitherto this program had failed because systems of ICL were either too weak (to provide a sound interpretation) or too strong (sometimes even inconsistent).

Reducibility of types in typed Lambda calculus

Pure Type Systems. PTSs, introduced as a generalisation of the type systems of Barendregts lambda-cube, provide a foundation for actual proof assistants, aiming at the mechanic verification of formal proofs. In this paper we consider simplifications of some of the rules of PTSs. This is of independent interest for PTSs as this produces more flexible PTS-like systems, but it will also help, in a later paper, to bridge the gap between PTSs and systems of Illative Combinatory Logic. First we consider a simplification of the start and weakening rules of PTSs. which allows contexts to be sets of statements, and a generalisation of the conversion rule. The resulting Set-modified PTSs or SPTSs, though essentially equivalent to PTSs, are closer to standard logical systems. A simplification of the abstraction rule results in Abstraction-modified PTSs or APTSs. These turn out to be equivalent to standard PTSs if and only if a condition (*) holds. Finally we consider SAPTSs which have both modifications.

Simple Types λA

Abstract In the 1960s, mathematicians from Eastern Europe showed that every admissible rule in the intuitionistic implicational calculus is derivable. We shall prove the corresponding proposition for the simply typed lambda calculus: The type μ1 → μ2 → · · · μm → v is inhabited iff for each substitution s of types for the variables in μ1, ...,μm, v the inhabitation of s(μ1),..., s(μm) implies the inhabitation of s(v).

Lambda calculus with types

Abstract Consider types built up from a base type 0 using the operation →. A type σ is reducible to a type τ, notation σ ≤ τ, iff there exists a closed term M in σ → τ such that for all closed N 1 , N 2 in α we have N 1 = βη N 2 ⇔ MN 1 = βη MN 2 . Two types are equivalent iff each is reducible to the other. In ( Statman, 1980 , in “To H. B. Curry: Essays on Combinatory Logic, Lambda Calculus and Formalism” ( J. P. Seldin and J. R. Hindley, Eds. ), pp. 511–534, Academic Press, New York/London) is shown that the equivalence classes of types are well ordered in type ω + 2 or ω + 3. The paper does not decide if it is ω + 2 or ω + 3 because it is not clear whether μ ≡ (((0 → 0) → 0) → 0) → 0 → 0 and ν ≡ (0 → 0) → (0 → 0) → 0 → 0 are equivalent. We show that μ and ν are not equivalent and conclude that the equivalence classes are ordered in type ω + 3.

Typed lambda calculus

Definition 3.0.7 (Redexes) The redexes in L 0 and their reducts are defined below. • cf (v) is a redex, and each defined value of cf (v) is a reduct of cf (v). • if (true, e 1 , e 2) is a redex, and its reduct is e 1. • if (false, e 1 , e 2) is a redex, and its reduct is e 2. • fst(v 1 , v 2) is a redex, and its reduct is v 1. • snd(v 1 , v 2) is a redex, and its reduct is v 2. • (λx.e)v is a redex, and its reduct is e[x := v]. • fix f.e is a redex, and its reduct is e[f := fix f.e]. For instance, if we assume that + represents the usual addition function on integers, then 1 + 2 is a redex and 3 is the only reduct of 1 + 2. More interestingly, we may also assume the existence of a nullary constant function random such that random () is a redex and every natural number is a reduct of random ().