Theoretical Pearls:Representing ‘undefined’ in lambda calculus

1992 ◽  
Vol 2 (3) ◽  
pp. 367-374 ◽  
Author(s):  
Henk Barendregt
Keyword(s):  
Normal Form ◽  
Recursive Function ◽  
Lambda Calculus ◽  
Recursion Theory ◽  
Partial Recursive Function ◽  
Normalization Theorem ◽  
Image Position

AbstractLet ψ be a partial recursive function (of one argument) with λ-defining termF∈Λ°. This meansThere are several proposals for whatF⌜n⌝ should be in case ψ(n) is undefined: (1) a term without a normal form (Church); (2) an unsolvable term (Barendregt); (3) an easy term (Visser); (4) a term of order 0 (Statman).These four possibilities will be covered by one ‘master’ result of Statman which is based on the ‘Anti Diagonal Normalization Theorem’ of Visser (1980). That ingenious theorem about precomplete numerations of Ershov is a powerful tool with applications in recursion theory, metamathematics of arithmetic and lambda calculus.

Theoretical Pearls: Self-interpretation in lambda calculus

1991 ◽  
Vol 1 (2) ◽  
pp. 229-233 ◽  
Author(s):  
Henk Barendregt
Keyword(s):  
Normal Form ◽  
Programming Languages ◽  
Lambda Calculus ◽  
Direct Consequence ◽  
Functional Program ◽  
Computer Scientists ◽  
Strange Loop ◽  
Reduction Machine ◽  
Image Position ◽  
Computable Functions

Programming languages which are capable of interpreting themselves have been fascinating computer scientists. Indeed, if this is possible then a ‘strange loop’ (in the sense of Hofstadter, 1979) is involved. Nevertheless, the phenomenon is a direct consequence of the existence of universal languages. Indeed, if all computable functions can be captured by a language, then so can the particular job of interpreting the code of a program of that language. Self-interpretation will be shown here to be possible in lambda calculus.The set of λ-terms, notation Λ, is defined by the following abstract syntaxwhereis the set {v, v′, v″, v′″,…} of variables. Arbitrary variables are usually denoted by x, y,z,… and λ-terms by M,N,L,…. A redex is a λ-term of the formthat is, the result of substituting N for (the free occurrences of) x in M. Stylistically, it can be said that λ-terms represent functional programs including their input. A reduction machine executes such terms by trying to reduce them to normal form; that is, redexes are continuously replaced by their contracta until hopefully no more redexes are present. If such a normal form can be reached, then this is the output of the functional program; otherwise, the program diverges.

scholarly journals Realizability interpretation of PA by iterated limiting PCA

2014 ◽  
Vol 24 (6) ◽  
Author(s):  
YOHJI AKAMA
Keyword(s):  
Normal Form ◽  
Recursive Function ◽  
Inductive Limit ◽  
Partial Function ◽  
Partial Functions ◽  
Double Negation ◽  
Partial Recursive Function ◽  
Form Theorem ◽  
Normal Form Theorem

For any partial combinatory algebra (PCA for short) $\mathcal{A}$, the class of $\mathcal{A}$-representable partial functions from $\mathbb{N}$ to $\mathcal{A}$ quotiented by the filter of cofinite sets of $\mathbb{N}$ is a PCA such that the representable partial functions are exactly the limiting partial functions of $\mathcal{A}$-representable partial functions (Akama 2004). The n-times iteration of this construction results in a PCA that represents any n-iterated limiting partial recursive function, and the inductive limit of the PCAs over all n is a PCA that represents any arithmetical partial function. Kleene's realizability interpretation over the former PCA interprets the logical principles of double negation elimination for Σ0n-formulae, and over the latter PCA, it interprets Peano's arithmetic (PA for short). A hierarchy of logical systems between Heyting's arithmetic (HA for short) and PA is used to discuss the prenex normal form theorem, relativised independence-of-premise schemes, and the statement ‘PA is an unbounded extension of HA’.

J. C. Shepherdson. Algorithmic procedures, generalized Turing algorithms, and elementary recursion theory. Harvey Friedman's research on the foundations of mathematics, edited by L. A. Harrington, M. D. Morley, A. S̆c̆edrov, and S. G. Simpson, Studies in logic and the foundations of mathematics, vol. 117, North-Holland, Amsterdam, New York, and Oxford, 1985, pp. 285–308. - J. C. Shepherdson. Computational complexity of real functions. Harvey Friedman's research on the foundations of mathematics, edited by L. A. Harrington, M. D. Morley, A. S̆c̆edrov, and S. G. Simpson, Studies in logic and the foundations of mathematics, vol. 117, North-Holland, Amsterdam, New York, and Oxford, 1985, pp. 309–315. - A. J. Kfoury. The pebble game and logics of programs. Harvey Friedman's research on the foundations of mathematics, edited by L. A. Harrington, M. D. Morley, A. S̆c̆edrov, and S. G. Simpson, Studies in logic and the foundations of mathematics, vol. 117, North-Holland, Amsterdam, New York, and Oxford, 1985, pp. 317–329. - R. Statman. Equality between functionals revisited. Harvey Friedman's research on the foundations of mathematics, edited by L. A. Harrington, M. D. Morley, A. S̆c̆edrov, and S. G. Simpson, Studies in logic and the foundations of mathematics, vol. 117, North-Holland, Amsterdam, New York, and Oxford, 1985, pp. 331–338. - Robert E. Byerly. Mathematical aspects of recursive function theory. Harvey Friedman's research on the foundations of mathematics, edited by L. A. Harrington, M. D. Morley, A. S̆c̆edrov, and S. G. Simpson, Studies in logic and the foundations of mathematics, vol. 117, North-Holland, Amsterdam, New York, and Oxford, 1985, pp. 339–352.

1990 ◽  
Vol 55 (2) ◽  
pp. 876-878
Author(s):  
J. V. Tucker
Keyword(s):  
New York ◽  
Computational Complexity ◽  
Recursive Function ◽  
Function Theory ◽  
Recursion Theory ◽  
Foundations Of Mathematics ◽  
Real Functions ◽  
Pebble Game ◽  
Logics Of Programs

Refinements of Vaught's normal from theorem

1979 ◽  
Vol 44 (3) ◽  
pp. 289-306 ◽  
Author(s):  
Victor Harnik
Keyword(s):  
Normal Form ◽  
Boolean Combination ◽  
Form Theorem ◽  
Central Notion ◽  
The Matrix ◽  
Normal Form Theorem ◽  
Skolem Theorem ◽  
Image Position ◽  
Countable Models

The central notion of this paper is that of a (conjunctive) game-sentence, i.e., a sentence of the formwhere the indices ki, ji range over given countable sets and the matrix conjuncts are, say, open -formulas. Such game sentences were first considered, independently, by Svenonius [19], Moschovakis [13]—[15] and Vaught [20]. Other references are [1], [3]—[5], [10]—[12]. The following normal form theorem was proved by Vaught (and, in less general forms, by his predecessors).Theorem 0.1. Let L = L0(R). For every -sentence ϕ there is an L0-game sentence Θ such that ⊨′ ∃Rϕ ↔ Θ.(A word about the notations: L0(R) denotes the language obtained from L0 by adding to it the sequence R of logical symbols which do not belong to L0; “⊨′α” means that α is true in all countable models.)0.1 can be restated as follows.Theorem 0.1′. For every-sentence ϕ there is an L0-game sentence Θ such that ⊨ϕ → Θ and for any-sentence ϕ if ⊨ϕ → ϕ and L′ ⋂ L ⊆ L0, then ⊨ Θ → ϕ.(We sketch the proof of the equivalence between 0.1 and 0.1′.0.1 implies 0.1′. This is obvious once we realize that game sentences and their negations satisfy the downward Löwenheim-Skolem theorem and hence, ⊨′α is equivalent to ⊨α whenever α is a boolean combination of and game sentences.

Normalization theorems for full first order classical natural deduction

1991 ◽  
Vol 56 (1) ◽  
pp. 129-149 ◽  
Author(s):  
Gunnar Stålmarck
Keyword(s):  
Normal Form ◽  
Natural Deduction ◽  
Full System ◽  
Strong Normalization ◽  
Logical Constants ◽  
Restricted Version ◽  
First Order ◽  
Form Theorem ◽  
Normalization Theorem ◽  
Normal Form Theorem

In this paper we prove the strong normalization theorem for full first order classical N.D. (natural deduction)—full in the sense that all logical constants are taken as primitive. We also give a syntactic proof of the normal form theorem and (weak) normalization for the same system.The theorem has been stated several times, and some proofs appear in the literature. The first proof occurs in Statman [1], where full first order classical N.D. (with the elimination rules for ∨ and ∃ restricted to atomic conclusions) is embedded in a system for second order (propositional) intuitionistic N.D., for which a strong normalization theorem is proved using strongly impredicative methods.A proof of the normal form theorem and (weak) normalization theorem occurs in Seldin [1] as an extension of a proof of the same theorem for an N.D.-system for the intermediate logic called MH.The proof of the strong normalization theorem presented in this paper is obtained by proving that a certain kind of validity applies to all derivations in the system considered.The notion “validity” is adopted from Prawitz [2], where it is used to prove the strong normalization theorem for a restricted version of first order classical N.D., and is extended to cover the full system. Notions similar to “validity” have been used earlier by Tait (convertability), Girard (réducibilité) and Martin-Löf (computability).In Prawitz [2] the N.D. system is restricted in the sense that ∨ and ∃ are not treated as primitive logical constants, and hence the deductions can only be seen to be “natural” with respect to the other logical constants. To spell it out, the strong normalization theorem for the restricted version of first order classical N.D. together with the well-known results on the definability of the rules for ∨ and ∃ in the restricted system does not imply the normalization theorem for the full system.

On a Normal Form of the Orthogonal Transformation III

1958 ◽  
Vol 1 (3) ◽  
pp. 183-191 ◽  
Author(s):  
Hans Zassenhaus
Keyword(s):  
Normal Form ◽  
Linear Space ◽  
Bilinear Form ◽  
Matrix Equation ◽  
Orthogonal Transformation ◽  
The Matrix ◽  
Image Position

Under the assumptions of case of theorem 1 we derive from (3.32) the matrix equationso that there corresponds the matrix B to the bilinear form4.1on the linear space4.2and fP,μ, is symmetric if ɛ = (-1)μ+1, anti-symmetric if ɛ = (-1)μ.The last statement remains true in the case a) if P is symmetric irreducible because in that case fP,μ is 0.

A universal partial recursive function

1991 ◽  
Vol 49 (2) ◽  
pp. 186-189
Author(s):  
E. A. Polyakov
Keyword(s):  
Recursive Function ◽  
Partial Recursive Function

scholarly journals Near-rings of polynomials over groups

1985 ◽  
Vol 28 (1) ◽  
pp. 1-7 ◽  
Author(s):  
J. D. P. Meldrum ◽  
G. Pilz ◽  
Y. S. So
Keyword(s):  
Normal Form ◽  
Polynomial Functions ◽  
Image Position

The set G[x] of polynomials over a group (G, + )—as well as the polynomial functions P(G) on (G, +) form near-rings with respect to addition and composition (substitution). See [1] for polynomials and [2] for near-rings. A number of results on G[x] can be deduced from [2].Due to [1], the polynomials in G[x] can uniquely be represented in the following “normal form”:

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