A system of lambda-calculus possessing facilities for typing and abbreviating: part II : formal description

A system of lambda-calculus possessing facilities for typing

and abbreviating

Citation for published version (APA):

Nederpelt, R. P. (1980). A system of lambda-calculus possessing facilities for typing and abbreviating: part II : formal description. (Eindhoven University of Technology : Dept of Mathematics : memorandum; Vol. 8011). Technische Hogeschool Eindhoven.

Document status and date: Published: 01/01/1980

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Department of Mathematics

Memorandum 1980-11

June 1980

A system of lambda-calculus possessing facilities

for typing and abbreviating

Part II: Formal description

University of Technology Department of Mathematics PO Box 513, Eindhoven The Netherlands by R.P. Nederpelt

- 1

-Contents

- Abstract and introduction

- Formal description a) alphabet

b) terms, term parts c) meta-variables d) subterms

e) weight

f) closed terms

g) well-structured terms h) normable terms and norms i) one-step ~-reduction

j) one-step ~-reduction

k) one-step S-reduction

- References

Abstract and introduction

This is a sequel to "A system lambda-calculus ... , Part I" (see reference [lJ) , in which an unorthodox lambda-calculus system was informally explained. We s"all now give a formal description of this system. In fact, we

des-cribe a number of systems simultaneously: the ordinary system, its so-called S-extension and its S-~-~-extension.

Moreover, we define subsystems of the ordinary system, viz. the systems of closed terms, of well-structured terms and of normable terms.

If any definition is not self-evident, we refer to Part I for an explanation. More comments may be found in [2J.

a. alphabet

The

aZp

hab

e

t

consists of the following symbols: (,),A,o,a, all natural numbers and O. The alphabet of the a-extension contains, apart from these symbols, the symbol a. The a-~-~-extension has an alphabet that contains, moreover, the function symbols ~ and ~, having double, in-teger subscripts, and -to The symbols A,o,a and a will be called

markers;

~ and ~ are

reference

tran8fo~ing

mappings.

b. terms, term parts

Te~s and

term part8

are symbol strings constructed from the alphabet according to the following recursive rules:

(1) (i) The empty string is a term (it will be rendered invisibly; sometimes, however, i t will be denoted by

0,

for the sake of convenience),

(ii) each term part is a term,

(iii) if A and B are terms, then concatenation AB is a term.

(2) (i) I f A is a term, then (AA), (AO), (Aa) (and (Aa) in the a- and

a-~-~-extensions) are term parts (called

A-parts, a-parts,

a-parts

and

a-parts

respectively),

(ii) if n E IN and wEN u .{O}, then (n,w) is a term part (call~d

a

ref-part) .

In the a-~-~-extension, one may insert in any term, at any place,

function-symb~ls ~k,i or ~k,i' for integer k and i, or the symbol

t.

c. meta-variables

We use the following meta-variables: A,B, . . . ,F,G, . . . for terms,

a for markers,

S, . . . for symbol strings

(including primed or indexed variants).

In the following definitions, existential quantifiers are sometimes omitted; the conventions for adding these quantifiers are as usual. Symbol-for-symbol identity is denoted by

=.

- 3

-d. sub terms

Let F and G be terms. We say that G is a subterm of F followed by

s2' if F

=

s1GS2' where either S2 is empty, or S2

=

as;. If the place of the occurrence of G in F is obvious, it suffices to say that G is a subterm of F (or G sub F).

(It may occur that a term G occurs inside a term F, without G being a subterm according to the above definition. In order to avoid mis-understandings, one may refer to the above-mentioned subterms as

pro

pe

r

subterms).

e. weight

Let A be a term, where A

=

S1SS2. Let (n

1,w1), ... ,(nk,wk) be all ref-parts occurring in S, and a

1, ••• ,,\ all markers occurring in S. Then the

k

weight of S (denoted

S)

is defined as £ +

L

w .• (It follows that the i=l ~

weight of ref-part (n,w) is w.)

f. closed terms

A term A is closed if for each partition of (symbol string) A of the form A

=

S1 (n,w)S2' the following holds: there is a partitioning of Sl of the form S - S'(Ba)S" such that

1

(i) S" n - 1, and

(ii) either: a - A,W = 0 and (n,w) sub A,

or: a

=

a (or S in the S-extension) and w

=

B.

(In the above circumstances we say that (n,w) is

bound

by (Ba).)

g. well-structured terms

Terms in each of the present systems have a natural tree representation (cf [lJ, section g). For a natural interpretation of a term A in any of the systems, the following

domination property

is convenient: if ref-part

(n,w) is bound by (Ba), then there is a path leading from the vertex marked (n,w) via the vertex marked a (in the axil) to the root. This is a

customary property for lambda-calculus systems: the vertex representing a binding variable dominates all vertices representing bound instances of this variable.

Terms obeying the domination property will be called

well-structuped.

h. normable terms and norms

The norm is a partial mapping from terms to terms. Our definitions of norms and normability will in principle concern subterms occurring at a certain place in a given closed, well-structured term. We shall, however, not push formalization so far as to precisely mention the place of occurrence.

Let F be a closed, well-structured term. If G sub F and

(i) G - (AA)B with A and B normable, then G is normable, and G - (AA)B, (ii) G - (Acr)B (or (AS)B in the S-extension), with A and B normable,

then G is normable, and G = B,

(iii) G =(M)B with A and B normable, and

{

B is

_-

empty or B - (Co)D, then G is normable, and G -

(AO)S,

(AA)E,

B - then G is normable, and G - E,

(iv) G = (n,w)B with (n,w) bound in F by (AA) or (Acr) (or (AS) in the

a-extension) and A,B and AB normable, then G is normable, and

G

=

AB.

A rlosed, well-structured term in the ordinary (non-extended) system that is also normable, will be called

inter.pretable.

Such a term resem-bles terms in ordinary "A-typed A-calculusses" (see e.g. [2]), but for the abbrevation of segments.

i. one-step ~-reduction

We define > , a relation on the ~-S-~-extension.

~

The effect of ~ is the following. When

reference transforming mapping

~k,£ encounters a ref-part (n,w), then reference number n changes into n + £ (only) if n > k. The first index, k., of )Jk,R.' records the weight: in passing (n,w), ~k,R. changes into ~k+w,R.; in passing a, )Jk,R. changes

- 5

-into ~k+1,~ The reference transforming mapping ~k,~ vanishes on encoun-tering t (and symbol t vanishes as well) or at the end of the term. See also [ 1 J, section c.

The rules for ~-reduction are the following. (Note: the rules in this and the following section have to be read in the obvious way; rule 2 of

~-reduction should be, for example: let F be an extended term and

F :: Sl~k,~ (Aa)S2' then F > ~ Sl (~k,~Aa)S2.)

1. _{~k,~(n,w) > ~ { (n + ~, w) ~k+w , ~} if n > k

(n,w)~k+w,~ if n :s; _k

(reference transformation) (pseudo-transformation) 2. ~k, ~ (Aa) ;> _~ (~k,R,Aa) ( intrusion)

3. _(A~k,R,a) >

~ (Aa) ~k+1, R, (evasion)

4. _~k,~ >

~

0

at the end of the term (natural dissolution) 5. ~k,R,t >

₀

~ (forced dissolution)

j . one-step p-reduction

The relation >?jl is defined on the ~-qJ-t3-extension.

The effect of qJ is the following. When

weight transfoPming mapping

qJk,R, encounters a ref-part (n,w), then weight w changes into w + R, (only) if n

=

k. The first index, k, of qJk,R,' records weights just as this is the case with ~k,~. The weight transforming mapping qJk,R, vanishes at the end of the term.

Th~ rules for qJ-reduction are the following. 1. CPk, ~ (n,w) > _qJ {(n,w + ~)~ _{w,~ +w,~} nCPk n

(n,w)qJk n

+w,~

if n

=

k (weight transformation) if n ~ k (pseudo-transformation)

2. _{CPk, R, (Aa)} > _(cpk,R,Aa) ( intrusion)

cP

3. _(ACPk,R,a) > _(Aa)qJk+1,~ (evasion)

qJ

4. CPk,~ >

0

at the end of the term (natural dissolution) cP

k. one-step S-reduction

The relation > S is defined on the S-extension. In the rules for S-reduc-tion, the

reference transforming mappings

TI, v and

w

appear. The effects of these mappings is the following.

When mapping TIk,~ encounters ref-part (n,w), i t affects reference number n as follows: if k + 1 ~ n ~ k + ~ + 1, then n changes into n - 1; if n

=

k, then n changes into k + ~ + 1. Hence, the effect of TIk,~ can be described as a permutation (k + ~ + 1, k + ~, ... ,k) .

When mapping vk,~ encounters ref-part (n,w), then it changes n into k + 1 + ~

(only) if n = k + 1. Finally, mapping w

k , ""m n has a complicated effect. In encountering ref-part

(n,w), if affects reference number n as follows. If k + 1 ~ n ~ k + m, then it changes n into n + ~ + 1; if k + m + 1 ~ n ~ k + ~ + m +1, then it changes n into n - m. Hence, segments [k + 1,k + ~ + 1] and [k + ~ + 2,k + ~ + m + 1] are "interchanged".

Each of the mappings TI, v and

w

can be expressed by means of ~:

~k+~+m+1,-~-m-1~k+2~+m+2,m~k,~+1 The rules for S-reduction are the following.

-Oa. (B<S)A(cA) >S(BS)A(CS)\lO,C+A

Ob. (Ba) >13 (BI3)

i f A :: 0 and B - C

1a. (BS) (l,w) >Q B(~O BS)

two

"'B

" ,w , ,w if w

=

B lb. (BS) (n,w) >S(n -

B -

1,w) (~O,w BS) t w '" O,B,w if n > 2. (BS) (Da) _> 13 ( (B 13 ) Da) Ql 1 , 'B+ 1 3. (A (BS) a) > 13 (Aa) (~O, 1 B 13 )

_-r

_{7T 1 , 'BQl 1 , -}

s-

₁ B + 1

4. (BS) _>S

₀

at the end of the term

( o-t..-mutation) (a-mutation) ( replacement) (pseudo-replacement) (intrusion) (evasion) (natural dissolution)

7

-References

[1J NEDERPELT, R.P. A system of lambda-calculus possessing facilities

for typing and abbreviating. Part I: Informal intro-duction. Memorandum 1979-02, Eindhoven University of Technology, Dept. of Math., March 1979.

[2J NEDERPELT, R.P. An approach to theorem proving on the basis of a

typed lamda-calculus. Proceedings of the 5th Confe-rence on Automated Deduction, les Arcs, France, 1980. Lecture Notes in Computer Science no 87, Berlin (1980), 182-194.