some results on the length of proofs - American Mathematical Society

TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 177, March 197 3

SOMERESULTS ON THE LENGTH OF PROOFS BY

R. J. PARIKH ABSTRACT. Given a theory T, let \-^A most

k lines".

We consider

full induction

but addition

that \-k A is decidable analogue

lengthOjof

theorem

sented.

system

E.g.

Now

Thus

ourselves

one

to tormulate

a criterion of the

schemata

include tuitionistic

of (the

shortest)

theorems

as

an axiom

by

results,

theories means

rules

formalised of

"not

the

number

so nice"

second

of

Formalisations

type formalisations

length to

or else

approach.

in some language

a finite

of inference.

type and Gentzen

the

either

theories

and here

formula

is pre-

reduces

we have

"nice" take

the

of a given

the system

of particular

We shall

regarding

proof

on the way in which

distinguishes

theory.

consider

schematic

2. Schematic

systems.

explained

to define

we expand symbols

questions

An

corollary.

of the

axioms, of this

of classical

axiom kind

will

and in-

arithmetic.

generally order

consider

formalisations

which

calculus

and

of the

same

we shall

Hilbert

length

to particular

lower predicate

is an easy

we shall

strongly

with We show

under a weak enrule.

of proofs

in order to get significant

formalisations In particular

the

arithmetic

being ternary relations.

on the length

paper

depends

adjoining

of some proofs.

confine

In this

proofs.

in a formal

PA* of Peano

and multiplication

for PA* and hence PA* is closed

of Gödel's

1. Introduction.

mean "A has a proof in T of at

a formulation

Since

the notion

the

axiom

in the literature

notation

and emphasize

schemata

of a schematic of the

and rules

of inference

with the help of formula system

predicate

substitution

variables,

we do the obvious,

calculus

as the central

to include idea.

are

in namely,

metamathematical

Precise

details

follow: (a) Notation

of the predicate

(i) variables: Received

calculus:

x, y, z, x,, • • •,

by the editors

December

AMS (MOS) subject classifications Key words

and phrases.

(!) Throughout See [Pa^j

for related

this

paper

results

Peano

(1969). Primary 1018, 0230.

arithmetic,

the length where

2, 1970. length

of a proof

the length

of proofs,

will mean

is taken

Presburger

the number

to be the number

arithmetic. of lines.

of symbols.

Copyright © 1973, American Mathematical Society

License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use

R. J. PARIKH

30 (ii) constants:

c, c.,

(iii) rc-ary predicate

(iv)

function

c..,•••

,

symbols

symbols:

LMarch

for n > Û: F, G(x), f/(x,

y), • • • ,

/, g, h, /j, • • • ,

(v) logical symbols:

-., &, V, —», V, 3 etc.,

(vi) brackets. Remark.

We do not assume

that

all the

apparatus

mentioned

in (ii)—>(v)

is present.

(b) Metanotation: (i) metavariables:

u, v, w, u., • • • ,

(ii) term variables:

r, s, t, r,, • • •,

(iii) ra-ary predicate We can assume

above,

but

will

variables,

terms.

of the function

Terms

which

and will be denoted Atomic

stants

formulae

Formulae

terms.

will'be

x

for n > 0: P, that the

Q{x), R(x, y),- •

n in (iii)

w-ary predicate

is bounded

variables

for each

metavariables :,,•••,/

certain

predicate

objects

used

will always

variables

to study be such

S will induce

will

zz,,•••, I

the formal

system conflict. called

letters

regular

and con-

from these

if they

assignment

zz , of regular n

formulae

and the variables

1. Let the last step i-,,§!»••■,

together

(£.,

K| associated with them. We

are no common

Then the sequence

m = 0.

the rule

formulae.

or these

ÏJ.3í¿;gÍ.---.g¿;K1;...,3;{,...; so that there

of applying

are obtained

by

restrictions. substitution

o for

this

sequence,

i.e.

such

that

§(?;.)= S(§j),S(K.)= S(£.) and S(l) = A. Then

by induction

corresponding

hypothesis,

subproof

o(oL.) by the given

each

and, moreover,

rule.

But

S(M.) will be provable

with the

ö(3H) = A will be provable

ö(K.) = b{x .). This

shows

there

from the

is a proof

as

required. Conversely

mises the

in the

last

suppose

last

formula

stitutions

substitution

S,, ö

to avoid

with

conflicts,

arithmetic

A, M and axioms

formulae logical

with

saying

two systems, of logical symbols.)

T,

in PA* will often

3\,"

Hence

there

of the pre-

subproof,

will

exist

and

sub-

have

ô , • ■• , S,, S'

+, • replaced

all

into one

functions. system

the

logical

be longer.)

schema,

induction

complexity

of confined

is the number

A is provable

Presburger

with respect

arithmetic It is known

to closed

of

in T

{PA, PA* are of equal

multiplication.

PA*

predicates

If T is either with

r T A will mean

of inference.

PA*) is complete

schematic

by ternary

mean

strength,

will mean [Pr]

formulae

that

PA

in which

not appear.

Lemma B. Let 3^,- • • , "5m, §j,- •-,§;, that

each

the metavariables

represent

T,

S, +, 0 but without does

with

< /. (Here

of rules

with also

rule.

Since

Then

corresponding

will mean one of the usual

will

complexity

arithmetic multiplication

last

the

S, +, • and the full induction

system

but proofs

(and hence

PA

with

For any theory

< k applications

with

we can combine

that these

then

as required.

Q.E.D.

In the following,

of Peano

the

for all these.

will mean the corresponding

these

is a proof

be provable

S as required.

Notation.

systems

will

A provable

§.,•••,

been renamed

there

rule

•, ¡?m, §,,•••,§,

are formulae

AA^x), • ■■, Ap{x), R be such of PA*,

AA_x), • ■ ■, A Ax) are


to

1973]

SOME RESULTS ON THE LENGTH OF PROOFS

regular there

is

formulae

33

of PA*, m = I + p, and R is a set of restrictions.

is a formula

equivalent

B{x)

to

of Presburger

the

arithmetic

proposition,

"there

such

is

that,

Then

for all

a substitution

n,

B(n)

& obeying

R such

that S(3\) = S(§.) for i < / and S(3:/ .) = A.{n) for i 1. Then there

in J ., §.

Case

symbol

1. \).

to it by the

variables.

two pairs

P'(-),

procedure then

condition

§'(-)

on P',

P is maximal

in P

3\ = P(-),

§"(-)•

where

u decreases.

predicate

P , P

g¿ = §'.(-)

Similarly

on P are

implication

P and every

for predicate

easily

translated

(If we cannot

corresponding,

say,

are

two new

-* §"(-)

variables in

carry

to

by the terms

functional

combinations

are handled

conjunction,

similarly.

is of the form (Vx)Çj' . The reduction

is free for y in P" becomes

"y

is similar,

is not free in P"


of

out this

a

exist.)

2. Q.

"a

and

formula

the pair

The restrictions

because

All truth Case

Replace

to P.

d cannot

that

is not atomic.

equivalent

restrictions

a P such

is of the form §'. —» §".

predicate equivalent

exists

but the

or "x

does

34

R. J. PARIKH

not occur in a""

[March

if y / x. The condition

"o

is free for x in P"

number

u = 1. The

is dropped,

etc. After a finite

Now we can also

of steps

assume

M£) .) are all essentially occurring

atomic.

in R. If there

leftmost

atomic

i = 1, ■• • , n, in whose

to each pair

loss

consideration,

P is

F.

a fixed

Thus

S(P(yI,...,yi))

it falls.

be all the variables o (P) = the

by all the quantifiers

Then

o

is a solution

that

involved.

For

the predicate

variables

symbol

not covered

formula

A can

be assigned

we can assume

for each

predicate

variable

without

P that

= (ß1xI)...(ß/t)Fp(ö1(v:1...)yl),...)9g(yl,...,yi)) Fp

on the y .. Given

is known,

two such

they will be equal iff the corresponding The argument

so far applied

S and 0. The remainder

and the

6. are certain

(not obviously

inequivalent)

terms

to arbitrary

of the argument

where

Moreover,

the

terms

formulae,

are equal. schematic

(below)

systems

applies

6, 6', 6"

ate of the form Sq0

possible

variables

can

containing

only to PA*.

An atomic formula in PA* must be of the form (i) 6 = 6, or (iii) M{6, 6,6")

a variable.

(3x.),

if o is.

symbols

To all predicate

(Vx),

regular

where the Q. ate quantifiers,

depending

are all atomic.

that the o(J\),

o, we can define

preceded

£j = F(—), we can assume

variable

of generality.

• •, x

that we know the predicate

J. = P(-),

for the predicate by this

scope

lj.

of generality

a substitution

of o{P)

assume

loss

For let x,,-

exists

subformula

We can also

we get

without

(ii) A(0, 6,6)

or Sqx where

be restricted

to

x is

a finite

set

of possibilities. Thus whether

a substitution

is given

completely

it is =, A or M and for each

specifying

the values

q . where

by specifying

6 whether

(a) for each

it is Sq0 or Sqx and

Fp (b)

6. = S lQ ot S 1x ot the term variable

t .=

SQlQ ot Sq\. Now, part (a) consists For each duces

of these

possibilities,

to the various

arithmetic.

Thus

of choosing

the condition

numbers

q . satisfying

if we consider

all

a particular

n, we get an equivalence:

where

B . is a formula

each

one among finitely

Theorem

to length

ways

o exists

v(j)

equations

= o(§ .) re-

in Presburger

in which

ö

can

exist

for

iff B An) V •• • V B (n)

arithmetic.

B j(x) V ... V Br(x). This proves Lemma B. 4. Applications

for o to give certain

possible

in Presburger

many possibilities.

Now take

B{x) to be

Q.E.D.

of proofs.

1. (a) // is uniformly

decidahle

in k, A if \-

* A holds

or not.

(b) Let B{x) be a formula of PA*, k £ N. Then the set \n\ r PA* B(n)\ is a finite

union

of arithmetic

(linear)

progressions.


1973]

SOME RESULTS ON THE LENGTH OF PROOFS

Proof,

(a) is an immediate

Presburger

arithmetic

from Lemmas

that

consequence

is decidable.

To see

A and B that there

i.e.

Theorem depending

a finite

2. Let

such

We confine

§ 1 » • ■• » b

A, R)

be decomposed

by [GS], the set progressions.

for every

I can be found

J l? • • • , j

\n\B

in) is true!

is

Q.E.D.

analysis

exists

a number

I

< k ,

U of k lines

for the sequence

a

The rest

such

from k, k .

and in-

(3*^ • • • « 3"m> K>

, K i s the sequence

are only finitely

formulae.

It follows

arithmetic

A of complexity

effectively

» a •«••"»

c{A) < k , there

into atomic

formula

to a particular

of substitutions

where

A. Since

that,

ourselves

the existence

by Lemma

the following.

B (x) of Presburger

T be PA or PA*, k, k £ N. Then there

\-Z-A iff (-* A. Moreover,

Proof.

However,

union of arithmetic

only on k, k

vestigate

A, B and the fact that

(b) we notice

is a formula

\~pa* B(n) iff B in) is true.

semilinear,

of Lemmas

35

many ways

of the argument

provided

in which

is quite

A can

similar

to

that in Lemma B. (In fact suppose axiom

J.»

schema

a-

or ^

3

3

comes

or in a schematic

is at most

a.

was

been

x,

atomic,

is

a.

m = k — 1, the initial

of Lemma

complexity

than

of any formula

rule of inference,

B reduces

m(u)

then the maximum

greater

essentially

complexity

Since

. Now the analysis

1. If the maximum

reduction have

the maximum

value

of any

of u is uQ =

the value

of u until

at a certain

at a preceding

= k since

as axiom,

Then the complexity

of any formula

complexity

2x + 1. Now mil)

occuring

stage

the images

and miu + l) < 2miu) + 1. We immediately

it be-

stage could

in this not

can be taken

see that

to be

/ =

m(«0)