Single Axioms for the Left Group And

Single Axioms for the Left Group

and Right Group Calculi

Wil liam W McCune

Mathematics and Computer Science Division

Argonne National Lab oratory

Argonne Illinois

USA

phone email mccunemcsanlgov

March

Introduction

In J A Kalman presents axiomatizations of the left group tautologies and the right

group tautologies In this pap er we sharp en those results by showing that Kalmans axiom

atizations are dep endent and by giving other simpler axiomatizations including ones that

consist of single formulas

A left group formula is an expression constructed from variables and a binary function

symbol E A left group formula is a left group tautology i is valid in multiplicative

group theory when E x y is interpreted as x y The left group calculus consists of left

group formulas and the inference rules variable instantiation and mo dus p onens where E

is treated as implication ie from and E infer An axiomatization of the left

group calculus is a nite set of left group tautologies from which every left group tautology

This work was supp orted by the Applied Mathematical Sciences subprogram of the Oce of Energy Research US Department of Energy under Contract WEng

is derivable in the left group calculus A single axiom for the left group tautologies is a left

group formula such that fg is an axiomatization of the left group calculus

There are analogous denitions for the right group calculus in which E x y is inter

preted as x y Ordinary mo dus p onens rather than reverse mo dus p onens is used for

the right group calculus This is discussed in Section

The inference rule used for the pro ofs in this pap er is C A Merediths condensed

detachment which uses unication to combine the op erations of mo dus p onens and

instantiation consider premises and E in which variables have b een renamed

if necessary so that they have no variables in common if and unify then infer the

most general corresp onding instance of Every formula that can b e derived by mo dus

p onens and instantiation either can b e derived by condensed detachment or is a instance

of a formula that can b e derived by condensed detachment See the pro of of Theorem

b elow for simple examples of the application of condensed detachment

Section contains condensed detachment pro ofs that three L L and L of Kalmans

ve axioms for the left group calculus are dep endent on the remaining two axioms It is

then shown that

E E E E x y z E E u v E E E w v E w u s E z E E y x s S

is a single axiom for the left group calculus Several other simple axiomatizations which are

not single axioms are also given Section contains condensed detachment pro ofs that four

R R R and R of Kalmans ve axioms for the right group calculus are dep endent

on the remaining axiom R

E x E x E E y z E E y u E z u R

Five other single axioms for the right group calculus are also given without pro of

Single axioms are known for the equivalential calculus in which E can b e

interpreted as the in Bo olean groups and for the Lcalculus resp ectively R

1 1

calculus in which E can b e interpreted as resp ectively in Ab elian

groups Prior to the work rep orted in this pap er no single axioms for the left group or right

group calculi were known to the author The new single axioms answer questions raised by

C A Meredith in p

We made extensive use of the automated theoremproving program Otter in ob

taining the new axiomatizations Theoremproving programs have b een used to study given

candidate axiomatizations in related areas for example but here the goal was

to nd simpler axiomatizations Otter was used to generate candidate axiomatizations

to search for pro ofs that the candidates are in fact axiomatizations and to search for de

p endencies in axiomatizations

The Left Group Calculus

From here on formulas are written in Polish notation In the pro ofs that follow the

justication mn indicates that formula m is the ma jor premise E and that n is the

minor premise which unies with The result is the corresp onding instance of Variables

are renamed starting with x y z u v w s t The formula numbers indicate p osition in the

sequence of formulas retained by Otter

Kalmans axiomatization of the left group calculus consists of the following ve

axioms

E E E xE E y y xz z L

E E E E E xy E xz E y z uu L

E E E E E E xy E xz uE E y z uv v L

E E E E xy z uE E E xv z E E y v u L

E E E xE E y xz E E uxv E E E E xy uz v L

Theorem The pair of formulas fLLg axiomatizes the left group calculus

Proof Otter The following condensed detachment pro of derives L L which is a

generalization of L and L from fLLg

E E E E E xy E xz E y z uu L

E E E E E E xy E xz uE E y z uv v L

E E E xy z E E uy E E xuz

E E xy E E E z uxE E v uE E z v y

E E xy E E E E z uE z v xE E uv y

E E xE y z E E E y uxE uz

E E xy E E E E E E z uE z v w E E uv w xy

E E E xy E E E z uE z v w E E sy E E xsE E uv w

E E E xy E E E E E z uE z v w E E uv w sE E ty E E xts

E E E E xy E xz E E uv E uw E E y z E v w

E E E E xy E xz E E E E E uv E uwsE E v wstE E y z t

E E E E E xy z uE z E xv E uE y v

E E E E xy z E E uy v E z E E uxv

E E E xy E E E E z uE z v E uv E xw E y w

E E E xy E E z xuE E z y u

E E xE E E y z uv E E E uE y w xE E z w v

E E E E xy z E E uy E E xuv E z v

E E xE E y z uE E E z v xE E y v u

E E xy E E E E E z uv E E w uE E z w v xy

E E E E xy z uE E E xv z E E y v u L

E xE E y z E E z y x

E xE y E E E E z uy E uz x

E xE E E E y z E E z y uux

E E E xy z E E E E uv E E v uxy z

E E E xE E y z uE E v xw E E E E z y v uw L

E E E xE E y y xz z L

Theorem The pair of formulas fLPg axiomatizes the left group calculus

E E E E E xy E xz E y z uu L

E E E xy z E E uy E E xuz P

Proof Otter P is a left group tautology which can b e veried by rewriting E

to and reducing to The following condensed detachment pro of derives L from

fLPg

E E E E E xy E xz E y z uu L

E E E xy z E E uy E E xuz P

E E xE y z E E E y uxE uz

E E E E E xy z uE z E xv E uE y v

E E E E xy z E E uy v E z E E uxv

E E E xy E E z xuE E z y u

E xE E y z E E z y x

E E E E xy z E E uy E E xuv E z v

E xE E y z E E uy E E z ux

E E E xy E E y xz z

E xE y E E z uE E E E uv y E v z x

E E E xy E E z uE E uz E xv E y v

E E xE E y z uE E E z y xu

E E E xE y z E xE E z y uu

E E xE y E E z uv E E E y E uz xv

E xE E E E y z E y uE z ux

E xE E E E y z E y uv E E v E z ux

E E E E E E xy E xz uE E y z uv v L

Theorem Formula S is a single axiom for the left group calculus

E E E E xy z E E uv E E E w v E wusE z E E y xs S

Proof Otter S is a left group tautology which can b e veried by rewriting E to

and reducing to The following condensed detachment pro of derives P and L

from S

E E E E xy z E E uv E E E wv E w usE z E E y xs S

E E E xy E E E z y E z xuE E E v wE E sv E sw u

E E E E xy E xz E E E uE E v wE v sE uE w s tE E y z t

E E E E xE E y z E y uE xE z uE E E v w E v stE E w st

E E E xy z E E E uxE uy z

E E E E xy E xz uE E y z u

E E E xE y z E xuE E E v y E v z u

E E E xy z E E uy E E xuz P

E E E E E xy E xz E y z uu L

Six other axiomatizations of the left group calculus were also discovered with the assis

tance of Otter The axiomatizations include formulas from the following list

E E E E E xy E xz E y z uu L

E E E xy z E E uy E E xuz P

E xE E E E y z E y uE z ux P

E xE E y z E E z y x Q

E E xy E E z xE z y Q

E E E xy E E y xz z Q

E E E xy E xz E y z Q

Each of the sets fLPg fLQQg fPQg fPQg fQQQg fQQQg is an

axiomatization of the left group calculus Pro ofs can b e found in

The Right Group Calculus

Kalmans axiomatization of the right group tautologies consists of the following ve

axioms

E xE xE E y E z z y R

E xE xE E y z E E y uE z u R

E xE xE E y E z uE y E E z v E uv R

E E E xE y z E uE y v E xE uE z v R

E E xE y E z E uv E E xE v z E E y E v uv R

Let the mirror image of a formula b e obtained by rewriting each o ccurrence of E to E

Note that each of the ve axioms RR is the mirror image after renaming variables of

the the corresp onding axiom in LL When the inference rule used with the right group

tautologies is reverse mo dus p onens it is easy to see that the resulting calculus is isomorphic

to the left group calculus However Kalman states without pro of that ordinary mo dus

p onens can also b e used with RR to axiomatize the right group tautologies We sketch

a pro of of this result here

Theorem From formulas RR one can derive al l right group tautologies with instan

tiation and ordinary modus ponens

Proof sketch Otter We show that from RR in fact from just R and ordinary

condensed detachment we can derive reverse mo dus p onens We do this by assuming E

and for constants and and deriving Once we have reverse mo dus p onens we can

derive all right group tautologies

E xE xE E y z E E y uE z u R

E E E xy E E xz E y z

E E xy E E xz E y z

E E E xy z E E E xuE y uz

E E xE x

E E E xE y xE y

E E E xy E z y E E xuE z u

E E E E xy E E z xy E z

E E E xy z E E xy z

For the right group calculus we use ordinary mo dus p onens rather than reverse mo dus

p onens in order to have a system that is substantially dierent from the left group calculus

In addition it app ears that the right group calculus has axiomatizations that are simpler

than the left group calculus has

Theorem Formula R is a single axiom for the right group calculus

Proof Otter The following ordinary condensed detachment pro of derives R R R

and R which is a generalization of R from R

E xE xE E y z E E y uE z u R

E E xE xE E y z E E y uE z uE E v wE E v sE ws

E E xy E E xz E y z

E E E xy z E E E xuE y uz

E E E xy E E xz E y z E E uv E E uwE v w

E E xy E E xE E z uE E z v E uv y

E E E E xy z E uz E E E xv E y v u

E E E E xy z E E E xuE y uz E E v wE E v sE ws

E E E xy E z y E E xuE z u

E E E xy E z y E E xz E E uv E E uwE v w

E E E xy E E z uE E z v E uv E E xw E y w

E E E xy E z y E xz

E E E E xy z E E uE y v z E E xv u

E E E xE E y z E uz v E E xE y uv

E E E E xy E z y uE E xz u

E E E xy E z E y uE E xuz

E E xE E y z E uz E xE y u

E E E xE y z E uy E xE uz

E E E xy z E E xuE z E uy

E E E xy E E z uE y uE xz

E E xE y z E E xE uz E y u

E E E xE E y y z z x

E E E E xE y z E uy v E E xE uz v

E E E E xy z E uz E E xv E uE v y

E E E E xy z uE E E xv E z E v y u

E E E xy z E E E xuv E z E v E y u

E E xE E y z E E y uE z ux

E E E xE y z E uE y v E E xE v z u

E E E xE E y z uE uE E y v E z v x

E E E xE y z E z y x

E E E E xE y z E uy E z ux

E E E E xE y z uv E xE v E E z y u

E E E xy E E z uE y E uz x

E E E E xE y E z uE E v uy E z v x

E xE xE E y E z uE y E E z v E uv R

E xE xE E y E z z y R

E E E xE E y z E uE z y v E xE v u

E E xE y E z uE xE E y E v uE z v

E E xE y E z E uv E xE E y E v uz

E E E xE y z E uE y v E xE uE z v R

E E xE y E z E uv E E xE w z E E y E v uw R

One might conjecture that a set of formulas axiomatizes the left group calculus if and

only if its set of mirror images axiomatizes the right group calculus On the contrary

although R is a single axiom for the right group calculus L cannot b e a single axiom for

the left group calculus b ecause it do es not ordinary condensed detach with itself

Each of the following formulas is also a single axiom for the right group calculus with

ordinary mo dus p onens Pro ofs can b e found in

E E xE y z E xE E y uE z u S

E xE xE E E y z E uz E y u S

E E xE y z E E xE uz E y u S

E E xE E y z E E y uE z ux S

E E xE E E y z E uz E y ux S

The Role of Otter

The program Otter is a generalpurp ose resolutionparamo dulation theorem prover

for rstorder logic with equality The main consideration in the design of Otter was the

ability to quickly explore large search spaces rather than the use of heuristics to carefully

control the searches

We used Otter in two ways to obtain these results First to nd the multiformula

axiomatizations listed at the end of Section we iterated as follows take a known axiom

atization replace a complex axiom say with a set of simpler tautologies then search

for a pro of of if a pro of is found search for dep endencies in the new axiomatization

Second to nd the single axioms we generated large sets of tautologies and with each

searched for a known axiomatization The main metho d for generating the large sets of

candidate single axioms was to enumerate tautologies not containing instances of E x x

Most tautologies contain instances of E x x but the interesting axiomatizations usually

do not Approximately Otter searches were run consuming ab out four days of

computer time Another pap er contains a detailed presentation of the use of Otter to

obtain the results presented in this pap er

Acknowledgments I wish to thank Dana Scott for suggesting these calculi as challenges

for Otter Larry Wos for collab orating on the formulation of search strategies applied

by Otter in related areas John Kalman for discussions on the topic and a referee for

substantially improving to this pap er

References

J A Kalman Substitutionanddetachment systems related to Ab elian groups In A

Spectrum of Mathematics Essays Presented to H G Forder pages Auckland

University Press

J A Kalman Axiomatizations of logics with values in groups J London Math

Society

J A Kalman A shortest single axiom for the classical equivalential calculus Notre

Dame J Formal Logic

J A Kalman Condensed detachment as a rule of inference Studia Logica

LXI I

E J Lemmon C A Meredith D Meredith A N Prior and I Thomas Calculi of

pure strict implication Technical rep ort Canterbury University College Christchurch

Reprinted in Philosophical Logic Reidel

W McCune Otter Users Guide Tech Rep ort ANL Argonne National

Lab oratory Argonne IL March

W McCune Automated discovery of new axiomatizations of the left group and right

group calculi Preprint MCSP Mathematics and Computer Science Division

Argonne National Lab oratory Argonne IL

C A Meredith and A N Prior Equational logic Notre Dame J Formal Logic

D Meredith In memoriam Carew Arthur Meredith Notre Dame J

Formal Logic

J G Peterson Shortest single axioms for the classical equivalential calculus Notre

Dame J Formal Logic

J Lukasiewicz Selected Works NorthHolland Edited by L Borkowski

L Wos S Winker R Vero B Smith and L Henschen Questions concerning p ossible

shortest single axioms in equivalential calculus An application of automated theorem

proving to innite domains Notre Dame J Formal Logic