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 1 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 1 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 0 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 0 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 1 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 1 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 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 E 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 0 and R which is a generalization of R from R E xE xE E y z E E y uE z u R E