Computer-Aided Studies of All Possible Shortest Single Axioms for the Equivalential Calculus

It has been known since 1939 that for a formula to be a single axiom for the equivalential calculus i t s length must be at least 11, and that single axioms of this length exist . Also, a single axiom of length 11 must have the two-property. There are 630 formulas with the two-property and of length 11. With computer assistance, the authors have shown that 612 of these 630 formulas are not single axioms. The main object of this paper is to outline the methods used to obtain these results. This paper logical ly precedes a recent paper of L. Wos which announces computer-assisted proofs that a further 5 of the 630 formulas are not single axioms, and should serve as an introduction to the method of schemata mentioned in that paper. 1. THE EQUIVALENTIAL CALCULUS, CONDENSED DETACHMENT, AND THE TWO-PROPERTY We consider formulas bu i l t up from variables p, q, r, . . . and a single binary connective E, and wri t ten in Polish notation. The "length" of such a formula is the number of occurrences in it of E's and variables. Such a formula is called an "equivalen t ia l tautology" i f i t holds in the logical matrix i .e . if this matrix sat isf ies i t . For example, Epp and EEpqEqp are equivalential tautologies, but Epq and EEpqEpp are not. The matrix (1) is essentially the truth table for material equivalence (p->-q)&(q-»-p) ( in Polish notation: KCpqCqp ). A set S of equivalential tautologies is called a "deductive axiomatization" of the equivalential calculus EC if every equivalential tautology is derivable by the rules of substitution and modus ponens from the formulas in S; if S has just one element x then x is called a "single axiom" for EC. It is known (Kalman, 1983, Theorem 1) that a formula is derivable from formulas S by substitut ion and modus ponens if and only if it is a substi tut ion instance of a formula derivable from the formulas S by condensed detachment. Here condensed detachment is the rule which, when applied to two formulas x=Euv and y having no variables in common, produces the formula w if u and y are unif iable and w=o(v) for some most general uni f ier o of u and y; w is then unique to within variance ( i . e . a formula w' may be produced by applying condensed detachment to x and y if and only if w' is a variant of w), and it is customary to write w»Dxy ; if x and y have variables in common, and y' is a variant of y having no variables in common with x, then Dxy is defined if and only if Dxy' is defined, and if Dxy' is defined we set Dxy«Dxy' . Thus, if S is a deductive axiomatization of EC, and T is a set of equivalential tautologies such that every formula in S is a substitution instance of a formula derivable by condensed detachment from the formulas in T, then T w i l l be a deductive axiomatization of EC; in part icular , if x is a single axiom for EC, and x is derivable by condensed detachment from the equivalential tautology y, then y w i l l be a single axiom for EC. It is known (cf. (Lukasiewicz, 1939, §8)) that a shortest single axiom x for EC has length 11, and has the "two-property" (Belnap, 1976) that every variable which occurs in x occurs exactly twice in x; also, EEpqEErqEpr is known to be a shortest single axiom for EC (Lukasiewicz, 1939). It is easily seen that there are 630 formulas of length 11 with the twoproperty. However not a l l of these 630 formulas are single axioms for EC; for instance, the formula EEpqEEqrEpr is not a single axiom for EC. The main object of this paper is to discuss how computers may be used to help show that formulas such as EEpqEEqrEpr are not single axioms for EC. 2. THEOREM-GENERATING PROGRAM TG We i l l us t ra te in §§3 and 6 how a theoremgenerating program TG, which was or ig inal ly developed as a tool for showing that part icular formulas are derivable from others by condensed detachment, was used to show syntactical ly that part icular formulas are not so derivable. Given a f i n i t e set S of formulas, the program TG generates the set Th(S) of a l l formulas which may be derived by condensed detachment from the formulas in S. In general, Th(S) is i n f i n i t e ; if Th(S) is f i n i t e , then S is not a deductive axiomat izat ion of EC. We i l l us t ra te in §3 how this may be exploited to show syntactical ly that 286 of the 630 formulas are not single axioms for EC. In these 286 cases, using minimal instead of arb i t rary substitutions enables us to reduce an i n f i n i t e set of derivable formulas to a f i n i t e set. 934 J. Kalman and J. Peterson Although TG is not an interactive program, the user can specify changes, to take place during a run, in certain parameters which control the length of retained formulas and how formulas are selected to be used in subsequent condensed detachments. During each run, s ta t is t ics of how many condensed detachments have fai led and how many have produced formulas which were too long to retain are regular ly produced. At the end of each run, a l i s t of a l l the derived formulas, sorted by length and lexicographically for formulas of the same length, is printed out. With these aids, the user can sometimes discover syntactic properties possessed by each of the formulas in a part icular set Th(S). We i l l us t ra te in §6 how this can be exploited to show syntactically that 11 of the 630 formulas are not single axioms for EC. 3. FIRST SYNTACTIC METHOD: FINITENESS Consider for example the question whether the formula x = EEEpqErpEqr is a single axiom for EC. When the formula x is given as input to the program TG, the formulas Dxx=y=EEEpqpq and Dyx=z=Epp are generated; the program also determines that Dxz=z, Dzx=x, Dzy=y, and Dzz=z, and that a l l other combinations Dst with s, t E {x,y,z} are undefined. Since in part icular the known single axiom EEpqEErqEpr for EC is not a substitut ion instance of any of x, y, or z, it follows that x cannot be a single axiom for EC. In general we may say that a formula x is "of f i n i t e type Fn" if (to within variance) the set Th({x}) of formulas generated by x is a f i n i t e set with n elements; thus EEEpqErpEqr is of f i n i t e type F3. Using TG, we easily f ind that 286 of the 630 formulas are of f i n i t e type Fn for some n = 1, 2, 3, 4, 5, 7, 8; since a very large number of u n i f i cations is involved, computer assistance is invaluable here. It is possible that more than 286 of the 630 formulas are of f i n i t e type. 4. MATRIX-TESTING PROGRAM MT In showing that part icular formulas are not single axioms for EC, it is useful to have ava i l able a program MT which, given a logical matrix such as (1) and a part icular formula such as EEpqEqp, determines whether or not the matrix sat isf ies the formula. More generally, given a par t ia l l y completed matrix M and a f i n i t e set S of formulas, MT can search for a l l ways ( i f any) of completing M so that it sat isf ies a l l the formulas in S; in part icular , if M is the void matrix of a given size, MT w i l l search for a l l matrices of that size which satisfy a l l the formulas in S. The purpose of the program MT is similar to that of the interactive program TESTER writ ten by Nuel D. Belnap, Jr . at the University of Pittsburgh and now in use by logicians at a number of univers i t ies in the United States, Br i ta in and Austral ia. 5. SEMANTIC METHOD: LOGICAL MATRICES A logical matrix M is said to be "normal" if it has the property that whenever a, b E M are such that a and Eab are designated, it follows that b is designated. For example, is a normal logical matrix; (2) is essentially the truth table for material implication. The matrix (1) is also normal. It is known that a formula x is not a single axiom for EC if and only if there exists a normal logical matrix M, possibly of i n f i n i t e size, such that x holds in M but some equivalential tautology (e.g. some single axiom for EC) does not hold in M. For example, easy calculations show that the formula EEpqEEqrEpr holds in the matrix (2), but the formula EEpqEErqEpr does not; it follows that EEpqEEqrEpr is not a single axiom for EC. With the help of a collection of 14 logical matrices, the authors have shown that, of the 630 286 = 344 formulas remaining for considerat ion after eliminating 286 formulas of f i n i t e type, 315 are not single axioms for EC. Of the 14 matrices, one (the matrix (2)) is of size 2, 7 are of size 3, 3 are of size 4, 2 are of size 8, and one is of size 10. The program MT is very useful for checking these results, but has not been of great use in finding the matrices: the 6 matrices of sizes 4, 8, and 10 were in fact a l l found by hand. There are 4 matrices of size 4, and a straightforward search to f ind which of these satisfy a given formula of length 11 with the two-property would be far beyond the capacity of MT running on existing computers. 6. SECOND SYNTACTIC METHOD: FORM OF GENERATED FORMULAS Of the 344 315 = 29 formulas now remaining for consideration, Peterson showed how 11 could be rejected as single axioms for EC by syntactic arguments based on the form of the formulas generated when these formulas are given as input to the program TG. Consider for example the question whether the formula x=EpEEEpqEqrr is a single axiom for EC. Examination of the output when x is given as input to the program TG reveals that, apart from the or ig inal formula x, a l l the formulas generated are of the form