Contains reports on three research projects

Lehman, A. B.

Rogers, H., Jr.

Schützenberger, M. P.

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XXII. MATHEMATICSProf. H. Rogers, Jr.Dr. M. P. SchiitzenbergerA. B. LehmanA. ON EFFECTIVE COMPUTABILITYThe theory of general recursive computability (1) concerns computations that can becarried out by a Turing machine ("T-machine," a finite digital computer with indefinitelylarge memory). A function (integers into integers) computable on a T-machine is calleda "recursive" function. In general, all true assertions of the form "T-machine M com-putes a function" (as M varies) cannot be effectively listed (and any formal logic isincomplete with respect to them). There has been interest in various "natural" sub-families of the recursive functions, for which T-machines can be effectively listed. Theprimitive recursive functions are such a subfamily. We suggest the following largerfamily: the recursive functions with (each of) which we can associate a T-machine Mwith the property that the assertion "T-machine M computes a function" is provable asa combinatorial statement within ordinary number theory; we require, of course, thatM compute the function with which it is associated. We call these functions "p-functions"(provable functions); "p-l-l-functions" and "p-permutations" are similarly defined.The last form a group (the improvement over known properties of the primitive recursivefunctions) and give a stronger isomorphism between sets of integers than that given bythe group of all recursive permutations. Many of the isomorphism results of recursivefunction theory can be strengthened to isomorphism under p-permutations. Other resultsinclude that the recursive sets (a set is recursive if membership in it can be decided by aT-machine) fall into an infinity of isomorphism types of which one is minimum (i.e.,most effective) relative to an appropriate ordering by reducibility. We also observe(by simple extension of known results) that the sets of provable theorems of any twoconsistent mathematical systems containing elementary number theory are isomorphicin this strong sense.H. Rogers, Jr.References1. S. C. Kleene, Introduction to Metamathematics (D. Van Nostrand Company, Inc.,New York, 1952).B. A NEW INEQUALITY IN COMMUNICATION THEORYConsider a continuous signal x which is transmitted by making it a parameter onwhich a second random variate y depends. After receiving y, x is estimated in sucha way as to minimize the rms error on both the a priori distribution of x and the155(XXII. MATHEMATICS)conditional distribution of y, for fixed x. Inequalities on the rms error are well knownin mathematical statistics, but they deal exclusively with the special case in which theexact a priori distribution of x is unknown, that is to say, when this distribution ismost unfavorable.By extending the argument, with the help of a fundamental identity in communicationtheory, a lower bound to the root mean square error was found, which reduces to theknown bound in the special case mentioned above. In the general case, the approximationif far more sharp. If y is obtained by adding a random noise to x, it can be provedthat the bound is attained if and only if both the noise and the signal x are normallydistributed.M. P. SchiitzenbergerC. VALUATED ALGEBRASAn abstraction of some relationships common to lattice theory and electric net-works has given rise to a new subfield of abstract algebras, called, at present, "val-uated algebras." The following is an abstract of some of the basic definitions and thedirection of research. They will be presented later in more detail.Let S be a set that is closed under binary operations U and n , and let G be acommutative group under +. A map I1 of S into G satisfying laU(bU a) i = I(aU b)Ucl,lan(bNc)i = I(anb)Oc , and laUbI + lanbI = lai+lbI is called a "valuation." Thisvaluation is a generalization of lattice valuations (1) and network impedance, but is notto be confused with certain other mathematical valuations (2).This definition leads directly to an implicit function theorem (series-parallel recip-rocity) and consequently to some theorems on valuation congruences. In particular,valuation congruences on U or n are valuation congruences on both simultaneously.Proceeding further along the line of lattice valuations, we are led to several problemsin lattices and network transformations. One of the specialized results gives a basisfor a theory of information (3).Self-valuations, valuations in which the U, N and + operations are defined on thesame set, are sometimes extendable beyond series-parallel graphs (polynomials) intogeneral graphs or generalized networks (4). If U (or n ) distributes over +, we are ledto Boolean rings and switching algebra. If + distributes over U (or n ), we are led todivision semirings, L-groups (1), and resistor networks. A simple extendable self-valuation not satisfying either distributive law is derivable from the solution ofC. E. Shannon' s switching game.A. B. LehmanReferences1. G. Birkhoff, Lattice Theory, Vol. 25 (American Mathematical Society Colloquium Publications, New York, rev. ed., 1948).2. O. F. G. Schilling, The Theory of Valuations, Mathematical Surveys No. 4 (American Mathematical Society, New York, 1950).3. T. Von Randow, Lattice Theory of Information (unpublished).4. C. E. Shannon, Network Rings, 1948 (unpublished).156

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