J. Richard Buchi

On a Decision Method in Restricted Second Order Arithmetic

Let SC be the interpreted formalism which makes use of individual variables t, x, y, z,... ranging over natural numbers, monadic predicate variables q( ), r( ), s( ), i( ),... ranging over arbitrary sets of natural numbers, the individual symbol 0 standing for zero, the function symbol ′ denoting the successor function, propositional connectives, and quantifiers for both types of variables. Thus SC is a fraction of the restricted second order theory of natural numbers, or of the first order theory of real numbers. In fact, if predicates on natural numbers are interpreted as binary expansions of real numbers, it is easy to see that SC is equivalent to the first order theory of [Re, +, Pw, Nn], whereby Re, Pw, Nn are, respectively, the sets of non-negative reals, integral powers of 2, and natural numbers.

Turing-Machines and the Entscheidungsproblem

Let Q be the set of all sentences of elementary quantification theory (without equality). In its semantic version Hilbert’s Entscheidungsproblem for a class X ⊇ Q of sentences is, [X]: To find a method which for every S ∈ X yields a decision as to whether or not S is satisfiable.

Symposium on Decision Problems: On a Decision Method in Restricted Second Order Arithmetic

Publisher Summary In this chapter, SC is a fraction of the restricted second order theory of natural numbers or of the first order theory of real numbers. This chapter discusses the definability in SC and outlines an effective method for deciding the truth of sentences in SC. A congruence of finite rank on words is in congruence with the finite partition of concatenation; a multi-periodic set of words is a union of congruence classes of a congruence of finite rank. These concepts are related to that of a finite automaton and turn out to be the key to an investigation of SC. The results concerning SC may be viewed as an application of the theory of finite automata to logic. In turn, SC arises quite naturally as a condition-language on finite automata or sequential circuits and “sequential calculus” is an appropriate name for SC. The significance of the decision method for SC is that it provides a method for deciding whether or not the input (i)-to-output (u) transformation of a proposed circuit A (i, r, u) satisfies a condition C (i, u) stated in SC.

Representation of Complete Lattices by Sets

Throughout this paper L denotes a complete lattice. That is a set of elements a, b, c, ⋯ partly ordered by a relation «⊂» such that to every class (aν) of elements aν in L there exists a greatest lower bound \( \mathop{ \cup }\limits_{\nu } \) aν in L. Then also the least upper bound \( \mathop{ \cup }\limits_{\nu } \)aν exists in L and furthermore there is a greatest element e in L, namely the g. l. b. of the empty subclass of L, and there exists the smallest element 0 in L, the g. l. b. of all elements of L.

The Theory of Proportionality as an Abstraction of Group Theory.

In this paper we present an axiomatic theory denoted by T which is an abstraction of group theory in the sense that projective geometry is an abstraction of affine geometry. Extending to group theory, F. Klein’s idea, that a geometry is determined by its group of automorphisms, we describe T as the theory whose group of automorphisms is generated by the automorphisms and translations of group theory. In axiomatic terms the relationship between the two theories is characterized by the fact that from T we can arrive at group theory by distinguishing an element, i. e., by introducing a name for the group identity. This situation is analogous to that in geometry, where affine geometry is derived from projective geometry by the introduction of a name for the lines at infinity.

Invariants of the Anti-Automorphisms of a Group

The background for this paper is provided by Klein’s work presented in the Erlangerprogram [1], and more recent developments of these ideas as well as their application outside the field of geometry [2; 3; 4; 5; 6].

INVESTIGATION OF THE EQUIVALENCE OF THE AXIOM OF CHOICE AND ZORN'S LEMMA FROM THE VIEWPOINT OF THE HIERARCHY OF TYPES

It is a well known fact that Zermelo’s Axiom of Choice and Zorn’s Lemma are equivalent logical assumptions. However, an investigation from the viewpoint of the hierarchy of types reveals a complication in the nature of this equivalence. In a type-theoretical formalism both postulates enter as spectra (ZA α ) and (ZL α ) of formulas, where ZA α is Zermelo’s Axiom and ZL α is Zorn’s Lemma stated for variables of a fixed type α. The complication has its origin in the fact that, although the formula ZA α implies ZL α , in order to deduce ZA α it seems to be necessary to assert Zorn’s Lemma ZL α for a type β which is higher than α. In other words, we don’t know a proof assuring the equivalence of the two formulas ZA α and ZL α . Therefore the equivalence of Zermelo’s Axiom and Zorn’s Lemma has to be understood in the sense that the assertion of the formulas ZA α for all types α implies every one of the formulas ZL β and conversely the assertion of the spectrum of formulas (ZL α ) implies every one of the formulas ZA β .