Showing papers on "Recursively enumerable language published in 1967"

Recursion, Metarecursion, and Inclusion

Abstract: In this paper it will be shown that the ordering of the recursively enumerable (r.e.) sets under inclusion modulo finite differences (m.f.d.), the ordering of the II1 1 1 sets under inclusion m.f.d., and the ordering of the metarecursively enumerable (meta-r.e.) sets under inclusion m.f.d. are all distinct. In fact, it will be shown that the three orderings are pairwise elementarily inequivalent when interpreted in the obvious way in a first order language with one binary relation “≥.” Our result answers a question of Hartley Rogers, Jr. [3, p. 203]. All necessary background material may be found in [2] and [4].

On a theorem of lachlan and martin

Abstract: In [3] we raised the following question: does there exist a recursively enumerable degree d such that 0(n)< <(n) < O(n+1) for all n ?0? This question was answered affirmatively by Lachlan [I] and by Martin [21. Lachlan's proof combines a familiar priority argument with the fixed point theorem of Kleene. Martin's proof is a new form of priority argument based on Theorem 3 of section 6 of [3]. In this paper we give a vanishingly short proof of the Lachlan-Martin result without any use of priority. Our argument is an exercise in the fixed point theorem. We exploit, possibly for the first time, a uniformity concealed in most proofs of the fixed point theorem. Let A be an arbitrary set of natural numbers, and let WA, WA, W2, be a standard simultaneous enumeration of all sets recursively enumerable in A. For each e ? 0, let

Recursively Enumerable Degrees and the Degrees Less Than 0(1)

Abstract: Publisher Summary This chapter presents results to strengthen Shoenfield's result by proving that there is a degree a between 0 and 0( 1 ), which is incomparable with every recursively enumerable (r.e.) degree between 0 and 0 ( 1 ). Shoenfield proved that there is a degree between 0 and 0 ( 1 ), which is not recursively enumerable. The chapter focuses on a string of a finite sequence of ones and twos. The number of elements of a string σ will be denoted by length ( σ ), the n -th element by σ ( n ), and the initial segment of σ , which has length n by σ [ n ]. Because a set T is identified with its representing function and this in turn can be considered as an infinite sequence of ones and twos, σ ⊂ T to mean that σ represents T on the segment of the integers, which has length equal to length ( σ ). There exist a number of obvious recursive enumerations of the set of all strings and recursion theory can be proved with strings just as easily as with integers. This is usually done indirectly by explicitly representing strings by integers.

Remarks on an Infinitary Language with Constructive Formulas

Abstract: (I) extensions obtained by allowing higher order variables, and (II) extensions obtained by allowing infinitely long formulas. Amongst the extensions of type (II) can be found the languages Las which are obtained by modifying the formation rules for the formulas so that conjunctions/ disjunctions of less than a formulas and quantifications over sequences of individual variables of length smaller than fi are allowed. Because no second (or higher) order variables occur amongst the formulas of La8, La8 is sometimes called a first-order language, however it was noticed by Scott in [S] that L.,., (and hence Lag a > W1, ft > w1) behaves, in comparison to L.,C much more closely to a second-order language. Results of Kino/Takeuti (cf. Theorems 2 and 3 in [K/T]) give further support to Scott's remark. One of the advantages of the extensions of type (I) over those of type (II) is that the formulas are constructively defined (even though sometimes the set of formulas is not recursive). Kino/Takeuti have introduced a method of obtaining an extension of type (II) in which the formulas are (in a certain sense) constructively defined. Loosely speaking, they considered the subset of LC1.1 consisting of those formulas A e LC1 such that the conjunctions/disjunctions occurring in A are of recursively enumerable sets of formulas and the quantifications are over recursively enumerable sets of individual variables (we shall denote the language of Kino/Takeuti by 'CL<,,1,,'). Theorems 2 and 3 of [K/T] show that CL1,,J,, coupled with the usual satisfaction relation (as in [T2]), is a very powerful formalism. The price paid for having such a powerful system is the complexity of the set Tr,,l,, of (Gddel numbers of) the valid sentences of CL1,,i W,. A simple modification of Tarski's method in [TI] shows that Tr,,,1,,1 is not analytical, in fact Scott's incompleteness proof for

Predicates definable over transformational derivations by intersection with regular languages

Abstract: The main result of this paper is that a regular-based transformational grammar which meets the condition on recoverability of deletions and which uses filter and postcyclic transformations can generate any recursively enumerable language. From this follows immediately that context free or context sensitive based transformational grammars which meet the condition on recoverability of deletions and which use filter and postcyclic transformations can generate any recursively enumerable language. These problems are investigated with a view to their implications for the current framework of transformational theory of grammars of natural languages.

A Remark on Post Normal Systems

Abstract: Let N(T) be the normal system of Post which corresponds to the Thue system, T, as in Martin Davis, Computability and Unsolvability (McGraw-Hill, New York, 1958), pp. 98-100. It is proved that for any recursively enumerable degree of unsolvability, D, there exists a normal system, NT(D), such that the decision problem for NT(D) is of degree D. Define a generalized normal system as a normal system without initial assertion. For such a GN the decision problem is to determine for any enunciations A and B whether or not A and B are equivalent in GN. Thus the generalized system corresponds more naturally to algebraic problems. It is proved that for any recursively enumerable degree of unsolvability, D, there exists a generalized normal system, GNT(D), such that the decision problem for GNT(D), such that the decision problem for GNT(D) is of degree D.

Nonaxiomatizability results for infinitary systems

Abstract: Methods are being developed for treating questions of decidability in fewer than Q steps, Q being a regular2 nondenumerable cardinal. In this paper we consider the set-theoretical predicate Taut(x), "x is a tautology," taken in the infinitary sense. In case x is hereditarily finite there is no question that it is decidable in finitely many steps. But what if x is infinite? We are not assuming that x is in any way constructive or even that the propositional formulas can be well-ordered, so it is not appropriate to treat this predicate as one of natural numbers or even as one of ordinals. There is available however the notion of X1-predicate of sets, studied by Levy in [4], which reduces to a notion equivalent to recursively enumerable predicate of natural numbers when the universe is restricted to the hereditarily finite sets. The El-predicates retain their original meaning when the universe is restricted to Ha, the set of sets hereditarily of power less than Q2 (Theorem 1.4). Call a predicate on Ho a Zn-predicate if it is the restriction to Ha of a E1-predicate, a HIe-predicate if its complement with respect to H. is a V-predicate. There is little doubt that a subset C of H0 whose elements are generable in fewer than n steps by some effective procedure, is EQ. This is because it always seems to be possible to attach to an effective procedure a Zermelo-Fraenkel formula W(v0, vl) in such a way that if if is satisfied by (x0, x1) in a transitive set, then x1 is the output for input x0, and conversely, if x1 is the output for input x0, then there is a transitive set in which (x0, x1) satisfies W. Thus x1 is the output for x0 iff (3t)(Trans(t) A x0, x1 E t A }= 'e(xo, xj)). But the satisfaction predicate relative to a set is II 1 r) . (See Levy [4]. The calculus of Appendix I suffices for this.) Thus the condition, ''x1 is the output for x0," is El. But then the predicate x1 E C for x1 E H0 is equivalent to (3x0) (xl is the output for x0), a 1,-predicate. By Theorem 1.4, C is El. Returning to the predicate Taut(x) and its restriction Q-Taut(x) to x Ec Ha, the results in this paper specify the Q for which Q-Taut is El (assuming the continuum hypothesis (CH)) and specify also the Q for which Q-Taut is En, given a parameter in H0 (assuming the generalized continuum hypothesis (GCH)). These results plus the completeness theorems in [2] completely settle the question of ?-effective axiomatizability for ?2-propositional calculi assuming GCH, for it is precisely in the positive cases, the cases where Q-Taut is En given a parameter in Ho, that

Infinite subclasses of recursively enumerable classes

Abstract: P. R. Young [1 ] has constructed an infinite recursively enumerable (r.e.) class with no proper infinite r.e. subclasses, and has asked if infinite r.e. classes with m+1 infinite r.e. subclasses exist for every m>O. It can further be asked what is the most general partially ordered set we can represent by the infinite r.e. subclasses of such a class (under inclusion). These questions are answered by the theorem below. The author wishes to thank A. H. Lachlan for his guidance and encouragement. Our construction is based on a formulation of Young's due to Lachlan.

No recursively enumerable set is the union of finitely many immune retraceable sets

Abstract: In what follows, by retracing function we will always mean special retracing function: a partial recursive function f whose range is contained in its domain and which is associated with a unique point b such that for each x in the domain of f, f(x) x and for some n, fn (x) = b. Suppose R were a recursively enumerable set which was the union of n immune retraceable sets with n as small as possible. Then R must be infinite by definition of immunity. Let R=S1U .. USn where each Si is retraced by partial recursive function fi to basepoint bi. We observe that if T is any infinite recursive subset of R, for each i, Tn'Si is immune retraceable. For let B= {bi, b2, * * , bn} and T* = TUB. Then T* = S* U ... US* wrhere St = T*flSi. Now St is retraceable for it is retraced by the function f* where domain ft = T n Dom f, and xEDom f =>ft (x) is the largest element y of T* such that for some m f7(x) =y. Hence St could only be immune or finite, but the minimality of n insures that it must be immune. But B is finite so our conclusion holds without adjunction of B. Next, we use the previous observation to show that the intersection of the domains of the fi is infinite. For if this were not the case we could choose a largest subcollection of the fi whose domains had infinite intersection and enumerate the intersection of their domains. Tlhis infinite recursively enumerable subset of R contains only finitely imany members of the domain of some fj not in the subcollection hence only finitely many members of the corresponding Sj. But this set contains an infinite recursive subset with finite intersection with Sj contradicting our observation. Since the intersection of the domains of thefi is infinite recursively enumerable we may choose an infinite recursive subset of this intersection which (by our observation) is the union of n immune retraceable sets; or equivalently, assume that R is recursive and each fi has R as domain. If, for some n, fn(x) =y we will write i: x->y. If this happens for no n, we write i: x-f-y. We notice that if yE Sj then {z|j:zy} is finite. For if it were infinite, since if it is disjoint from Sj by the defi-

Some theorems on extensions of arithmetic

Abstract: ?1. Introduction. In this paper we study some of the consequences of adding as new axioms to consistent re (recursively enumerable) extensions T of Peano arithmetic P sentences undecidable in Tof the fl10 variety, as are those one naturally comes upon. The question arises: how much can two extensions "differ" when each is obtained by adding just once distinct fl10 sentences undecidable in T? We shall show that if T is as stated above, we can find extensions T, and T2, constructed very simply from T, having the property that if (A, B) is any pair of re sets with recursive intersection, there can be found a flI1 formula of P which represents A in T, and B in T2. Another question concerns the relative strength of flI1 undecidable sentences. In this connection, the position of an undecidable sentence which asserts the consistency of Tin a direct, legitimate way is of special interest; it is known that there are plenty of undecidable 1110 sentences S which are stronger than the consistency statement in the sense that the latter becomes provable when S is adjoined as a new axiom, whereas the reverse is not the case. Also, the completeness result of Turing, established in his fundamental paper [9], implies that a consistency statement for any system in a recursive progression of theories based on reflection principle I of [6, p. 289], becomes provable at level W + 1. In fact, this holds true of progressions which proceed by the successive addition of consistency statements, [6, p. 287], and hence in some progressions in which the new axioms are always flH1 sentences. We shall show, on the other hand, that there is a

CREATIVE AND WEAKLY CREATIVE SEQUENCES OF r.e. SETS

Abstract: 1. In [1] Cleave introduced the notion of a creative sequence of r.e. (recursively enumerable) sets and proved that all such sequences are r. (recursively) isomorphic and 1-1 universal for the class of all r.e. sequences of r.e. sets. In [2] and [3] Lachlan introduced an alternate definition and proved its equivalency with the definition of Cleave. A sequence of r.e. sets Eo, E1, is called r.e. iff there is an r. function g such that Ei=wg(i) for every iEEN, where