This paper is the first one in a sequel of papers resulting from the authors Habilitationsschrift [22] which are devoted to determine the growth in proofs of standard parts of analysis. A hierarchy (GnA # )n#I N of systems of arithmetic in all finite types is introduced whose definable objects of type 1 = 0(0) correspond to the Grzegorczyk hierarchy of primitive recursive functions. We establish the following extraction rule for an extension of GnA # by quantifier--free choice AC--qf and analytical axioms # having the form #x # #y ## sx#z # F0 (including also a `non-- standard' axiom F - which does not hold in the full set--theoretic model but in the strongly majorizable functionals): From a proof GnA # +AC--qf + # # #u 1 , k 0 #v ## tuk#w 0 A0(u, k, v, w) one can extract a uniform bound # such that #u 1 , k 0 #v ## tuk#w # #ukA0 (u, k, v, w) holds in the full set--theoretic type structure. In case n = 2 (resp. n = 3) #uk is a polynomial (resp. an elementary recursive function) in k, u M := #x. max(u0, . . . , ux). In the present paper we show that for n # 2, GnA # +AC--qf+F - proves a generalization of the binary Knig's lemma yielding new conservation results since the conclusion of the above rule can be verified in G max(3,n) A # in this case. In a subsequent paper we will show that many important ine#ective analytical principles and theorems can be proved already in G2A # +AC--qf+# for suitable #. 1

Building upon the known generalized-quantifier-based first-order characterization of LOGCFL, we lay the groundwork for a deeper investigation. Specifically, we examine subclasses of LOGCFL arising from varying the arity and nesting of groupoidal quantifiers. Our work extends the elaborate theory relating monoidal quantifiers to NC and its subclasses. In the absence of the BIT predicate, we resolve the main issues: we show in particular that no single outermost unary groupoidal quantifier with FO can capture all the context-free languages, and we obtain the surprising result that a variant of Greibach's "hardest context-free language" is LOGCFL-complete under quantifier-free BIT-free projections. We then prove that FO with unary groupoidal quantifiers is strictly more expressive with the BIT predicate than without. Considering a particular groupoidal quantifier, we prove that first-order logic with majority of pairs is strictly more expressive than first-order with major...

We propose an improved definition of the complexity of a formal axiomatic system: this is now taken to be the minimum size of a selfdelimiting program for enumerating the set of theorems of the formal system. Using this new definition, we show (a) that no formal system of complexity n can exhibit a specific object with complexity greater than n + c, and (b) that a formal system of complexity n can determine at most n + c scattered bits of the halting probability\Omega . We also present a short, self-contained proof of (b).

this paper how a general and rigorous definition of a string generative system, based on the above triple, can be developed in logical terms. In such a perspective, we indicate how all the important aspects of the usual systems can be formally and uniformly described. A rewriting relation ) can be logically represented if it is possible to determine a suitable first order model

We survey de nability and decidability issues related to rst-order fragments of arithmetic, with a special emphasis on Presburger and Skolem arithmetic and their (un)decidable extensions.

For integer k 0, let srm(n O(1) ; k) denote the collection of relations computable by a stack register machine with stack registers bounded by a polynomial p(n) in the input n, and work registers bounded by k. Let nsrm(n O(1) ; k) denote the analogous class accepted by nondeterministic stack register machines. In this paper, nondeterminism is shown to provide no additional power. Specifically, nsrm(n O(1) ; 0) = srm(n O(1) ; 0) nsrm(n O(1) ; 1) = srm(n O(1) ; 1) nsrm(n O(1) ; k) = srm(n O(1) ; k); for k 4 srm(n O(1) ; k) = alintime ; for k 4:

We present the information-theoretic incompleteness theorems that arise in a theory of program-size complexity based on something close to real LISP. The complexity of a formal axiomatic system is defined to be the minimum size in characters of a LISP definition of the proofchecking function associated with the formal system. Using this concrete and easy to understand definition, we show (a) that it is difficult to exhibit complex S-expressions, and (b) that it is difficult to determine the bits of the LISP halting probability\Omega LISP . We also construct improved versions\Omega 0 LISP and\Omega 00 LISP of the LISP halting probability that asymptotically have maximum possible LISP complexity. Copyright c fl 1992, Elsevier Science Publishing Co., Inc., reprinted by permission. 2 G. J. Chaitin 1. Introduction The main incompleteness theorems of myAlgorithmic Information Theory monograph [1] are reformulated and proved here using a concrete and easy-to-understand definition ...

Building upon the known generalized-quantifier-based first-order characterization of LOGCFL, we lay the groundwork for a deeper investigation. Specifically, we examine subclasses of LOGCFL arising from varying the arity and nesting of groupoidal quantifiers. Our work extends the elaborate theory relating monoidal quantifiers to NC 1 and its subclasses. In the absence of the BIT predicate, we resolve the main issues: we show in particular that no single outermost unary groupoidal quantifier with FO can capture all the context-free languages, and we obtain the surprising result that a variant of Greibach's "hardest context-free language" is LOGCFL-complete under quantifier-free BIT-free projections. We then prove that FO with unary groupoidal quantifiers is strictly more expressive with the BIT predicate than without. Considering a particular groupoidal quantifier, we prove that first-order logic with majority of pairs is strictly more expressive than first-order with major...

) V. Manca June 23, 1997 ADDR: Corso Italia 40,56125 Pisa,Italy. TEL: +39-50-887111. FAX: +39-50-887226 Logical Formalizations of Syntactical Properties (Extended Abstract) Vincenzo Manca Universit`a di Pisa Dipartimento di Informatica Corso Italia, 40 - 56125 Pisa - Italy e-mail: mancav@di.unipi.it 1 Introduction Assume the `marvelous' 7 logical symbols: !;:;;;$;8;9 with the standard syntactical and semantical first order logical notions (equality predicate = is assumed in its usual usage) that can be found in introductory treatises or basic chapters of textbooks in mathematical logic (cf., for example, [1], [6], [15], [16], [3], [2] [13]): 1. variables and constants 2. predicates and functors 3. first order formulae and terms 4. free variables, bound variables, and substitutions 5. connective truth tables VI Tarragona Seminar on Formal Syntax and Semantics, University Rovira i Virgili, Tarragona, Spain, October 14-18 1996 (Monday Talk). 6. first order structures,...

We establish the fact that for dimension four there are exactly six diagonal normalized polynomials as has been conjectured for some time. They are determined on the grounds of both theoretical as well as computational results.