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Mathematically Strong Subsystems of Analysis With Low Rate of Growth of Provably Recursive Functionals
, 1995
"... 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 ..."
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Cited by 39 (23 self)
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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 quantifierfree choice ACqf 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 settheoretic model but in the strongly majorizable functionals): From a proof GnA # +ACqf + # # #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 settheoretic 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 # +ACqf+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 # +ACqf+# for suitable #. 1
The Descriptive Complexity Approach to LOGCFL
, 1998
"... Building upon the known generalizedquantifierbased firstorder 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 ..."
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Cited by 14 (5 self)
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Building upon the known generalizedquantifierbased firstorder 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 contextfree languages, and we obtain the surprising result that a variant of Greibach's "hardest contextfree language" is LOGCFLcomplete under quantifierfree BITfree 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 firstorder logic with majority of pairs is strictly more expressive than firstorder with major...
String Rewriting and Metabolism: A Logical Perspective
 Computing with BioMolecules. Theory and Experiments
, 1998
"... 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 ..."
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Cited by 5 (2 self)
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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
Arithmetic, firstorder logic, and counting quantifiers
 ACM TRANS. COMPUT. LOG
, 2004
"... This paper gives a thorough overview of what is known about firstorder logic with counting quanti ers and with arithmetic predicates. As a main theorem we show that Presburger arithmetic is closed under unary counting quantifiers. Precisely, this means that for every firstorder formula '(y; ~ ..."
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Cited by 4 (1 self)
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This paper gives a thorough overview of what is known about firstorder logic with counting quanti ers and with arithmetic predicates. As a main theorem we show that Presburger arithmetic is closed under unary counting quantifiers. Precisely, this means that for every firstorder formula '(y; ~z) over the signature f<;+g there is a firstorder formula (x; ~z) which expresses over the structure hN;<;+i (respectively, over initial segments of this structure) that the variable x is interpreted exactly by the number of possible interpretations of the variable y for which the formula '(y; ~z) is satised. Applying this theorem, we obtain an easy proof of Ruhl's result that reachability (and similarly, connectivity) in nite graphs is not expressible in firstorder logic with unary counting quantiers and addition. Furthermore, the above result on Presburger arithmetic helps to show the failure of a particular version of the Crane Beach conjecture.
A Survey of Arithmetical Definability
, 2002
"... We survey definability and decidability issues related to firstorder fragments of arithmetic, with a special emphasis on Presburger and Skolem arithmetic and their (un)decidable extensions. ..."
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Cited by 2 (0 self)
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We survey definability and decidability issues related to firstorder fragments of arithmetic, with a special emphasis on Presburger and Skolem arithmetic and their (un)decidable extensions.
Lisp ProgramSize Complexity II
, 1992
"... We present the informationtheoretic incompleteness theorems that arise in a theory of programsize 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 associa ..."
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We present the informationtheoretic incompleteness theorems that arise in a theory of programsize 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 Sexpressions, 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 easytounderstand definition ...
Nondeterministic Stack Register Machines
, 1996
"... 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 r ..."
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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:
The Ackermann functions are not optimal, but by how much
 J. Symbolic Logic
"... By taking a closer look at the construction of an Ackermann function we see that between any primitive recursive degree and its Ackermann modification there is a dense chain of primitive recursive degrees. ..."
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By taking a closer look at the construction of an Ackermann function we see that between any primitive recursive degree and its Ackermann modification there is a dense chain of primitive recursive degrees.
The Diagonal Polynomials of Dimension Four
"... 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. ..."
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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.
Theories of arithmetics in finite models
"... We investigate theories of initial segments of the standard models for arithmetics. It is easy to see that if the ordering relation is definable in the standard model then the decidability results can be transferred from the infinite model into the finite models. On the contrary we show that the Σ2– ..."
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We investigate theories of initial segments of the standard models for arithmetics. It is easy to see that if the ordering relation is definable in the standard model then the decidability results can be transferred from the infinite model into the finite models. On the contrary we show that the Σ2–theory of multiplication is undecidable in finite models. We show that this result is optimal by proving that the Σ1–theory of multiplication and order is decidable in finite models as well as in the standard model. We show also that the exponentiation function is definable in finite models by a formula of arithmetic with multiplication and that one can define in finite models the arithmetic of addition and multiplication with the concatenation operation. We consider also the spectrum problem. We show that the spectrum of arithmetic with multiplication and arithmetic with exponentiation is strictly contained in the spectrum of arithmetic with addition and multiplication. 1