Papers by Salvatore Caporaso
Acta Informatica, Aug 1, 1998
If uniform coding (Gödelization) of potentially infinite sequences of numbers can be performed in... more If uniform coding (Gödelization) of potentially infinite sequences of numbers can be performed in PSPACEF, then PSPACE = EXPTIME = EXPSPACE = 2-EXPTIME, and, for all p, we have p-EXPSPACE = p+1-EXPTIME; if it can be performed in LINSPACEF, we also have cx ); the proof fails, when relativized to oracle-TM's. A by-product of this research is that PTIMEF is not closed under number-theoretic, limited, course-of-values recursion.
Archive for Mathematical Logic, Dec 1, 1978
arXiv (Cornell University), Sep 23, 2006
A decidable transfinite hierarchy is defined by assigning ordinals to the programs of an imperati... more A decidable transfinite hierarchy is defined by assigning ordinals to the programs of an imperative language. It singles out: the classes TIMEF(n c ) and TIMEF(n c ); the finite Grzegorczyk classes at and above the elementary level, and the Σ k -IND fragments of PA. Limited operators, diagonalization, and majorization functions are not used.
arXiv (Cornell University), Dec 8, 2007
The recursion theorem in the weak form {e}(z) = x(e, z) (universal function not needed) and in Ro... more The recursion theorem in the weak form {e}(z) = x(e, z) (universal function not needed) and in Rogers form φ φ f (n) (z) = φn(z), and Rice theorem are proved a first time using programs in C, and a second time with scripts in Bash.
Information Processing Letters, 2006
A language is defined by closure under safe iteration and under a new form of safe diagonalizatio... more A language is defined by closure under safe iteration and under a new form of safe diagonalization that, unlike other forms of diagonalization used in literature to define sub-recursive hierarchies, is constructive and decidable. By counting the nesting levels of these schemes, an ordinal is assigned to each program. This yields a hierarchy T α (α < ω ω ) that singles-out the complexity classes DTIMEF(n cn d +e ) for all c, d, e ≥ 0.
International Journal of Foundations of Computer Science, Sep 1, 1996
We introduce a class of safe Turing machines which execute structured while, if-then- else progra... more We introduce a class of safe Turing machines which execute structured while, if-then- else programs and operate on stacks and on a read-only input tape. A hierarchy is obtained by taking as Si the class of all functions computed by programs of loop-depth i. The main result is that S1 equals Lintime and S2 equals Polytime while, for i≥3, we have that Si equals the i-th Grzegorczyk class. By adding to the language a non-deterministic construct choose we take S2 into a class equivalent to NP. This gives a syntactical characterization in a pure-machine model of the mentioned classes.
Corr, Sep 23, 2006
A decidable transfinite hierarchy is defined by assigning ordinals to the programs of an imperati... more A decidable transfinite hierarchy is defined by assigning ordinals to the programs of an imperative language. It singles out: the classes TIMEF(n c ) and TIMEF(n c ); the finite Grzegorczyk classes at and above the elementary level, and the Σ k -IND fragments of PA. Limited operators, diagonalization, and majorization functions are not used.
Lecture Notes in Computer Science, 2000
A resource-free characterization of some complexity classes is given by means of the predicative ... more A resource-free characterization of some complexity classes is given by means of the predicative recursion and constructive diagonal-ization schemes, and of restrictions to substitution. Among other classes, we predicatively harmonize in the same hierarchy ...
Theoretical Computer Science, 2001
Characterizations of PTIME, PSPACE, the polynomial hierarchy and its elements are given, which ar... more Characterizations of PTIME, PSPACE, the polynomial hierarchy and its elements are given, which are decidable (membership can be decided by syntactic inspection to the constructions), predicative (according to points of view by Leivant and others), and are obtained by means of increasing restrictions to course-of-values recursion on trees (represented in a dialect of Lisp).
The recursion theorem in the weak form {e}(z) = x(e,z) (universal function not needed) and in Rog... more The recursion theorem in the weak form {e}(z) = x(e,z) (universal function not needed) and in Rogers form ��f(n)(z) = �n(z), and Rice theorem are proved a first time using programs in C, and a second time with scripts in Bash.
Arxiv preprint arXiv:0712.1279, 2007
Abstract: The recursion theorem in the weak form {e}(z)= x (e, z)(universal function not needed) ... more Abstract: The recursion theorem in the weak form {e}(z)= x (e, z)(universal function not needed) and in Rogers form {n}(z)={{x}(n)}(z) and Rice theorem are proved a first time using programs in C, and a second time with scripts in Bash.
Journal of Functional Programming, 2001
We harmonize many time-complexity classes DTIMEF(f(n)) (f(n) [ges ] n) with the PR functions (at ... more We harmonize many time-complexity classes DTIMEF(f(n)) (f(n) [ges ] n) with the PR functions (at and above the elementary level) in a transfinite hierarchy of classes of functions [Tscr ]α. Class [Tscr ]α is obtained by means of unlimited operators, namely: a variant Π of the predicative or safe recursion scheme, introduced by Leivant, and by Bellantoni and Cook, if α is a successor; and constructive diagonalization if α is a limit. Substitution (SBST) is discarded because the time complexity classes are not closed under this scheme. [Tscr ]α is a structure for the PR functions finer than [Escr ]α, to the point that we have [Tscr ]ε0 = [Escr ]3 (elementary functions). Although no explicit use is made of hierarchy functions, it is proved that f(n) ∈ [Tscr ]α implies f(n) [les ] nGα(n), where Gα belongs to the slow growing hierarchy (of functions) studied, in particular, by Girard and Wainer.
International Journal of Foundations of Computer Science, 1996
We introduce a class of safe Turing machines which execute structured while, if-then- else progra... more We introduce a class of safe Turing machines which execute structured while, if-then- else programs and operate on stacks and on a read-only input tape. A hierarchy is obtained by taking as Si the class of all functions computed by programs of loop-depth i. The main result is that S1 equals Lintime and S2 equals Polytime while, for i≥3, we have that Si equals the i-th Grzegorczyk class. By adding to the language a non-deterministic construct choose we take S2 into a class equivalent to NP. This gives a syntactical characterization in a pure-machine model of the mentioned classes.
Information Processing Letters, 2006
A language is defined by closure under safe iteration and under a new form of safe diagonalizatio... more A language is defined by closure under safe iteration and under a new form of safe diagonalization that, unlike other forms of diagonalization used in literature to define sub-recursive hierarchies, is constructive and decidable. By counting the nesting levels of these schemes, an ordinal is assigned to each program. This yields a hierarchy T α (α < ω ω ) that singles-out the complexity classes DTIMEF(n cn d +e ) for all c, d, e ≥ 0.
Proceedings of the 6th WSEAS …, 2007
We extend the Implicit Computational Complexity program, promoted by Leivant and by other scholar... more We extend the Implicit Computational Complexity program, promoted by Leivant and by other scholars, to all complexity classes DTIMESPACEF(f (n), g(n)), between DTIMEF(n) and DSPACEF(n n c ). Let clps(α, n) denote the result of replacing ω by n in Cantor normal form for α < ω ω ω . A hierarchy T S αβ is defined by means of a very restricted form of substitution, and of two un-limited operators (simultaneous predicative recurrence and constructive diagonalization), and it is proved that DTIMESPACEF(n clps (β,n) , n clps(α,n) ) = T S αβ . For example DTIMESPACEF(n 2 , n n ) = T S ω ω ,2 .
Archiv für mathematische Logik und Grundlagenforschung, 1980
ABSTRACT
Archiv für Mathematische Logik und Grundlagenforschung, 1978
Acta Informatica, 1998
If uniform coding (Gödelization) of potentially infinite sequences of numbers can be performed in... more If uniform coding (Gödelization) of potentially infinite sequences of numbers can be performed in PSPACEF, then PSPACE = EXPTIME = EXPSPACE = 2-EXPTIME, and, for all p, we have p-EXPSPACE = p+1-EXPTIME; if it can be performed in LINSPACEF, we also have cx ); the proof fails, when relativized to oracle-TM's. A by-product of this research is that PTIMEF is not closed under number-theoretic, limited, course-of-values recursion.
J. Univers. Comput. Sci., 2000
A class LT 0 of functions computable in a proper sub-class of Lintime is de ned, and formalized i... more A class LT 0 of functions computable in a proper sub-class of Lintime is de ned, and formalized in a system LT0 of monadic and atomic (quanti er-free) logic. In spite of its poor computational complexity power and logical apparatus, this system has enough power to describe its own proof-predicate. Therefore it might qualify as smallest known system in which Godel-like diagonalization can be applied. A proof is given that the identically true functions of LT 0 are productive. Hence this incompleteness phenomenon doesn't depend on the technicalities adopted to show it.
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Papers by Salvatore Caporaso