## Subrecursion as Basis for a Feasible Programming Language (1994)

Venue: | Proceedings of CSL'94, number 933 in LNCS |

Citations: | 9 - 8 self |

### BibTeX

@INPROCEEDINGS{Voda94subrecursionas,

author = {Paul J. Voda},

title = {Subrecursion as Basis for a Feasible Programming Language},

booktitle = {Proceedings of CSL'94, number 933 in LNCS},

year = {1994},

pages = {324--338},

publisher = {Springer Verlag}

}

### OpenURL

### Abstract

We are motivated by finding a good basis for the semantics of programming languages and investigate small classes in subrecursive hierarchies of functions. We do this with the help of pairing functions because in this way we can explore the amazing coding powers of S-expressions of LISP within the domain of natural numbers. In the process of doing this we introduce a missing stage in Grzegorczyk-based hierarchies which solves the longstanding open problem of what is the precise relation between the small recursive classes and those of complexity theory. 1 Introduction We investigate subrecursive hierarchies based on pairing functions and solve a longstanding open problem in small recursive classes of what is the relationship between these and computational complexity classes (see [11]). The problem is solved by discovering that there is a missing stage in Grzegorczyk-based hierarchies [7, 11]. The motivation for this research comes from our search for a good programming langu...

### Citations

534 | Concrete Mathematics
- Graham, Knuth, et al.
- 1994
(Show Context)
Citation Context ...e length of the interval is the number of ways an expression consisting of n binary infix operators can be parenthesized. These numbers are known as Catalan numbers C (n) = 1 n+1 \Delta ` 2n n ' (see =-=[6]-=-). By convention C (0) = 1 and this is also the length of the closed interval [0; 0] consisting of the numbers of the pair size 0. By a straightforward manipulation of binomial coefficients we get C (... |

150 |
The intrinsic computational difficulty of functions
- Cobham
- 1964
(Show Context)
Citation Context ...i, etc. The Turing machine characterization of PO 2 can be also proved from the theorem 22 by the result of Ritchie [10] that E 2 consists of n-ary functions Turing computable in linear space. Cobham =-=[2, 11]-=- was the first one to characterize the n-ary functions computable in polynomial time by an inductively defined class based on the bit size recursion with the length of iteration d(x). Let us see why t... |

114 |
Computability and Unsolvability
- Davis
- 1958
(Show Context)
Citation Context ... gives the length of the finite sequence coded by x. We have Len(x)sjxj. There are many semi-suitable pairing functions. For instance, if we offset the standard recursion-theoretic pairing function J =-=[3]-=- by one, i.e. if J(x; y) = ((x + y) \Delta (x + y + 1)) \Xi 2 + x + 1 we get a semi-suitable pairing function. The function J is not good for our purposes as it is not a suitable pairing function sati... |

59 |
Subrecursion: Functions and Hierarchies
- Rose
- 1984
(Show Context)
Citation Context ...ubrecursive hierarchies based on pairing functions and solve a longstanding open problem in small recursive classes of what is the relationship between these and computational complexity classes (see =-=[11]-=-). The problem is solved by discovering that there is a missing stage in Grzegorczyk-based hierarchies [7, 11]. The motivation for this research comes from our search for a good programming language. ... |

42 |
Some classes of recursive functions
- Grzegorczyk
- 1953
(Show Context)
Citation Context ... classes of what is the relationship between these and computational complexity classes (see [11]). The problem is solved by discovering that there is a missing stage in Grzegorczyk-based hierarchies =-=[7, 11]-=-. The motivation for this research comes from our search for a good programming language. We restrict our attention to declarative programming where we construct computable functions over some inducti... |

38 |
Classes of predictably computable functions
- Ritchie
- 1963
(Show Context)
Citation Context ... on the order of jxj. Now we see that the class P 2 contains +; : \Gamma; \Delta; \Xi, etc. The Turing machine characterization of PO 2 can be also proved from the theorem 22 by the result of Ritchie =-=[10]-=- that E 2 consists of n-ary functions Turing computable in linear space. Cobham [2, 11] was the first one to characterize the n-ary functions computable in polynomial time by an inductively defined cl... |

24 | Finitary inductively presented logics
- FEFERMAN
- 1989
(Show Context)
Citation Context ...main has been also investigated with the help of an imperative language for complexity purposes for instance in [5], or as a meta-language for the study of formal systems for constructive mathematics =-=[4]-=-. Since the domain of S-expressions with a single atom is denumerable it seems natural to identify it with the set of natural numbers. Functions of our programming language will become recursive funct... |

21 |
Constant Time Factors Do Matter
- Jones
(Show Context)
Citation Context ...logy I and Trilogy II based on S-expressions with a single atom 0 (Nil) [13, 1]. This domain has been also investigated with the help of an imperative language for complexity purposes for instance in =-=[5]-=-, or as a meta-language for the study of formal systems for constructive mathematics [4]. Since the domain of S-expressions with a single atom is denumerable it seems natural to identify it with the s... |

11 |
Einfuhrung in die Komplexitatstheorie
- Reischuk
- 1990
(Show Context)
Citation Context ...ited unary iteration, i.e. P 2:5 = PO 2:5 . PH = PSPACE iff PM 2:5 is closed under limited unary iteration, i.e. PM 2:5 = PO 2:5 . Proof: If P = NP then the polynomial hierachy collapses: P = PH (see =-=[9]-=-) and then P 2:5s= PM 2:5s. Vice versa, if the latter identity holds then by the theorems 24, 27, and 28 we have P = PH and hence P = NP . Similarly, P = PSPACE iff P 2:5s= PO 2:5sand PH = PSPACE iff ... |

9 |
Types of Trilogy
- Voda
- 1988
(Show Context)
Citation Context ...zingly powerful, domain specified as words. We have designed and implemented two practical declarative programming languages Trilogy I and Trilogy II based on S-expressions with a single atom 0 (Nil) =-=[13, 1]-=-. This domain has been also investigated with the help of an imperative language for complexity purposes for instance in [5], or as a meta-language for the study of formal systems for constructive mat... |

3 |
Types as Values Polymorphism
- Borovansky, Voda
- 1993
(Show Context)
Citation Context ...zingly powerful, domain specified as words. We have designed and implemented two practical declarative programming languages Trilogy I and Trilogy II based on S-expressions with a single atom 0 (Nil) =-=[13, 1]-=-. This domain has been also investigated with the help of an imperative language for complexity purposes for instance in [5], or as a meta-language for the study of formal systems for constructive mat... |

1 |
Small Grzegorczyk classes and limited minimum
- Harrow
- 1975
(Show Context)
Citation Context ...nimum in the class PM 2 . This class but without pair iteration is the class M 2 of the minimum hierarchy of Harrow whose predicates are constructive arithmetic, or bounded arithmetic predicates (see =-=[8, 11]-=-). Theorem 29 P = NP iff P 2:5 is closed under bounded unary minimum, i.e. P 2:5 = PM 2:5 . P = PSPACE iff P 2:5 is closed under limited unary iteration, i.e. P 2:5 = PO 2:5 . PH = PSPACE iff PM 2:5 i... |

1 |
The Polynomial-Time Hierachy, Theor
- Stockmeyer
- 1977
(Show Context)
Citation Context ...tic by pair iteration as the latter is exponentially shorter than the length of binary codes. We now characterize the class PM 2:5s. The class of languages PH is the union of the polynomial hierachy (=-=[12]-=-). The languages in PH are accepted by alternating Turing machines in polynomial time with constantly many alternations. We have P ae NP;coNP ae PH. A language L ae \Sigma in PH is such that w 2 L $ Q... |