ON THE CONTEXT-FREENESS OF THE θ-EXPANSIONS OF THE INTEGERS

1999 ◽  
Vol 09 (03n04) ◽  
pp. 347-350 ◽  
Author(s):  
CHRISTIANE FROUGNY ◽  
BORIS SOLOMYAK
Keyword(s):  
Positive Integer ◽  
Real Number ◽  
Context Free Language ◽  
Pisot Unit ◽  
Nombre Réel ◽  
Positive Integers ◽  
Context Free ◽  
Free Language

Let θ>1 be a nonintegral real number such that the θ-expansion of every positive integer is finite. If the set of θ-expansions of all the positive integers is a context-free language, then θ must be a quadratic Pisot unit. Résumé: Soit θ>1 un nombre réel non entier tel que le θ-développement de tout entier positif soit fini. Si l'on suppose que l'enslembe des θ-développements des entiers positifs forme un langage algébrique, alors θ doit être un nombre de Pisot quadratique unitaire.

AUTOMATIC CONVERSION FROM FIBONACCI REPRESENTATION TO REPRESENTATION IN BASE φ, AND A GENERALIZATION

1999 ◽  
Vol 09 (03n04) ◽  
pp. 351-384 ◽  
Author(s):  
CHRISTIANE FROUGNY ◽  
JACQUES SAKAROVITCH
Keyword(s):  
Positive Integer ◽  
Fibonacci Numbers ◽  
Golden Mean ◽  
Context Free Language ◽  
Nous Montrons ◽  
Finite Sum ◽  
Positive Integers ◽  
Context Free ◽  
Free Language

Every positive integer can be written as a sum of Fibonacci numbers; it can also be written as a (finite) sum of (positive and negative) powers of the golden mean φ. We show that there exists a letter-to-letter finite two-tape automaton that maps the Fibonacci representation of any positive integer onto its φ-expansion, provided the latter is folded around the radix point. As a corollary, the set of φ-expansions of the positive integers is a linear context-free language. These results are actually proved in the more general case of quadratic Pisot units. Résumé: Tout nombre entier positif peut s'écrire comme une somme de nombres de Fibonacci; tout entier peut également s'écrire comme une somme (finie) de puissances (positives et négatives) du "nombre d'or" φ. Nous montrons qu'il existe un automate à deux bandes, fini et lettre-à-lettre, qui envoie la représentation d'un entier en base de Fibonacci sur sa représentation dans la base φ modulo le fait qu'on a replié cette dernière autour du point décimal. On en déduit que l'ensemble des représentations des entiers en base φ est un langage context-free linéaire. Tous ces résultats sont en fait établis dans le cas général où la base considérée est un nombre de Pisot quadratique unitaire.

WHEN CHURCH-ROSSER BECOMES CONTEXT FREE

2007 ◽  
Vol 18 (06) ◽  
pp. 1293-1302 ◽  
Author(s):  
MARTIN KUTRIB ◽  
ANDREAS MALCHER
Keyword(s):  
Linear Time ◽  
Membership Problem ◽  
Closure Properties ◽  
Context Free Language ◽  
Arithmetic Hierarchy ◽  
Context Free ◽  
Free Language

We investigate the intersection of Church-Rosser languages and (strongly) context-free languages. The intersection is still a proper superset of the deterministic context-free languages as well as of their reversals, while its membership problem is solvable in linear time. For the problem whether a given Church-Rosser or context-free language belongs to the intersection we show completeness for the second level of the arithmetic hierarchy. The equivalence of Church-Rosser and context-free languages is Π1-complete. It is proved that all considered intersections are pairwise incomparable. Finally, closure properties under several operations are investigated.

scholarly journals ON INTEGER SEQUENCES GENERATED BY LINEAR MAPS

2009 ◽  
Vol 51 (2) ◽  
pp. 243-252
Author(s):  
ARTŪRAS DUBICKAS
Keyword(s):  
Lower Bound ◽  
Positive Integer ◽  
Real Number ◽  
Constant Independent ◽  
Integer Sequences ◽  
Real Numbers ◽  
Complexity Function ◽  
Linear Maps ◽  
Coprime Integers ◽  
Positive Integers

AbstractLetx0<x1<x2< ⋅⋅⋅ be an increasing sequence of positive integers given by the formulaxn=⌊βxn−1+ γ⌋ forn=1, 2, 3, . . ., where β > 1 and γ are real numbers andx0is a positive integer. We describe the conditions on integersbd, . . .,b0, not all zero, and on a real number β > 1 under which the sequence of integerswn=bdxn+d+ ⋅⋅⋅ +b0xn,n=0, 1, 2, . . ., is bounded by a constant independent ofn. The conditions under which this sequence can be ultimately periodic are also described. Finally, we prove a lower bound on the complexity function of the sequenceqxn+1−pxn∈ {0, 1, . . .,q−1},n=0, 1, 2, . . ., wherex0is a positive integer,p>q> 1 are coprime integers andxn=⌈pxn−1/q⌉ forn=1, 2, 3, . . . A similar speculative result concerning the complexity of the sequence of alternatives (F:x↦x/2 orS:x↦(3x+1)/2) in the 3x+1 problem is also given.

Layered Region Based Flow-Sensitive Demand-Driven Alias Analysis

2014 ◽  
Vol 577 ◽  
pp. 917-920
Author(s):  
Long Pang ◽  
Xiao Hong Su ◽  
Pei Jun Ma ◽  
Ling Ling Zhao
Keyword(s):  
Program Analysis ◽  
Control Flow ◽  
Alias Analysis ◽  
Reachability Problem ◽  
Analysis Algorithm ◽  
Context Free Language ◽  
Context Free ◽  
Free Language ◽  
Demand Driven Analysis

The pointer alias is indispensable for program analysis. Comparing to point-to set, it’s more efficient to formulate the alias as the context free language (CFL) reachability problem. However, the precision is limited to flow-insensitivity. To solve this problem, we propose a flow sensitive, demand-driven analysis algorithm for answering may-alias queries. First the partial single static assignment is used to discriminate the address-taken pointers. Then the order of control flow is encoded in the level linearization code to ease comparison. Finally, the query of alias in demand driven is converted into the search of CFL reachability with feasible flows. The experiments demonstrate the effectiveness of the proposed approach.

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