HOLONOMIC GENERATING FUNCTIONS AND CONTEXT FREE LANGUAGES

1992 ◽  
Vol 03 (02) ◽  
pp. 181-191 ◽  
Author(s):  
A. BERTONI ◽  
P. MASSAZZA ◽  
N. SABADINI
Keyword(s):  
Generating Function ◽  
Generating Functions ◽  
Equivalence Problem ◽  
Context Free Language ◽  
Context Free ◽  
Free Language

In this paper we give some undecidability and decidability results about context-free languages. First, we prove that the problem of deciding whether a context-free language which admits a holonomic generating function is Turing equivalent to the finiteness question for r.e. sets. Second, we show that the Equivalence Problem is decidable for a suitable class of languages, called LCLR.

STURMIAN WORDS AND AMBIGUOUS CONTEXT-FREE LANGUAGES

1990 ◽  
Vol 01 (03) ◽  
pp. 309-323 ◽  
Author(s):  
FILIPPO MIGNOSI
Keyword(s):  
Number Theory ◽  
Generating Function ◽  
Rational Number ◽  
Asymptotic Density ◽  
Gap Theorem ◽  
Context Free Language ◽  
Sturmian Words ◽  
Context Free ◽  
Free Language

If x is a rational number, 0<x≤1, then A(x)c is a context-free language, where A(x) is the set of factors of the infinite Sturmian words with asymptotic density of 1’s smaller than or equal to x. We also prove a “gap” theorem i.e. A(x) can never be an unambiguous co-context-free language. The “gap” theorem is established by proving that the counting generating function of A(x) is transcendental. We show some links between Sturmian words, combinatorics and number theory.

scholarly journals The Insertion Encoding of Permutations

10.37236/1944 ◽  
2005 ◽  
Vol 12 (1) ◽  
Author(s):  
Michael H. Albert ◽  
Steve Linton ◽  
Nik Ruškuc
Keyword(s):  
Polynomial Time ◽  
Generating Functions ◽  
Regular Language ◽  
Necessary Conditions ◽  
Polynomial Time Algorithms ◽  
Context Free Language ◽  
Context Free ◽  
Free Language

We introduce the insertion encoding, an encoding of finite permutations. Classes of permutations whose insertion encodings form a regular language are characterized. Some necessary conditions are provided for a class of permutations to have insertion encodings that form a context free language. Applications of the insertion encoding to the evaluation of generating functions for classes of permutations, construction of polynomial time algorithms for enumerating such classes, and the illustration of bijective equivalence between classes are demonstrated.

scholarly journals Permutations Generated by a Depth 2 Stack and an Infinite Stack in Series are Algebraic

10.37236/4571 ◽  
2015 ◽  
Vol 22 (2) ◽  
Author(s):  
Murray Elder ◽  
Geoffrey Lee ◽  
Andrew Rechnitzer
Keyword(s):  
Generating Function ◽  
Context Free Grammar ◽  
Context Free Language ◽  
In Series ◽  
Simple Variant ◽  
Context Free ◽  
Free Language

We prove that the class of permutations generated by passing an ordered sequence $12\dots n$ through a stack of depth 2 and an infinite stack in series is in bijection with an unambiguous context-free language, where a permutation of length $n$ is encoded by a string of length $3n$. It follows that the sequence counting the number of permutations of each length has an algebraic generating function. We use the explicit context-free grammar to compute the generating function:\[\sum_{n\geq 0} c_n t^n = \frac{(1+q)\left(1+5q-q^2-q^3-(1-q)\sqrt{(1-q^2)(1-4q-q^2)}\right)}{8q}\]where $c_n$ is the number of permutations of length $n$ that can be generated, and $q \equiv q(t) = \frac{1-2t-\sqrt{1-4t}}{2t}$ is a simple variant of the Catalan generating function. This in turn implies that $c_n^{1/n} \to 2+2\sqrt{5}$.

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.

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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