Size Analysis with Indexed Families of max 0 - Polynomials ?
نویسندگان
چکیده
Our previous work studied a size-aware type system for functional programs with non-monotonic polynomial size dependencies. In that approach output sizes depended only on input sizes. That is rather restrictive since in many cases the size of the output can differ for different input data of the same size. In this paper we remove that limitation by presenting value-sensitive size dependencies via indexed families of polynomials. When a program has a family of polynomials as its output-on-input size dependency, then for each actual input value there is an index such that the corresponding polynomial in the family precisely defines the size of the output. We introduce and study a novel size-aware type system in which size annotations are indexed families of max0-polynomials; max0-polynomials extend polynomials with the max0-operation to prevent negative sizes. We prove soundness of the type system and we give several decidability results for different variants of annotations.
منابع مشابه
Polynomial Size Analysis with Families of Piecewise Polynomials for Functional Programs over Lists ? With soundness proof and examples in detail
Size analysis can play an important role in optimising memory management and in preventing failure due to memory exhaustion. Static size analysis can be performed using size-aware type systems. In size-aware type systems types express output-on-input size dependencies where sizes of outputs depend on sizes of inputs but they do not depend directly on the actual values. We present a novel type s...
متن کاملSome Families of Graphs whose Domination Polynomials are Unimodal
Let $G$ be a simple graph of order $n$. The domination polynomial of $G$ is the polynomial $D(G, x)=sum_{i=gamma(G)}^{n} d(G,i) x^{i}$, where $d(G,i)$ is the number of dominating sets of $G$ of size $i$ and $gamma(G)$ is the domination number of $G$. In this paper we present some families of graphs whose domination polynomials are unimodal.
متن کاملModified degenerate Carlitz's $q$-bernoulli polynomials and numbers with weight ($alpha ,beta $)
The main goal of the present paper is to construct some families of the Carlitz's $q$-Bernoulli polynomials and numbers. We firstly introduce the modified Carlitz's $q$-Bernoulli polynomials and numbers with weight ($_{p}$. We then define the modified degenerate Carlitz's $q$-Bernoulli polynomials and numbers with weight ($alpha ,beta $) and obtain some recurrence relations and other identities...
متن کاملA new scalar product for nonsymmetric Jack polynomials
Jack polynomials are a remarkable family of polynomials in n variables x = (x1, · · · , xn) with coefficients in the field F := Q(α) where α is an indeterminate. They arise naturally in several statistical, physical, combinatorial, and representation theoretic considerations. The symmetric polynomials ([M1], [St], [LV], [KS]) Jλ = J (α) λ are indexed by partitions λ = (λ1, · · · , λn) where λ1 ...
متن کاملTwo-step Darboux Transformations and Exceptional Laguerre Polynomials
It has been recently discovered that exceptional families of SturmLiouville orthogonal polynomials exist, that generalize in some sense the classical polynomials of Hermite, Laguerre and Jacobi. In this paper we show how new families of exceptional orthogonal polynomials can be constructed by means of multiple-step algebraic Darboux transformations. The construction is illustrated with an examp...
متن کامل