Program synthesis has seen many new applications in recent years, in large part thanks to the introduction of SyGuS. However, no existing SyGuS solvers have support for synthesizing recursive functions. We introduce a...
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The theory of recursive functions is related in a well-known way to the notion of least fixed points, by endowing a set of partial functions with an ordering in terms of their domain of definition. When terms in the p...
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In this paper we study several partition relations, defined by Saharon Shelah, and relate them to the Hales-Jewett numbers. In particular we give an upper bound for the Hales-Jewett numbers using the primitive recursi...
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This paper details an algorithm for a binary, primitive recursive functionthat apparently computes, for any $i$ and $n$, $f_i\\left(i,n\ight)$. Thealgorithm works by exploiting the fact that in the formal system descr...
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OCaml function calls consume space on the system stack. Operating systems set default limits on the stack space which are much lower than the available memory. If a program runs out of stack space, they get the dreade...
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OCaml function calls consume space on the system stack. Operating systems set default limits on the stack space which are much lower than the available memory. If a program runs out of stack space, they get the dreaded "Stack Overflow" exception – they crash. As a result, OCaml programmers have to be careful, when they write recursive functions, to remain in the so-called tail-recursive fragment, using tail calls that do not consume stack space. This discipline is a source of difficulties for both beginners and experts. Beginners have to be taught recursion, and then tail-recursion. Experts disagree on the "right" way to write ***. The direct version is beautiful but not tail-recursive, so it crashes on larger inputs. The naive tail-recursive transformation is (slightly) slower than the direct version, and experts may want to avoid that cost. Some libraries propose horrible implementations, unrolling code by hand, to compensate for this performance loss. In general, tail-recursion requires the programmer to manually perform sophisticated program transformations. In this work we propose an implementation of "Tail Modulo Cons" (TMC) for OCaml. TMC is a program transformation for a fragment of non-tail-recursive functions, that rewrites them in destination-passing style. The supported fragment is smaller than other approaches such as continuation-passing-style, but the performance of the transformed code is on par with the direct, non-tail-recursive version. Many useful functions that traverse a recursive datastructure and rebuild another recursive structure are in the TMC fragment, in particular *** (and List.{filter,append}, etc.). Finally those functions can be written in a way that is beautiful, correct on all inputs, and efficient. In this work we give a novel modular, compositional definition of the TMC transformation. We discuss the design space of user-interface choices: what degree of control for the user, when to warn or fail when the transformation
We obtain an asymptotic formula with an error term O(X 1 2 +Ε) for the smoothed first moment of quadratic twists of modular L-functions. We also give a similar result for the smoothed first moment of the first deriva...
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Given a simple recursive function, we show how to extract from it a reversible and an classical iterative part. Those parts can synchronously cooperate under a Producer/Consumer pattern in order to implement the origi...
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We present a method for synthesizing recursive functions that satisfy both a functional specification and an asymptotic resource bound. Prior methods for synthesis with a resource metric require the user to specify a ...
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In this paper, by virtue of a determinantal formula for derivatives of the ratio between two differentiable functions, in view of the Faà di Bruno formula, and with the help of several identities and closed-form ...
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In this paper, by virtue of a determinantal formula for derivatives of the ratio between two differentiable functions, in view of the Faà di Bruno formula, and with the help of several identities and closed-form formulas for the partial Bell polynomials Bn,k, the author (1) establishes thirteen Maclaurin series expansions of the functions ln ex +1 ln ex −1 2 ,, ln cosh x, x ln sinh x (exx −1 )r [ ln(1 x + x) ]r, x for r = ± 12 and r ∈ R in terms of the Dirichlet eta function η(1 − 2k), the Riemann zeta function ζ(1 − 2k), and the Stirling numbers of the first and second kinds s(n, k) and S(n, k). (2) presents four determinantal expressions and three recursive relations for the Bernoulli numbers B2n. (3) finds out three closed-form formulas for the Bernoulli numbers B2n and the generalized Bernoulli numbers Bn(r) in terms of the Stirling numbers of the second kind S(n, k), and deduce two combinatorial identities for the Stirling numbers of the second kind S(n, k). (4) acquires two combinatorial identities, which can be regarded as diagonal recursive relations, involving the Stirling numbers of the first and second kinds s(n, k) and S(n, k). (5) recovers an integral representation and a closed-form formula, and establish an alternative explicit and closed-form formula, for the Bernoulli numbers of the second kind bn in terms of the Stirling numbers of the first kind s(n, k). (6) obtains three identities connecting the Stirling numbers of the first and second kinds s(n, k) and S(n, k). The most highlights of this paper include the unification (exx−1)r of the generating functions of the Bernoulli numbers Bn and the Stirling numbers of the second kind S(n, k), the unification [ ln(1+xx) ]r of the generating functions of the Bernoulli numbers of the second kind bn and the Stirling numbers of the first kind s(n, k), and the disclosure of the transformations between these two *** Codes Primary 05A15, Secondary 03D20, 11B73, 11B83, 26A09, 33B10, 41A58 Copyright
In this paper, we reconstruct explicitly the generating function of genus-zero K-theoretic permutation-invariant Gromov-Witten invariants, known as the big J-function, for any partial flag variety. The reconstruction ...
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