On formal aspects of the epistemic approach to paraconsistency

This paper reviews the central points and presents some recent developments of the epistemic approach to paraconsistency in terms of the preservation of evidence. Two formal systems are surveyed, the basic logic of evidence (BLE) and the logic of evidence and truth (LET J ), designed to deal, respectively, with evidence and with evidence and truth. While BLE is equivalent to Nelson’s logic N4, it has been conceived for a different purpose. Adequate valuation semantics that provide decidability are given for both BLE and LET J . The meanings of the connectives of BLE and LET J , from the point of view of preservation of evidence, is explained with the aid of an inferential semantics. A formalization of the notion of evidence for BLE as proposed by M. Fitting is also reviewed here. As a novel result, the paper shows that LET J is semantically characterized through the so-called Fidel structures. Some opportunities for further research are also discussed

Two-Loop Helicity Amplitudes for Gluon-Gluon Scattering in QCD and Supersymmetric Yang-Mills Theory

We present the two-loop helicity amplitudes for the scattering of two gluons into two gluons in QCD, which are relevant for next-to-next-to-leading order corrections to jet production at hadron colliders. We give the results in the `t Hooft-Veltman and four-dimensional helicity variants of dimensional regularization. Summing our expressions over helicities and colors, and converting to conventional dimensional regularization, gives results in complete agreement with those of Glover, Oleari and Tejeda-Yeomans. We also present the amplitudes for 2 to 2 scattering in pure N=1 supersymmetric Yang-Mills theory.Comment: 55 pages, 3 figures, corrected remark below eq. (4.33), other minor changes, version appearing in JHE

Recent Symbolic Summation Methods to Solve Coupled Systems of Differential and Difference Equations

We outline a new algorithm to solve coupled systems of differential equations in one continuous variable $x$ (resp. coupled difference equations in one discrete variable $N$) depending on a small parameter $\epsilon$: given such a system and given sufficiently many initial values, we can determine the first coefficients of the Laurent-series solutions in $\epsilon$ if they are expressible in terms of indefinite nested sums and products. This systematic approach is based on symbolic summation algorithms in the context of difference rings/fields and uncoupling algorithms. The proposed method gives rise to new interesting applications in connection with integration by parts (IBP) methods. As an illustrative example, we will demonstrate how one can calculate the $\epsilon$-expansion of a ladder graph with 6 massive fermion lines

The Complete O(αs2)O(\alpha_s^2) Non-Singlet Heavy Flavor Corrections to the Structure Functions g1,2ep(x,Q2)g_{1,2}^{ep}(x,Q^2), F1,2,Lep(x,Q2)F_{1,2,L}^{ep}(x,Q^2), F1,2,3ν(νˉ)(x,Q2)F_{1,2,3}^{\nu(\bar{\nu})}(x,Q^2) and the Associated Sum Rules

We calculate analytically the flavor non-singlet $O(\alpha_s^2)$ massive Wilson coefficients for the inclusive neutral current non-singlet structure functions $F_{1,2,L}^{ep}(x,Q^2)$ and $g_{1,2}^{ep}(x,Q^2)$ and charged current non-singlet structure functions $F_{1,2,3}^{\nu(\bar{\nu})p}(x,Q^2)$, at general virtualities $Q^2$ in the deep-inelastic region. Numerical results are presented. We illustrate the transition from low to large virtualities for these observables, which may be contrasted to basic assumptions made in the so-called variable flavor number scheme. We also derive the corresponding results for the Adler sum rule, the unpolarized and polarized Bjorken sum rules and the Gross-Llewellyn Smith sum rule. There are no logarithmic corrections at large scales $Q^2$ and the effects of the power corrections due to the heavy quark mass are of the size of the known $O(\alpha_s^4)$ corrections in the case of the sum rules. The complete charm and bottom corrections are compared to the approach using asymptotic representations in the region $Q^2 \gg m_{c,b}^2$. We also study the target mass corrections to the above sum rules.Comment: 50 pages LATEX, 35 figure

Recovery operators, paraconsistency and duality

There are two foundational, but not fully developed, ideas in paraconsistency, namely, the duality between paraconsistent and intuitionistic paradigms, and the introduction of logical operators that express meta-logical notions in the object language. The aim of this paper is to show how these two ideas can be adequately accomplished by the Logics of Formal Inconsistency (LFIs) and by the Logics of Formal Undeterminedness (LFUs). LFIs recover the validity of the principle of explosion in a paraconsistent scenario, while LFUs recover the validity of the principle of excluded middle in a paracomplete scenario. We introduce definitions of duality between inference rules and connectives that allow comparing rules and connectives that belong to different logics. Two formal systems are studied, the logics mbC and mbD, that display the duality between paraconsistency and paracompleteness as a duality between inference rules added to a common core– in the case studied here, this common core is classical positive propositional logic (CPL + ). The logics mbC and mbD are equipped with recovery operators that restore classical logic for, respectively, consistent and determined propositions. These two logics are then combined obtaining a pair of logics of formal inconsistency and undeterminedness (LFIUs), namely, mbCD and mbCDE. The logic mbCDE exhibits some nice duality properties. Besides, it is simultaneously paraconsistent and paracomplete, and able to recover the principles of excluded middle and explosion at once. The last sections offer an algebraic account for such logics by adapting the swap-structures semantics framework of the LFIs the LFUs. This semantics highlights some subtle aspects of these logics, and allows us to prove decidability by means of finite non-deterministic matrices

A toolbox to solve coupled systems of differential and difference equations

We present algorithms to solve coupled systems of linear differential equations, arising in the calculation of massive Feynman diagrams with local operator insertions at 3-loop order, which do {\it not} request special choices of bases. Here we assume that the desired solution has a power series representation and we seek for the coefficients in closed form. In particular, if the coefficients depend on a small parameter \ep (the dimensional parameter), we assume that the coefficients themselves can be expanded in formal Laurent series w.r.t.\ \ep and we try to compute the first terms in closed form. More precisely, we have a decision algorithm which solves the following problem: if the terms can be represented by an indefinite nested hypergeometric sum expression (covering as special cases the harmonic sums, cyclotomic sums, generalized harmonic sums or nested binomial sums), then we can calculate them. If the algorithm fails, we obtain a proof that the terms cannot be represented by the class of indefinite nested hypergeometric sum expressions. Internally, this problem is reduced by holonomic closure properties to solving a coupled system of linear difference equations. The underlying method in this setting relies on decoupling algorithms, difference ring algorithms and recurrence solving. We demonstrate by a concrete example how this algorithm can be applied with the new Mathematica package \texttt{SolveCoupledSystem} which is based on the packages \texttt{Sigma}, \texttt{HarmonicSums} and \texttt{OreSys}. In all applications the representation in $x$-space is obtained as an iterated integral representation over general alphabets, generalizing Poincar\'{e} iterated integrals