Soundness and completeness proofs by coinductive methods

We show how codatatypes can be employed to produce compact, high-level proofs of key results in logic: the soundness and completeness of proof systems for variations of first-order logic. For the classical completeness result, we first establish an abstract property of possibly infinite derivation trees. The abstract proof can be instantiated for a wide range of Gentzen and tableau systems for various flavors of first-order logic. Soundness becomes interesting as soon as one allows infinite proofs of first-order formulas. This forms the subject of several cyclic proof systems for first-order logic augmented with inductive predicate definitions studied in the literature. All the discussed results are formalized using Isabelle/HOL’s recently introduced support for codatatypes and corecursion. The development illustrates some unique features of Isabelle/HOL’s new coinductive specification language such as nesting through non-free types and mixed recursion–corecursion

Neurocognition and quality of life after reinitiating antiretroviral therapy in children randomized to planned treatment interruption

Objective: Understanding the effects of antiretroviral treatment (ART) interruption on neurocognition and quality of life (QoL) are important for managing unplanned interruptions and planned interruptions in HIV cure research. Design: Children previously randomized to continuous (continuous ART, n=41) vs. planned treatment interruption (PTI, n=47) in the Pediatric European Network for Treatment of AIDS (PENTA) 11 study were enrolled. At study end, PTI children resumed ART. At 1 and 2 years following study end, children were assessed by the coding, symbol search and digit span subtests of Wechsler Intelligence Scale for Children (6-16 years old) or Wechsler Adult Intelligence Scale ( 6517 years old) and by Pediatrics QoL questionnaires for physical and psychological QoL. Transformed scaled scores for neurocognition and mean standardized scores for QoL were compared between arms by t-test and Mann-Whitney U test, respectively. Scores indicating clinical concern were compared (<7 for neurocognition and <70 for QoL tests). Results: Characteristics were similar between arms with a median age of 12.6 years, CD4 + of 830 cells/\u3bcl and HIV RNA of 1.7 log 10 copies/ml. The median cumulative ART exposure was 9.6 in continuous ART vs. 7.7 years in PTI (P=0.02). PTI children had a median of 12 months off ART and had resumed ART for 25.2 months at time of first assessment. Neurocognitive scores were similar between arms for all tests. Physical and psychological QoL scores were no different. About 40% had low neurocognitive and QoL scores indicating clinical concern. Conclusion: No differences in information processing speed, sustained attention, short-term memory and QoL functioning were observed between children previously randomized to continuous ART vs. PTI in the PENTA 11 trial

Error Bounds for Polynomial Optimization over the Hypercube using Putinar type Representations

7 pages, 1 tableInternational audienceConsider the optimization problem $p_{\min, Q} := \min_{\mathbf{x} \in Q} p(\mathbf{x})$, where $p$ is a degree $m$ multivariate polynomial and $Q := [0, 1]^n$ is the hypercube. We provide explicit degree and error bounds for the sums of squares approximations of $p_{\min, Q}$ corresponding to the Positivstellensatz of Putinar. Our approach uses Bernstein multivariate approximation of polynomials, following the methodology of De Klerk and Laurent to provide error bounds for Schmüdgen type positivity certificates over the hypercube. We give new bounds for Putinar type representations by relating the quadratic module and the preordering associated with the polynomials $g_i := x_i (1 - x_i), \: i=1,\dots,n$, describing the hypercube $Q$