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Nicholas Pischke

Publications and source records attributed to Nicholas Pischke.

28 records · Page 2Linked to original sources
arXiv2023

Rates of convergence for the asymptotic behavior of second-order Cauchy problems

We provide a quantitative version of a result due to Poffald and Reich on the asymptotic behavior of solutions of a second-order Cauchy problem generated by an accretive operator in the form of a rate of convergence. This quantitative result is then used to generalize a result of Xu on the asymptotic behavior of almost-orbits of the solution semigroup of a first-order Cauchy problem to this second-order case.

math.AP↗
arXiv2022

Logical metatheorems for accretive and (generalized) monotone set-valued operators

Accretive and monotone operator theory are central branches of nonlinear functional analysis and constitute the abstract study of set-valued mappings between function spaces. This paper deals with the computational properties of certain large classes of operators, namely accretive and (generalized) monotone set-valued ones. In particular, we develop (and extend) for this field the theoretical framework of proof mining, a program in mathematical logic that seeks to extract computational information from prima facie `non-computational' proofs from the mainstream literature. To this end, we establish logical metatheorems that guarantee and quantify the computational content of theorems pertaining to accretive and (generalized) monotone set-valued operators. On one hand, our results unify a number of recent case studies, while they also provide characterizations of central analytical notions in terms of proof theoretic ones on the other, which provides a crucial perspective on needed quantitative assumptions in future applications of proof mining to these branches.

math.LO↗
arXiv2022

Quantitative results on algorithms for zeros of differences of monotone operators in Hilbert space

We provide quantitative information in the form of a rate of metastability in the sense of T. Tao and (under a metric regularity assumption) a rate of convergence for an algorithm approximating zeros of differences of maximally monotone operators due to A. Moudafi by using techniques from `proof mining', a subdiscipline of mathematical logic. For the rate of convergence, we provide an abstract and general result on the construction of rates of convergence for quasi-Fejér monotone sequences with metric regularity assumptions, generalizing previous results for Fejér monotone sequences due to U. Kohlenbach, G. López-Acedo and A. Nicolae.

math.FA↗
arXiv2021

A note on public announcements in standard Gödel modal logic

We study public announcement operators in the context of standard Gödel modal logic as introduced by Caicedo and Rodriguez. Over that base logic, admitting a natural semantics over [0,1]-valued generalizations of modal Kripke models, we exhibit three possible semantic interpretations of the public announcement operator, all equivalent in a classical setting, and show that these logics are all different in terms of expressive strength. We provide partial completeness results for these logics via Hilbert-style calculi.

math.LO↗
arXiv2021

On Infinitary Gödel logics

We study propositional and first-order Gödel logics over infinitary languages which are motivated semantically by corresponding interpretations into the unit interval [0,1]. We provide infinitary Hilbert-style calculi for the particular (propositional and first-order) cases with con-/disjunctions of countable length and prove corresponding completeness theorems by extending the usual Lindenbaum-Tarski construction to the infinitary case for a respective algebraic semantics via complete linear Heyting algebras. We provide infinitary hypersequent calculi and prove corresponding cut-elimination theorems in the Schütte-Tait-style. Initial observations are made regarding truth-value sets other than [0,1].

math.LO↗
arXiv2021

A note on strong axiomatization of Gödel Justification Logic

Justification logics are special kinds of modal logics which provide a framework for reasoning about epistemic justifications. For this, they extend classical boolean propositional logic by a family of necessity-style modal operators "t:", indexed over t by a corresponding set of justification terms, which thus explicitly encode the justification for the necessity assertion in the syntax. With these operators, one can therefore not only reason about modal effects on propositions but also about dynamics inside the justifications themselves. We replace this classical boolean base with Gödel logic, one of the three most prominent fuzzy logics, i.e. special instances of many-valued logics, taking values in the unit interval [0,1], which are intended to model inference under vagueness. We extend the canonical possible-world semantics for justification logic to this fuzzy realm by considering fuzzy accessibility- and evaluation-functions evaluated over the minimum t-norm and establish strong completeness theorems for various fuzzy analogies of prominent extensions for basic justification logic.

math.LO↗
arXiv2021

Quantitative analysis of a subgradient-type method for equilibrium problems

We use techniques originating from the subdiscipline of mathematical logic called `proof mining' to provide rates of metastability and - under a metric regularity assumption - rates of convergence for a subgradient-type algorithm solving the equilibrium problem in convex optimization over fixed-point sets of firmly nonexpansive mappings. The algorithm is due to H. Iiduka and I. Yamada who in 2009 gave a noneffective proof of its convergence. This case study illustrates the applicability of the logic-based abstract quantitative analysis of general forms of Fejér monotonicity as given by the second author in previous papers.

math.OC↗
arXiv2020

On Intermediate Justification Logics

We study abstract intermediate justification logics, that is arbitrary intermediate propositional logics extended with a subset of specific axioms of (classical) justification logics. For these, we introduce various semantics by combining either Heyting algebras or Kripke frames with the usual semantic machinery used by Mkrtychev's, Fitting's or Lehmann's and Studer's models for classical justification logics. We prove unified completeness theorems for all intermediate justification logics and their corresponding semantics using a respective propositional completeness theorem of the underlying intermediate logic. Further, by a modification of a method of Fitting, we prove unified realization theorems for a large class of intermediate justification logics and accompanying intermediate modal logics.

math.LO↗
arXiv2018

Dynamic extensions for the logic of knowing why with public announcements of formulas

In this paper, we address the logic of knowing why, an example of a non-standard epistemic logic dealing with justified knowledge via a new epistemic operator, under the extensions with ideas from dynamic epistemic logic, namely public announcements. Through the additional notions present in the knowing why context, we consider two possible variants, namely the extensions by (i): public announcements of a formula and by (ii): public announcements of reasons, although the deeper analysis of the latter is left for future work. We consider another logical operator, the conditional knowing-why operator, for which we study the applications to the axiomatization of public announcements as well as the solely framework. At the end, we consider the logical expressivity of these different logics in comparison to each other, and thus we show one of the main problems with the usual process of proving completeness through translation in the context of logics with public announcements.

math.LO↗