Cut-Simulation and Impredicativity

We investigate cut-elimination and cut-simulation in impredicative (higher-order) logics. We illustrate that adding simple axioms such as Leibniz equations to a calculus for an impredicative logic -- in our case a sequent calculus for classical type theory -- is like adding cut. The phenomenon equally applies to prominent axioms like Boolean- and functional extensionality, induction, choice, and description. This calls for the development of calculi where these principles are built-in instead of being treated axiomatically.Comment: 21 page

In defense of mechanism

In Life Itself and in Essays on Life Itself, Robert Rosen (1991, 2000) argued that machines were, in principle, incapable of modeling the defining feature of living systems, which he claimed to be the existence of closed causal loops. Rosen's argument has been used to support critiques of computational models in ecological psychology. This article shows that Rosen's attack on mechanism is fundamentally misconceived. It is, in fact, of the essence of a mechanical system that it contains closed causal loops. Moreover, Rosen's epistemology is based on a strong form of indirect realism and his arguments, if correct, would call into question some of the fundamental principles of ecological psychology

Analytic Tableaux for Simple Type Theory and its First-Order Fragment

We study simple type theory with primitive equality (STT) and its first-order fragment EFO, which restricts equality and quantification to base types but retains lambda abstraction and higher-order variables. As deductive system we employ a cut-free tableau calculus. We consider completeness, compactness, and existence of countable models. We prove these properties for STT with respect to Henkin models and for EFO with respect to standard models. We also show that the tableau system yields a decision procedure for three EFO fragments

Classical and Intuitionistic Arithmetic with Higher Order Comprehension Coincide on Inductive Well-Foundedness

Assume that we may prove in Classical Functional Analysis that a primitive recursive relation R is well-founded, using the inductive definition of well-founded. In this paper we prove that such a proof of well-foundation may be made intuitionistic. We conclude that if we are able to formulate any mathematical problem as the inductive well-foundation of some primitive recursive relation, then intuitionistic and classical provability coincide, and for such a statement of well-foundation we may always find an intuitionistic proof if we may find a proof at all. The core of intuitionism are the methods for computing out data with given properties from input data with given properties: these are the results we are looking for when we do constructive mathematics. Proving that a primitive recursive relation R is inductively well-founded is a more abstract kind of result, but it is crucial as well, because once we proved that R is inductively well-founded, then we may write programs by induction over R. This is the way inductive relation are currently used in intuitionism and in proof assistants based on intuitionism, like Coq. In the paper we introduce the comprehension axiom for Functional Analysis in the form of introduction and elimination rules for predicates of types Prop, Nat->Prop, ..., in order to use Girard\u27s method of candidates for impredicative arithmetic

Robert Rosen and Relational System Theory: An Overview

Relational system theory is the science of organization and function. It is the study of how systems are organized which is based on their functions and the relations between their functions. The science was originally developed by Nicolas Rashevsky, and further developed by Rashevsky’s student Robert Rosen, and continues to be developed by Rosen’s student A. H. Louie amongst others. Due to its revolutionary character, it is often misunderstood, and to some, controversial. We will mainly be focusing on Rosen’s contributions to this science. The formal and conceptual setting for Rosen’s relational system theory is category theory. Rosen was the first to apply category theory to scientific problems, outside of pure mathematics, and the first to think about science from the point of view of category theory. We will provide an overview of Rosen’s theory of modeling, complexity, anticipation, and organism. We will present the foundations of this science and the philosophical motivations behind it along with conceptual clarification and historical context. The purpose of this dissertation is to present Rosen’s ideas to a wider audience

Quantitative Storytelling: Science, Narratives, and Uncertainty in Nexus Innovations

Innovations are central instruments of sustainability policies. They project future visions onto technological solutions and enable win-win framings of complex sustainability issues. Yet, they also create new problems by interconnecting different resources such as water, food, and energy, what is known as the â WEF nexus.â In this paper, we apply a new approach called Quantitative Storytelling (QST) to the assessment of four innovations with a strong nexus component in EU policy: biofuels, shale gas, electric vehicles, and alternative water resources. Recognizing irreducible pluralism and uncertainties, QST inspects the relationships between the narratives used to frame sustainability issues and the evidence on those issues. Our experiences outlined two rationales for implementing QST. First, QST can be used to question dominant narratives that promote certain innovations despite evidence against their effectiveness. Second, QST can offer avenues for pluralistic processes of co-creation of alternative narratives and imaginaries. We reflect on the implementation of QST and on the role played by different uncertainties throughout these processes. Our experiences suggest that while the role of nexus assessments using both numbers and narratives may not be instrumental in directly inducing policy change, they are valuable means to open discussions on innovations outside of dominant nexus imaginaries. © The Author(s) 2021.The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This research has been funded by the European Union’s H2020 project MAGIC: Moving towards Adaptive Governance in Complexity (MAGIC GA No. 689669); European Union’s FP7 project IANEX: Integrated Assessment of the Nexus: The Case of Hydraulic Fracturing, Marie Curie International Outgoing Fellowship GA No. 623593; and Spanish Ministry of Science’s Juan de la Cierva Fellowship (IJC2019-038847-I/AEI/10.13039/501100011033)

The Higher-Order Prover Leo-II.

Leo-II is an automated theorem prover for classical higher-order logic. The prover has pioneered cooperative higher-order-first-order proof automation, it has influenced the development of the TPTP THF infrastructure for higher-order logic, and it has been applied in a wide array of problems. Leo-II may also be called in proof assistants as an external aid tool to save user effort. For this it is crucial that Leo-II returns proof information in a standardised syntax, so that these proofs can eventually be transformed and verified within proof assistants. Recent progress in this direction is reported for the Isabelle/HOL system.The Leo-II project has been supported by the following grants: EPSRC grant EP/D070511/1 and DFG grants BE/2501 6-1, 8-1 and 9-1.This is the final version of the article. It first appeared from Springer via http://dx.doi.org/10.1007/s10817-015-9348-y

Domain-Aware Session Types

We develop a generalization of existing Curry-Howard interpretations of (binary) session types by relying on an extension of linear logic with features from hybrid logic, in particular modal worlds that indicate domains. These worlds govern domain migration, subject to a parametric accessibility relation familiar from the Kripke semantics of modal logic. The result is an expressive new typed process framework for domain-aware, message-passing concurrency. Its logical foundations ensure that well-typed processes enjoy session fidelity, global progress, and termination. Typing also ensures that processes only communicate with accessible domains and so respect the accessibility relation. Remarkably, our domain-aware framework can specify scenarios in which domain information is available only at runtime; flexible accessibility relations can be cleanly defined and statically enforced. As a specific application, we introduce domain-aware multiparty session types, in which global protocols can express arbitrarily nested sub-protocols via domain migration. We develop a precise analysis of these multiparty protocols by reduction to our binary domain-aware framework: complex domain-aware protocols can be reasoned about at the right level of abstraction, ensuring also the principled transfer of key correctness properties from the binary to the multiparty setting