Infinity

Prose by Eric Baugh. Finalist in the 2018 Manuscripts Prose Contest

Complexity Results for Modal Dependence Logic

Modal dependence logic was introduced recently by V\"a\"an\"anen. It enhances the basic modal language by an operator =(). For propositional variables p_1,...,p_n, =(p_1,...,p_(n-1);p_n) intuitively states that the value of p_n is determined by those of p_1,...,p_(n-1). Sevenster (J. Logic and Computation, 2009) showed that satisfiability for modal dependence logic is complete for nondeterministic exponential time. In this paper we consider fragments of modal dependence logic obtained by restricting the set of allowed propositional connectives. We show that satisfibility for poor man's dependence logic, the language consisting of formulas built from literals and dependence atoms using conjunction, necessity and possibility (i.e., disallowing disjunction), remains NEXPTIME-complete. If we only allow monotone formulas (without negation, but with disjunction), the complexity drops to PSPACE-completeness. We also extend V\"a\"an\"anen's language by allowing classical disjunction besides dependence disjunction and show that the satisfiability problem remains NEXPTIME-complete. If we then disallow both negation and dependence disjunction, satistiability is complete for the second level of the polynomial hierarchy. In this way we completely classify the computational complexity of the satisfiability problem for all restrictions of propositional and dependence operators considered by V\"a\"an\"anen and Sevenster.Comment: 22 pages, full version of CSL 2010 pape

Changing a semantics: opportunism or courage?

The generalized models for higher-order logics introduced by Leon Henkin, and their multiple offspring over the years, have become a standard tool in many areas of logic. Even so, discussion has persisted about their technical status, and perhaps even their conceptual legitimacy. This paper gives a systematic view of generalized model techniques, discusses what they mean in mathematical and philosophical terms, and presents a few technical themes and results about their role in algebraic representation, calibrating provability, lowering complexity, understanding fixed-point logics, and achieving set-theoretic absoluteness. We also show how thinking about Henkin's approach to semantics of logical systems in this generality can yield new results, dispelling the impression of adhocness. This paper is dedicated to Leon Henkin, a deep logician who has changed the way we all work, while also being an always open, modest, and encouraging colleague and friend.Comment: 27 pages. To appear in: The life and work of Leon Henkin: Essays on his contributions (Studies in Universal Logic) eds: Manzano, M., Sain, I. and Alonso, E., 201

The k-variable property is stronger than H-dimension k

Accepted versio

Building the economic evidence case for social prescribing

Understanding the economic impact of social prescribing remains an urgent priority for the National Academy of Social Prescribing (NASP). As yet it is unclear how much data exists within the different systems to enable economic analyses of the impact of social prescribing schemes to be conducted. The complexity in understanding the economic impact of social prescribing—and indeed all non-clinical community-based approaches to health—is compounded by the multisector nature of social prescribing. Furthermore, a variety of approaches are being used to test similar but different understandings of both cost and value, including social value, cost, benefit and economic value. There are a growing range of reports and peer-reviewed publications that focus on the impact of social prescribing on health and social care demand, some of which have economic analyses and some which remain as potential data sets for economic analyses. At least one third of all outcomes (if not more) are directly related to the social determinants of health1,2 (SDH) which are not taken into account with economic analysis focused only on health service usage. This range of outcomes experienced by service users2-4 is driving many researchers to conduct economic analyses that attempt to assign value to outcomes beyond the health sector, for instance using social return on investment (SROI) and proxy values. Other researchers have discussed the evolution in economic analyses at length and suggest additional components to existing methodologies, e.g., multi criteria decision analysis (MCDA) to account for additional complexity of social prescribing5. Further developments are also being trialled such as the Wellbeing-adjusted Life Years (WELLBY) to understand the economic value attached to wellbeing6, as opposed to the Quality Adjusted Life Years (QALY), which reports the economic value of quality of life. We are entering an era of providing personalised support to people in integrated and multidisciplinary systems with different local population needs. As such, there is a need to evolve the approaches to determining cost and value of social prescribing, and to reach agreements on methodologies that all sectors are willing to accept as sound approaches. Furthermore, as discussed by McDaid and colleagues in 2019 7, there is a need to move beyond the immediate benefits of social prescribing and to explore the longer-term benefits of sustained engagement in non-clinical activities and provision of support to address issues linked to the SDH. This would enable more data to inform the preventative role and economic impact that social prescribing may have, which is currently an evidence gap. This rapid scoping review was commissioned by NASP and additional roundtables were supported by the National Centre for Creative Health and UKRI/AHRC’s ‘Mobilising Community Assets to Tackle Health Inequalities’ research programme (led by University College London). It aims to provide an update to the first economic evidence review from NASP and explore economic data and health and social care usage data in more detail. This rapid scoping review aims to ascertain: What the current literature indicates in terms of cost or value of social prescribing schemes or parts of the social prescribing scheme. If there are potential data sets that report the impact of social prescribing on health service usage that could have economic analysis applied to them. Stakeholder opinions on the methodological approaches for creating the current economic evaluation evidence base for social prescribing and potential future developments that are needed. How these findings can inform a larger programme of research that is needed to establish the economic impact and value of social prescribing across all relevant sectors in the community. As this report contains three separate elements to it, each element will be reported with methods and results, and then key themes will be brought together with recommendations

A logic road from special relativity to general relativity

We present a streamlined axiom system of special relativity in first-order logic. From this axiom system we "derive" an axiom system of general relativity in two natural steps. We will also see how the axioms of special relativity transform into those of general relativity. This way we hope to make general relativity more accessible for the non-specialist

Dependence Logic with Generalized Quantifiers: Axiomatizations

We prove two completeness results, one for the extension of dependence logic by a monotone generalized quantifier Q with weak interpretation, weak in the meaning that the interpretation of Q varies with the structures. The second result considers the extension of dependence logic where Q is interpreted as "there exists uncountable many." Both of the axiomatizations are shown to be sound and complete for FO(Q) consequences.Comment: 17 page

Complexity of equational theory of relational algebras with standard projection elements

The class $\mathsf{TPA}$ of t rue p airing a lgebras is defined to be the class of relation algebras expanded with concrete set theoretical projection functions. The main results of the present paper is that neither the equational theory of $\mathsf{TPA}$ nor the first order theory of $\mathsf{TPA}$ are decidable. Moreover, we show that the set of all equations valid in $\mathsf{TPA}$ is exactly on the $\Pi ^1_1$ level. We consider the class $\mathsf{TPA}^-$ of the relation algebra reducts of $\mathsf{TPA}$’s, as well. We prove that the equational theory of $\mathsf{TPA}^-$ is much simpler, namely, it is recursively enumerable. We also give motivation for our results and some connections to related work

Modal Ω-Logic: Automata, Neo-Logicism, and Set-Theoretic Realism

This essay examines the philosophical significance of $\Omega$-logic in Zermelo-Fraenkel set theory with choice (ZFC). The duality between coalgebra and algebra permits Boolean-valued algebraic models of ZFC to be interpreted as coalgebras. The modal profile of $\Omega$-logical validity can then be countenanced within a coalgebraic logic, and $\Omega$-logical validity can be defined via deterministic automata. I argue that the philosophical significance of the foregoing is two-fold. First, because the epistemic and modal profiles of $\Omega$-logical validity correspond to those of second-order logical consequence, $\Omega$-logical validity is genuinely logical, and thus vindicates a neo-logicist conception of mathematical truth in the set-theoretic multiverse. Second, the foregoing provides a modal-computational account of the interpretation of mathematical vocabulary, adducing in favor of a realist conception of the cumulative hierarchy of sets

Theory of Concepts

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