Global semantic typing for inductive and coinductive computing

Inductive and coinductive types are commonly construed as ontological (Church-style) types, denoting canonical data-sets such as natural numbers, lists, and streams. For various purposes, notably the study of programs in the context of global semantics, it is preferable to think of types as semantical properties (Curry-style). Intrinsic theories were introduced in the late 1990s to provide a purely logical framework for reasoning about programs and their semantic types. We extend them here to data given by any combination of inductive and coinductive definitions. This approach is of interest because it fits tightly with syntactic, semantic, and proof theoretic fundamentals of formal logic, with potential applications in implicit computational complexity as well as extraction of programs from proofs. We prove a Canonicity Theorem, showing that the global definition of program typing, via the usual (Tarskian) semantics of first-order logic, agrees with their operational semantics in the intended model. Finally, we show that every intrinsic theory is interpretable in a conservative extension of first-order arithmetic. This means that quantification over infinite data objects does not lead, on its own, to proof-theoretic strength beyond that of Peano Arithmetic. Intrinsic theories are perfectly amenable to formulas-as-types Curry-Howard morphisms, and were used to characterize major computational complexity classes Their extensions described here have similar potential which has already been applied

Deciding Entailments in Inductive Separation Logic with Tree Automata

Separation Logic (SL) with inductive definitions is a natural formalism for specifying complex recursive data structures, used in compositional verification of programs manipulating such structures. The key ingredient of any automated verification procedure based on SL is the decidability of the entailment problem. In this work, we reduce the entailment problem for a non-trivial subset of SL describing trees (and beyond) to the language inclusion of tree automata (TA). Our reduction provides tight complexity bounds for the problem and shows that entailment in our fragment is EXPTIME-complete. For practical purposes, we leverage from recent advances in automata theory, such as inclusion checking for non-deterministic TA avoiding explicit determinization. We implemented our method and present promising preliminary experimental results

Strong Normalization for the Parameter-Free Polymorphic Lambda Calculus Based on the Omega-Rule.

Following Aehlig, we consider a hierarchy F^p= { F^p_n }_{n in Nat} of parameter-free subsystems of System F, where each F^p_n corresponds to ID_n, the theory of n-times iterated inductive definitions (thus our F^p_n corresponds to the n+1th system of Aehlig). We here present two proofs of strong normalization for F^p_n, which are directly formalizable with inductive definitions. The first one, based on the Joachimski-Matthes method, can be fully formalized in ID_n+1. This provides a tight upper bound on the complexity of the normalization theorem for System F^p_n. The second one, based on the Godel-Tait method, can be locally formalized in ID_n. This provides a direct proof to the known result that the representable functions in F^p_n are provably total in ID_n. In both cases, Buchholz\u27 Omega-rule plays a central role

Meditators' Non-academic Definition of Mindfulness

Mindfulness has been defined differently in academic scientific contexts and in Buddhist academic contexts. An under-studied area is that of lay (non-academic) theories of mindfulness. The goal of this article is to identify, organize, analyze in detail, and provide themes from the meditators' definitions of mindfulness. Possible differences and similarities of the collected definitions of mindfulness with the scientific-academic definitions and with the academic-Buddhist definitions are also checked. A qualitative and inductive thematic analysis on the definitions of mindfulness offered by the participants was carried out. The sample consisted of 326 meditators who offered a definition of mindfulness through an open question. Seven themes were identified: (1) mindfulness defined as attention/awareness; (2) mindfulness defined as a non-evaluative attitude; (3) mindfulness defined as strategy; (4) mindfulness defined from a theoretical analysis; (5) mindfulness defined as a psycho-affective-spiritual state; (6) mindfulness defined as personal development; and (7) lack of understanding of mindfulness. From these themes, it can be deduced that the definitions collected share more patterns of meaning with the scientific-academic definition of mindfulness than with the academic-Buddhist one. The findings of this study provide new insights into the complexity and heterogeneity of the definition of mindfulness. What has been discovered may indicate the complexity of the mindfulness construct itself. The online version contains supplementary material available at 10.1007/s12671-022-01899-3

Meditators’ Non-academic Definition of Mindfulness

Objectives Mindfulness has been defined differently in academic scientific contexts and in Buddhist academic contexts. An under-studied area is that of lay (non-academic) theories of mindfulness. The goal of this article is to identify, organize, analyze in detail, and provide themes from the meditators' definitions of mindfulness. Possible differences and similarities of the collected definitions of mindfulness with the scientific-academic definitions and with the academic-Buddhist definitions are also checked. Methods A qualitative and inductive thematic analysis on the definitions of mindfulness offered by the participants was carried out. Results The sample consisted of 326 meditators who offered a definition of mindfulness through an open question. Seven themes were identified: (1) mindfulness defined as attention/awareness; (2) mindfulness defined as a non-evaluative attitude; (3) mindfulness defined as strategy; (4) mindfulness defined from a theoretical analysis; (5) mindfulness defined as a psycho-affective-spiritual state; (6) mindfulness defined as personal development; and (7) lack of understanding of mindfulness. From these themes, it can be deduced that the definitions collected share more patterns of meaning with the scientific-academic definition of mindfulness than with the academic-Buddhist one. Conclusions The findings of this study provide new insights into the complexity and heterogeneity of the definition of mindfulness. What has been discovered may indicate the complexity of the mindfulness construct itself.Open Access funding provided thanks to the CRUE-CSIC agreement with Springer Nature. Open Access funding provided by EHU/UPV agreement with Springer Nature

On the Complexity of Model Expansion

Abstract. We study the complexity of model expansion (MX), which is the prob-lem of expanding a given finite structure with additional relations to produce a finite model of a given formula. This is the logical task underlying many prac-tical constraint languages and systems for representing and solving search prob-lems, and our work is motivated by the need to provide theoretical foundations for these. We present results on both data and combined complexity of MX for several fragments and extensions of FO that are relevant for this purpose, in par-ticular the guarded fragment GFk of FO and extensions of FO and GFk with inductive definitions. We present these in the context of the two closely related, but more studied, problems of model checking and finite satisfiability. To obtain results on FO(ID), the extension of FO with inductive definitions, we provide translations between FO(ID) with FO(LFP), which are of independent interest.