On Causal Equivalence by Tracing in String Rewriting

We introduce proof terms for string rewrite systems and, using these, show that various notions of equivalence on reductions known from the literature can be viewed as different perspectives on the notion of causal equivalence. In particular, we show that permutation equivalence classes (as known from the lambda-calculus and term rewriting) are uniquely represented both by trace graphs (known from physics as causal graphs) and by so-called greedy multistep reductions (as known from algebra). We present effective maps from the former to the latter, topological multi-sorting TM, and vice versa, the proof term algebra [[ ]].Comment: In Proceedings TERMGRAPH 2022, arXiv:2303.1421

Correspondence-based analogies for choosing problem representations

Mathematics and computing students learn new concepts and fortify their expertise by solving problems. The representation of a problem, be it through algebra, diagrams, or code, is key to understanding and solving it. Multiple-representation interactive environments are a promising approach, but the task of choosing an appropriate representation is largely placed on the user. We propose a new method to recommend representations based on correspondences: conceptual links between domains. Correspondences can be used to analyse, identify, and construct analogies even when the analogical target is unknown. This paper explains how correspondences build on probability theory and Gentner's structure-mapping framework; proposes rules for semi-automated correspondence discovery; and describes how correspondences can explain and construct analogies

Normal forms and normal theories in conditional rewriting

this is the author’s version of a work that was accepted for publication in Journal of Logical and Algebraic Methods in Programming. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Journal of Logical and Algebraic Methods in Programming vol. 85 (2016) DOI 10.1016/j.jlamp.2015.06.001We present several new concepts and results on conditional term rewriting within the general framework of order-sorted rewrite theories (OSRTs), which support types, subtypes and rewriting modulo axioms, and contains the more restricted framework of conditional term rewriting systems (CTRSs) as a special case. The concepts shed light on several subtle issues about conditional rewriting and conditional termination. We point out that the notions of irreducible term and of normal form, which coincide for unconditional rewriting, have been conflated for conditional rewriting but are in fact totally different notions. Normal form is a stronger concept. We call any rewrite theory where all irreducible terms are normal forms a normal theory. We argue that normality is essential to have good executability and computability properties. Therefore we call all other theories abnormal, freaks of nature to be avoided. The distinction between irreducible terms and normal forms helps in clarifying various notions of strong and weak termination. We show that abnormal theories can be terminating in various, equally abnormal ways; and argue that any computationally meaningful notion of strong or weak conditional termination should be a property of normal theories. In particular we define the notion of a weakly operationally terminating (or weakly normalizing) OSRT, discuss several evaluation mechanisms to compute normal forms in such theories, and investigate general conditions under which the rewriting-based operational semantics and the initial algebra semantics of a confluent, weakly normalizing OSRT coincide thanks to a notion of canonical term algebra. Finally, we investigate appropriate conditions and proof methods to ensure that a rewrite theory is normal; and characterize the stronger property of a rewrite theory being operationally terminating in terms of a natural generalization of the notion of quasidecreasing order. (C) 2015 Elsevier Inc. All rights reserved.We thank the anonymous referees for their constructive criticism and helpful comments. This work has been partially supported by NSF grant CNS 13-19109. Salvador Lucas' research was developed during a sabbatical year at UIUC and was also supported by the EU (FEDER), Spanish MINECO projects TIN2010-21062-C02-02 and TIN 2013-45732-C4-1-P, and GV grant BEST/2014/026 and project PROMETEO/2011/052.Lucas Alba, S.; Meseguer, J. (2016). Normal forms and normal theories in conditional rewriting. Journal of Logical and Algebraic Methods in Programming. 85(1):67-97. https://doi.org/10.1016/j.jlamp.2015.06.001S679785

Hopeful Monsters : A Note on Multiple Conclusions

Arguments, the story goes, have one or more premises and only one conclusion. A contentious generalisation allows arguments with several disjunctively connected conclusions. Contentious as this generalisation may be, I will argue nevertheless that it is justified. My main claim is that multiple conclusions are epiphenomena of the logical connectives: some connectives determine, in a certain sense, multiple-conclusion derivations. Therefore, such derivations are completely natural and can safely be used in proof-theoretic semantics.info:eu-repo/semantics/publishedVersio

Understanding second language acquisition in relation to intentionality, epistemology and cognitive processes in an academic context: A realist perspective

This study investigated the nature of the second language acquisition (SLA) of eight international PhD students that occurred as they undertook the literature review when planning their doctoral research projects at a New Zealand university. While the social, cultural approaches have been seen as broadening the field of SLA by introducing diverse epistemologies (Ortega, 2012), their conceptions of the mind and cognition of the second language speaker as a social product, and of SLA as the outcome of social processes (see Atkinson, 2011a), appear to be problematic in terms of understanding the central mechanisms of SLA. This study addresses this issue by explicating the mind and cognition of the participants and their SLA from a phenomenological realist perspective. Central to the theoretical framework for the study are Husserl’s (1970) realist ontology and epistemology, Widdowson’s (1983, 1990) theory of language learning through the negotiation of meaning and Bruce’s (2008a) identification of extra-linguistic and linguistic areas of genre knowledge in an English-medium academic context while developing their literature reviews (LR). Interpretative phenomenological analysis (Smith, Flowers & Larkin, 2009) was used as the overarching methodology to investigate the ontology of the participants, their epistemological processing, and their acquisition of academic English. This involved analysing monthly interviews with individual participants and supplementary data collected during a six-month conditional enrolment period. In addition, an analysis of the actual LR texts of five participants was undertaken to examine the textual outcomes of the LR process. This analysis focused, in particular, on genre knowledge and logicality as critical elements of the academic competence that the participants were engaged in developing. The findings of the study suggest that intentionality as a cognitive process was what enabled the participants to engage with social processes in the course of their SLA. Thought and language, as two separate entities in the knowledge of the participants, seemed to interact hierarchically in the process of undertaking the LR. Their SLA seemed to occur through this hierarchical thought-language operation, which involved participants drawing on and using new linguistic and procedural resources to express their thought. In addition, the thought of the participants seemed not to be constrained or regulated by linguistic and rhetorical systems (either from their first language or English). Rather, in their efforts to engage with disciplinary knowledge when processing and communicating the meaning of academic texts intended by the authors, participants went beyond such cultural frames. Significantly, this meaning-uncovering intentionality appeared to facilitate parallel SLA. While advice and feedback from other members of their particular community appeared to be important to their SLA, it was not evident that social interaction was the central, facilitative process. Moreover, the overall findings of the study suggest a need to expand the scope of what constitutes SLA in academic contexts. That is, developing competence in using academic English appeared to involve not only the acquisition of linguistic resources, but also developing extra-linguistic genre knowledge that ensures textual coherence, in order to communicate intended thought including logicality and criticality successfully

The Logical Writings of Karl Popper

This open access book is the first ever collection of Karl Popper's writings on deductive logic. Karl R. Popper (1902-1994) was one of the most influential philosophers of the 20th century. His philosophy of science ("falsificationism") and his social and political philosophy ("open society") have been widely discussed way beyond academic philosophy. What is not so well known is that Popper also produced a considerable work on the foundations of deductive logic, most of it published at the end of the 1940s as articles at scattered places. This little-known work deserves to be known better, as it is highly significant for modern proof-theoretic semantics. This collection assembles Popper's published writings on deductive logic in a single volume, together with all reviews of these papers. It also contains a large amount of unpublished material from the Popper Archives, including Popper's correspondence related to deductive logic and manuscripts that were (almost) finished, but did not reach the publication stage. All of these items are critically edited with additional comments by the editors. A general introduction puts Popper's work into the context of current discussions on the foundations of logic. This book should be of interest to logicians, philosophers, and anybody concerned with Popper's work

Axiomatic Architecture of Scientific Theories

The received concepts of axiomatic theory and axiomatic method, which stem from David Hilbert, need a systematic revision in view of more recent mathematical and scientific axiomatic practices, which do not fully follow in Hilbert’s steps and re-establish some older historical patterns of axiomatic thinking in unexpected new forms. In this work I motivate, formulate and justify such a revised concept of axiomatic theory, which for a variety of reasons I call constructive, and then argue that it can better serve as a formal representational tool in mathematics and science than the received concept