Confluence of Conditional Rewriting in Logic Form

We characterize conditional rewriting as satisfiability in a Herbrand-like model of terms where variables are also included as fresh constant symbols extending the original signature. Confluence of conditional rewriting and joinability of conditional critical pairs is characterized similarly. Joinability of critical pairs is then translated into combinations of (in)feasibility problems which can be efficiently handled by a number of automatic tools. This permits a more efficient use of standard results for proving confluence of conditional term rewriting systems, most of them relying on auxiliary proofs of joinability of conditional critical pairs, perhaps with additional syntactical and (operational) termination requirements on the system. Our approach has been implemented in a new system: CONFident . Its ability to (dis)prove confluence of conditional term rewriting systems is witnessed by means of some benchmarks comparing our tool with existing tools for similar purposes

Derivational Complexity and Context-Sensitive Rewriting

[EN] Context-sensitive rewriting is a restriction of rewriting where reduction steps are allowed on specific arguments mu(f) subset of {1, ..., k} of k-ary function symbols f only. Terms which cannot be further rewritten in this way are called mu-normal forms. For left-linear term rewriting systems (TRSs), the so-called normalization via mu-normalization procedure provides a systematic way to obtain normal forms by the stepwise computation and combination of intermediate mu-normal forms. In this paper, we show how to obtain bounds on the derivational complexity of computations using this procedure by using bounds on the derivational complexity of context-sensitive rewriting. Two main applications are envisaged: Normalization via mu-normalization can be used with non-terminating TRSs where the procedure still terminates; on the other hand, it can be used to improve on bounds of derivational complexity of terminating TRSs as it discards many rewritings.Partially supported by the EU (FEDER), and projects RTI2018-094403-B-C32 and PROMETEO/2019/098.Lucas Alba, S. (2021). Derivational Complexity and Context-Sensitive Rewriting. Journal of Automated Reasoning. 65(8):1191-1229. https://doi.org/10.1007/s10817-021-09603-11191122965

Applications and extensions of context-sensitive rewriting

[EN] Context-sensitive rewriting is a restriction of term rewriting which is obtained by imposing replacement restrictions on the arguments of function symbols. It has proven useful to analyze computational properties of programs written in sophisticated rewriting-based programming languages such asCafeOBJ, Haskell, Maude, OBJ*, etc. Also, a number of extensions(e.g., to conditional rewritingor constrained equational systems) and generalizations(e.g., controlled rewritingor forbidden patterns) of context-sensitive rewriting have been proposed. In this paper, we provide an overview of these applications and related issues. (C) 2021 Elsevier Inc. All rights reserved.Partially supported by the EU (FEDER), and projects RTI2018-094403-B-C32 and PROMETEO/2019/098.Lucas Alba, S. (2021). Applications and extensions of context-sensitive rewriting. Journal of Logical and Algebraic Methods in Programming. 121:1-33. https://doi.org/10.1016/j.jlamp.2021.10068013312

The origins of the halting problem

[EN] The halting problem is a prominent example of undecidable problem and its formulation and undecidability proof is usually attributed to Turing's 1936 landmark paper. Copeland noticed in 2004, though, that it was so named and, apparently, first stated in a 1958 book by Martin Davis. We provide additional arguments partially supporting this claim as follows: (i) with a focus on computable (real) numbers with infinitely many digits (e.g., pi), in his paper Turing was not concerned with halting machines; (ii) the two decision problems considered by Turing concern the ability of his machines to produce specific kinds of outputs, rather than reaching a halting state, something which was missing from Turing's notion of computation; and (iii) from 1936 to 1958, when considering the literature of the field no paper refers to any "halting problem" of Turing Machines until Davis' book. However, there were important preliminary contributions by (iv) Church, for whom termination was part of his notion of computation (for the lambda-calculus), and (v) Kleene, who essentially formulated, in his 1952 book, what we know as the halting problem now.Partially supported by the EU (FEDER), and projects RTI2018-094403-B-C32, PROMETEO/2019/098.Lucas Alba, S. (2021). The origins of the halting problem. Journal of Logical and Algebraic Methods in Programming. 121:1-9. https://doi.org/10.1016/j.jlamp.2021.1006871912

Using Well-Founded Relations for Proving Operational Termination

[EN] In this paper, we study operational termination, a proof theoretical notion for capturing the termination behavior of computational systems. We prove that operational termination can be characterized at different levels by means of well- founded relations on specific formulas which can be obtained from the considered system. We show how to obtain such well-founded relations from logical models which can be automatically generated using existing tools.Partially supported by the EU (FEDER), Projects TIN2015-69175-C4-1-R, and GV PROMETEOII/2015/013.Lucas Alba, S. (2020). Using Well-Founded Relations for Proving Operational Termination. Journal of Automated Reasoning. 64(2):167-195. https://doi.org/10.1007/s10817-019-09514-2S167195642Alarcón, B., Gutiérrez, R., Lucas, S., Navarro-Marset, R.: Proving termination properties with MU-TERM. 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Use of the terms "Wellbeing" and "Quality of Life" in health sciences: A conceptual framework

Background and Objectives: The assessment of wellbeing is a top priority in health sciences. The aim of this paper is to review the history of the concept of wellbeing and “Quality of Life” (QoL), and to understand the theories and assumptions that guided this field in order to provide a conceptual framework that may eventually facilitate the development of a formal synset (grouping of synonyms and semantically similar terms) of health-related wellbeing Methods: The history of the concept of wellbeing and QoL was reviewed in order to provide a conceptual framework. Results: Huge differences exist on the definition of “Wellbeing” and its relationship with QoL, “Happiness” and “Functioning” in the health context. From a dimensional perspective, health related wellbeing could be regarded as an overarching construct characterised by asymmetrical polarity, where “wellbeing” embeds the concept of “ill-being” as “health” incorporates de concept of “disease”. Conclusions: A common conceptual framework of these terms may eventually facilitate the development of a formal synset of health-related wellbeing. This terminological clarification should be part of a new taxonomy of health-related wellbeing based on the International Classification of Functioning, Disability and Health (ICF) framework that may facilitate knowledge transfer across different sectors and semantic interoperability for care management and planningThe research leading to these results has received funding from the European Community’s Seventh Framework Programme under grant agreement numbers 223071 (COURAGE in Europe) and 282586 (ROAMER), from the Instituto de Salud Carlos III-FIS research grant number PS09/00295, and from the Spanish Ministry of Science and Innovation ACI-Promociona (ACI2009-1010 and ACI- 2011-1080). The study was supported by the Centro de Investigación Biomédica en Red de Salud Mental (CIBERSAM), Instituto de Salud Carlos II

Innermost Termination of Context-Sensitive Rewriting

Innermost context-sensitive rewriting (CSR) has been proved useful for modeling the computational behavior of programs of algebraic languages like Maude, OBJ, etc, which incorporate an innermost strategy which is used to break down the nondeterminism which is inherent to reduction relations. Furthermore, innermost termination of rewriting is often easier to prove than termination. Thus, under appropriate conditions, a useful strategy for proving termination of rewriting is trying to prove termination of innermost rewriting. This phenomenon has also been investigated for context-sensitive rewriting. Up to now, only few transformation-based methods have been proposed and used to (specifically) prove termination of innermost CSR. Powerful and e cient techniques for proving (innermost) termination of (unrestricted) rewriting like the dependency pair framework have not been considered yet. In this work, we investigate the adaptation of the dependency pair framework to innermost CSR. We provide a suitable notion of innermost context-sensitive dependency pair and show how to extend and adapt the main notions which conform the framework (chain, termination problem, processor, etc.). Thanks to the innermost context-sensitive dependency pairs, we can now use powerful techniques for proving termination of innermost CSR. This is made clear by means of some benchmarks showing that our techniques dramatically improve over previously existing transformational techniques, thus establishing the new state-of-the-art in the area. We have implemented them as part of the termination tool MU-TERM.Alarcón Jiménez, B.; Lucas, S. (2011). Innermost Termination of Context-Sensitive Rewriting. http://hdl.handle.net/10251/1079