{"title":"A note on functional relations in a certain class of implicative expansions of FDE related to Brady's 4-valued logic BN4","authors":"G. Robles, J. Méndez","doi":"10.1093/jigpal/jzad004","DOIUrl":"https://doi.org/10.1093/jigpal/jzad004","url":null,"abstract":"","PeriodicalId":304915,"journal":{"name":"Log. J. IGPL","volume":"57 2","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-02-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133071661","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
E. D. L. Cal, M. Fáñez, Mario Villar, J. Villar, Víctor M. González
There are many real-world applications like healthcare systems, job monitoring, well-being and personal fitness tracking, monitoring of elderly and frail people, assessment of rehabilitation and follow-up treatments, affording Fall Detection (FD) and ADL (Activity of Daily Living) identification, separately or even at a time. However, the two main drawbacks of these solutions are that most of the times, the devices deployed are obtrusive (devices worn on not quite common parts of the body like neck, waist and ankle) and the poor battery life. Thus, this work proposes a low-power classification algorithm based on an Ensemble of KNN and K-Means algorithms (EKMeans) to identify Falls and High-Intensity ADL events such as running, jogging and climbing up stairs. The input of EKMeans are triaxial accelerometer data gathered from wrist-wearable devices. The proposal will be validated on the Fall&ADL publicly available datasets UMAFall, UCIFall and FallAllD, considering two kinds of activity labelling: Two-Class and Multi-Class. An exhaustive comparative study between our proposal, and the baseline algorithms KNN and a feed-forward Neural Network (NN) is deployed, where EKMeans outperformed clearly the Specificity (ADL classification) of the KNN and NN for the three datasets. Finally, a comparative battery consumption study has been included deploying the analyzed algorithms in a WearOS smartwatch, where EKMeans drains the battery from 100% to 0% in 27.45 hours, saving 5% and 21% concerning KNN and NN, respectively. Keywords: Human Activity Recognition, ADL Identification, Fall Detection TS Clustering, TS Classification, Wearable Devices, Low-Power HAR.
{"title":"A low-power HAR method for fall and high-intensity ADLs identification using wrist-worn accelerometer devices","authors":"E. D. L. Cal, M. Fáñez, Mario Villar, J. Villar, Víctor M. González","doi":"10.1093/jigpal/jzac025","DOIUrl":"https://doi.org/10.1093/jigpal/jzac025","url":null,"abstract":"\u0000 There are many real-world applications like healthcare systems, job monitoring, well-being and personal fitness tracking, monitoring of elderly and frail people, assessment of rehabilitation and follow-up treatments, affording Fall Detection (FD) and ADL (Activity of Daily Living) identification, separately or even at a time. However, the two main drawbacks of these solutions are that most of the times, the devices deployed are obtrusive (devices worn on not quite common parts of the body like neck, waist and ankle) and the poor battery life. Thus, this work proposes a low-power classification algorithm based on an Ensemble of KNN and K-Means algorithms (EKMeans) to identify Falls and High-Intensity ADL events such as running, jogging and climbing up stairs. The input of EKMeans are triaxial accelerometer data gathered from wrist-wearable devices. The proposal will be validated on the Fall&ADL publicly available datasets UMAFall, UCIFall and FallAllD, considering two kinds of activity labelling: Two-Class and Multi-Class. An exhaustive comparative study between our proposal, and the baseline algorithms KNN and a feed-forward Neural Network (NN) is deployed, where EKMeans outperformed clearly the Specificity (ADL classification) of the KNN and NN for the three datasets. Finally, a comparative battery consumption study has been included deploying the analyzed algorithms in a WearOS smartwatch, where EKMeans drains the battery from 100% to 0% in 27.45 hours, saving 5% and 21% concerning KNN and NN, respectively. Keywords: Human Activity Recognition, ADL Identification, Fall Detection TS Clustering, TS Classification, Wearable Devices, Low-Power HAR.","PeriodicalId":304915,"journal":{"name":"Log. J. IGPL","volume":"17 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128102851","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Saad Alabdulsalam, T. Duong, Kim-Kwang Raymond Choo, Nhien-An Le-Khac
In an Internet of Things (IoT) environment, IoT devices are typically connected through different network media types such as mobile, wireless and wired networks. Due to the pervasive nature of such devices, they are a potential evidence source in both civil litigation and criminal investigations. It is, however, challenging to identify and acquire forensic artefacts from a broad range of devices, which have varying storage and communication capabilities. Hence, in this paper, we first propose an IoT network architecture for the forensic purpose that uses machine learning algorithms to autonomously detect IoT devices. Then we posit the importance of focusing on the links between different IoT devices (e.g. whether one device is controlled or can be accessed from another device in the system), and design an approach to do so. Specifically, our approach adopts a graph for modelling IoT communications’ message flows to facilitate the identification of correlated network traffic based on the direction of the network and the associated attributes. To demonstrate how such an approach can be deployed in practice, we provide a proof of concept using two IoT controllers to generate 480 commands for controlling two IoT devices in a smart home environment and achieve an accuracy rate of 98.3% for detecting the links between devices. We also evaluate the proposed autonomous discovering of IoT devices and their activities in a TCP network by using real-world measurements from a public dataset of a popular off-the-shelf smart home deployed in two different locations. We selected 39 out of 81 different IoT devices for this evaluation.
{"title":"An efficient IoT forensic approach for the evidence acquisition and analysis based on network link","authors":"Saad Alabdulsalam, T. Duong, Kim-Kwang Raymond Choo, Nhien-An Le-Khac","doi":"10.1093/jigpal/jzac012","DOIUrl":"https://doi.org/10.1093/jigpal/jzac012","url":null,"abstract":"\u0000 In an Internet of Things (IoT) environment, IoT devices are typically connected through different network media types such as mobile, wireless and wired networks. Due to the pervasive nature of such devices, they are a potential evidence source in both civil litigation and criminal investigations. It is, however, challenging to identify and acquire forensic artefacts from a broad range of devices, which have varying storage and communication capabilities. Hence, in this paper, we first propose an IoT network architecture for the forensic purpose that uses machine learning algorithms to autonomously detect IoT devices. Then we posit the importance of focusing on the links between different IoT devices (e.g. whether one device is controlled or can be accessed from another device in the system), and design an approach to do so. Specifically, our approach adopts a graph for modelling IoT communications’ message flows to facilitate the identification of correlated network traffic based on the direction of the network and the associated attributes. To demonstrate how such an approach can be deployed in practice, we provide a proof of concept using two IoT controllers to generate 480 commands for controlling two IoT devices in a smart home environment and achieve an accuracy rate of 98.3% for detecting the links between devices. We also evaluate the proposed autonomous discovering of IoT devices and their activities in a TCP network by using real-world measurements from a public dataset of a popular off-the-shelf smart home deployed in two different locations. We selected 39 out of 81 different IoT devices for this evaluation.","PeriodicalId":304915,"journal":{"name":"Log. J. IGPL","volume":"55 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-02-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125070103","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Abductive conclusions are drawn in a special, co-hortative mood (Peirce’s ‘investigand’). Abductive conclusions are representative interpretants that represent abduction (or retroduction) as a form of reasoning that can convey a general conception of the truth. The truth is not asserted; abduction merely delivers the idea of a matter of course, rendering that idea comparatively simple and natural, hence assuring us of its justified assertibility. Hence abductive reasoning is at home in addressing ‘How Possible’-questions in science. Abductive reasoning concerns the question of how things might, could or would conceivably be such that they can be plausibly asserted. Peirce took all reasoning to be diagrammatic and representable using the graphical method of logic. Yet no examples have previously been found in his large manuscript corpus of what such non-deductive graphs might look like. This paper proposes a new interpretation of a sole exception, a sketch of two graphs from a rejected page from 1903, which might be the only surviving example of Peirce’s abductive graphs. The proposed interpretation takes them to be representative interpretants of this special inverse type of inference.
{"title":"Abduction and diagrams","authors":"A. Pietarinen","doi":"10.1093/JIGPAL/JZZ034","DOIUrl":"https://doi.org/10.1093/JIGPAL/JZZ034","url":null,"abstract":"\u0000 Abductive conclusions are drawn in a special, co-hortative mood (Peirce’s ‘investigand’). Abductive conclusions are representative interpretants that represent abduction (or retroduction) as a form of reasoning that can convey a general conception of the truth. The truth is not asserted; abduction merely delivers the idea of a matter of course, rendering that idea comparatively simple and natural, hence assuring us of its justified assertibility. Hence abductive reasoning is at home in addressing ‘How Possible’-questions in science. Abductive reasoning concerns the question of how things might, could or would conceivably be such that they can be plausibly asserted. Peirce took all reasoning to be diagrammatic and representable using the graphical method of logic. Yet no examples have previously been found in his large manuscript corpus of what such non-deductive graphs might look like. This paper proposes a new interpretation of a sole exception, a sketch of two graphs from a rejected page from 1903, which might be the only surviving example of Peirce’s abductive graphs. The proposed interpretation takes them to be representative interpretants of this special inverse type of inference.","PeriodicalId":304915,"journal":{"name":"Log. J. IGPL","volume":"29 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-07-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129869793","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A version of intuitionistic type theory is extended with opposite types, allowing a different formalization of negation and obtaining a paraconsistent type theory ($textsf{PTT} $). The rules for opposite types in $textsf{PTT} $ are based on the rules of the so-called constructible falsity. A propositions-as-types correspondence between the many-sorted paraconsistent logic $textsf{PL}_textsf{S} $ (a many-sorted extension of López-Escobar’s refutability calculus presented in natural deduction format) and $textsf{PTT} $ is proven. Moreover, a translation of $textsf{PTT} $ into intuitionistic type theory is presented and some properties of $textsf{PTT} $ are discussed.
{"title":"Type Theory with Opposite Types: A Paraconsistent Type Theory","authors":"J. C. Agudelo, Andrés Sicard-Ramírez","doi":"10.1093/JIGPAL/JZAB022","DOIUrl":"https://doi.org/10.1093/JIGPAL/JZAB022","url":null,"abstract":"\u0000 A version of intuitionistic type theory is extended with opposite types, allowing a different formalization of negation and obtaining a paraconsistent type theory ($textsf{PTT} $). The rules for opposite types in $textsf{PTT} $ are based on the rules of the so-called constructible falsity. A propositions-as-types correspondence between the many-sorted paraconsistent logic $textsf{PL}_textsf{S} $ (a many-sorted extension of López-Escobar’s refutability calculus presented in natural deduction format) and $textsf{PTT} $ is proven. Moreover, a translation of $textsf{PTT} $ into intuitionistic type theory is presented and some properties of $textsf{PTT} $ are discussed.","PeriodicalId":304915,"journal":{"name":"Log. J. IGPL","volume":"1 4","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-07-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131752132","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The need for a ‘many-valued logic’ in linguistics has been evident since the 1970s, but there was lack of clarity as to whether it should come from the family of fuzzy logics or from the family of probabilistic logics. In this regard, Fine [14] and Kamp [26] pointed out undesirable effects of fuzzy logic (the failure of idempotency and coherence) which kept two generations of linguists and philosophers at arm’s length. (Another unwanted feature of fuzzy logic is the property of truth functionality.) While probabilistic logic is not fraught by the same problems, its lack of constructiveness, i.e. its inability to compose complex truth degrees from atomic truth degrees, did not make it more attractive to linguists either. In the absence of a clear perspective in ‘many-valued logic’, scholars chose to proliferate ontologies grafted atop the classical bivalent logic: ontologies for truth, individuals, events, situations, possible worlds and degrees. The result has been a collection of incompatible classical logics. In this paper, I present sample logic, in particular its semantics (not its axiomatization). Sample logics is a member of the family of probabilistic logics, which is constructive without being truth functional. More specifically, I integrate all the important linguistic data on which the classical logics are predicated. The concepts of (in)dependency and conditional (in)dependency are the cornerstones of sample logic.
{"title":"Sample logic","authors":"M. Gerner","doi":"10.1093/jigpal/jzab021","DOIUrl":"https://doi.org/10.1093/jigpal/jzab021","url":null,"abstract":"\u0000 The need for a ‘many-valued logic’ in linguistics has been evident since the 1970s, but there was lack of clarity as to whether it should come from the family of fuzzy logics or from the family of probabilistic logics. In this regard, Fine [14] and Kamp [26] pointed out undesirable effects of fuzzy logic (the failure of idempotency and coherence) which kept two generations of linguists and philosophers at arm’s length. (Another unwanted feature of fuzzy logic is the property of truth functionality.) While probabilistic logic is not fraught by the same problems, its lack of constructiveness, i.e. its inability to compose complex truth degrees from atomic truth degrees, did not make it more attractive to linguists either. In the absence of a clear perspective in ‘many-valued logic’, scholars chose to proliferate ontologies grafted atop the classical bivalent logic: ontologies for truth, individuals, events, situations, possible worlds and degrees. The result has been a collection of incompatible classical logics. In this paper, I present sample logic, in particular its semantics (not its axiomatization). Sample logics is a member of the family of probabilistic logics, which is constructive without being truth functional. More specifically, I integrate all the important linguistic data on which the classical logics are predicated. The concepts of (in)dependency and conditional (in)dependency are the cornerstones of sample logic.","PeriodicalId":304915,"journal":{"name":"Log. J. IGPL","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-07-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"128394069","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This is the second of a series of papers that investigate fragments of quasi-Nelson logic (QNL) from an algebraic logic standpoint. QNL, recently introduced as a common generalization of intuitionistic and Nelson’s constructive logic with strong negation, is the axiomatic extension of the substructural logic $FL_{ew}$ (full Lambek calculus with exchange and weakening) by the Nelson axiom. The algebraic counterpart of QNL (quasi-Nelson algebras) is a class of commutative integral residuated lattices (a.k.a. $FL_{ew}$-algebras) that includes both Heyting and Nelson algebras and can be characterized algebraically in several alternative ways. The present paper focuses on the algebraic counterpart (a class we dub quasi-Nelson implication algebras, QNI-algebras) of the implication–negation fragment of QNL, corresponding to the connectives that witness the algebraizability of QNL. We recall the main known results on QNI-algebras and establish a number of new ones. Among these, we show that QNI-algebras form a congruence-distributive variety (Cor. 3.15) that enjoys equationally definable principal congruences and the strong congruence extension property (Prop. 3.16); we also characterize the subdirectly irreducible QNI-algebras in terms of the underlying poset structure (Thm. 4.23). Most of these results are obtained thanks to twist representations for QNI-algebras, which generalize the known ones for Nelson and quasi-Nelson algebras; we further introduce a Hilbert-style calculus that is algebraizable and has the variety of QNI-algebras as its equivalent algebraic semantics.
{"title":"Fragments of Quasi-Nelson: The Algebraizable Core","authors":"U. Rivieccio","doi":"10.1093/JIGPAL/JZAB023","DOIUrl":"https://doi.org/10.1093/JIGPAL/JZAB023","url":null,"abstract":"\u0000 This is the second of a series of papers that investigate fragments of quasi-Nelson logic (QNL) from an algebraic logic standpoint. QNL, recently introduced as a common generalization of intuitionistic and Nelson’s constructive logic with strong negation, is the axiomatic extension of the substructural logic $FL_{ew}$ (full Lambek calculus with exchange and weakening) by the Nelson axiom. The algebraic counterpart of QNL (quasi-Nelson algebras) is a class of commutative integral residuated lattices (a.k.a. $FL_{ew}$-algebras) that includes both Heyting and Nelson algebras and can be characterized algebraically in several alternative ways. The present paper focuses on the algebraic counterpart (a class we dub quasi-Nelson implication algebras, QNI-algebras) of the implication–negation fragment of QNL, corresponding to the connectives that witness the algebraizability of QNL. We recall the main known results on QNI-algebras and establish a number of new ones. Among these, we show that QNI-algebras form a congruence-distributive variety (Cor. 3.15) that enjoys equationally definable principal congruences and the strong congruence extension property (Prop. 3.16); we also characterize the subdirectly irreducible QNI-algebras in terms of the underlying poset structure (Thm. 4.23). Most of these results are obtained thanks to twist representations for QNI-algebras, which generalize the known ones for Nelson and quasi-Nelson algebras; we further introduce a Hilbert-style calculus that is algebraizable and has the variety of QNI-algebras as its equivalent algebraic semantics.","PeriodicalId":304915,"journal":{"name":"Log. J. IGPL","volume":"25 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-07-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129753018","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
U. Andrews, Peter M. Gerdes, S. Lempp, Joseph S. Miller, N. Schweber
Combinatorial operations on sets are almost never well defined on Turing degrees, a fact so obvious that counterexamples are worth exhibiting. The case we focus on is the symmetric-difference operator; there are pairs of (nonzero) degrees for which the symmetric-difference operation is well defined. Some examples can be extracted from the literature, e.g. from the existence of nonzero degrees with strong minimal covers. We focus on the case of incomparable r.e. degrees for which the symmetric-difference operation is well defined.
{"title":"Computability and the Symmetric Difference Operator","authors":"U. Andrews, Peter M. Gerdes, S. Lempp, Joseph S. Miller, N. Schweber","doi":"10.1093/JIGPAL/JZAB017","DOIUrl":"https://doi.org/10.1093/JIGPAL/JZAB017","url":null,"abstract":"\u0000 Combinatorial operations on sets are almost never well defined on Turing degrees, a fact so obvious that counterexamples are worth exhibiting. The case we focus on is the symmetric-difference operator; there are pairs of (nonzero) degrees for which the symmetric-difference operation is well defined. Some examples can be extracted from the literature, e.g. from the existence of nonzero degrees with strong minimal covers. We focus on the case of incomparable r.e. degrees for which the symmetric-difference operation is well defined.","PeriodicalId":304915,"journal":{"name":"Log. J. IGPL","volume":"1 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-06-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132086820","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We obtain an effective embedding of the classical predicate logic into the logic of partial quasiary predicates. The embedding has the property that an image of a non-theorem of the classical logic is refutable in a model of the logic of partial quasiary predicates that has the same cardinality as the classical countermodel of the non-theorem. Therefore, we also obtain an embedding of the classical predicate logic of finite models into the logic of partial quasiary predicates over finite structures. As a consequence, we prove that the logic of partial quasiary predicates is undecidable—more precisely, $varSigma ^0_1$-complete—over arbitrary structures and not recursively enumerable—more precisely, $varPi ^0_1$-complete—over finite structures.
{"title":"Undecidability of the Logic of Partial Quasiary Predicates","authors":"M. Rybakov, D. Shkatov","doi":"10.1093/JIGPAL/JZAB018","DOIUrl":"https://doi.org/10.1093/JIGPAL/JZAB018","url":null,"abstract":"\u0000 We obtain an effective embedding of the classical predicate logic into the logic of partial quasiary predicates. The embedding has the property that an image of a non-theorem of the classical logic is refutable in a model of the logic of partial quasiary predicates that has the same cardinality as the classical countermodel of the non-theorem. Therefore, we also obtain an embedding of the classical predicate logic of finite models into the logic of partial quasiary predicates over finite structures. As a consequence, we prove that the logic of partial quasiary predicates is undecidable—more precisely, $varSigma ^0_1$-complete—over arbitrary structures and not recursively enumerable—more precisely, $varPi ^0_1$-complete—over finite structures.","PeriodicalId":304915,"journal":{"name":"Log. J. IGPL","volume":"13 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-05-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121792988","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The lambda calculus is a universal programming language. It can represent the computable functions, and such offers a formal counterpart to the point of view of functions as rules. Terms represent functions and this allows for the application of a term/function to any other term/function, including itself. The calculus can be seen as a formal theory with certain pre-established axioms and inference rules, which can be interpreted by models. Dana Scott proposed the first non-trivial model of the extensional lambda calculus, known as $ D_{infty }$, to represent the $lambda $-terms as the typical functions of set theory, where it is not allowed to apply a function to itself. Here we propose a construction of an $infty $-groupoid from any lambda model endowed with a topology. We apply this construction for the particular case $D_{infty }$, and we see that the Scott topology does not provide enough information about the relationship between higher homotopies. This motivates a new line of research focused on the exploration of $lambda $-models with the structure of a non-trivial $infty $-groupoid to generalize the proofs of term conversion (e.g., $beta $-equality, $eta $-equality) to higher-proofs in $lambda $-calculus.
{"title":"∞-Groupoid Generated by an Arbitrary Topological λ-Model","authors":"Daniel O. Martínez-Rivillas, R. D. Queiroz","doi":"10.1093/JIGPAL/JZAB015","DOIUrl":"https://doi.org/10.1093/JIGPAL/JZAB015","url":null,"abstract":"\u0000 The lambda calculus is a universal programming language. It can represent the computable functions, and such offers a formal counterpart to the point of view of functions as rules. Terms represent functions and this allows for the application of a term/function to any other term/function, including itself. The calculus can be seen as a formal theory with certain pre-established axioms and inference rules, which can be interpreted by models. Dana Scott proposed the first non-trivial model of the extensional lambda calculus, known as $ D_{infty }$, to represent the $lambda $-terms as the typical functions of set theory, where it is not allowed to apply a function to itself. Here we propose a construction of an $infty $-groupoid from any lambda model endowed with a topology. We apply this construction for the particular case $D_{infty }$, and we see that the Scott topology does not provide enough information about the relationship between higher homotopies. This motivates a new line of research focused on the exploration of $lambda $-models with the structure of a non-trivial $infty $-groupoid to generalize the proofs of term conversion (e.g., $beta $-equality, $eta $-equality) to higher-proofs in $lambda $-calculus.","PeriodicalId":304915,"journal":{"name":"Log. J. IGPL","volume":"44 14","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-04-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"114034974","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：