83 found
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  1.  69
    Reasoning with logical bilattices.Ofer Arieli & Arnon Avron - 1996 - Journal of Logic, Language and Information 5 (1):25--63.
    The notion of bilattice was introduced by Ginsberg, and further examined by Fitting, as a general framework for many applications. In the present paper we develop proof systems, which correspond to bilattices in an essential way. For this goal we introduce the notion of logical bilattices. We also show how they can be used for efficient inferences from possibly inconsistent data. For this we incorporate certain ideas of Kifer and Lozinskii, which happen to suit well the context of our work. (...)
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  2.  91
    Natural 3-valued logics—characterization and proof theory.Arnon Avron - 1991 - Journal of Symbolic Logic 56 (1):276-294.
  3. Simple Consequence Relations.Arnon Avron - unknown
    We provide a general investigation of Logic in which the notion of a simple consequence relation is taken to be fundamental. Our notion is more general than the usual one since we give up monotonicity and use multisets rather than sets. We use our notion for characterizing several known logics (including Linear Logic and non-monotonic logics) and for a general, semantics-independent classi cation of standard connectives via equations on consequence relations (these include Girard's \multiplicatives" and \additives"). We next investigate the (...)
     
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  4. The method of hypersequents in the proof theory of propositional non-classical logics.Arnon Avron - 1977 - In Wilfrid Hodges (ed.), Logic. New York: Penguin Books. pp. 1-32.
    Until not too many years ago, all logics except classical logic (and, perhaps, intuitionistic logic too) were considered to be things esoteric. Today this state of a airs seems to have completely been changed. There is a growing interest in many types of nonclassical logics: modal and temporal logics, substructural logics, paraconsistent logics, non-monotonic logics { the list is long. The diversity of systems that have been proposed and studied is so great that a need is felt by many researchers (...)
     
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  5.  84
    The Semantics and Proof Theory of Linear Logic.Arnon Avron - 1988 - Theoretical Computer Science 57 (2):161-184.
    Linear logic is a new logic which was recently developed by Girard in order to provide a logical basis for the study of parallelism. It is described and investigated in Gi]. Girard's presentation of his logic is not so standard. In this paper we shall provide more standard proof systems and semantics. We shall also extend part of Girard's results by investigating the consequence relations associated with Linear Logic and by proving corresponding str ong completeness theorems. Finally, we shall investigate (...)
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  6.  14
    The value of the four values.Ofer Arieli & Arnon Avron - 1998 - Artificial Intelligence 102 (1):97-141.
  7. A constructive analysis of RM.Arnon Avron - 1987 - Journal of Symbolic Logic 52 (4):939 - 951.
  8.  31
    Multi-valued Calculi for Logics Based on Non-determinism.Arnon Avron & Beata Konikowska - 2005 - Logic Journal of the IGPL 13 (4):365-387.
    Non-deterministic matrices are multiple-valued structures in which the value assigned by a valuation to a complex formula can be chosen non-deterministically out of a certain nonempty set of options. We consider two different types of semantics which are based on Nmatrices: the dynamic one and the static one . We use the Rasiowa-Sikorski decomposition methodology to get sound and complete proof systems employing finite sets of mv-signed formulas for all propositional logics based on such structures with either of the above (...)
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  9.  40
    Four-Valued Paradefinite Logics.Ofer Arieli & Arnon Avron - 2017 - Studia Logica 105 (6):1087-1122.
    Paradefinite logics are logics that can be used for handling contradictory or partial information. As such, paradefinite logics should be both paraconsistent and paracomplete. In this paper we consider the simplest semantic framework for introducing paradefinite logics. It consists of the four-valued matrices that expand the minimal matrix which is characteristic for first degree entailments: Dunn–Belnap matrix. We survey and study the expressive power and proof theory of the most important logics that can be developed in this framework.
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  10. Non-deterministic Matrices and Modular Semantics of Rules.Arnon Avron - unknown
    We show by way of example how one can provide in a lot of cases simple modular semantics for rules of inference, so that the semantics of a system is obtained by joining the semantics of its rules in the most straightforward way. Our main tool for this task is the use of finite Nmatrices, which are multi-valued structures in which the value assigned by a valuation to a complex formula can be chosen non-deterministically out of a certain nonempty set (...)
     
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  11. What is relevance logic?Arnon Avron - 2014 - Annals of Pure and Applied Logic 165 (1):26-48.
    We suggest two precise abstract definitions of the notion of ‘relevance logic’ which are both independent of any proof system or semantics. We show that according to the simpler one, R → source is the minimal relevance logic, but R itself is not. In contrast, R and many other logics are relevance logics according to the second definition, while all fragments of linear logic are not.
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  12. Cut-free ordinary sequent calculi for logics having generalized finite-valued semantics.Arnon Avron, Jonathan Ben-Naim & Beata Konikowska - 2007 - Logica Universalis 1 (1):41-70.
    . The paper presents a method for transforming a given sound and complete n-sequent proof system into an equivalent sound and complete system of ordinary sequents. The method is applicable to a large, central class of (generalized) finite-valued logics with the language satisfying a certain minimal expressiveness condition. The expressiveness condition decrees that the truth-value of any formula φ must be identifiable by determining whether certain formulas uniformly constructed from φ have designated values or not. The transformation preserves the general (...)
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  13.  59
    On an implication connective of RM.Arnon Avron - 1986 - Notre Dame Journal of Formal Logic 27 (2):201-209.
  14.  70
    On modal systems having arithmetical interpretations.Arnon Avron - 1984 - Journal of Symbolic Logic 49 (3):935-942.
  15. Non-deterministic Semantics for Logics with a Consistency Operator.Arnon Avron - unknown
    In order to handle inconsistent knowledge bases in a reasonable way, one needs a logic which allows nontrivial inconsistent theories. Logics of this sort are called paraconsistent. One of the oldest and best known approaches to the problem of designing useful paraconsistent logics is da Costa’s approach, which seeks to allow the use of classical logic whenever it is safe to do so, but behaves completely differently when contradictions are involved. Da Costa’s approach has led to the family of logics (...)
     
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  16.  54
    Maximal and Premaximal Paraconsistency in the Framework of Three-Valued Semantics.Ofer Arieli, Arnon Avron & Anna Zamansky - 2011 - Studia Logica 97 (1):31 - 60.
    Maximality is a desirable property of paraconsistent logics, motivated by the aspiration to tolerate inconsistencies, but at the same time retain from classical logic as much as possible. In this paper we introduce the strongest possible notion of maximal paraconsistency, and investigate it in the context of logics that are based on deterministic or non-deterministic three-valued matrices. We show that all reasonable paraconsistent logics based on three-valued deterministic matrices are maximal in our strong sense. This applies to practically all three-valued (...)
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  17.  32
    Proof Systems for 3-valued Logics Based on Gödel’s Implication.Arnon Avron - 2022 - Logic Journal of the IGPL 30 (3):437-453.
    The logic $G3^{<}_{{{}^{\scriptsize{-}}}\!\!\textrm{L}}$ was introduced in Robles and Mendéz as a paraconsistent logic which is based on Gödel’s 3-valued matrix, except that Kleene–Łukasiewicz’s negation is added to the language and is used as the main negation connective. We show that $G3^{<}_{{{}^{\scriptsize{-}}}\!\!\textrm{L}}$ is exactly the intersection of $G3^{\{1\}}_{{{}^{\scriptsize{-}}}\!\!\textrm{L}}$ and $G3^{\{1,0.5\}}_{{{}^{\scriptsize{-}}}\!\!\textrm{L}}$, the two truth-preserving 3-valued logics which are based on the same truth tables. We then construct a Hilbert-type system which has for $\to $ as its sole rule of inference, and is (...)
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  18.  96
    Whither relevance logic?Arnon Avron - 1992 - Journal of Philosophical Logic 21 (3):243 - 281.
  19.  69
    A Non-deterministic View on Non-classical Negations.Arnon Avron - 2005 - Studia Logica 80 (2-3):159-194.
    We investigate two large families of logics, differing from each other by the treatment of negation. The logics in one of them are obtained from the positive fragment of classical logic (with or without a propositional constant ff for “the false”) by adding various standard Gentzen-type rules for negation. The logics in the other family are similarly obtained from LJ+, the positive fragment of intuitionistic logic (again, with or without ff). For all the systems, we provide simple semantics which is (...)
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  20.  27
    Self-Extensional Three-Valued Paraconsistent Logics.Arnon Avron - 2017 - Logica Universalis 11 (3):297-315.
    A logic \ is called self-extensional if it allows to replace occurrences of a formula by occurrences of an \-equivalent one in the context of claims about logical consequence and logical validity. It is known that no three-valued paraconsistent logic which has an implication can be self-extensional. In this paper we show that in contrast, there is exactly one self-extensional three-valued paraconsistent logic in the language of \ for which \ is a disjunction, and \ is a conjunction. We also (...)
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  21.  73
    Relevant entailment--semantics and formal systems.Arnon Avron - 1984 - Journal of Symbolic Logic 49 (2):334-342.
  22.  55
    Weyl Reexamined: “Das Kontinuum” 100 Years Later.Arnon Avron - 2020 - Bulletin of Symbolic Logic 26 (1):26-79.
    Hermann Weyl was one of the greatest mathematicians of the 20th century, with contributions to many branches of mathematics and physics. In 1918 he wrote a famous book, “Das Kontinuum”, on the foundations of mathematics. In that book he described mathematical analysis as a ‘house built on sand’, and tried to ‘replace this shifting foundation with pillars of enduring strength’. In this paper we reexamine and explain the philosophical and mathematical ideas that underly Weyl’s system in “Das Kontinuum”, and show (...)
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  23.  39
    Self-extensional three-valued paraconsistent logics have no implications.Arnon Avron & Jean-Yves Beziau - 2016 - Logic Journal of the IGPL 25 (2):183-194.
    A proof is presented showing that there is no paraconsistent logics with a standard implication which have a three-valued characteristic matrix, and in which the replacement principle holds.
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  24.  30
    The Normal and Self-extensional Extension of Dunn–Belnap Logic.Arnon Avron - 2020 - Logica Universalis 14 (3):281-296.
    A logic \ is called self-extensional if it allows to replace occurrences of a formula by occurrences of an \-equivalent one in the context of claims about logical consequence and logical validity. It is known that no three-valued paraconsistent logic which has an implication can be self-extensional. In this paper we show that in contrast, the famous Dunn–Belnap four-valued logic has exactly one self-extensional four-valued extension which has an implication. We also investigate the main properties of this logic, determine the (...)
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  25. Gentzen-type systems, resolution and tableaux.Arnon Avron - 1993 - Journal of Automated Reasoning 10:265-281.
    In advanced books and courses on logic (e.g. Sm], BM]) Gentzen-type systems or their dual, tableaux, are described as techniques for showing validity of formulae which are more practical than the usual Hilbert-type formalisms. People who have learnt these methods often wonder why the Automated Reasoning community seems to ignore them and prefers instead the resolution method. Some of the classical books on AD (such as CL], Lo]) do not mention these methods at all. Others (such as Ro]) do, but (...)
     
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  26.  21
    A paraconsistent view on B and S5.Arnon Avron & Anna Zamansky - 2016 - In Lev Beklemishev, Stéphane Demri & András Máté (eds.), Advances in Modal Logic, Volume 11. CSLI Publications. pp. 21-37.
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  27.  46
    Paraconsistency, paracompleteness, Gentzen systems, and trivalent semantics.Arnon Avron - 2014 - Journal of Applied Non-Classical Logics 24 (1-2):12-34.
    A quasi-canonical Gentzen-type system is a Gentzen-type system in which each logical rule introduces either a formula of the form , or of the form , and all the active formulas of its premises belong to the set . In this paper we investigate quasi-canonical systems in which exactly one of the two classical rules for negation is included, turning the induced logic into either a paraconsistent logic or a paracomplete logic, but not both. We provide a constructive coherence criterion (...)
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  28.  20
    Multiplicative Conjunction as an Extensional Conjunction.Arnon Avron - 1997 - Logic Journal of the IGPL 5 (2):181-208.
    We show that the rule that allows the inference of A from A ⊗ B is admissible in many of the basic multiplicative systems. By adding this rule to these systems we get, therefore, conservative extensions in which the tensor behaves as classical conjunction. Among the systems obtained in this way the one derived from RMIm has a particular interest. We show that this system has a simple infinite-valued semantics, relative to which it is strongly complete, and a nice cut-free (...)
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  29. On Negation, Completeness and Consistency.Arnon Avron - unknown
    We have avoided here the term \false", since we do not want to commit ourselves to the view that A is false precisely when it is not true. Our formulation of the intuition is therefore obviously circular, but this is unavoidable in intuitive informal characterizations of basic connectives and quanti ers.
     
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  30.  45
    The middle ground-ancestral logic.Liron Cohen & Arnon Avron - 2019 - Synthese 196 (7):2671-2693.
    Many efforts have been made in recent years to construct formal systems for mechanizing general mathematical reasoning. Most of these systems are based on logics which are stronger than first-order logic. However, there are good reasons to avoid using full second-order logic for this task. In this work we investigate a logic which is intermediate between FOL and SOL, and seems to be a particularly attractive alternative to both: ancestral logic. This is the logic which is obtained from FOL by (...)
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  31.  32
    Proof Systems for Reasoning about Computation Errors.Arnon Avron & Beata Konikowska - 2009 - Studia Logica 91 (2):273-293.
    In the paper we examine the use of non-classical truth values for dealing with computation errors in program specification and validation. In that context, 3-valued McCarthy logic is suitable for handling lazy sequential computation, while 3-valued Kleene logic can be used for reasoning about parallel computation. If we want to be able to deal with both strategies without distinguishing between them, we combine Kleene and McCarthy logics into a logic based on a non-deterministic, 3-valued matrix, incorporating both options as a (...)
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  32.  50
    Gentzenizing Schroeder-Heister's natural extension of natural deduction.Arnon Avron - 1989 - Notre Dame Journal of Formal Logic 31 (1):127-135.
  33.  23
    Quasi-canonical systems and their semantics.Arnon Avron - 2018 - Synthese 198 (S22):5353-5371.
    A canonical Gentzen-type system is a system in which every rule has the subformula property, it introduces exactly one occurrence of a connective, and it imposes no restrictions on the contexts of its applications. A larger class of Gentzen-type systems which is also extensively in use is that of quasi-canonical systems. In such systems a special role is given to a unary connective \ of the language. Accordingly, each application of a logical rule in such systems introduces either a formula (...)
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  34. Logical Non-determinism as a Tool for Logical Modularity: An Introduction.Arnon Avron - unknown
    It is well known that every propositional logic which satisfies certain very natural conditions can be characterized semantically using a multi-valued matrix ([Los and Suszko, 1958; W´ ojcicki, 1988; Urquhart, 2001]). However, there are many important decidable logics whose characteristic matrices necessarily consist of an infinite number of truth values. In such a case it might be quite difficult to find any of these matrices, or to use one when it is found. Even in case a logic does have a (...)
     
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  35.  54
    What is a logical system?Arnon Avron - 1994 - In Dov M. Gabbay (ed.), What is a logical system? New York: Oxford University Press.
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  36.  22
    Combining classical logic, paraconsistency and relevance.Arnon Avron - 2005 - Journal of Applied Logic 3 (1):133-160.
  37. (1 other version)Many-valued non-deterministic semantics for first-order logics of formal (in)consistency.Arnon Avron - manuscript
    A paraconsistent logic is a logic which allows non-trivial inconsistent theories. One of the oldest and best known approaches to the problem of designing useful paraconsistent logics is da Costa’s approach, which seeks to allow the use of classical logic whenever it is safe to do so, but behaves completely differently when contradictions are involved. da Costa’s approach has led to the family of Logics of Formal (In)consistency (LFIs). In this paper we provide non-deterministic semantics for a very large family (...)
     
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  38. Tonk- A Full Mathematical Solution.Arnon Avron - unknown
    There is a long tradition (See e.g. [9, 10]) starting from [12], according to which the meaning of a connective is determined by the introduction and elimination rules which are associated with it. The supporters of this thesis usually have in mind natural deduction systems of a certain ideal type (explained in Section 3 below). Unfortunately, already the handling of classical negation requires rules which are not of that type. This problem can be solved in the framework of multiple-conclusion Gentzen-type (...)
     
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  39.  25
    Paraconsistency, self-extensionality, modality.Arnon Avron & Anna Zamansky - 2020 - Logic Journal of the IGPL 28 (5):851-880.
    Paraconsistent logics are logics that, in contrast to classical and intuitionistic logic, do not trivialize inconsistent theories. In this paper we take a paraconsistent view on two famous modal logics: B and S5. We use for this a well-known general method for turning modal logics to paraconsistent logics by defining a new negation as $\neg \varphi =_{Def} \sim \Box \varphi$. We show that while that makes both B and S5 members of the well-studied family of paraconsistent C-systems, they differ from (...)
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  40.  34
    The Problematic Nature of Gödel’s Disjunctions and Lucas-Penrose’s Theses.Arnon Avron - 2020 - Studia Semiotyczne 34 (1):83-108.
    We show that the name “Lucas-Penrose thesis” encompasses several different theses. All these theses refer to extremely vague concepts, and so are either practically meaningless, or obviously false. The arguments for the various theses, in turn, are based on confusions with regard to the meaning of these vague notions, and on unjustified hidden assumptions concerning them. All these observations are true also for all interesting versions of the much weaker thesis known as “Gö- del disjunction”. Our main conclusions are that (...)
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  41.  26
    A New Approach to Predicative Set Theory.Arnon Avron - 2010 - In Ralf Schindler (ed.), Ways of Proof Theory. De Gruyter. pp. 31-64.
    We suggest a new framework for the Weyl-Feferman predicativist program by constructing a formal predicative set theory P ZF which resembles ZF , and is suitable for mechanization. The basic idea is that the predicatively acceptable instances of the comprehension schema are those which determine the collections they define in an absolute way, independent of the extension of the “surrounding universe”. The language of P ZF is type-free, and it reflects real mathematical practice in making an extensive use of statically (...)
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  42.  20
    Rexpansions of nondeterministic matrices and their applications in nonclassical logics.Arnon Avron & Yoni Zohar - 2019 - Review of Symbolic Logic 12 (1):173-200.
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  43.  29
    The Classical Constraint on Relevance.Arnon Avron - 2014 - Logica Universalis 8 (1):1-15.
    We show that as long as the propositional constants t and f are not included in the language, any language-preserving extension of any important fragment of the relevance logics R and RMI can have only classical tautologies as theorems . This property is not preserved, though, if either t or f is added to the language, or if the contraction axiom is deleted.
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  44.  94
    Encoding modal logics in logical frameworks.Arnon Avron, Furio Honsell, Marino Miculan & Cristian Paravano - 1998 - Studia Logica 60 (1):161-208.
    We present and discuss various formalizations of Modal Logics in Logical Frameworks based on Type Theories. We consider both Hilbert- and Natural Deduction-style proof systems for representing both truth (local) and validity (global) consequence relations for various Modal Logics. We introduce several techniques for encoding the structural peculiarities of necessitation rules, in the typed -calculus metalanguage of the Logical Frameworks. These formalizations yield readily proof-editors for Modal Logics when implemented in Proof Development Environments, such as Coq or LEGO.
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  45. 5-valued Non-deterministic Semantics for The Basic Paraconsistent Logic mCi.Arnon Avron - 2008 - Studies in Logic, Grammar and Rhetoric 14 (27).
    One of the most important paraconsistent logics is the logic mCi, which is one of the two basic logics of formal inconsistency. In this paper we present a 5-valued characteristic nondeterministic matrix for mCi. This provides a quite non-trivial example for the utility and effectiveness of the use of non-deterministic many-valued semantics.
     
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  46.  38
    On purely relevant logics.Arnon Avron - 1986 - Notre Dame Journal of Formal Logic 27 (2):180-194.
  47.  55
    (3 other versions)Relevance and paraconsistency---a new approach. II. The formal systems.Arnon Avron - 1990 - Notre Dame Journal of Formal Logic 31 (2):169-202.
  48.  16
    A Constructive Analysis of $mathbf{RM}$.Arnon Avron - 1987 - Journal of Symbolic Logic 52 (4):939-951.
  49.  54
    Breaking the Tie: Benacerraf’s Identification Argument Revisited.Arnon Avron & Balthasar Grabmayr - 2023 - Philosophia Mathematica 31 (1):81-103.
    Most philosophers take Benacerraf’s argument in ‘What numbers could not be’ to rebut successfully the reductionist view that numbers are sets. This philosophical consensus jars with mathematical practice, in which reductionism continues to thrive. In this note, we develop a new challenge to Benacerraf’s argument by contesting a central premise which is almost unanimously accepted in the literature. Namely, we argue that — contra orthodoxy — there are metaphysically relevant reasons to prefer von Neumann ordinals over other set-theoretic reductions of (...)
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  50. A note on the structure of bilattices.Arnon Avron - unknown
    The notion of a bilattice was rst introduced by Ginsburg (see Gin]) as a general framework for a diversity of applications (such as truth maintenance systems, default inferences and others). The notion was further investigated and applied for various purposes by Fitting (see Fi1]- Fi6]). The main idea behind bilattices is to use structures in which there are two (partial) order relations, having di erent interpretations. The two relations should, of course, be connected somehow in order for the mathematical structure (...)
     
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