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MAZZA, DAMIANO. "Infinitary affine proofs." Mathematical Structures in Computer Science 27, no. 5 (July 7, 2015): 581–602. http://dx.doi.org/10.1017/s0960129515000298.
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Even though the multiplicative–additive fragment of linear logic forbids structural rules in general, is does admit a bounded form of exponential modalities enjoying a bounded form of structural rules. The approximation theorem, originally proved by Girard, states that if full linear logic proves a propositional formula, then the multiplicative–additive fragment proves every bounded approximation of it. This may be understood as the fact that multiplicative–additive linear logic is somehow dense in full linear logic. Our goal is to give a technical formulation of this informal remark. We introduce a Cauchy-complete space of infinitary affine term-proofs and we show that it yields a fully complete model of multiplicative exponential polarised linear logic, in the style of Girard's ludics. Moreover, the subspace of finite term-proofs, which is a model of multiplicative polarised linear logic, is dense in the space of all term-proofs.
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Lafont, Yves. "The undecidability of second order linear logic without exponentials." Journal of Symbolic Logic 61, no. 2 (June 1996): 541–48. http://dx.doi.org/10.2307/2275674.
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AbstractRecently, Lincoln, Scedrov and Shankar showed that the multiplicative fragment of second order intuitionistic linear logic is undecidable, using an encoding of second order intuitionistic logic. Their argument applies to the multiplicative-additive fragment, but it does not work in the classical case, because second order classical logic is decidable. Here we show that the multiplicative-additive fragment of second order classical linear logic is also undecidable, using an encoding of two-counter machines originally due to Kanovich. The faithfulness of this encoding is proved by means of the phase semantics.
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Cockett, J. R. B., and C. A. Pastro. "A Language For Multiplicative-additive Linear Logic." Electronic Notes in Theoretical Computer Science 122 (March 2005): 23–65. http://dx.doi.org/10.1016/j.entcs.2004.06.049.
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O'Hearn, Peter W., and David J. Pym. "The Logic of Bunched Implications." Bulletin of Symbolic Logic 5, no. 2 (June 1999): 215–44. http://dx.doi.org/10.2307/421090.
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AbstractWe introduce a logic BI in which a multiplicative (or linear) and an additive (or intuitionistic) implication live side-by-side. The propositional version of BI arises from an analysis of the proof-theoretic relationship between conjunction and implication; it can be viewed as a merging of intuitionistic logic and multiplicative intuitionistic linear logic. The naturality of BI can be seen categorically: models of propositional BI's proofs are given by bicartesian doubly closed categories, i.e., categories which freely combine the semantics of propositional intuitionistic logic and propositional multiplicative intuitionistic linear logic. The predicate version of BI includes, in addition to standard additive quantifiers, multiplicative (or intensional) quantifiers and which arise from observing restrictions on structural rules on the level of terms as well as propositions. We discuss computational interpretations, based on sharing, at both the propositional and predicate levels.
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CHAUDHURI, KAUSTUV. "Expressing additives using multiplicatives and subexponentials." Mathematical Structures in Computer Science 28, no. 5 (November 21, 2016): 651–66. http://dx.doi.org/10.1017/s0960129516000293.
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Subexponential logic is a variant of linear logic with a family of exponential connectives – called subexponentials – that are indexed and arranged in a pre-order. Each subexponential has or lacks associated structural properties of weakening and contraction. We show that a classical propositional multiplicative subexponential logic (MSEL) with one unrestricted and two linear subexponentials can encode the halting problem for two register Minsky machines, and is hence undecidable. We then show how the additive connectives can be directly simulated by giving an encoding of propositional multiplicative additive linear logic (MALL) in an MSEL with one unrestricted and four linear subexponentials.
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Hughes, Dominic J. D., and Rob J. Van Glabbeek. "Proof nets for unit-free multiplicative-additive linear logic." ACM Transactions on Computational Logic 6, no. 4 (October 2005): 784–842. http://dx.doi.org/10.1145/1094622.1094629.
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CHAUDHURI, KAUSTUV, JOËLLE DESPEYROUX, CARLOS OLARTE, and ELAINE PIMENTEL. "Hybrid linear logic, revisited." Mathematical Structures in Computer Science 29, no. 8 (April 22, 2019): 1151–76. http://dx.doi.org/10.1017/s0960129518000439.
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HyLL (Hybrid Linear Logic) is an extension of intuitionistic linear logic (ILL) that has been used as a framework for specifying systems that exhibit certain modalities. In HyLL, truth judgements are labelled by worlds (having a monoidal structure) and hybrid connectives (at and ↓) relate worlds with formulas. We start this work by showing that HyLL's axioms and rules can be adequately encoded in linear logic (LL), so that one focused step in LL will correspond to a step of derivation in HyLL. This shows that any proof in HyLL can be exactly mimicked by a LL focused derivation. Another extension of LL that has extensively been used for specifying systems with modalities is Subexponential Linear Logic (SELL). In SELL, the LL exponentials (!, ?) are decorated with labels representing locations, and a pre-order on such labels defines the provability relation. We propose an encoding of HyLL into SELL⋒ (SELL plus quantification over locations) that gives better insights about the meaning of worlds in HyLL. More precisely, we identify worlds as locations, and show that a flat subexponential structure is sufficient for representing any world structure in HyLL. This shows that HyLL's monoidal structure is not reflected in LL derivations, hence not increasing the expressiveness of LL, from a proof theoretical point of view. We conclude by proposing the notion of fixed points in multiplicative additive HyLL (μHyMALL), which can be encoded into multiplicative additive linear logic with fixed points (μMALL). As an application, we propose encodings of Computational Tree Logic (CTL) into both μMALL and μHyMALL. In the former, states are represented as atoms in the linear context, hence reflecting a more operational view of CTL connectives. In the latter, worlds represent states of the transition system, thus exhibiting a pleasant similarity with the semantics of CTL.
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Okada, Mitsuhiro, and Kazushige Terui. "The finite model property for various fragments of intuitionistic linear logic." Journal of Symbolic Logic 64, no. 2 (June 1999): 790–802. http://dx.doi.org/10.2307/2586501.
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AbstractRecently Lafont [6] showed the finite model property for the multiplicative additive fragment of linear logic (MALL) and for affine logic (LLW), i.e., linear logic with weakening. In this paper, we shall prove the finite model property for intuitionistic versions of those, i.e. intuitionistic MALL (which we call IMALL), and intuitionistic LLW (which we call ILLW). In addition, we shall show the finite model property for contractive linear logic (LLC), i.e., linear logic with contraction. and for its intuitionistic version (ILLC). The finite model property for related substructural logics also follow by our method. In particular, we shall show that the property holds for all of FL and GL−-systems except FLc and of Ono [11], that will settle the open problems stated in Ono [12].
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Hamano, Masahiro. "A MALL geometry of interaction based on indexed linear logic." Mathematical Structures in Computer Science 30, no. 10 (November 2020): 1025–53. http://dx.doi.org/10.1017/s0960129521000062.
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AbstractWe construct a geometry of interaction (GoI: dynamic modelling of Gentzen-style cut elimination) for multiplicative-additive linear logic (MALL) by employing Bucciarelli–Ehrhard indexed linear logic MALL(I) to handle the additives. Our construction is an extension to the additives of the Haghverdi–Scott categorical formulation (a multiplicative GoI situation in a traced monoidal category) for Girard’s original GoI 1. The indices are shown to serve not only in their original denotational level, but also at a finer grained dynamic level so that the peculiarities of additive cut elimination such as superposition, erasure of subproofs, and additive (co-) contraction can be handled with the explicit use of indices. Proofs are interpreted as indexed subsets in the category Rel, but without the explicit relational composition; instead, execution formulas are run pointwise on the interpretation at each index, with respect to symmetries of cuts, in a traced monoidal category with a reflexive object and a zero morphism. The sets of indices diminish overall when an execution formula is run, corresponding to the additive cut-elimination procedure (erasure), and allowing recovery of the relational composition. The main theorem is the invariance of the execution formulas along cut elimination so that the formulas converge to the denotations of (cut-free) proofs.
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Bucciarelli, Antonio, and Thomas Ehrhard. "On phase semantics and denotational semantics in multiplicative–additive linear logic." Annals of Pure and Applied Logic 102, no. 3 (April 2000): 247–82. http://dx.doi.org/10.1016/s0168-0072(99)00040-8.
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Joinet, Jean-Baptiste, Harold Schellinx, and Lorenzo Tortora De Falco. "SN and CR for free-style LKtq: linear decorations and simulation of normalization." Journal of Symbolic Logic 67, no. 1 (March 2002): 162–96. http://dx.doi.org/10.2178/jsl/1190150036.
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AbstractThe present report is a, somewhat lengthy, addendum to [4], where the elimination of cuts from derivations in sequent calculus for classical logic was studied ‘from the point of view of linear logic’. To that purpose a formulation of classical logic was used, that - as in linear logic - distinguishes between multiplicative and additive versions of the binary connectives.The main novelty here is the observation that this type-distinction is not essential: we can allow classical sequent derivations to use any combination of additive and multiplicative introduction rules for each of the connectives, and still have strong normalization and confluence of tq-reductions.We give a detailed description of the simulation of tq-reductions by means of reductions of the interpretation of any given classical proof as a proof net of PN2 (the system of second order proof nets for linear logic), in which moreover all connectives can be taken to be of one type, e.g., multiplicative.We finally observe that dynamically the different logical cuts, as determined by the four possible combinations of introduction rules, are independent: it is not possible to simulate them internally, i.e.. by only one specific combination, and structural rules.
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Lafont, Yves. "The finite model property for various fragments of linear logic." Journal of Symbolic Logic 62, no. 4 (December 1997): 1202–8. http://dx.doi.org/10.2307/2275637.
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To show that a formula A is not provable in propositional classical logic, it suffices to exhibit a finite boolean model which does not satisfy A. A similar property holds in the intuitionistic case, with Kripke models instead of boolean models (see for instance [11]). One says that the propositional classical logic and the propositional intuitionistic logic satisfy a finite model property. In particular, they are decidable: there is a semi-algorithm for provability (proof search) and a semi-algorithm for non provability (model search). For that reason, a logic which is undecidable, such as first order logic, cannot satisfy a finite model property.The case of linear logic is more complicated. The full propositional fragment LL has a complete semantics in terms of phase spaces [2, 3], but it is undecidable [9]. The multiplicative additive fragment MALL is decidable, in fact PSPACE-complete [9], but the decidability of the multiplicative exponential fragment MELL is still an open problem. For affine logic, that is, linear logic with weakening, the situation is somewhat better: the full propositional fragment LLW is decidable [5].Here, we show that the finite phase semantics is complete for MALL and for LLW, but not for MELL. In particular, this gives a new proof of the decidability of LLW. The noncommutative case is mentioned, but not handled in detail.
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Metcalfe, George, and Franco Montagna. "Substructural fuzzy logics." Journal of Symbolic Logic 72, no. 3 (September 2007): 834–64. http://dx.doi.org/10.2178/jsl/1191333844.
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AbstractSubstructural fuzzy logics are substructural logics that are complete with respect to algebras whose lattice reduct is the real unit interval [0, 1]. In this paper, we introduce Uninorm logic UL as Multiplicative additive intuitionistic linear logic MAILL extended with the prelinearity axiom ((A → B) ∧ t) V ((B → A)∧ t). Axiomatic extensions of UL include known fuzzy logics such as Monoidal t-norm logic MIX and Gödel logic G, and new weakening-free logics. Algebraic semantics for these logics are provided by subvarieties of (representable) pointed bounded commutative residuated lattices. Gentzen systems admitting cut-elimination are given in the framework of hypersequents. Completeness with respect to algebras with lattice reduct [0, 1] is established for UL and several extensions using a two-part strategy. First, completeness is proved for the logic extended with Takeuti and Titani's density rule. A syntactic elimination of the rule is then given using a hypersequent calculus. As an algebraic corollary, it follows that certain varieties of residuated lattices are generated by their members with lattice reduct [0, 1].
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Blute, R. F., and P. J. Scott. "The shuffle Hopf algebra and noncommutative full completeness." Journal of Symbolic Logic 63, no. 4 (December 1998): 1413–36. http://dx.doi.org/10.2307/2586659.
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AbstractWe present a full completeness theorem for the multiplicative fragment of a variant of noncommutative linear logic, Yetter's cyclic linear logic (CyLL). The semantics is obtained by interpreting proofs as dinatural transformations on a category of topological vector spaces, these transformations being equivariant under certain actions of a noncocommutative Hopf algebra called the shuffle algebra Multiplicative sequents are assigned a vector space of such dinaturals, and we show that this space has as a basis the denotations of cut-free proofs in CyLL + MIX. This can be viewed as a fully faithful representation of a free *-autonomous category, canonically enriched over vector spaces.This paper is a natural extension of the authors' previous work, “Linear Läuchli Semantics”, where a similar theorem is obtained for the commutative logic MLL + MIX. In that paper, we interpret proofs as dinaturals which are invariant under certain actions of the additive group of integers. Here we also present a simplification of that work by showing that the invariance criterion is actually a consequence of dinaturality. The passage from groups to Hopf algebras in this paper corresponds to the passage from commutative to noncommutative logic. However, in our noncommutative setting, one must still keep the invariance condition on dinaturals.
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Kou, Junke, Qinmei Huang, and Huijun Guo. "Pointwise Wavelet Estimations for a Regression Model in Local Hölder Space." Axioms 11, no. 9 (September 10, 2022): 466. http://dx.doi.org/10.3390/axioms11090466.
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This paper considers an unknown functional estimation problem in a regression model with multiplicative and additive noise. A linear wavelet estimator is first constructed by a wavelet projection operator. The convergence rate under the pointwise error of linear wavelet estimators is studied in local Hölder space. A nonlinear wavelet estimator is provided by the hard thresholding method in order to obtain an adaptive estimator. The convergence rate of the nonlinear estimator is the same as the linear estimator up to a logarithmic term. Finally, it should be pointed out that the convergence rates of two wavelet estimators are consistent with the optimal convergence rate on pointwise nonparametric estimation.
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GIAMBERARDINO, PAOLO DI. "Jump from parallel to sequential proofs: exponentials." Mathematical Structures in Computer Science 28, no. 7 (December 5, 2016): 1204–52. http://dx.doi.org/10.1017/s0960129516000414.
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In previous works, by importing ideas from game semantics (notably Faggian–Maurel–Curien'sludics nets), we defined a new class of multiplicative/additive polarized proof nets, calledJ-proof nets. The distinctive feature of J-proof nets with respect to other proof net syntaxes, is the possibility of representing proof nets which are partially sequentialized, by usingjumps(that is, untyped extra edges) as sequentiality constraints. Starting from this result, in the present work, we extend J-proof nets to the multiplicative/exponential fragment, in order to take into account structural rules: More precisely, we replace the familiar linear logic notion of exponential box with a less restricting one (calledcone) defined by means of jumps. As a consequence, we get a syntax for polarized nets where, instead of a structure of boxes nested one into the other, we have one of cones which can bepartially overlapping. Moreover, we define cut-elimination for exponential J-proof nets, proving, by a variant of Gandy's method, that even in case of ‘superposed’ cones, reduction enjoys confluence and strong normalization.
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Hryhorenko, Ihor, Serhii Kondrashov, and Aleksandr Opryshkin. "FORMATION OF TEST IMPACTS FOR THE FIRST LEVEL OF THE INFORMATION AND MEASUREMENT SYSTEM." Bulletin of the National Technical University «KhPI» Series: New solutions in modern technologies, no. 1(15) (May 5, 2023): 19–26. http://dx.doi.org/10.20998/2413-4295.2023.01.03.
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The issue of increasing the efficiency of coffee grinding systems thanks to non-dismantling test control of metrological characteristics of information and measurement systems remains relevant today, despite the fact that both domestic and foreign scientists continue to deal with this issue. Test control methods are widely used in various industries, and grinding systems are no exception. In addition to the factors affecting the coffee grinding process, it is necessary to take into account the change in the metrological characteristics of both the individual components of the equipment and the information and measurement system as a whole. Since the grinding process is dynamic, it is necessary to carry out test control of the equipment without stopping work. There is a need for non-dismantling test control by forming additive and multiplicative test effects at the first level of the information and measurement system. The difficulty is that when building a mathematical model of the process, it is practically impossible to take into account dynamic changes on the object, which can be both linear and non-linear. The specified feature of the grinding process makes it necessary to move to the creation of a situational system with fuzzy logic, which can help in determining the optimal relationship between the characteristics of the technological equipment to ensure the minimum value of the dynamic error. The Mamdani fuzzy inference construction method was chosen in the work to create a situational system with fuzzy logic. This algorithm is most common when it is necessary to help a metrological engineer in making a decision. With the help of a graphical user interface, a situational system with fuzzy logic was built for the first level of the information and measurement system, which is able to determine such parameters that will ensure the minimum value of the dynamic error of the grinding system when working out the action of additive and multiplicative test effects. The use of the presented approach can be extended to the class of problems of determining the optimal ratio between disparate parameters to obtain the best result.
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Gallinaro, Francesco Paolo. "Around Exponential-Algebraic Closedness." Bulletin of Symbolic Logic 29, no. 2 (June 2023): 300. http://dx.doi.org/10.1017/bsl.2022.46.
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AbstractWe present some results related to Zilber’s Exponential-Algebraic Closedness Conjecture, showing that various systems of equations involving algebraic operations and certain analytic functions admit solutions in the complex numbers. These results are inspired by Zilber’s theorems on raising to powers.We show that algebraic varieties which split as a product of a linear subspace of an additive group and an algebraic subvariety of a multiplicative group intersect the graph of the exponential function, provided that they satisfy Zilber’s freeness and rotundity conditions, using techniques from tropical geometry.We then move on to prove a similar theorem, establishing that varieties which split as a product of a linear subspace and a subvariety of an abelian variety A intersect the graph of the exponential map of A (again under the analogues of the freeness and rotundity conditions). The proof uses homology and cohomology of manifolds.Finally, we show that the graph of the modular j-function intersects varieties which satisfy freeness and broadness and split as a product of a Möbius subvariety of a power of the upper-half plane and a complex algebraic variety, using Ratner’s orbit closure theorem to study the images under j of Möbius varieties.Abstract prepared by Francesco Paolo GallinaroE-mail: francesco.gallinaro@mathematik.uni-freiburg.deURL: https://etheses.whiterose.ac.uk/31077/
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Zimmermann, Ernst. "Natural Deduction Bottom Up." Journal of Logic, Language and Information 30, no. 3 (March 16, 2021): 601–31. http://dx.doi.org/10.1007/s10849-021-09329-8.
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AbstractThe paper introduces a new type of rules into Natural Deduction, elimination rules by composition. Elimination rules by composition replace usual elimination rules in the style of disjunction elimination and give a more direct treatment of additive disjunction, multiplicative conjunction, existence quantifier and possibility modality. Elimination rules by composition have an enormous impact on proof-structures of deductions: they do not produce segments, deduction trees remain binary branching, there is no vacuous discharge, there is only few need of permutations. This new type of rules fits especially to substructural issues, so it is shown for Lambek Calculus, i.e. intuitionistic non-commutative linear logic and to its extensions by structural rules like permutation, weakening and contraction. Natural deduction formulated with elimination rules by composition from a complexity perspective is superior to other calculi.
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O'HEARN, PETER. "On bunched typing." Journal of Functional Programming 13, no. 4 (June 25, 2003): 747–96. http://dx.doi.org/10.1017/s0956796802004495.
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We study a typing scheme derived from a semantic situation where a single category possesses several closed structures, corresponding to different varieties of function type. In this scheme typing contexts are trees built from two (or more) binary combining operations, or in short, bunches. Bunched typing and its logical counterpart, bunched implications, have arisen in joint work of the author and David Pym. The present paper gives a basic account of the type system, and then focusses on concrete models that illustrate how it may be understood in terms of resource access and sharing. The most basic system has two context-combining operations, and the structural rules of Weakening and Contraction are allowed for one but not the other. This system includes a multiplicative, or substructural, function type −∗ alongside the usual (additive) function type $\rightarrow$; it is dubbed the $\alpha\lambda$-calculus after its binders, $\alpha$ for the $\alpha$dditive binder and $\lambda$ for the multiplicative, or $\lambda$inear, binder. We show that the features of this system are, in a sense, complementary to calculi based on linear logic; it is incompatible with an interpretation where a multiplicative function uses its argument once, but perfectly compatible with a reading based on sharing of resources. This sharing interpretation is derived from syntactic control of interference, a type-theoretic method of controlling sharing of storage, and we show how bunch-based management of Contraction can be used to provide a more flexible type system for interference control.
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Troquard, Nicolas. "Individual resource games and resource redistributions." Journal of Logic and Computation 30, no. 5 (May 28, 2020): 1023–62. http://dx.doi.org/10.1093/logcom/exaa031.
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Abstract To introduce agent-based technologies in real-world systems, one needs to acknowledge that the agents often have limited access to resources. They have to seek after resource objectives and compete for those resources. We introduce a class of resource games where resources and preferences are specified with the language of a resource-sensitive logic. The agents are endowed with a bag of resources and try to achieve a resource objective. For each agent, an action consists in making available a part of their endowed resources. All the resources made available can be used towards the agents’ objectives. We study three decision problems, the first of which is deciding whether an action profile is a Nash equilibrium: when all the agents have chosen an action, it is a Nash Equilibrium if no agent has an incentive to change their action unilaterally. When dealing with resources, interesting questions arise as to whether some equilibria can be eliminated or constructed by a central authority by redistributing the available resources among the agents. In our economies, division of property in divorce law exemplifies how a central authority can redistribute the resources of individuals and why they would desire to do so. We thus study two related decision problems: (i) rational elimination: given an action profile’s outcome, can the endowed resources be redistributed so that it is not the outcome of a Nash equilibrium? (ii) Rational construction: given an action profile’s outcome, can the endowed resources be redistributed so that it is the outcome of a Nash equilibrium? Among other results, we prove that all three problems are $\mathsf{PSPACE}$-complete when the resources are described in the very expressive language of the propositional multiplicative and additive linear logic. We also identify a new modest fragment of linear logic that we call MULT, suitable to represent multisets and reason about the inclusion and equality of bags of resources. We show that when the resources are described in MULT, the problem of deciding whether a profile is a Nash equilibrium is in $\textsf{PTIME}$.
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Firth, David. "1. Overcoming the Reference Category Problem in the Presentation of Statistical Models." Sociological Methodology 33, no. 1 (August 2003): 1–18. http://dx.doi.org/10.1111/j.0081-1750.2003.t01-1-00125.x.
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Effects of categorical variables in statistical models typically are reported in terms of comparison either with a reference category or with a suitably defined “mean effect,” for reasons of parameter identification. A conventional presentation of estimates and standard errors, but without the full variance-covariance matrix, does not allow subsequent readers either to make inference on a comparison of interest that is not presented or to compare or combine results from different studies where the same variables but different reference levels are used. It is shown how an alternative presentation, in terms of “quasi standard errors,” overcomes this problem in an economical and intuitive way. A primary application is the reporting of effects of categorical predictors, often called factors, in linear and generalized linear models, hazard models, multinomial-response models, generalized additive models, etc. Other applications include the comparison of coefficients between related regression equations—for example, log-odds ratios in a multinomial logit model—and the presentation of multipliers or “scores” in models with multiplicative interaction structure.
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Piazza, Mario, Gabriele Pulcini, and Matteo Tesi. "Linear logic in a refutational setting." Journal of Logic and Computation, August 10, 2023. http://dx.doi.org/10.1093/logcom/exad048.
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Abstract Sequent-style refutation calculi with non-invertible rules are challenging to design because multiple proof-search strategies need to be simultaneously verified. In this paper, we present a refutation calculus for the multiplicative–additive fragment of linear logic ($\textsf{MALL}$) whose binary rule for the multiplicative conjunction $(\otimes )$ and the unary rule for the additive disjunction $(\oplus )$ fail invertibility. Specifically, we design a cut-free hypersequent calculus $\textsf{HMALL}$, which is equivalent to $\textsf{MALL}$, and obtained by transforming the usual tree-like shape of derivations into a parallel and linear structure. Next, we develop a refutation calculus $\overline{\textsf{HMALL}}$ based on the calculus $\textsf{HMALL}$. As far as we know, this is also the first refutation calculus for a substructural logic. Finally, we offer a fractional semantics for $\textsf{MALL}$—whereby its formulas are interpreted by a rational number in the closed interval [0, 1] —thus extending to the substructural landscape the project of fractional semantics already pursued for classical and modal logics.
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Díaz-Caro, Alejandro, and Gilles Dowek. "A linear linear lambda-calculus." Mathematical Structures in Computer Science, May 31, 2024, 1–35. http://dx.doi.org/10.1017/s0960129524000197.
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Abstract We present a linearity theorem for a proof language of intuitionistic multiplicative additive linear logic, incorporating addition and scalar multiplication. The proofs in this language are linear in the algebraic sense. This work is part of a broader research program aiming to define a logic with a proof language that forms a quantum programming language.
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Fussner, Wesley, and Simon Santschi. "Interpolation in Linear Logic and Related Systems." ACM Transactions on Computational Logic, July 25, 2024. http://dx.doi.org/10.1145/3680284.
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We prove that there are continuum-many axiomatic extensions of the full Lambek calculus with exchange that have the deductive interpolation property. Further, we extend this result to both classical and intuitionistic linear logic as well as their multiplicative-additive fragments. None of the logics we exhibit have the Craig interpolation property, but we show that the exhibited extensions of classical and intuitionistic linear logic all enjoy a guarded form of Craig interpolation. We also give continuum-many axiomatic extensions of classical linear logic without the deductive interpolation property.
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Płaczek, Paweł. "Sequent systems for consequence relations of cyclic linear logics." Bulletin of the Section of Logic, April 24, 2024. http://dx.doi.org/10.18778/0138-0680.2024.06.
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Linear Logic is a versatile framework with diverse applications in computer science and mathematics. One intriguing fragment of Linear Logic is Multiplicative-Additive Linear Logic (MALL), which forms the exponential-free component of the larger framework. Modifying MALL, researchers have explored weaker logics such as Noncommutative MALL (Bilinear Logic, BL) and Cyclic MALL (CyMALL) to investigate variations in commutativity. In this paper, we focus on Cyclic Nonassociative Bilinear Logic (CyNBL), a variant that combines noncommutativity and nonassociativity. We introduce a sequent system for CyNBL, which includes an auxiliary system for incorporating nonlogical axioms. Notably, we establish the cut elimination property for CyNBL. Moreover, we establish the strong conservativeness of CyNBL over Full Nonassociative Lambek Calculus (FNL) without additive constants. The paper highlights that all proofs are constructed using syntactic methods, ensuring their constructive nature. We provide insights into constructing cut-free proofs and establishing a logical relationship between CyNBL and FNL.
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HYVERNAT, PIERRE. "A linear category of polynomial diagrams." Mathematical Structures in Computer Science 24, no. 1 (May 17, 2013). http://dx.doi.org/10.1017/s0960129512001016.
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We present a categorical model for intuitionistic linear logic in which objects are polynomial diagrams and morphisms aresimulation diagrams. The multiplicative structure (tensor product and its adjoint) can be defined in any locally cartesian closed category, but the additive (product and coproduct) and exponential (-comonoid comonad) structures require additional properties and are only developed in the categorySet, where the objects and morphisms have natural interpretations in terms of games, simulation and strategies.
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KUZNETSOV, STEPAN. "COMPLEXITY OF THE INFINITARY LAMBEK CALCULUS WITH KLEENE STAR." Review of Symbolic Logic, July 22, 2020, 1–27. http://dx.doi.org/10.1017/s1755020320000209.
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Abstract We consider the Lambek calculus, or noncommutative multiplicative intuitionistic linear logic, extended with iteration, or Kleene star, axiomatised by means of an $\omega $ -rule, and prove that the derivability problem in this calculus is $\Pi _1^0$ -hard. This solves a problem left open by Buszkowski (2007), who obtained the same complexity bound for infinitary action logic, which additionally includes additive conjunction and disjunction. As a by-product, we prove that any context-free language without the empty word can be generated by a Lambek grammar with unique type assignment, without Lambek’s nonemptiness restriction imposed (cf. Safiullin, 2007).
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Rastin, Sepideh J., David A. Rhoades, Chris Rollins, Matthew C. Gerstenberger, Annemarie Christophersen, and Kiran K. S. Thingbaijam. "Spatial Distribution of Earthquake Occurrence for the New Zealand National Seismic Hazard Model 2022." Bulletin of the Seismological Society of America, May 31, 2024. http://dx.doi.org/10.1785/0120230173.
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ABSTRACT We develop candidate hybrid models representing the spatial distribution of earthquake occurrence in New Zealand over the next 100 yr. These models are used within the onshore/near-shore, shallow component of the distributed seismicity model within the New Zealand National Seismic Hazard Model 2022. They combine a variety of spatially gridded covariates based on smoothed seismicity, strain rates, and proximity to mapped faults and plate boundaries in both multiplicative and additive hybrids. They were optimized against a standardized catalog of New Zealand earthquakes with magnitude M ≥ 4.95 and hypocentral depth ≤40 km from 1951 to 2020. We extract smoothed seismicity covariates using three different methods. The additive models are linear combinations of earthquake likelihood models derived from individual covariates. We choose three preferred hybrid models based on the information gain statistics, consideration of the ongoing Canterbury sequence and regions of low seismicity, and inclusion of the most informative covariates. Since the hazard model is designed for the next 100 yr, the preferred hybrid models are also combined with 20-year earthquake forecasts from the “Every Earthquake a Precursor According to Scale” model. Thus, in total, six hybrid spatial distribution candidates are advanced for sensitivity analyses and expert elicitation for inclusion in the final logic tree for the New Zealand National Seismic Hazard Model.
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Hössjer, Ola, Ingrid Kockum, Lars Alfredsson, Anna Karin Hedström, Tomas Olsson, and Magnus Lekman. "A General Framework for and New Normalization of Attributable Proportion." Epidemiologic Methods 6, no. 1 (December 23, 2016). http://dx.doi.org/10.1515/em-2015-0028.
Full textAbstract:
AbstractA unified theory is developed for attributable proportion (AP) and population attributable fraction (PAF) of joint effects, marginal effects or interaction among factors. We use a novel normalization with a range between –1 and 1 that gives the traditional definitions of AP or PAF when positive, but is different when they are negative. We also allow for an arbitrary number of factors, both those of primary interest and confounders, and quantify interaction as departure from a given model, such as a multiplicative, additive odds or disjunctive one. In particular, this makes it possible to compare different types of threeway or higher order interactions. Effect parameters are estimated on a linear or logit scale in order to find point estimates and confidence intervals for the various versions of AP and PAF, for prospective or retrospective studies. We investigate the accuracy of three confidence intervals; two of which use the delta method and a third bootstrapped interval. It is found that the delta method with logit type transformations, and the bootstrap, perform well for a wide range of models. The methodology is also applied to a multiple sclerosis (MS) data set, with smoking and two genetic variables as risk factors.