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The differential calculus of causal functions [article]

David Sprunger, Bart Jacobs
2019 arXiv   pre-print
Sprunger and B. Jacobs 23:11 The recurrence rule So far, causal differential calculus is rather similar to traditional differential calculus.  ... 
arXiv:1904.10611v1 fatcat:ofbgmlugibdmvegp46zkzp7ysi

Differentiable Causal Computations via Delayed Trace [article]

David Sprunger, Shin-ya Katsumata
2019 arXiv   pre-print
We investigate causal computations taking sequences of inputs to sequences of outputs where the nth output depends on the first n inputs only. We model these in category theory via a construction taking a Cartesian category C to another category St(C) with a novel trace-like operation called "delayed trace", which misses yanking and dinaturality axioms of the usual trace. The delayed trace operation provides a feedback mechanism in St(C) with an implicit guardedness guarantee. When C is
more » ... with a Cartesian differential operator, we construct a differential operator for St(C) using an abstract version of backpropagation through time, a technique from machine learning based on unrolling of functions. This obtains a swath of properties for backpropagation through time, including a chain rule and Schwartz theorem. Our differential operator is also able to compute the derivative of a stateful network without requiring the network to be unrolled.
arXiv:1903.01093v1 fatcat:si6vhlhhjfbvlff4oq6tj66srq

Neural Nets via Forward State Transformation and Backward Loss Transformation [article]

Bart Jacobs, David Sprunger
2018 arXiv   pre-print
This article studies (multilayer perceptron) neural networks with an emphasis on the transformations involved --- both forward and backward --- in order to develop a semantical/logical perspective that is in line with standard program semantics. The common two-pass neural network training algorithms make this viewpoint particularly fitting. In the forward direction, neural networks act as state transformers. In the reverse direction, however, neural networks change losses of outputs to losses
more » ... inputs, thereby acting like a (real-valued) predicate transformer. In this way, backpropagation is functorial by construction, as shown earlier in recent other work. We illustrate this perspective by training a simple instance of a neural network.
arXiv:1803.09356v1 fatcat:nu77qfcssfcsfejw6gp2keliii

Quantitative bisimulations using coreflections and open morphisms [article]

Jérémy Dubut, Ichiro Hasuo, Shin-ya Katsumata, David Sprunger
2018 arXiv   pre-print
We investigate a canonical way of defining bisimilarity of systems when their semantics is given by a coreflection, typically in a category of transition systems. We use the fact, from Joyal et al., that coreflections preserve open morphisms situations in the sense that a coreflection induces a path subcategory in the category of systems in such a way that open bisimilarity with respect to the induced path category coincides with usual bisimilarity of their semantics. We prove that this method
more » ... s particularly well-suited for systems with quantitative information: we canonically recover the path category of probabilistic systems from Cheng et al., and of timed systems from Nielsen et al., and, finally, we propose a new canonical path category for hybrid systems.
arXiv:1809.09278v1 fatcat:47n4uw6ozfeeffhci47ydoxk6m

Eigenvalues and Transduction of Morphic Sequences: Extended Version [article]

David Sprunger, William Tune, Jörg Endrullis, Lawrence S. Moss
2014 arXiv   pre-print
We study finite state transduction of automatic and morphic sequences. Dekking proved that morphic sequences are closed under transduction and in particular morphic images. We present a simple proof of this fact, and use the construction in the proof to show that non-erasing transductions preserve a condition called alpha-substitutivity. Roughly, a sequence is alpha-substitutive if the sequence can be obtained as the limit of iterating a substitution with dominant eigenvalue alpha. Our results
more » ... ulminate in the following fact: for multiplicatively independent real numbers alpha and beta, if v is an alpha-substitutive sequence and w is a beta-substitutive sequence, then v and w have no common non-erasing transducts except for the ultimately periodic sequences. We rely on Cobham's theorem for substitutions, a recent result of Durand.
arXiv:1406.1754v1 fatcat:do4e65vmmbeozn7ubsajwh3wva

Functorial String Diagrams for Reverse-Mode Automatic Differentiation [article]

Mario Alvarez-Picallo, Dan R. Ghica, David Sprunger, Fabio Zanasi
2021 arXiv   pre-print
., Edinburgh, Scotland, EH3 8BL, United Kingdom, dan.ghica@gmail.com, Computer Science and University of Birmingham, Birmingham, England, B15 2TT, United Kingdom; David Sprunger, Computer Science, University  ... 
arXiv:2107.13433v1 fatcat:2bnsqea5nvgxhhsakuwlhwq6oi

Relational Differential Dynamic Logic [article]

Juraj Kolčák, Ichiro Hasuo, Jérémy Dubut, Shin-ya Katsumata, David Sprunger, Akihisa Yamada
2020 arXiv   pre-print
In the field of quality assurance of hybrid systems (that combine continuous physical dynamics and discrete digital control), Platzer's differential dynamic logic (dL) is widely recognized as a deductive verification method with solid mathematical foundations and sophisticated tool support. Motivated by benchmarks provided by our industry partner, we study a relational extension of dL, aiming to formally prove statements such as "an earlier deployment of the emergency brake decreases the
more » ... on speed." A main technical challenge here is to relate two states of two dynamics at different time points. Our main contribution is a theory of suitable simulations (a relational extension of differential invariants that are central proof methods in dL), and a derived technique of time stretching. The latter features particularly high applicability, since the user does not have to synthesize a simulation out of the air. We derive new inference rules for dL from these notions, and demonstrate their use over a couple of automotive case studies.
arXiv:1903.00153v2 fatcat:qgnadja5wvcyflcscryt4ahely

Fibrational Bisimulations and Quantitative Reasoning [chapter]

David Sprunger, Shin-ya Katsumata, Jérémy Dubut, Ichiro Hasuo
2018 Lecture Notes in Computer Science  
Bisimulation and bisimilarity are fundamental notions in comparing state-based systems. Their extensions to a variety of systems have been actively pursued in recent years, a notable direction being quantitative extensions. In this paper we present an abstract categorical framework for such extended (bi)simulation notions. We use coalgebras as system models and fibrations for organizing predicatesfollowing the seminal work by Hermida and Jacobs-but our focus is on the structural aspect of
more » ... ional frameworks. Specifically we use morphisms of fibrations as well as canonical liftings of functors via Kan extensions. We apply this categorical framework by deriving some known properties of the Hausdorff pseudometric and approximate bisimulation in control theory.
doi:10.1007/978-3-030-00389-0_11 fatcat:hwm2ozmnbzcvdf5gdetl6zx75y

Linearization of Automatic Arrays and Weave Specifications

David Sprunger
2013 Electronical Notes in Theoretical Computer Science  
Grabmayer, Endrullis, Hendriks, Klop, and Moss [6] developed a method for defining automatic sequences in terms of 'zip specifications' and proved that a sequence is automatic [2] iff it has a zip specification where all zip terms have the same arity. This paper begins by investigating a similar definitional scheme for the higher-dimensional counterpart of automatic sequences, automatic arrays. In the course of establishing the results required for this machinery, we find an isomorphism-closely
more » ... related to the z-order curve [8]-between a final coalgebra for arrays and the standard final coalgebra for sequences . This isomorphism preserves automaticity properties: an array is k, l-automatic iff its corresponding sequence is kl-automatic. The former notion of automaticity (k, l-automatic, note the comma) is defined for arrays as in [2] , and the latter notion is the standard notion of automaticity for sequences. It also provides a convenient way to translate between stream zip specifications and array zip specifications.
doi:10.1016/j.entcs.2013.09.021 fatcat:atngl4vx6vf4jc7t7fkzmexyki

Relational Differential Dynamic Logic [chapter]

Juraj Kolčák, Jérémy Dubut, Ichiro Hasuo, Shin-ya Katsumata, David Sprunger, Akihisa Yamada
2020 Lecture Notes in Computer Science  
In the field of quality assurance of hybrid systems, Platzer's differential dynamic logic (dL) is widely recognized as a deductive verification method with solid mathematical foundations and sophisticated tool support. Motivated by case studies provided by our industry partner, we study a relational extension of dL, aiming to formally prove statements such as "an earlier engagement of the emergency brake yields a smaller collision speed." A main technical challenge is to combine two dynamics,
more » ... that the powerful inference rules of dL (such as the differential invariant rules) can be applied to such relational reasoning, yet in such a way that we relate two different time points. Our contributions are a semantical theory of time stretching, and the resulting synchronization rule that expresses time stretching by the syntactic operation of Lie derivative. We implemented this rule as an extension of KeYmaera X, by which we successfully verified relational properties of a few models taken from the automotive domain.
doi:10.1007/978-3-030-45190-5_11 fatcat:y77k3agaerei7ituxh2ss77xu4

Full abstraction for digital circuits [article]

Dan R. Ghica, George Kaye, David Sprunger
2022 arXiv   pre-print
This paper refines the existing axiomatic semantics of digital circuits with delay and feedback, in which circuits are constructed as morphisms in a freely generated cartesian traced (dataflow) category. First, we give a cleaner presentation, making a clearer distinction between syntax and semantics, including a full formalisation of the semantics as stream functions. As part of this effort, we refocus the categorical framework through the lens of string diagrams, which not only makes reading
more » ... uations more intuitive but removes bureaucracy such as associativity from proofs. We also extend the existing framework with a new axiom, inspired by the Kleene fixed-point theorem, which allows circuits with non-delay-guarded feedback, typically handled poorly by traditional methodologies, to be replaced with a series of finitely iterated circuits. This eliminates the possibility of infinitely unfolding a circuit; instead, one can always reduce a circuit to some (possibly undefined) value. To fully characterise the stream functions that correspond to digital circuits, we examine how the behaviour of the latter can be modelled using Mealy machines. By establishing that the translation between sequential circuits and Mealy machines preserves their behaviour, one can observe that circuits always implement monotone stream functions with finite stream derivatives.
arXiv:2201.10456v3 fatcat:5j3idlsqsnbhxag37jq2v4ww2y

A Complete Logic for Behavioural Equivalence in Coalgebras of Finitary Set Functors [chapter]

David Sprunger
2016 Lecture Notes in Computer Science  
doi:10.1007/978-3-319-40370-0_10 fatcat:ure6ah7mvjhene4ejd7pc7dcma

Neural Nets via Forward State Transformation and Backward Loss Transformation

Bart Jacobs, David Sprunger
2019 Electronical Notes in Theoretical Computer Science  
This article studies (multilayer perceptron) neural networks with an emphasis on the transformations involved -both forward and backward -in order to develop a semantic/logical perspective that is in line with standard program semantics. The common two-pass neural network training algorithms make this viewpoint particularly fitting. In the forward direction, neural networks act as state transformers, using Kleisli composition for the multiset monad -for the linear parts of network layers. In
more » ... reverse direction, however, neural networks change losses of outputs to losses of inputs, thereby acting like a (real-valued) predicate transformer. In this way, backpropagation is functorial by construction, as shown in other works recently. We illustrate this perspective by training a simple instance of a neural network.
doi:10.1016/j.entcs.2019.09.009 fatcat:upnn42qgdfamriubf5yty66ee4

Esotropia and Exotropia Preferred Practice Pattern®

David K. Wallace, Stephen P. Christiansen, Derek T. Sprunger, Michele Melia, Katherine A. Lee, Christie L. Morse, Michael X. Repka
2018 Ophthalmology (Rochester, Minn.)  
The Esotropia and Exotropia PPP was then sent for review to additional internal and external groups and individuals in July 2017. All those returning comments were required to provide disclosure of relevant relationships with industry to have their comments considered (indicated with an asterisk below). Members of the Pediatric Ophthalmology/Strabismus Preferred Practice Pattern Panel reviewed and discussed these comments and determined revisions to the document.
doi:10.1016/j.ophtha.2017.10.007 pmid:29108746 fatcat:3qrao755qnhs3ivneag2q6esca

Relational differential dynamic logic

Juraj Kolčák, Ichiro Hasuo, Jérémy Dubut, Shin-ya Katsumata, David Sprunger, Akihisa Yamada
2019 Proceedings of the 22nd ACM International Conference on Hybrid Systems Computation and Control - HSCC '19  
In the field of quality assurance of hybrid systems, Platzer's differential dynamic logic (dL) is widely recognized as a deductive verification method with solid mathematical foundations and sophisticated tool support. Motivated by case studies provided by our industry partner, we study a relational extension of dL, aiming to formally prove statements such as "an earlier engagement of the emergency brake yields a smaller collision speed." A main technical challenge is to combine two dynamics,
more » ... that the powerful inference rules of dL (such as the differential invariant rules) can be applied to such relational reasoning, yet in such a way that we relate two different time points. Our contributions are a semantical theory of time stretching, and the resulting synchronization rule that expresses time stretching by the syntactic operation of Lie derivative. We implemented this rule as an extension of KeYmaera X, by which we successfully verified relational properties of a few models taken from the automotive domain.
doi:10.1145/3302504.3313362 dblp:conf/hybrid/KolcakHDKSY19 fatcat:hkxyv3cxhvhuvctuloyvhonzxm
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