IA Scholar Query: Applications of Metric Coinduction
https://scholar.archive.org/
Internet Archive Scholar query results feedeninfo@archive.orgWed, 07 Sep 2022 00:00:00 GMTfatcat-scholarhttps://scholar.archive.org/help1440Realizations of countable Borel equivalence relations
https://scholar.archive.org/work/gywr3w5s2beghjxb4nvpun652a
We study topological realizations of countable Borel equivalence relations, including realizations by continuous actions of countable groups, with additional desirable properties. Some examples include minimal realizations on any perfect Polish space, realizations as K_σ relations, and realizations by continuous actions on the Baire space. We also consider questions related to realizations of specific important equivalence relations, like Turing and arithmetical equivalence. We focus in particular on the problem of realization by continuous actions on compact spaces and more specifically subshifts. This leads to the study of properties of subshifts, including universality of minimal subshifts, and a characterization of amenability of a countable group in terms of subshifts. Moreover we consider a natural universal space for actions and equivalence relations and study the descriptive and topological properties in this universal space of various properties, like, e.g., compressibility, amenability or hyperfiniteness.Joshua Frisch, Alexander Kechris, Forte Shinko, Zoltán Vidnyánszkywork_gywr3w5s2beghjxb4nvpun652aWed, 07 Sep 2022 00:00:00 GMTThe syntactic side of autonomous categories enriched over generalised metric spaces
https://scholar.archive.org/work/7xoahceecbajfbkgzmjgnlz63u
Programs with a continuous state space or that interact with physical processes often require notions of equivalence going beyond the standard binary setting in which equivalence either holds or does not hold. In this paper we explore the idea of equivalence taking values in a quantale V, which covers the cases of (in)equations and (ultra)metric equations among others. Our main result is the introduction of a V-equational deductive system for linear λ-calculus together with a proof that it is sound and complete. In fact we go further than this, by showing that linear λ-theories based on this V-equational system form a category that is equivalent to a category of autonomous categories enriched over 'generalised metric spaces'. If we instantiate this result to inequations, we get an equivalence with autonomous categories enriched over partial orders. In the case of (ultra)metric equations, we get an equivalence with autonomous categories enriched over (ultra)metric spaces. We additionally show that this syntax-semantics correspondence extends to the affine setting. We use our results to develop examples of inequational and metric equational systems for higher-order programming in the setting of real-time, probabilistic, and quantum computing.Fredrik Dahlqvist, Renato Neveswork_7xoahceecbajfbkgzmjgnlz63uWed, 31 Aug 2022 00:00:00 GMTGenerating circuits with generators
https://scholar.archive.org/work/t2kg427zwzewrocmurivhnhiku
The most widely used languages and methods used for designing digital hardware fall into two rough categories. One of them, register transfer level (RTL), requires specifying each and every component in the designed circuit. This gives the designer full control, but burdens the designer with many trivial details. The other, the high-level synthesis (HLS) method, allows the designer to abstract the details of hardware away and focus on the problem being solved. This method however cannot be used for a class of hardware design problems because the circuit's clock is also abstracted away. We present YieldFSM, a hardware description language that uses the generator abstraction to represent clock-level timing in a digital circuit. It represents a middle ground between the RTL and HLS approaches: the abstraction level is higher than in RTL, but thanks to explicit information about clock-level timing, it can be used in applications where RTL is traditionally used. We also present the YieldFSM compiler, which uses methods developed by the functional programming community -- including continuation-passsing style translation and defunctionalization -- to translate YieldFSM programs to Mealy machines. It is implemented using Template Haskell and the Clash functional hardware description language. We show that this approach leads to short and conceptually simple hardware descriptions.Marek Materzokwork_t2kg427zwzewrocmurivhnhikuMon, 29 Aug 2022 00:00:00 GMTOn Feller continuity and full abstraction
https://scholar.archive.org/work/3mxaqxc32zdhxoo4vapqzpe3da
We study the nature of applicative bisimilarity in λ-calculi endowed with operators for sampling from contin- uous distributions. On the one hand, we show that bisimilarity, logical equivalence, and testing equivalence all coincide with contextual equivalence when real numbers can be manipulated through continuous functions only. The key ingredient towards this result is a notion of Feller-continuity for labelled Markov processes, which we believe of independent interest, giving rise a broad class of LMPs for which coinductive and logically inspired equivalences coincide. On the other hand, we show that if no constraint is put on the way real numbers are manipulated, characterizing contextual equivalence turns out to be hard, and most of the aforementioned notions of equivalence are even unsound.Gilles Barthe, Raphaëlle Crubillé, Ugo Dal Lago, Francesco Gavazzowork_3mxaqxc32zdhxoo4vapqzpe3daMon, 29 Aug 2022 00:00:00 GMTLimits of real numbers in the binary signed digit representation
https://scholar.archive.org/work/dzomtnxzazct5f6axl3yytkqde
We extract verified algorithms for exact real number computation from constructive proofs. To this end we use a coinductive representation of reals as streams of binary signed digits. The main objective of this paper is the formalisation of a constructive proof that real numbers are closed with respect to limits. All the proofs of the main theorem and the first application are implemented in the Minlog proof system and the extracted terms are further translated into Haskell. We compare two approaches. The first approach is a direct proof. In the second approach we make use of the representation of reals by a Cauchy-sequence of rationals. Utilizing translations between the two represenation and using the completeness of the Cauchy-reals, the proof is very short. In both cases we use Minlog's program extraction mechanism to automatically extract a formally verified program that transforms a converging sequence of reals, i.e.~a sequence of streams of binary signed digits together with a modulus of convergence, into the binary signed digit representation of its limit. The correctness of the extracted terms follows directly from the soundness theorem of program extraction. As a first application we use the extracted algorithms together with Heron's method to construct an algorithm that computes square roots with respect to the binary signed digit representation. In a second application we use the convergence theorem to show that the signed digit representation of real numbers is closed under multiplication.Franziskus Wiesnet, Nils Köppwork_dzomtnxzazct5f6axl3yytkqdeThu, 18 Aug 2022 00:00:00 GMTOn the Reinhardt Conjecture and Formal Foundations of Optimal Control
https://scholar.archive.org/work/rxdljdx5hnf7jmix4rh3hxse2u
We describe a reformulation (following Hales (2017)) of a 1934 conjecture of Reinhardt on pessimal packings of convex domains in the plane as a problem in optimal control theory. Several structural results of this problem including its Hamiltonian structure and Lax pair formalism are presented. General solutions of this problem for constant control are presented and are used to prove that the Pontryagin extremals of the control problem are constrained to lie in a compact domain of the state space. We further describe the structure of the control problem near its singular locus, and prove that we recover the Pontryagin system of the multi-dimensional Fuller optimal control problem (with two dimensional control) in this case. We show how this system admits logarithmic spiral trajectories when the control set is the circumscribing disk of the 2-simplex with the associated control performing an infinite number of rotations on the boundary of the disk in finite time. We also describe formalization projects in foundational optimal control viz., model-based and model-free Reinforcement Learning theory. Key ingredients which make these formalization novel viz., the Giry monad and contraction coinduction are considered and some applications are discussed.Koundinya Vajjhawork_rxdljdx5hnf7jmix4rh3hxse2uMon, 08 Aug 2022 00:00:00 GMTDeciding All Behavioral Equivalences at Once: A Game for Linear-Time–Branching-Time Spectroscopy
https://scholar.archive.org/work/jjeatgzwwfhehd6uawfx5uer4y
We introduce a generalization of the bisimulation game that finds distinguishing Hennessy-Milner logic formulas from every finitary, subformula-closed language in van Glabbeek's linear-time--branching-time spectrum between two finite-state processes. We identify the relevant dimensions that measure expressive power to yield formulas belonging to the coarsest distinguishing behavioral preorders and equivalences; the compared processes are equivalent in each coarser behavioral equivalence from the spectrum. We prove that the induced algorithm can determine the best fit of (in)equivalences for a pair of processes.Benjamin Bisping, David N. Jansen, Uwe Nestmannwork_jjeatgzwwfhehd6uawfx5uer4yMon, 08 Aug 2022 00:00:00 GMTDifferentials and distances in probabilistic coherence spaces
https://scholar.archive.org/work/4bulxkufynb7xfovjemmcvxyey
In probabilistic coherence spaces, a denotational model of probabilistic functional languages, morphisms are analytic and therefore smooth. We explore two related applications of the corresponding derivatives. First we show how derivatives allow to compute the expectation of execution time in the weak head reduction of probabilistic PCF (pPCF). Next we apply a general notion of "local" differential of morphisms to the proof of a Lipschitz property of these morphisms allowing in turn to relate the observational distance on pPCF terms to a distance the model is naturally equipped with. This suggests that extending probabilistic programming languages with derivatives, in the spirit of the differential lambda-calculus, could be quite meaningful.Thomas Ehrhardwork_4bulxkufynb7xfovjemmcvxyeyThu, 04 Aug 2022 00:00:00 GMTVerifying a Sequent Calculus Prover for First-Order Logic with Functions in Isabelle/HOL
https://scholar.archive.org/work/euhmedglxbc2bkj7dlxr43jxya
We describe the design, implementation and verification of an automated theorem prover for first-order logic with functions. The proof search procedure is based on sequent calculus and we formally verify its soundness and completeness in Isabelle/HOL using an existing abstract framework for coinductive proof trees. Our analytic completeness proof covers both open and closed formulas. Since our deterministic prover considers only the subset of terms relevant to proving a given sequent, we do so as well when building a countermodel from a failed proof. Finally, we formally connect our prover with the proof system and semantics of the existing SeCaV system. In particular, the prover can generate human-readable SeCaV proofs which are also machine-verifiable proof certificates.Asta Halkjær From, Frederik Krogsdal Jacobsen, June Andronick, Leonardo de Mourawork_euhmedglxbc2bkj7dlxr43jxyaWed, 03 Aug 2022 00:00:00 GMTFixpoint Theory – Upside Down
https://scholar.archive.org/work/2cf3cwf3qjdkraekcnineyhch4
Knaster-Tarski's theorem, characterising the greatest fixpoint of a monotone function over a complete lattice as the largest post-fixpoint, naturally leads to the so-called coinduction proof principle for showing that some element is below the greatest fixpoint (e.g., for providing bisimilarity witnesses). The dual principle, used for showing that an element is above the least fixpoint, is related to inductive invariants. In this paper we provide proof rules which are similar in spirit but for showing that an element is above the greatest fixpoint or, dually, below the least fixpoint. The theory is developed for non-expansive monotone functions on suitable lattices of the form 𝕄^Y, where Y is a finite set and 𝕄 an MV-algebra, and it is based on the construction of (finitary) approximations of the original functions. We show that our theory applies to a wide range of examples, including termination probabilities, metric transition systems, behavioural distances for probabilistic automata and bisimilarity. Moreover it allows us to determine original algorithms for solving simple stochastic games.Paolo Baldan, Richard Eggert, Barbara König, Tommaso Padoanwork_2cf3cwf3qjdkraekcnineyhch4Mon, 25 Jul 2022 00:00:00 GMTOn Feller Continuity and Full Abstraction (Long Version)
https://scholar.archive.org/work/2zjvqmwnezcxvbowgjchaw3opm
We study the nature of applicative bisimilarity in λ-calculi endowed with operators for sampling from continuous distributions. On the one hand, we show that bisimilarity, logical equivalence, and testing equivalence all coincide with contextual equivalence when real numbers can be manipulated only through continuous functions. The key ingredient towards this result is a novel notion of Feller-continuity for labelled Markov processes, which we believe of independent interest, being a broad class of LMPs for which coinductive and logically inspired equivalences coincide. On the other hand, we show that if no constraint is put on the way real numbers are manipulated, characterizing contextual equivalence turns out to be hard, and most of the aforementioned notions of equivalence are even unsound.Gilles Barthe, Raphaëlle Crubillé, Ugo Dal Lago, Francesco Gavazzowork_2zjvqmwnezcxvbowgjchaw3opmThu, 21 Jul 2022 00:00:00 GMTHennessy-Milner Theorems via Galois Connections
https://scholar.archive.org/work/u4exkwmig5c2rlgvkachn7c6ci
We introduce a general and compositional, yet simple, framework that allows us to derive soundness and expressiveness results for modal logics characterizing behavioural equivalences or metrics (also known as Hennessy-Milner theorems). It is based on Galois connections between sets of (real-valued) predicates on the one hand and equivalence relations/metrics on the other hand and covers a part of the linear-time-branching-time spectrum, both for the qualitative case (behavioural equivalences) and the quantitative case (behavioural metrics). We derive behaviour functions from a given logic and give a condition, called compatibility, that characterizes under which conditions a logically induced equivalence/metric is induced by a fixpoint equation. In particular this framework allows us to derive a new fixpoint characterization of directed trace metrics.Harsh Beohar and Sebastian Gurke and Barbara König and Karla Messingwork_u4exkwmig5c2rlgvkachn7c6ciWed, 20 Jul 2022 00:00:00 GMTOn Pitts' Relational Properties of Domains
https://scholar.archive.org/work/vf7jou5vnvhwdi7peptxmjbp7q
Andrew Pitts' framework of relational properties of domains is a powerful method for defining predicates or relations on domains, with applications ranging from reasoning principles for program equivalence to proofs of adequacy connecting denotational and operational semantics. Its main appeal is handling recursive definitions that are not obviously well-founded: as long as the corresponding domain is also defined recursively, and its recursion pattern lines up appropriately with the definition of the relations, the framework can guarantee their existence. Pitts' original development used the Knaster-Tarski fixed-point theorem as a key ingredient. In these notes, I show how his construction can be seen as an instance of other key fixed-point theorems: the inverse limit construction, the Banach fixed-point theorem and the Kleene fixed-point theorem. The connection underscores how Pitts' construction is intimately tied to the methods for constructing the base recursive domains themselves, and also to techniques based on guarded recursion, or step-indexing, that have become popular in the last two decades.Arthur Azevedo de Amorimwork_vf7jou5vnvhwdi7peptxmjbp7qThu, 14 Jul 2022 00:00:00 GMTSplitting quasireductive supergroups and volumes of supergrassmannians
https://scholar.archive.org/work/nlux53aeirfwferrkdqyelabqm
We introduce the notion of splitting subgroups of quasireducitve supergroups, and explain their significance. For GL(m|n), Q(n), and defect one basic classical supergroups, we give explicit splitting subgroups. We further prove they are minimal up to conjugacy, except in the GL(m|n) case where it remains a conjecture. A key tool in the proof is the computation of the volumes of complex supergrassmannians, which is of interest in its own right.Vera Serganova, Alexander Shermanwork_nlux53aeirfwferrkdqyelabqmWed, 15 Jun 2022 00:00:00 GMTValue iteration is optic composition
https://scholar.archive.org/work/xdbvkviqrjdfxi32kk34a53tdu
Dynamic programming is a class of algorithms used to compute optimal control policies for Markov decision processes. Dynamic programming is ubiquitous in control theory, and is also the foundation of reinforcement learning. In this paper, we show that value improvement, one of the main steps of dynamic programming, can be naturally seen as composition in a category of optics, and intuitively, the optimal value function is the limit of a chain of optic compositions. We illustrate this with three classic examples: the gridworld, the inverted pendulum and the savings problem. This is a first step towards a complete account of reinforcement learning in terms of parametrised optics.Jules Hedges, Riu Rodríguez Sakamotowork_xdbvkviqrjdfxi32kk34a53tduThu, 09 Jun 2022 00:00:00 GMTStatic Analysis of Probabilistic Programs: An Algebraic Approach
https://scholar.archive.org/work/zlv4j3kxgndxrcvg7g3ukozhbu
Probabilistic programs are programs that can draw random samples from probability distributions and involve random control flows. They are becoming increasingly popular and have been applied in many areas such as algorithm design, cryptographic protocols, uncertainty modeling, and statistical inference. Formal reasoning about probabilistic programs comes with unique challenges, because it is usually not tractable to obtain the exact result distributions of probabilistic programs. This thesis focuses on an algebraic approach for static analysis of probabilistic programs. The thesis first provides a brief background on measure theory and introduces an imperative arithmetic probabilistic programming language Appl with a novel hyper-graph program model. Second, the thesis presents an algebraic denotational semantics for Appl that can be instantiated with different models of nondeterminism. The thesis also develops a new model of nondeterminism that involves nondeterminacy among state transformers and presents a domain-theoretic characterization of the new model. Based on the algebraic denotational semantics, the thesis proposes a general algebraic framework PMAF for designing, implementing, and proving the correctness of static analyses of probabilistic programs. The thesis also includes a concrete static analysis—central-moment analysis for cost accumulators in probabilistic programs—and elaborates implementation strategies to improve the usability and efficiency of the analysis. There is a gap between the general PMAF framework and the central-moment analysis, in the sense that the former is based on abstraction and iterative approximation, but the latter is based on constraint solving. The thesis provides some preliminary results on bridging the gap, via the development of novel regular hyper-path expressions, which finitely represent possibly-infinite hyperpaths on control-flow hyper-graphs of probabilistic programs without nondeterminism, and DMKAT algebraic structures, which can be used to interpret regular hyp [...]Di Wangwork_zlv4j3kxgndxrcvg7g3ukozhbuMon, 06 Jun 2022 00:00:00 GMTAn informal introduction to categorical representation theory and the local geometric Langlands program
https://scholar.archive.org/work/kr45f3mst5b65cmjibwmgxhr2u
We provide a motivated introduction to the theory of categorical actions of groups and the local geometric Langlands program. Along the way we emphasize applications, old and new, to the usual representation theory of reductive and affine Lie algebraGurbir Dhillonwork_kr45f3mst5b65cmjibwmgxhr2uSun, 29 May 2022 00:00:00 GMTProjected images of the Sierpinski tetrahedron and other fractal imaginary cubes
https://scholar.archive.org/work/qoxlxc3h3re2xaass5pqc5lnlm
A projected image of a Sierpinski tetrahedron has a positive measure if and only if the images O, P, Q, R of the four vertices satisfy p OP + q OQ + r OR = 0 for odd numbers p, q, r. This fact was essentially obtained by Kenyon in [7]. We reformulate his proof through the notion of projection of differenced digit set so that it could be applied to projections of other fractal objects. We study projections of H fractal and T fractal as well as Sierpinski tetrahedron, which are the fractal imaginary cubes of the first two levels. We characterize the directions from which these fractals are projected to sets with positive measures.Hideki Tsuikiwork_qoxlxc3h3re2xaass5pqc5lnlmFri, 27 May 2022 00:00:00 GMTStay Safe under Panic: Affine Rust Programming with Multiparty Session Types
https://scholar.archive.org/work/tugchb7qe5dsjegzotjcyz6twu
Communicating systems comprise diverse software components across networks. To ensure their robustness, modern programming languages such as Rust provide both strongly typed channels, whose usage is guaranteed to be affine (at most once), and cancellation operations over binary channels. For coordinating components to correctly communicate and synchronise with each other, we use the structuring mechanism from multiparty session types, extending it with affine communication channels and implicit/explicit cancellation mechanisms. This new typing discipline, affine multiparty session types (AMPST), ensures cancellation termination of multiple, independently running components and guarantees that communication will not get stuck due to error or abrupt termination. Guided by AMPST, we implemented an automated generation tool (MultiCrusty) of Rust APIs associated with cancellation termination algorithms, by which the Rust compiler auto-detects unsafe programs. Our evaluation shows that MultiCrusty provides an efficient mechanism for communication, synchronisation and propagation of the notifications of cancellation for arbitrary processes. We have implemented several usecases, including popular application protocols (OAuth, SMTP), and protocols with exception handling patterns (circuit breaker, distributed logging).Nicolas Lagaillardie, Rumyana Neykova, Nobuko Yoshidawork_tugchb7qe5dsjegzotjcyz6twuThu, 28 Apr 2022 00:00:00 GMTA Generic Approach to Quantitative Verification
https://scholar.archive.org/work/s4ktcdwunvgwtitzak6shoutim
This thesis is concerned with quantitative verification, that is, the verification of quantitative properties of quantitative systems. These systems are found in numerous applications, and their quantitative verification is important, but also rather challenging. In particular, given that most systems found in applications are rather big, compositionality and incrementality of verification methods are essential. In order to ensure robustness of verification, we replace the Boolean yes-no answers of standard verification with distances. Depending on the application context, many different types of distances are being employed in quantitative verification. Consequently, there is a need for a general theory of system distances which abstracts away from the concrete distances and develops quantitative verification at a level independent of the distance. It is our view that in a theory of quantitative verification, the quantitative aspects should be treated just as much as input to a verification problem as the qualitative aspects are. In this work we develop such a general theory of quantitative verification. We assume as input a distance between traces, or executions, and then employ the theory of games with quantitative objectives to define distances between quantitative systems. Different versions of the quantitative bisimulation game give rise to different types of distances, viz.~bisimulation distance, simulation distance, trace equivalence distance, etc., enabling us to construct a quantitative generalization of van Glabbeek's linear-time--branching-time spectrum. We also extend our general theory of quantitative verification to a theory of quantitative specifications. For this we use modal transition systems, and we develop the quantitative properties of the usual operators for behavioral specification theories.Uli Fahrenbergwork_s4ktcdwunvgwtitzak6shoutimSun, 24 Apr 2022 00:00:00 GMT