Coalgebraic completeness-via-canonicity for distributive substructural logics

We prove strong completeness of a range of substructural logics with respect to a natural poset-based relational semantics using a coalgebraic version of completeness-via-canonicity. By formalizing the problem in the language of coalgebraic logics, we develop a modular theory which covers a wide variety of different logics under a single framework, and lends itself to further extensions. Moreover, we believe that the coalgebraic framework provides a systematic and principled way to study the relationship between resource models on the semantics side, and substructural logics on the syntactic side.Comment: 36 page

The Positivication of Coalgebraic Logics

We present positive coalgebraic logic in full generality, and show how to obtain a positive coalgebraic logic from a boolean one. On the model side this involves canonically computing a endofunctor T\u27: Pos->Pos from an endofunctor T: Set->Set, in a procedure previously defined by the second author et alii called posetification. On the syntax side, it involves canonically computing a syntax-building functor L\u27: DL->DL from a syntax-building functor L: BA->BA, in a dual procedure which we call positivication. These operations are interesting in their own right and we explicitly compute posetifications and positivications in the case of several modal logics. We show how the semantics of a boolean coalgebraic logic can be canonically lifted to define a semantics for its positive fragment, and that weak completeness transfers from the boolean case to the positive case

How to write a coequation

There is a large amount of literature on the topic of covarieties, coequations and coequational specifications, dating back to the early seventies. Nevertheless, coequations have not (yet) emerged as an everyday practical specification formalism for computer scientists. In this review paper, we argue that this is partly due to the multitude of syntaxes for writing down coequations, which seems to have led to some confusion about what coequations are and what they are for. By surveying the literature, we identify four types of syntaxes: coequations-as-corelations, coequations-as-predicates, coequations-as-equations, and coequations-as-modal-formulas. We present each of these in a tutorial fashion, relate them to each other, and discuss their respective uses

An Internal Language for Categories Enriched over Generalised Metric Spaces

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 ?, which covers the cases of (in)equations and (ultra)metric equations among others. Our main result is the introduction of a ?-equational deductive system for linear ?-calculus together with a proof that it is sound and complete (in fact, an internal language) for a class of enriched autonomous categories. In the case of inequations, we get an internal language for autonomous categories enriched over partial orders. In the case of (ultra)metric equations, we get an internal language for autonomous categories enriched over (ultra)metric spaces. We use our results to obtain examples of inequational and metric equational systems for higher-order programs that contain real-time and probabilistic behaviour

Deterministic stream-sampling for probabilistic programming: semantics and verification

Probabilistic programming languages rely fundamentally on some notion of sampling, and this is doubly true for probabilistic programming languages which perform Bayesian inference using Monte Carlo techniques. Verifying samplers - proving that they generate samples from the correct distribution - is crucial to the use of probabilistic programming languages for statistical modelling and inference. However, the typical denotational semantics of probabilistic programs is incompatible with deterministic notions of sampling. This is problematic, considering that most statistical inference is performed using pseudorandom number generators. We present a higher-order probabilistic programming language centred on the notion of samplers and sampler operations. We give this language an operational and denotational semantics in terms of continuous maps between topological spaces. Our language also supports discontinuous operations, such as comparisons between reals, by using the type system to track discontinuities. This feature might be of independent interest, for example in the context of differentiable programming. Using this language, we develop tools for the formal verification of sampler correctness. We present an equational calculus to reason about equivalence of samplers, and a sound calculus to prove semantic correctness of samplers, i.e. that a sampler correctly targets a given measure by construction.Comment: Extended version of LiCS 2023 pape

The syntactic side of autonomous categories enriched over generalised metric spaces

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 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. Additionally, we 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.This work is financed by National Funds through FCT - Fundação para a Ciência e a Tecnologia, I.P. (Portuguese Foundation for Science and Technology) within project IBEX, with reference PTDC/CCI-COM/4280/2021. We are also thankful for the reviewers’ helpful feedback

Overregulation of Health Care: Musings on Disruptive Innovation Theory

Disruptive innovation theory provides one lens through which to describe how regulations may stifle innovation and increase costs. Basing their discussion on this theory, Curtis and Schulman consider some of the effects that regulatory controls may have on innovation in the health sector

Completeness via Canonicity for Distributive Substructural Logics: A Coalgebraic Perspective

We prove strong completeness of a range of substructural logics with respect to their relational semantics by completeness-via-canonicity. Specifically, we use the topological theory of canonical (in) equations in distributive lattice expansions to show that distributive substructural logics are strongly complete with respect to their relational semantics. By formalizing the problem in the language of coalgebraic logics, we develop a modular theory which covers a wide variety of different logics under a single framework, and lends itself to further extensions