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

A Complete V-Equational System for Graded lambda-Calculus

Modern programming frequently requires generalised notions of program equivalence based on a metric or a similar structure. Previous work addressed this challenge by introducing the notion of a V-equation, i.e. an equation labelled by an element of a quantale V, which covers inter alia (ultra-)metric, classical, and fuzzy (in)equations. It also introduced a V-equational system for the linear variant of lambda-calculus where any given resource must be used exactly once. In this paper we drop the (often too strict) linearity constraint by adding graded modal types which allow multiple uses of a resource in a controlled manner. We show that such a control, whilst providing more expressivity to the programmer, also interacts more richly with V-equations than the linear or Cartesian cases. Our main result is the introduction of a sound and complete V-equational system for a lambda-calculus with graded modal types interpreted by what we call a Lipschitz exponential comonad. We also show how to build such comonads canonically via a universal construction, and use our results to derive graded metric equational systems (and corresponding models) for programs with timed 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

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 in coalgebraic logics

This thesis aims to provide a suite of techniques to generate completeness re- sults for coalgebraic logics with axioms of arbitrary rank. We have chosen to investigate the possibility to generalize what is arguably one of the most suc- cessful methods to prove completeness results in ‘classical’ modal logic, namely completeness-via-canonicity. This technique is particularly well-suited to a coal- gebraic generalization because of its clean and abstract algebraic formalism. In the case of classical modal logic, it can be summarized in two steps, first it isolates the purely algebraic problem of canonicity, i.e. of determining when a variety of boolean Algebras with Operators (BAOs) is closed under canonical extension (i.e. canonical). Secondly, it connects the notion of canonical vari- eties to that of canonical models to explicitly build models, thereby proving completeness. The classical algebraic theory of canonicity is geared towards normal logics, or, in algebraic terms, BAOs (or generalizations thereof). Most coalgebraic log- ics are not normal, and we thus develop the algebraic theory of canonicity for Boolean Algebra with Expansions (BAEs), or more generally for Distributive Lattice Expansions (DLEs). We present new results about a class of expan- sions defined by weaker preservation properties than meet or join preservation, namely (anti)-k-additive and (anti-)k-multiplicative expansions. We show how canonical and Sahlqvist equations can be built from such operations. In order to connect the theory of canonicity in DLEs and BAEs to coalgebraic logic, we choose to work in the abstract formulation of coalgebraic logic. An abstract coalgebraic logic is defined by a functor L : BA → BA, and we can heuristically separate these logics in two classes. In the first class the functor L is relatively simple, and in particular can be interpreted as defining a BAE. This class includes the predicate lifting style of coalgebraic logics. In the second class the functor L can be very complicated and the whole theory requires a different approach. This class includes the nabla style of coalgebraic logics. For simple functors, we develop results on strong completeness and then prove strong completeness-via-canonicity in the presence of canonical frame con- ditions for strongly complete abstract coalgebraic logics. In particular we show coalgebraic completeness-via-canonicity for Graded Modal Logic, Intuitionistic Logic, the distributive full Lambek calculus, and the logic of trees of arbitrary branching degrees defined by the List functor. These results are to the best of our knowledge, new. For a complex functor L we use an indirect approach via the notion of functor presentation. This allows us to represent L as the quotient of a much simpler polynomial functor. Polynomial functors define BAEs and can thus be treated as objects in the first class of functors, in particular we can apply all the above mentioned techniques to the logics defined by such functors. We develop techniques that ensure that results obtained for the simple presenting logic can be transferred back to the complicated presented logic. We can then prove strong-completeness-via-canonicity in the presence of canonical frame conditions for coalgebraic logics which do not define a BAE, such as the nabla coalgebraic logics.Open Acces