Electronic Notes in Theoretical Computer Science最新文献

Indexed and fibred categorical concepts are widely used in computer science as models of logical systems and type theories. Here we focus on Hoare logic and show that a comprehensive categorical analysis of its axiomatic semantics needs the languages of indexed category and fibred category theory. The structural features of the language are presented in an indexed setting, while the logical features of deduction are modeled in the fibred one. Especially, Hoare triples arise naturally as special arrows in a fibred category over a syntactic category of programs, while deduction in the Hoare calculus can be characterized categorically by the heuristic deduction = generation of cartesian arrows + composition of arrows.

Populational Announcement Logic (PPAL), is a variant of the standard Public Announcement Logic (PAL) with a fuzzy-inspired semantics, where instead of specific agents we have populations and groups. The semantics and the announcement logic are defined, and an example is provided. We show validities analogous to PAL axioms and their proofs, and also provide a proof of decidability. We briefly talk about model checking and compare the framework against probabilistic logic. We conclude that the main advantage of PPAL over PAL is the flexibility to work with previously defined agents.

In this paper, we present a directed homotopy type theory for reasoning synthetically about (higher) categories and directed homotopy theory. We specify a new 'homomorphism' type former for Martin-Löf type theory which is roughly analogous to the identity type former originally introduced by Martin-Löf. The homomorphism type former is meant to capture the notions of morphism (from the theory of categories) and directed path (from directed homotopy theory) just as the identity type former is known to capture the notions of isomorphism (from the theory of groupoids) and path (from homotopy theory). Our main result is an interpretation of these homomorphism types into $C\mathrm{at}$, the category of small categories. There, the interpretation of each homomorphism type ${\mathrm{hom}}_C(a,b)$ is indeed the set of morphisms between the objects a and b of the category $C$. We end the paper with an analysis of the interpretation in $C\mathrm{at}$ with which we argue that our homomorphism types are indeed the directed version of Martin-Löf's identity types

This paper first investigates a form of frequentist learning that is often called Maximal Likelihood Estimation (MLE). It is redescribed as a natural transformation from multisets to distributions that commutes with marginalisation and disintegration. It forms the basis for the next, main topic: learning of hidden states, which is reformulated as learning along a channel. This topic requires a fundamental look at what data is and what its validity is in a particular state. The paper distinguishes two forms, denoted as ‘M’ for ‘multiple states’ and ‘C’ for ‘copied states’. It is shown that M and C forms exist for validity of data, for learning from data, and for learning along a channel. This M/C distinction allows us to capture two completely different examples from the literature which both claim to be instances of Expectation-Maximisation.

Applicative bisimiliarity is a coinductively-defined program equivalence in which programs are tested as argument-passing processes. Starting with the seminal work by Abramsky, applicative bisimiliarity has been proved to be a powerful technique for higher-order program equivalence. Recently, applicative bisimiliarity has also been generalised to lambda calculi with algebraic effects, and with discrete probabilistic choice in particular. In this paper, we show that applicative bisimiliarity behaves well in a lambda-calculus in which probabilistic choice is available in a more general form, namely through an operator for sampling of values from continuous distributions. Our main result shows that applicative bisimilarity is sound for contextual equivalence, hence providing a new reasoning principle for higher-order probabilistic languages.

Probabilistic programming is an increasingly popular formalism for modeling randomness and uncertainty. Designing semantic models for probabilistic programs has been extensively studied, but is technically challenging. Particular complications arise when trying to account for (i) unstructured control-flow, a natural feature in low-level imperative programs; (ii) general recursion, an extensively used programming paradigm; and (iii) nondeterminism, which is often used to represent adversarial actions in probabilistic models, and to support refinement-based development. This paper presents a denotational-semantics framework that supports the three features mentioned above, while allowing nondeterminism to be handled in different ways. To support both probabilistic choice and nondeterministic choice, the semantics is given over control-flow hyper-graphs. The semantics follows an algebraic approach: it can be instantiated in different ways as long as certain algebraic properties hold. In particular, the semantics can be instantiated to support nondeterminism among either program states or state transformers. We develop a new formalization of nondeterminism based on powerdomains over sub-probability kernels. Semantic objects in the powerdomain enjoy a notion we call generalized convexity, which is a generalization of convexity. As an application, the paper sketches an algebraic framework for static analysis of probabilistic programs, which has been proposed in a companion paper.

In order to reason about effects, we can define quantitative formulas to describe behavioural aspects of effectful programs. These formulas can for example express probabilities that (or sets of correct starting states for which) a program satisfies a property. Fundamental to this approach is the notion of quantitative modality, which is used to lift a property on values to a property on computations. Taking all formulas together, we say that two terms are equivalent if they satisfy all formulas to the same quantitative degree. Under sufficient conditions on the quantitative modalities, this equivalence is equal to a notion of Abramsky's applicative bisimilarity, and is moreover a congruence. We investigate these results in the context of Levy's call-by-push-value with general recursion and algebraic effects. For example, the results apply to (combinations of) nondeterministic choice, probabilistic choice, global store, and error.

Bisimulation is a concept that captures behavioural equivalence. It has been studied extensively on nonprobabilistic systems and on discrete-time Markov processes and on so-called continuous-time Markov chains. In the latter, time is continuous but the evolution still proceeds in jumps. We propose two definitions of bisimulation on continuous-time stochastic processes where the evolution is a flow through time. We show that they are equivalent and we show that when restricted to discrete-time, our concept of bisimulation encompasses the standard discrete-time concept. The concept we introduce is not a straightforward generalization of discrete-time concepts.

This paper extends the results of Hermida's thesis about logical predicates to more general logical relations and a wider collection of types. The extension of type constructors from types to logical relations is derived from an interpretation of those constructors on a model of predicate logic. This is then further extended to n-ary relations by pullback. Hermida's theory shows how right adjoints in the category of fibrations are composed from a combination of Cartesian lifting and a local adjunction. This result is generalised to make it more applicable to left adjoints, and then shown to be stable under pullback, deriving an account of n-ary relations from standard predicate logic. A brief discussion of lifting monads to predicates includes the existence of an initial such lifting, generalising existing results.

In flowchart languages, predicates play an interesting double role. In the textual representation, they are often presented as conditions, i.e., expressions which are easily combined with other conditions (often via Boolean combinators) to form new conditions, though they only play a supporting role in aiding branching statements choose a branch to follow. On the other hand, in the graphical representation they are typically presented as decisions, intrinsically capable of directing control flow yet mostly oblivious to Boolean combination.

While categorical treatments of flowchart languages are abundant, none of them provide a treatment of this dual nature of predicates. In the present paper, we argue that extensive restriction categories are precisely categories that capture such a condition/decision duality, by means of morphisms which, coincidentally, are also called decisions. Further, we show that having these categorical decisions amounts to having an internal logic: Analogous to how subobjects of an object in a topos form a Heyting algebra, we show that decisions on an object in an extensive restriction category form a De Morgan quasilattice, the algebraic structure associated with the (three-valued) weak Kleene logic $K_3^w$. Full classical propositional logic can be recovered by restricting to total decisions, yielding extensive categories in the usual sense, and confirming (from a different direction) a result from effectus theory that predicates on objects in extensive categories form Boolean algebras.

As an application, since (categorical) decisions are partial isomorphisms, this approach provides naturally reversible models of classical propositional logic and weak Kleene logic.