Electronic Notes in Theoretical Computer Science最新文献

The main goal of this article is to expose and relate different ways of interpreting the multiplicative fragment of classical linear logic in polycategories. Polycategories are known to give rise to models of classical linear logic in so-called representable polycategories with duals, which ask for the existence of various polymaps satisfying the different universal properties needed to define tensor, par, and negation. We begin by explaining how these different universal properties can all be seen as instances of a single notion of universality of a polymap parameterised by an input or output object, which also generalises the classical notion of universal multimap in a multicategory. We then proceed to introduce a definition of in-cartesian and out-cartesian polymaps relative to a refinement system (= strict functor) of polycategories, in such a way that universal polymaps can be understood as a special case. In particular, we obtain that a polycategory is a representable polycategory with duals if and only if it is bifibred over the terminal polycategory $1$. Finally, we present a Grothendieck correspondence between bifibrations of polycategories and pseudofunctors into MAdj, the (weak) 2-polycategory of multivariable adjunctions. When restricted to bifibrations over $1$ we get back the correspondence between *-autonomous categories and Frobenius pseudomonoids in MAdj that was recently observed by Shulman.

Parametrized fixed points are of particular interest to denotational semantics and are often given by “dagger operations” [Stephen L. Bloom and Zoltán Ésik, Fixed-Point Operations on ccc's. Part I, Theoretical Computer Science (ISSN 0304-3975) 155 (1996), 1–38, https://doi.org/10.1016/0304-3975(95)00010-0; Stephen L. Bloom and Zoltán Ésik, Iteration Theories. The Equational Logic of Iterative Processes, in: EATCS Monographs on Theoretical Computer Science, Springer-Verlag Berlin Heidelberg, ISBN 978-3-642-78034-9, 1993, xv+630 pp., https://doi.org/10.1007/978-3-642-78034-9; Stephen L. Bloom and Zoltán Ésik, Some Equational Laws of Initiality in 2CCC's, International Journal of Foundations of Computer Science 6 (1995) 95–118, https://doi.org/10.1142/S0129054195000081.]. Dagger operations that satisfy the Conway identities [Stephen L. Bloom and Zoltán Ésik, Fixed-Point Operations on ccc's. Part I, Theoretical Computer Science (ISSN 0304-3975) 155 (1996), 1–38, doi: https://doi.org/10.1016/0304-3975(95)00010-0.] are particularly useful, because these identities imply a large class of identities used in semantic reasoning. We generalize existing techniques to define dagger operations on ω-categories and on O-categories. These operations enjoy a 2-categorical structure that implies the Conway identities. We illustrate these operators by considering applications to the semantics of session-typed languages.

Continuous monads are an axiomatic class of submonads of the double power set monad. ρ-sets are an axiomatic generalization of directed sets. The ρ-generalization of continuous lattices arises as the algebras of a continuous monad and conversely. Each ρ-continuous poset has two topologies which respectively generalize the Scott and Lawson topologies. Each ρ-contnuous lattice is compact in the canonical topology if and only if the corresponding continuous monad contains the ultrafilter monad.

Weakest precondition transformers are useful tools in program verification. One of their key properties is compositionality, that is, the weakest precondition predicate transformer (wppt for short) associated to program f;g should be equal to the composition of the wppts associated to f and g. In this paper, we study the categorical structure behind wppts from a fibrational point of view. We characterize the wppts that satisfy compositionality as the ones constructed from the Cartesian lifting of a monad. We moreover show that Cartesian liftings of monads along lax slice categories bijectively correspond to Eilenberg-Moore monotone algebras. We then instantiate our techniques by deriving wppts for commonplace effects such as the maybe monad, the non-empty powerset monad, the counter monad or the distribution monad. We also show how to combine them to derive the wppts appearing in the literature of verification of probabilistic programs.

In this paper we recast the proof of Rossman's equirank homomorphism preservation theorem using comonadic formulations of bounded quantifier rank and variable count (and dually tree width and tree-depth), and work towards generalisation of it that simultaneously preserves quantifier rank and variable count. Along the way, we give an exposition of the required comonads, showing how their properties arise.

Monads can be interpreted as encoding formal expressions, or formal operations in the sense of universal algebra. We give a construction which formalizes the idea of “evaluating an expression partially”: for example, “2+3” can be obtained as a partial evaluation of “2+2+1”. This construction can be given for any monad, and it is linked to the famous bar construction [Saunders Mac Lane, Categories for the Working Mathematician, Springer, 2000, VII.6], of which it gives an operational interpretation: the bar construction is a simplicial set, and its 1-cells are partial evaluations.

We study the properties of partial evaluations for general monads. We prove that whenever the monad is weakly cartesian, partial evaluations can be composed via the usual Kan filler property of simplicial sets, of which we give an interpretation in terms of substitution of terms.

For the case of probability monads, partial evaluations correspond to what probabilists call conditional expectation of random variables, and partial evaluation relation is known as second-order stochastic dominance.

In terms of rewritings, partial evaluations give an abstract reduction system which is reflexive, confluent, and transitive whenever the monad is weakly cartesian. This manuscript is part of a work in progress on a general rewriting interpretation of the bar construction.

A domain-theoretic method for solving initial value problems (IVPs) is presented, together with proofs of soundness, completeness, and some results on the algebraic complexity of the method. While the common fixed-precision interval arithmetic methods are restricted by the precision of the underlying machine architecture, domain-theoretic methods may be complete, i.e., the result may be obtained to any degree of accuracy. Furthermore, unlike methods based on interval arithmetic which require access to the syntactic representation of the vector field, domain-theoretic methods only deal with the semantics of the field, in the sense that the field is assumed to be given via finitely-representable approximations, to within any required accuracy.

In contrast to the domain-theoretic first-order Euler method, the second-order method uses the local Lipschitz properties of the field. This is achieved by using a domain for Lipschitz functions, whose elements are consistent pairs that provide approximations of the field and its local Lipschitz properties. In the special case where the field is differentiable, the local Lipschitz properties are exactly the local differential properties of the field. In solving IVPs, Lipschitz continuity of the field is a common assumption, as a sufficient condition for uniqueness of the solution. While the validated methods for solving IVPs commonly impose further restrictions on the vector field, the second-order Euler method requires no further condition. In this sense, the method may be seen as the most general of its kind.

To avoid complicated notations and lengthy arguments, the results of the paper are stated for the second-order Euler method. Nonetheless, the framework, and the results, may be extended to any higher-order Euler method, in a straightforward way.

We formalise the well-known rules of partial differentiation in a version of equational logic with function variables and binding constructs. We prove the resulting theory is complete with respect to polynomial interpretations. The proof makes use of Severi's interpolation theorem that all multivariate Hermite problems are solvable. We also present a number of related results, such as decidability and equational completeness.

Bisimulation is a concept that captures behavioural equivalence of states in a transition system. In [Linan Chen, Florence Clerc, and Prakash Panangaden, Bisimulation for feller-dynkin processes, in: Proceedings of the Thirty-Fifth Conference on the Mathematical Foundations of Programming Semantics, Electronic Notes in Theoretical Computer Science 347 (2019) 45–63.], we proposed two equivalent definitions of bisimulation on continuous-time stochastic processes where the evolution is a flow through time. In the present paper, we develop the theory further: we introduce different concepts that correspond to different behavioural equivalences and compare them to bisimulation. In particular, we study the relation between bisimulation and symmetry groups of the dynamics. We also provide a game interpretation for two of the behavioural equivalences. We then compare those notions to their discrete-time analogues.

Algebras can be used to interpret the behaviour of effectful programs. In particular, we use Eilenberg-Moore algebras given over a complete lattices of truth values, which specify answers to queries about programs. The algebras can be used to formulate a quantitative logic of behavioural properties, specifying a congruent notion of program equivalence coinciding with a notion of applicative bisimilarity. Many combinations of effects can be interpreted using these algebras. In this paper, we specify a method of generically combining effects and the algebras used to interpret them. At the core of this method is the tensor of complete lattices, which combines the carrier sets of the algebras. We show that this tensor preserves complete distributivity of complete lattices. Moreover, the universal properties of this tensor can then be used to properly combine the Eilenberg-Moore algebras. We will apply this method to combine the effects of probability, global store, cost, nondeterminism, and error effects. We will then compare this method of combining effects with the more traditional method of combining equational theories using interaction laws.