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Selected Papers of the 9th Conference on Algebra and Coalgebra in Computer Science (CALCO 2021)
Editors: Fabio Gadducci, Alexandra Silva
1. Minimality Notions via Factorization Systems and Examples
For the minimization of state-based systems (i.e. the reduction of the number of states while retaining the system's semantics), there are two obvious aspects: removing unnecessary states of the system and merging redundant states in the system. In the present article, we relate the two minimization aspects on coalgebras by defining an abstract notion of minimality. The abstract notions minimality and minimization live in a general category with a factorization system. We will find criteria on the category that ensure uniqueness, existence, and functoriality of the minimization aspects. The proofs of these results instantiate to those for reachability and observability minimization in the standard coalgebra literature. Finally, we will see how the two aspects of minimization interact and under which criteria they can be sequenced in any order, like in automata minimization.
2. Stream processors and comodels
In 2009, Hancock, Pattinson and Ghani gave a coalgebraic characterisation of stream processors $A^\mathbb{N} \to B^\mathbb{N}$ drawing on ideas of Brouwerian constructivism. Their stream processors have an intensional character; in this paper, we give a corresponding coalgebraic characterisation of extensional stream processors, i.e., the set of continuous functions $A^\mathbb{N} \to B^\mathbb{N}$. Our account sites both our result and that of op. cit. within the apparatus of comodels for algebraic effects originating with Power-Shkaravska. Within this apparatus, the distinction between intensional and extensional equivalence for stream processors arises in the same way as the the distinction between bisimulation and trace equivalence for labelled transition systems and probabilistic generative systems.
3. Pushdown Automata and Context-Free Grammars in Bisimulation Semantics
The Turing machine models an old-fashioned computer, that does not interact with the user or with other computers, and only does batch processing. Therefore, we came up with a Reactive Turing Machine that does not have these shortcomings. In the Reactive Turing Machine, transitions have labels to give a notion of interactivity. In the resulting process graph, we use bisimilarity instead of language equivalence. Subsequently, we considered other classical theorems and notions from automata theory and formal languages theory. In this paper, we consider the classical theorem of the correspondence between pushdown automata and context-free grammars. By changing the process operator of sequential composition to a sequencing operator with intermediate acceptance, we get a better correspondence in our setting. We find that the missing ingredient to recover the full correspondence is the addition of a notion of state awareness.
4. A Coinductive Reformulation of Milner's Proof System for Regular Expressions Modulo Bisimilarity
Milner (1984) defined an operational semantics for regular expressions as finite-state processes. In order to axiomatize bisimilarity of regular expressions under this process semantics, he adapted Salomaa's proof system that is complete for equality of regular expressions under the language semantics. Apart from most equational axioms, Milner's system Mil inherits from Salomaa's system a non-algebraic rule for solving single fixed-point equations. Recognizing distinctive properties of the process semantics that render Salomaa's proof strategy inapplicable, Milner posed completeness of the system Mil as an open question. As a proof-theoretic approach to this problem we characterize the derivational power that the fixed-point rule adds to the purely equational part Mil$^-$ of Mil. We do so by means of a coinductive rule that permits cyclic derivations that consist of a finite process graph with empty steps that satisfies the layered loop existence and elimination property LLEE, and two of its Mil$^{-}$-provable solutions. With this rule as replacement for the fixed-point rule in Mil, we define the coinductive reformulation cMil as an extension of Mil$^{-}$. In order to show that cMil and Mil are theorem equivalent we develop effective proof transformations from Mil to cMil, and vice versa. Since it is located half-way in between bisimulations and proofs in Milner's system Mil, cMil may become a beachhead for a completeness proof of Mil. This […]
5. Sum and Tensor of Quantitative Effects
Inspired by the seminal work of Hyland, Plotkin, and Power on the combination of algebraic computational effects via sum and tensor, we develop an analogous theory for the combination of quantitative algebraic effects. Quantitative algebraic effects are monadic computational effects on categories of metric spaces, which, moreover, have an algebraic presentation in the form of quantitative equational theories, a logical framework introduced by Mardare, Panangaden, and Plotkin that generalises equational logic to account for a concept of approximate equality. As our main result, we show that the sum and tensor of two quantitative equational theories correspond to the categorical sum (i.e., coproduct) and tensor, respectively, of their effects qua monads. We further give a theory of quantitative effect transformers based on these two operations, essentially providing quantitative analogues to the following monad transformers due to Moggi: exception, resumption, reader, and writer transformers. Finally, as an application, we provide the first quantitative algebraic axiomatizations to the following coalgebraic structures: Markov processes, labelled Markov processes, Mealy machines, and Markov decision processes, each endowed with their respective bisimilarity metrics. Apart from the intrinsic interest in these axiomatizations, it is pleasing they have been obtained as the composition, via sum and tensor, of simpler quantitative equational theories.