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Register a new userMinimisation in Logical Form
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Added by
Dexter Kozen
Publication date
2020
fields
Informatics Engineering
and research's language is
English
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Stone-type dualities provide a powerful mathematical framework for studying properties of logical systems. They have recently been fruitfully explored in understanding minimisation of various types of automata. In Bezhanishvili et al. (2012), a dual equivalence between a category of coalgebras and a category of algebras was used to explain minimisation. The algebraic semantics is dual to a coalgebraic semantics in which logical equivalence coincides with trace equivalence. It follows that maximal quotients of coalgebras correspond to minimal subobjects of algebras. Examples include partially observable deterministic finite automata, linear weighted automata viewed as coalgebras over finite-dimensional vector spaces, and belief automata, which are coalgebras on compact Hausdorff spaces. In Bonchi et al. (2014), Brzozowskis double-reversal minimisation algorithm for deterministic finite automata was described categorically and its correctness explained via the duality between reachability and observability. This work includes generalisations of Brzozowskis algorithm to Moore and weighted automata over commutative semirings. In this paper we propose a general categorical framework within which such minimisation algorithms can be understood. The goal is to provide a unifying perspective based on duality. Our framework consists of a stack of three interconnected adjunctions: a base dual adjunction that can be lifted to a dual adjunction between coalgebras and algebras and also to a dual adjunction between automata. The approach provides an abstract understanding of reachability and observability. We illustrate the general framework on range of concrete examples, including deterministic Kripke frames, weighted automata, topological automata (belief automata), and alternating automata.
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Appel and McAllesters step-indexed logical relations have proven to be a simple and effective technique for reasoning about programs in languages with semantically interesting types, such as general recursive types and general reference types. However, proofs using step-indexed models typically involve tedious, error-prone, and proof-obscuring step-index arithmetic, so it is important to develop clean, high-level, equational proof principles that avoid mention of step indices. In this paper, we show how to reason about binary step-indexed logical relations in an abstract and elegant way. Specifically, we define a logic LSLR, which is inspired by Plotkin and Abadis logic for parametricity, but also supports recursively defined relations by means of the modal later operator from Appel, Melli`es, Richards, and Vouillons very modal model paper. We encode in LSLR a logical relation for reasoning relationally about programs in call-by-value System F extended with general recursive types. Using this logical relation, we derive a set of useful rules with which we can prove contextual equivalence and approximation results without counting steps.
While quantum computers are expected to yield considerable advantages over classical devices, the precise features of quantum theory enabling these advantages remain unclear. Contextuality--the denial of a notion of classical physical reality--has emerged as a promising hypothesis. Magic states are quantum resources critical for practically achieving universal quantum computation. They exhibit the standard form of contextuality that is known to enable probabilistic advantages in a variety of computational and communicational tasks. Strong contextuality is an extremal form of contextuality describing systems that exhibit logically paradoxical behaviour. Here, we consider special magic states that deterministically enable quantum computation. After introducing number-theoretic techniques for constructing exotic quantum paradoxes, we present large classes of strongly contextual magic states that enable deterministic implementation of gates from the Clifford hierarchy. These surprising discoveries bolster a refinement of the resource theory of contextuality that emphasises the computational power of logical paradoxes.
The main aim of the present paper is to use a proof system for hybrid modal logic to formalize what are called falsebelief tasks in cognitive psychology, thereby investigating the interplay between cognition and logical reasoning about belief. We consider two differe
Model checking timed automata becomes increasingly complex with the increase in the number of clocks. Hence it is desirable that one constructs an automaton with the minimum number of clocks possible. The problem of checking whether there exists a timed automaton with a smaller number of clocks such that the timed language accepted by the original automaton is preserved is known to be undecidable. In this paper, we give a construction, which for any given timed automaton produces a timed bisimilar automaton with the least number of clocks. Further, we show that such an automaton with the minimum possible number of clocks can be constructed in time that is doubly exponential in the number of clocks of the original automaton.
Intersection types are an essential tool in the analysis of operational and denotational properties of lambda-terms and functional programs. Among them, non-idempotent intersection types provide precise quantitative information about the evaluation of terms and programs. However, unlike simple or second-order types, intersection types cannot be considered as a logical system because the application rule (or the intersection rule, depending on the presentation of the system) involves a condition expressing that the proofs of premises satisfy a very strong uniformity condition: the underlying lambda-terms must be the same. Using earlier work introducing an indexed version of Linear Logic, we show that non-idempotent typing can be given a logical form in a system where formulas represent hereditarily indexed families of intersection types.
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