3,120 research outputs found
A categorical foundation for structured reversible flowchart languages: Soundness and adequacy
Structured reversible flowchart languages is a class of imperative reversible
programming languages allowing for a simple diagrammatic representation of
control flow built from a limited set of control flow structures. This class
includes the reversible programming language Janus (without recursion), as well
as more recently developed reversible programming languages such as R-CORE and
R-WHILE.
In the present paper, we develop a categorical foundation for this class of
languages based on inverse categories with joins. We generalize the notion of
extensivity of restriction categories to one that may be accommodated by
inverse categories, and use the resulting decisions to give a reversible
representation of predicates and assertions. This leads to a categorical
semantics for structured reversible flowcharts, which we show to be
computationally sound and adequate, as well as equationally fully abstract with
respect to the operational semantics under certain conditions
Fifty years of Hoare's Logic
We present a history of Hoare's logic.Comment: 79 pages. To appear in Formal Aspects of Computin
Simple characterizations for commutativity of quantum weakest preconditions
In a recent letter [Information Processing Letters~104 (2007) 152-158], it
has shown some sufficient conditions for commutativity of quantum weakest
preconditions. This paper provides some alternative and simple
characterizations for the commutativity of quantum weakest preconditions, i.e.,
Theorem 3.1, Theorem 3.2 and Proposition 3.3 in what follows. We also show that
to characterize the commutativity of quantum weakest preconditions in terms of
() is hard in the sense of Proposition 4.1 and Proposition 4.2.Comment: Re-written, comments are welcom
Condition/Decision Duality and the Internal Logic of Extensive Restriction Categories
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 . 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.Comment: 19 pages, including 6 page appendix of proofs. Accepted for MFPS XXX
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