682 research outputs found
The categorical limit of a sequence of dynamical systems
Modeling a sequence of design steps, or a sequence of parameter settings,
yields a sequence of dynamical systems. In many cases, such a sequence is
intended to approximate a certain limit case. However, formally defining that
limit turns out to be subject to ambiguity. Depending on the interpretation of
the sequence, i.e. depending on how the behaviors of the systems in the
sequence are related, it may vary what the limit should be. Topologies, and in
particular metrics, define limits uniquely, if they exist. Thus they select one
interpretation implicitly and leave no room for other interpretations. In this
paper, we define limits using category theory, and use the mentioned relations
between system behaviors explicitly. This resolves the problem of ambiguity in
a more controlled way. We introduce a category of prefix orders on executions
and partial history preserving maps between them to describe both discrete and
continuous branching time dynamics. We prove that in this category all
projective limits exist, and illustrate how ambiguity in the definition of
limits is resolved using an example. Moreover, we show how various problems
with known topological approaches are now resolved, and how the construction of
projective limits enables us to approximate continuous time dynamics as a
sequence of discrete time systems.Comment: In Proceedings EXPRESS/SOS 2013, arXiv:1307.690
A criterion for separating process calculi
We introduce a new criterion, replacement freeness, to discern the relative
expressiveness of process calculi. Intuitively, a calculus is strongly
replacement free if replacing, within an enclosing context, a process that
cannot perform any visible action by an arbitrary process never inhibits the
capability of the resulting process to perform a visible action. We prove that
there exists no compositional and interaction sensitive encoding of a not
strongly replacement free calculus into any strongly replacement free one. We
then define a weaker version of replacement freeness, by only considering
replacement of closed processes, and prove that, if we additionally require the
encoding to preserve name independence, it is not even possible to encode a non
replacement free calculus into a weakly replacement free one. As a consequence
of our encodability results, we get that many calculi equipped with priority
are not replacement free and hence are not encodable into mainstream calculi
like CCS and pi-calculus, that instead are strongly replacement free. We also
prove that variants of pi-calculus with match among names, pattern matching or
polyadic synchronization are only weakly replacement free, hence they are
separated both from process calculi with priority and from mainstream calculi.Comment: In Proceedings EXPRESS'10, arXiv:1011.601
Changing a semantics: opportunism or courage?
The generalized models for higher-order logics introduced by Leon Henkin, and
their multiple offspring over the years, have become a standard tool in many
areas of logic. Even so, discussion has persisted about their technical status,
and perhaps even their conceptual legitimacy. This paper gives a systematic
view of generalized model techniques, discusses what they mean in mathematical
and philosophical terms, and presents a few technical themes and results about
their role in algebraic representation, calibrating provability, lowering
complexity, understanding fixed-point logics, and achieving set-theoretic
absoluteness. We also show how thinking about Henkin's approach to semantics of
logical systems in this generality can yield new results, dispelling the
impression of adhocness. This paper is dedicated to Leon Henkin, a deep
logician who has changed the way we all work, while also being an always open,
modest, and encouraging colleague and friend.Comment: 27 pages. To appear in: The life and work of Leon Henkin: Essays on
his contributions (Studies in Universal Logic) eds: Manzano, M., Sain, I. and
Alonso, E., 201
Structured operational semantics and bisimulation as a congruence
AbstractIn this paper we are interested in general properties of classes of transition system specifications in Plotkin style. The discussion takes place in a setting of labelled transition systems. The states of the transition systems are terms generated by a single sorted signature and the transitions between states are defined by conditional rules over the syntax. It is argued that in this setting it is natural to require that strong bisimulation equivalence be a congruence on the states of the transition systems. A general format, called the tyft/tyxt format, is presented for the rules in a transition system specification, such that bisimulation is always a congruence when all the rules fit this format. With a series of examples it is demonstrated that the tyft/tyxt format cannot be generalized in any obvious way. Another series of examples illustrates the usefulness of our congruence theorem. Briefly we touch upon the issue of modularity of transition system specifications. It is argued that certain pathological tyft/tyxt rules (the ones which are not pure) can be disqualified because they behave badly with respect to modularization. Next we address the issue of full abstraction. We characterize the completed trace congruence induced by the operators in pure tyft/tyxt format as 2-nested simulation equivalence. The pure tyft/tyxt format includes the format given by de Simone (Theoret. Comput. Sci. 37, 245â267 (1985)) but is incomparable to the GSOS format of Bloom, Istrail, and Meyer (in âConference Record of the 15th Annual Symposium on Principles of Programming Languages, San Diego, California, 1988,â pp. 229â239). However, it turns out that 2-nested simulation equivalence strictly refines the completed trace congruence induced by the GSOS format
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