1,496 research outputs found
Reasoning about embedded dependencies using inclusion dependencies
The implication problem for the class of embedded dependencies is
undecidable. However, this does not imply lackness of a proof procedure as
exemplified by the chase algorithm. In this paper we present a complete
axiomatization of embedded dependencies that is based on the chase and uses
inclusion dependencies and implicit existential quantification in the
intermediate steps of deductions
Non-Deterministic Kleene Coalgebras
In this paper, we present a systematic way of deriving (1) languages of
(generalised) regular expressions, and (2) sound and complete axiomatizations
thereof, for a wide variety of systems. This generalizes both the results of
Kleene (on regular languages and deterministic finite automata) and Milner (on
regular behaviours and finite labelled transition systems), and includes many
other systems such as Mealy and Moore machines
A simple sequent calculus for nominal logic
Nominal logic is a variant of first-order logic that provides support for
reasoning about bound names in abstract syntax. A key feature of nominal logic
is the new-quantifier, which quantifies over fresh names (names not appearing
in any values considered so far). Previous attempts have been made to develop
convenient rules for reasoning with the new-quantifier, but we argue that none
of these attempts is completely satisfactory.
In this article we develop a new sequent calculus for nominal logic in which
the rules for the new- quantifier are much simpler than in previous attempts.
We also prove several structural and metatheoretic properties, including
cut-elimination, consistency, and equivalence to Pitts' axiomatization of
nominal logic
CCS Dynamic Bisimulation is Progressing
Weak Observational Congruence (woc) defined on CCS agents is not a bisimulation since it does not require two states reached by bisimilar computations of woc agents to be still woc, e.g.\ and are woc but and are not. This fact prevents us from characterizing CCS semantics (when is considered invisible) as a final algebra, since the semantic function would induce an equivalence over the agents that is both a congruence and a bisimulation. In the paper we introduce a new behavioural equivalence for CCS agents, which is the coarsest among those bisimulations which are also congruences. We call it Dynamic Observational Congruence because it expresses a natural notion of equivalence for concurrent systems required to simulate each other in the presence of dynamic, i.e.\ run time, (re)configurations. We provide an algebraic characterization of Dynamic Congruence in terms of a universal property of finality. Furthermore we introduce Progressing Bisimulation, which forces processes to simulate each other performing explicit steps. We provide an algebraic characterization of it in terms of finality, two characterizations via modal logic in the style of HML, and a complete axiomatization for finite agents. Finally, we prove that Dynamic Congruence and Progressing Bisimulation coincide for CCS agents. Thus the title of the paper
Realms: A Structure for Consolidating Knowledge about Mathematical Theories
Since there are different ways of axiomatizing and developing a mathematical
theory, knowledge about a such a theory may reside in many places and in many
forms within a library of formalized mathematics. We introduce the notion of a
realm as a structure for consolidating knowledge about a mathematical theory. A
realm contains several axiomatizations of a theory that are separately
developed. Views interconnect these developments and establish that the
axiomatizations are equivalent in the sense of being mutually interpretable. A
realm also contains an external interface that is convenient for users of the
library who want to apply the concepts and facts of the theory without delving
into the details of how the concepts and facts were developed. We illustrate
the utility of realms through a series of examples. We also give an outline of
the mechanisms that are needed to create and maintain realms.Comment: As accepted for CICM 201
Axioms and Decidability for Type Isomorphism in the Presence of Sums
We consider the problem of characterizing isomorphisms of types, or,
equivalently, constructive cardinality of sets, in the simultaneous presence of
disjoint unions, Cartesian products, and exponentials. Mostly relying on
results about polynomials with exponentiation that have not been used in our
context, we derive: that the usual finite axiomatization known as High-School
Identities (HSI) is complete for a significant subclass of types; that it is
decidable for that subclass when two types are isomorphic; that, for the whole
of the set of types, a recursive extension of the axioms of HSI exists that is
complete; and that, for the whole of the set of types, the question as to
whether two types are isomorphic is decidable when base types are to be
interpreted as finite sets. We also point out certain related open problems
The exp-log normal form of types
Lambda calculi with algebraic data types lie at the core of functional
programming languages and proof assistants, but conceal at least two
fundamental theoretical problems already in the presence of the simplest
non-trivial data type, the sum type. First, we do not know of an explicit and
implemented algorithm for deciding the beta-eta-equality of terms---and this in
spite of the first decidability results proven two decades ago. Second, it is
not clear how to decide when two types are essentially the same, i.e.
isomorphic, in spite of the meta-theoretic results on decidability of the
isomorphism.
In this paper, we present the exp-log normal form of types---derived from the
representation of exponential polynomials via the unary exponential and
logarithmic functions---that any type built from arrows, products, and sums,
can be isomorphically mapped to. The type normal form can be used as a simple
heuristic for deciding type isomorphism, thanks to the fact that it is a
systematic application of the high-school identities.
We then show that the type normal form allows to reduce the standard beta-eta
equational theory of the lambda calculus to a specialized version of itself,
while preserving the completeness of equality on terms. We end by describing an
alternative representation of normal terms of the lambda calculus with sums,
together with a Coq-implemented converter into/from our new term calculus. The
difference with the only other previously implemented heuristic for deciding
interesting instances of eta-equality by Balat, Di Cosmo, and Fiore, is that we
exploit the type information of terms substantially and this often allows us to
obtain a canonical representation of terms without performing sophisticated
term analyses
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