856 research outputs found
(Co-)Inductive semantics for Constraint Handling Rules
In this paper, we address the problem of defining a fixpoint semantics for
Constraint Handling Rules (CHR) that captures the behavior of both
simplification and propagation rules in a sound and complete way with respect
to their declarative semantics. Firstly, we show that the logical reading of
states with respect to a set of simplification rules can be characterized by a
least fixpoint over the transition system generated by the abstract operational
semantics of CHR. Similarly, we demonstrate that the logical reading of states
with respect to a set of propagation rules can be characterized by a greatest
fixpoint. Then, in order to take advantage of both types of rules without
losing fixpoint characterization, we present an operational semantics with
persistent. We finally establish that this semantics can be characterized by
two nested fixpoints, and we show the resulting language is an elegant
framework to program using coinductive reasoning.Comment: 17 page
Coinduction up to in a fibrational setting
Bisimulation up-to enhances the coinductive proof method for bisimilarity,
providing efficient proof techniques for checking properties of different kinds
of systems. We prove the soundness of such techniques in a fibrational setting,
building on the seminal work of Hermida and Jacobs. This allows us to
systematically obtain up-to techniques not only for bisimilarity but for a
large class of coinductive predicates modelled as coalgebras. By tuning the
parameters of our framework, we obtain novel techniques for unary predicates
and nominal automata, a variant of the GSOS rule format for similarity, and a
new categorical treatment of weak bisimilarity
Proof Relevant Corecursive Resolution
Resolution lies at the foundation of both logic programming and type class
context reduction in functional languages. Terminating derivations by
resolution have well-defined inductive meaning, whereas some non-terminating
derivations can be understood coinductively. Cycle detection is a popular
method to capture a small subset of such derivations. We show that in fact
cycle detection is a restricted form of coinductive proof, in which the atomic
formula forming the cycle plays the role of coinductive hypothesis.
This paper introduces a heuristic method for obtaining richer coinductive
hypotheses in the form of Horn formulas. Our approach subsumes cycle detection
and gives coinductive meaning to a larger class of derivations. For this
purpose we extend resolution with Horn formula resolvents and corecursive
evidence generation. We illustrate our method on non-terminating type class
resolution problems.Comment: 23 pages, with appendices in FLOPS 201
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