Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userSemantics of Higher-Order Recursion Schemes
167
0
0.0
(
0
)
Added by
Jiri Adamek
Publication date
2011
fields
Informatics Engineering
and research's language is
English
Authors
Jiri Adamek
Ask ChatGPT about the research
No Arabic abstract
Higher-order recursion schemes are recursive equations defining new operations from given ones called terminals. Every such recursion scheme is proved to have a least interpreted semantics in every Scotts model of lambda-calculus in which the terminals are interpreted as continuous operations. For the uninterpreted semantics based on infinite lambda-terms we follow the idea of Fiore, Plotkin and Turi and work in the category of sets in context, which are presheaves on the category of finite sets. Fiore et al showed how to capture the type of variable binding in lambda-calculus by an endofunctor Hlambda and they explained simultaneous substitution of lambda-terms by proving that the presheaf of lambda-terms is an initial Hlambda-monoid. Here we work with the presheaf of rational infinite lambda-terms and prove that this is an initial iterative Hlambda-monoid. We conclude that every guarded higher-order recursion scheme has a unique uninterpreted solution in this monoid.
rate research
Read More
This paper studies the logical properties of a very general class of infinite ranked trees, namely those generated by higher-order recursion schemes. We consider, for both monadic second-order logic and modal mu-calculus, three main problems: model-checking, logical reflection (aka global model-checking, that asks for a finite description of the set of elements for which a formula holds) and selection (that asks, if exists, for some finite description of a set of elements for which an MSO formula with a second-order free variable holds). For each of these problems we provide an effective solution. This is obtained thanks to a known connection between higher-order recursion schemes and collapsible pushdown automata and on previous work regarding parity games played on transition graphs of collapsible pushdown automata.
Higher inductive types are a class of type-forming rules, introduced to provide basic (and not-so-basic) homotopy-theoretic constructions in a type-theoretic style. They have proven very fruitful for the synthetic development of homotopy theory within type theory, as well as in formalizing ordinary set-level mathematics in type theory. In this article, we construct models of a wide range of higher inductive types in a fairly wide range of settings. We introduce the notion of cell monad with parameters: a semantically-defined scheme for specifying homotopically well-behaved notions of structure. We then show that any suitable model category has *weakly stable typal initial algebras* for any cell monad with parameters. When combined with the local universes construction to obtain strict stability, this specializes to give models of specific higher inductive types, including spheres, the torus, pushout types, truncations, the James construction, and general localisations. Our results apply in any sufficiently nice Quillen model category, including any right proper, simplicially locally cartesian closed, simplicial Cisinski model category (such as simplicial sets) and any locally presentable locally cartesian closed category (such as sets) with its trivial model structure. In particular, any locally presentable locally cartesian closed $(infty,1)$-category is presented by some model category to which our results apply.
We describe a mathematical structure that can give extensional denotational semantics to higher-order probabilistic programs. It is not limited to discrete probabilities, and it is compatible with integration in a way the models that have been proposed before are not. It is organised as a model of propositional linear logic in which all the connectives have intuitive probabilistic interpretations. In addition, it has least fixed points for all maps, so it can interpret recursion.
We develop a denotational semantics of Linear Logic with least and greatest fixed points in coherence spaces (where both fixed points are interpreted in the same way) and in coherence spaces with totality (where they have different interpretations). These constructions can be carried out in many different denotational models of LL (hypercoherences, Scott semantics, finiteness spaces etc). We also present a natural embedding of G{o}del System T in LL with fixed points thus enforcing the expressive power of this system as a programming language featuring both normalization and a huge expressive power in terms of data types.
Ong has shown that the modal mu-calculus model checking problem (equivalently, the alternating parity tree automaton (APT) acceptance problem) of possibly-infinite ranked trees generated by order-n recursion schemes is n-EXPTIME complete. We consider two subclasses of APT and investigate the complexity of the respective acceptance problems. The main results are that, for APT with a single priority, the problem is still n-EXPTIME complete; whereas, for APT with a disjunctive transition function, the problem is (n-1)-EXPTIME complete. This study was motivated by Kobayashis recent work showing that the resource usage verification of functional programs can be reduced to the model checking of recursion schemes. As an application, we show that the resource usage verification problem is (n-1)-EXPTIME complete.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
