Unifying Structured Recursion Schemes

AbstractFolds and unfolds have been understood as fundamental building blocks for total programming, and have been extended to form an entire zoo of specialised structured recursion schemes. A great number of these schemes were unified by the introduction of adjoint folds, but more exotic beasts such as recursion schemes from comonads proved to be elusive. In this paper, we show how the two canonical derivations of adjunctions from (co)monads yield recursion schemes of significant computational importance: monadic catamorphisms come from the Kleisli construction, and more astonishingly, the elusive recursion schemes from comonads come from the Eilenberg–Moore construction. Thus, we demonstrate that adjoint folds are more unifying than previously believed.</jats:p

The Recursion Scheme from the Cofree Recursive Comonad

AbstractWe instantiate the general comonad-based construction of recursion schemes for the initial algebra of a functor F to the cofree recursive comonad on F. Differently from the scheme based on the cofree comonad on F in a similar fashion, this scheme allows not only recursive calls on elements structurally smaller than the given argument, but also subsidiary recursions. We develop a Mendler formulation of the scheme via a generalized Yoneda lemma for initial algebras involving strong dinaturality and hint a relation to circular proofs à la Cockett, Santocanale

Relational semantics of linear logic and higher-order model-checking

In this article, we develop a new and somewhat unexpected connection between higher-order model-checking and linear logic. Our starting point is the observation that once embedded in the relational semantics of linear logic, the Church encoding of any higher-order recursion scheme (HORS) comes together with a dual Church encoding of an alternating tree automata (ATA) of the same signature. Moreover, the interaction between the relational interpretations of the HORS and of the ATA identifies the set of accepting states of the tree automaton against the infinite tree generated by the recursion scheme. We show how to extend this result to alternating parity automata (APT) by introducing a parametric version of the exponential modality of linear logic, capturing the formal properties of colors (or priorities) in higher-order model-checking. We show in particular how to reunderstand in this way the type-theoretic approach to higher-order model-checking developed by Kobayashi and Ong. We briefly explain in the end of the paper how his analysis driven by linear logic results in a new and purely semantic proof of decidability of the formulas of the monadic second-order logic for higher-order recursion schemes.Comment: 24 pages. Submitte

Terminal semantics for codata types in intensional Martin-L\"of type theory

In this work, we study the notions of relative comonad and comodule over a relative comonad, and use these notions to give a terminal coalgebra semantics for the coinductive type families of streams and of infinite triangular matrices, respectively, in intensional Martin-L\"of type theory. Our results are mechanized in the proof assistant Coq.Comment: 14 pages, ancillary files contain formalized proof in the proof assistant Coq; v2: 20 pages, title and abstract changed, give a terminal semantics for streams as well as for matrices, Coq proof files updated accordingl

Relational Semantics of Linear Logic and Higher-order Model Checking

In this article, we develop a new and somewhat unexpected connection between higher-order model-checking and linear logic. Our starting point is the observation that once embedded in the relational semantics of linear logic, the Church encoding of any higher-order recursion scheme (HORS) comes together with a dual Church encoding of an alternating tree automata (ATA) of the same signature. Moreover, the interaction between the relational interpretations of the HORS and of the ATA identifies the set of accepting states of the tree automaton against the infinite tree generated by the recursion scheme. We show how to extend this result to alternating parity automata (APT) by introducing a parametric version of the exponential modality of linear logic, capturing the formal properties of colors (or priorities) in higher-order model-checking. We show in particular how to reunderstand in this way the type-theoretic approach to higher-order model-checking developed by Kobayashi and Ong. We briefly explain in the end of the paper how this analysis driven by linear logic results in a new and purely semantic proof of decidability of the formulas of the monadic second-order logic for higher-order recursion schemes

Derived factorization categories of non-Thom--Sebastiani-type sums of potentials

We first prove semi-orthogonal decompositions of derived factorization categories arising from sums of potentials of gauged Landau-Ginzburg models, where the sums are not necessarily Thom--Sebastiani type. We then apply the result to the category \HMF^{L_f}(f) of maximally graded matrix factorizations of an invertible polynomial $f$ of chain type, and explicitly construct a full strong exceptional collection E_1,\hdots,E_{\mu} in \HMF^{L_f}(f) whose length $\mu$ is the Milnor number of the Berglund--H\"ubsch transpose $\widetilde{f}$ of $f$. This proves a conjecture, which postulates that for an invertible polynomial $f$ the category \HMF^{L_f}(f) admits a tilting object, in the case when $f$ is a chain polynomial. Moreover, by careful analysis of morphisms between the exceptional objects $E_i$, we explicitly determine the quiver with relations $(Q,I)$ which represents the endomorphism ring of the associated tilting object $\oplus_{i=1}^{\mu}E_i$ in \HMF^{L_f}(f), and in particular we obtain an equivalence \HMF^{L_f}(f)\cong \Db(\fmod kQ/I).Comment: Major improvements. The proof of the existence of a tilting object is added, and we compute the associated quiver with relations. 48 page

Verification of redecoration for infinite triangular matrices using coinduction

International audienceFinite triangular matrices with a dedicated type for the diagonal elements can be profitably represented by a nested data type, i. e., a heterogeneous family of inductive data types, while infinite triangular matrices form an example of a nested coinductive type, which is a heterogeneous family of coinductive data types. Redecoration for infinite triangular matrices is taken up from previous work involving the first author, and it is shown that redecoration forms a comonad with respect to bisimilarity. The main result, however, is a validation of the original algorithm against a model based on infinite streams of infinite streams. The two formulations are even provably equivalent, and the second is identified as a special instance of the generic cobind operation resulting from the well-known comultiplication operation on streams that creates the stream of successive tails of a given stream. Thus, perhaps surprisingly, the verification of redecoration is easier for infinite triangular matrices than for their finite counterpart. All the results have been obtained and are fully formalized in the current version of the Coq theorem proving environment where these coinductive datatypes are fully supported since the version 8.1, released in 2007. Nonetheless, instead of displaying the Coq development, we have chosen to write the paper in standard mathematical and type-theoretic language. Thus, it should be accessible without any specific knowledge about Coq

Stateful Runners of Effectful Computations

AbstractWhat structure is required of a set so that computations in a given notion of computation can be run statefully with this set as the state set? For running nondeterministic computations statefully, a resolver structure is needed; for interactive I/O computations, a “responder-listener” structure is necessary; to be able to serve stateful computations, the set must carry the structure of a lens. We show that, in general, to be a stateful runner of computations for a monad corresponding to a Lawvere theory (defined as a set equipped with a monad morphism between the given monad and the state monad for this set) is the same as to be a comodel of the theory, i.e., a coalgebra of the corresponding comonad. We work out a number of instances of this observation and also compare runners to handlers