Polymorphic Higher-Order Termination

We generalise the termination method of higher-order polynomial interpretations to a setting with impredicative polymorphism. Instead of using weakly monotonic functionals, we interpret terms in a suitable extension of System F_omega. This enables a direct interpretation of rewrite rules which make essential use of impredicative polymorphism. In addition, our generalisation eases the applicability of the method in the non-polymorphic setting by allowing for the encoding of inductive data types. As an illustration of the potential of our method, we prove termination of a substantial fragment of full intuitionistic second-order propositional logic with permutative conversions

Nominal Recursors as Epi-Recursors: Extended Technical Report

We study nominal recursors from the literature on syntax with bindings and compare them with respect to expressiveness. The term "nominal" refers to the fact that these recursors operate on a syntax representation where the names of bound variables appear explicitly, as in nominal logic. We argue that nominal recursors can be viewed as epi-recursors, a concept that captures abstractly the distinction between the constructors on which one actually recurses, and other operators and properties that further underpin recursion.We develop an abstract framework for comparing epi-recursors and instantiate it to the existing nominal recursors, and also to several recursors obtained from them by cross-pollination. The resulted expressiveness hierarchies depend on how strictly we perform this comparison, and bring insight into the relative merits of different axiomatizations of syntax. We also apply our methodology to produce an expressiveness hierarchy of nominal corecursors, which are principles for defining functions targeting infinitary non-well-founded terms (which underlie lambda-calculus semantics concepts such as B\"ohm trees). Our results are validated with the Isabelle/HOL theorem prover

Mathematical Logic: Proof theory, Constructive Mathematics

The workshop “Mathematical Logic: Proof Theory, Constructive Mathematics” was centered around proof-theoretic aspects of current mathematics, constructive mathematics and logical aspects of computational complexit

Automated Deduction – CADE 28

This open access book constitutes the proceeding of the 28th International Conference on Automated Deduction, CADE 28, held virtually in July 2021. The 29 full papers and 7 system descriptions presented together with 2 invited papers were carefully reviewed and selected from 76 submissions. CADE is the major forum for the presentation of research in all aspects of automated deduction, including foundations, applications, implementations, and practical experience. The papers are organized in the following topics: Logical foundations; theory and principles; implementation and application; ATP and AI; and system descriptions

Completeness of Flat Coalgebraic Fixpoint Logics

Modal fixpoint logics traditionally play a central role in computer science, in particular in artificial intelligence and concurrency. The mu-calculus and its relatives are among the most expressive logics of this type. However, popular fixpoint logics tend to trade expressivity for simplicity and readability, and in fact often live within the single variable fragment of the mu-calculus. The family of such flat fixpoint logics includes, e.g., LTL, CTL, and the logic of common knowledge. Extending this notion to the generic semantic framework of coalgebraic logic enables covering a wide range of logics beyond the standard mu-calculus including, e.g., flat fragments of the graded mu-calculus and the alternating-time mu-calculus (such as alternating-time temporal logic ATL), as well as probabilistic and monotone fixpoint logics. We give a generic proof of completeness of the Kozen-Park axiomatization for such flat coalgebraic fixpoint logics.Comment: Short version appeared in Proc. 21st International Conference on Concurrency Theory, CONCUR 2010, Vol. 6269 of Lecture Notes in Computer Science, Springer, 2010, pp. 524-53

Set systems: order types, continuous nondeterministic deformations, and quasi-orders

By reformulating a learning process of a set system L as a game between Teacher and Learner, we define the order type of L to be the order type of the game tree, if the tree is well-founded. The features of the order type of L (dim L in symbol) are (1) We can represent any well-quasi-order (wqo for short) by the set system L of the upper-closed sets of the wqo such that the maximal order type of the wqo is equal to dim L. (2) dim L is an upper bound of the mind-change complexity of L. dim L is defined iff L has a finite elasticity (fe for short), where, according to computational learning theory, if an indexed family of recursive languages has fe then it is learnable by an algorithm from positive data. Regarding set systems as subspaces of Cantor spaces, we prove that fe of set systems is preserved by any continuous function which is monotone with respect to the set-inclusion. By it, we prove that finite elasticity is preserved by various (nondeterministic) language operators (Kleene-closure, shuffle-closure, union, product, intersection,. . ..) The monotone continuous functions represent nondeterministic computations. If a monotone continuous function has a computation tree with each node followed by at most n immediate successors and the order type of a set system L is {\alpha}, then the direct image of L is a set system of order type at most n-adic diagonal Ramsey number of {\alpha}. Furthermore, we provide an order-type-preserving contravariant embedding from the category of quasi-orders and finitely branching simulations between them, into the complete category of subspaces of Cantor spaces and monotone continuous functions having Girard's linearity between them. Keyword: finite elasticity, shuffle-closur

A theory of agreements and protection

In this thesis we propose a theory of contracts. Contracts are modelled as interacting processes with an explicit association of obligations and objectives. Obligations are specified using event structures. In this model we formalise two fundamental notions of contracts, namely agreement and protection. These notions arise naturally by interpreting contracts as multi-player concurrent games. A participant agrees on a contract if she has a strategy to reach her objectives (or to make another participant sanctionable for a violation), whatever the moves of her counterparts. A participant is protected by a contract when she has a strategy to defend herself in all possible contexts, even in those where she has not reached an agreement. When obligations are represented using classical event structures, we show that agreement and protection mutually exclude each other for a wide class of contracts. To reconcile agreement with protection we propose a novel formalism for modelling contractual obligations: event structures with circular causality. We study this model from a foundational perspective, and we relate it with classical event structures. Using this model, we show how to construct contracts which guarantee both agreement and protection. We relate our contract model with Propositional Contract Logic, by establishing a correspondence between provability in the logic and the notions of agreement and strategies. This is a first step towards reducing the gap between two main paradigms for modelling contracts, that is the one which interprets them as interactive systems, and the one based on logic