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Existence results for Hilbert's problem 13th mean that any equation constructed by continue functions can be given solution represented as a superposition of continue functions of one variable or of continue functions of two variables.…
Complementary symmetric Rote sequences are binary sequences which have factor complexity $\mathcal{C}(n) = 2n$ for all integers $n \geq 1$ and whose languages are closed under the exchange of letters. These sequences are intimately linked…
One shortcoming of the chain rule is that it does not iterate: it gives the derivative of f(g(x)), but not (directly) the second or higher-order derivatives. We present iterated differentials and a version of the multivariable chain rule…
Milner (1984) defined an operational semantics for regular expressions as finite-state processes. In order to axiomatize bisimilarity of regular expressions under this process semantics, he adapted Salomaa's proof system that is complete…
Safety is a syntactic condition of higher-order grammars that constrains occurrences of variables in the production rules according to their type-theoretic order. In this paper, we introduce the safe lambda calculus, which is obtained by…
Basic proof-search tactics in logic and type theory can be seen as the root-first applications of rules in an appropriate sequent calculus, preferably without the redundancies generated by permutation of rules. This paper addresses the…
We prove that all axiomatic extensions of the full Lambek calculus with exchange can be axiomatized by formulas on the $\mathcal N_3$ level of the substructural hierarchy.
In Pure Inductive Logic, the rational principle of Predicate Exchangeability states that permuting the predicates in a given language L and replacing each occurrence of a predicate in an L-sentence $\phi$ according to this permutation…
We present a framework which allows a uniform approach to the recently introduced concept of pseudo-repetitions on words in the morphic case. This framework is at the same time more general and simpler. We introduce the concept of a…
We give a linear nested sequent calculus for the basic normal tense logic Kt. We show that the calculus enables backwards proof-search, counter-model construction and syntactic cut-elimination. Linear nested sequents thus provide the…
Regular sequences generalize the extensively studied automatic sequences. Let $S$ be an abstract numeration system. When the numeration language $L$ is prefix-closed and regular, a sequence is said to be $S$-regular if the module generated…
We use model-theoretic tools originating from stability theory to derive a result we call the Finitary Substitute Lemma, which intuitively says the following. Suppose we work in a stable graph class C, and using a first-order formula {\phi}…
We study a family of equivalence relations on $S_n$, the group of permutations on $n$ letters, created in a manner similar to that of the Knuth relation and the forgotten relation. For our purposes, two permutations are in the same…
This paper is a brief and informal presentation of cirquent calculus, a novel proof system for resource-conscious logics. As such, it is a refinement of sequent calculus with mechanisms that allow to explicitly account for the possibility…
We revisit the notion of intuitionistic equivalence and formal proof representations by adopting the view of formulas as exponential polynomials. After observing that most of the invertible proof rules of intuitionistic (minimal)…
Continuation Calculus (CC), introduced by Geron and Geuvers, is a simple foundational model for functional computation. It is closely related to lambda calculus and term rewriting, but it has no variable binding and no pattern matching. It…
We introduce a type and effect system, for an imperative object calculus, which infers "sharing" possibly introduced by the evaluation of an expression, represented as an equivalence relation among its free variables. This direct…
Fischer provided a new type of binomial determinant for the number of alternating sign matrices involving the third root of unity. In this paper we prove that her formula, when replacing the third root of unity by an indeterminate $q$, is…
We introduce the algorithm MASSA which takes classical modal formulas in input, and, when successful, effectively generates: (a) (analytic) geometric rules of the labelled calculus G3K, and (b) cut-free derivations (of a certain `canonical'…
We bring forward a logical system of transition algebras that enhances many-sorted first-order logic using features from dynamic logics. The sentences we consider include compositions, unions, and transitive closures of transition…