We introduce a category of vector spaces modelling full propositional linear logic, similar to probabilistic coherence spaces and to Koethe sequences spaces. Its objects are rigged sequence spaces, Banach spaces of sequences, with norms defined from pairing with finite sequences, and morphisms are bounded linear maps, continuous in a suitable topology. The main interest of the work is that our model gives a realization of the free linear logic exponentials construction.

Differential categories are now an established abstract setting for differentiation. However, not much attention has been given to the process which is inverse to differentiation: integration. This paper presents the parallel development for integration by axiomatizing an integral transformation, sA: !A → !A ⊗ A, in a symmetric monoidal category with a coalgebra modality. When integration is combined with differentiation, the two fundamental theorems of calculus are expected to hold (in a suitable sense): a differential category with integration which satisfies these two theorems is called a calculus category.

Modifying an approach to antiderivatives by T. Ehrhard, we define having antiderivatives as the demand that a certain natural transformation, K: !A → !A, is invertible. We observe that a differential category having antiderivatives, in this sense, is always a calculus category.

When the coalgebra modality is monoidal, it is natural to demand an extra coherence between integration and the coalgebra modality. In the presence of this extra coherence, we show that a calculus category with a monoidal coalgebra modality has its integral transformation given by antiderivatives and, thus, that the integral structure is uniquely determined by the differential structure.

The paper finishes by providing a suite of separating examples. Examples of differential categories, integral categories and calculus categories based on both monoidal and (mere) coalgebra modalities are presented. In addition, differential categories which are not integral categories are discussed and vice versa.

We interpret finite types as domains over nonflat inductive base types in order to bring out the finitary core that seems to be inherent in the concept of totality. We prove a strong version of the Kreisel density theorem by providing a total compact element as a witness, a result that we cannot hope to have if we work with flat base types. To this end, we develop tools that deal adequately with possibly inconsistent finite sets of information. The classical density theorem is reestablished via a ‘finite density theorem,’ and corollaries are obtained, among them Berger's separation property.

Software watermarking is a software protection technique used to defend the intellectual property of proprietary code. In particular, software watermarking aims at preventing software piracy by embedding a signature, i.e. an identifier reliably representing the owner, in the code. When an illegal copy is made, the owner can claim his/her identity by extracting the signature. It is important to hide the signature in the program in order to make it difficult for the attacker to detect, tamper or remove it. In this work, we present a formal framework for software watermarking, based on program semantics and abstract interpretation, where attackers are modelled as abstract interpreters. In this setting, we can prove that the ability to identify signatures can be modelled as a completeness property of the attackers in the abstract interpretation framework. Indeed, hiding a signature in the code corresponds to embed it as a semantic property that can be retrieved only by attackers that are complete for it. Any abstract interpreter that is not complete for the property specifying the signature cannot detect, tamper or remove it. We formalize in the proposed framework the major quality features of a software watermarking technique: secrecy, resilience, transparence and accuracy. This provides a unifying framework for interpreting both watermarking schemes and attacks, and it allows us to formally compare the quality of different watermarking techniques. Indeed, a large number of watermarking techniques exist in the literature and they are typically evaluated with respect to their secrecy, resilience, transparence and accuracy to attacks. Formally identifying the attacks for which a watermarking scheme is secret, resilient, transparent or accurate can be a complex and error-prone task, since attacks and watermarking schemes are typically defined in different settings and using different languages (e.g. program transformation vs. program analysis), complicating the task of comparing one against the others.