A mathematical denotation is proposed for the notion of complex software systems whose behavior is specified by rigorous formalisms. Complex systems are described in a recursive way as an interconnection of subsystems by means of architectural connectors. In order to consider the largest family of specification formalisms and architectural connectors, this denotation is essentially formalism, specification and connector independent. For this, we build our denotation on Goguen's institution theory. We then denote in this abstract framework, complexity by the notion of property emergence.

Abstract. In the course of understanding biological regulatory networks (BRN), scientists usually start by studying small BRNs that they believe to be of particular importance to represent a biological function, and then, embed them in the whole network. Such a reduction can lead to neglect relevant regulations and to study a network whose properties can be very different from the properties of this network viewed as a part of the whole. In this paper we study, from a logical point of view, the preservation of properties inherited from small BRNs. The signature of BRN, constituted by a graph, is one of the distinctive features on which embeddings can be defined which leads us to give a first condition on the subgraphs ensuring the preservation of properties of the embedded graphs.

Abstract. In the course of understanding biological regulatory networks (BRN), scientists usually start by studying small BRNs that they believe to be of particular importance to represent a biological function, and then, embed them in a larger network. Such a reduction can lead to neglect relevant regulations and to study a network whose properties can be very different from the properties of this network viewed as a part of the whole. In this paper we study, from a logical point of view, on which conditions concerning both networks, properties can be inherited by BRNs from sub-BRNs. We give some conditions on the nature of the network embeddings ensuring that dynamic properties on the embedded sub-BRNs are preserved at the level of the whole BRN.

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