Abstract
Kleene algebra with tests (KAT) was introduced as an algebraic structure to model and reason about classic imperative programs, i.e. sequences of discrete transitions guarded by Boolean tests. This paper introduces two generalisations of this structure able to express programs as weighted transitions and tests with outcomes in non necessarily bivalent truth spaces: graded Kleene algebra with tests (GKAT) and a variant where tests are also idempotent (I-GKAT). In this context, and in analogy to Kozen's encoding of Propositional Hoare Logic (PHL) in KAT we discuss the encoding of a graded PHL in I-GKAT and of its while-free fragment in GKAT. Moreover, to establish semantics for these structures four new algebras are defined: FSET(T), FREL(K,T) and FLANG(K,T) over complete residuated lattices K and T, and M (n, A) over a GKAT or I-GKAT A. As a final exercise, the paper discusses some program equivalence proofs in a graded context.

Abstract
Dynamic logic is a powerful framework for reasoning about imperative programs. This paper extends previous work [9] on the systematic generation of dynamic logics from the propositional to the equational case, to capture ‘full-fledged’ imperative programs. The generation process is parametric on a structure specifying a notion of ‘weight’ assigned to programs. The paper introduces also a notion of bisimilarity on models of the generated logics, which is shown to entail modal equivalence with respect to the latter.

Abstract
Although Finite-State Automata (FSA) have been successfully used in modeling and control of Discrete Event Systems (DESs), they are limited to represent complex and advanced features of DESs, such as context recognition and switching. The literature has suggested that a FSA can nevertheless be enriched with parameters properly collected from the modeled system, so that this favors design and control. A parameter can be embedded either on transitions or states. However, each approach is structured within a specific framework, so that their comparison and integration are not straightforward and they may lead to different control solutions, modeled, computed and implemented using distinct strategies. In this paper, we show how to combine advantages from parameters in modeling and control of DESs. Each approach is structured and their advantages are identified and exemplified. Then, we propose a conversion method that allows to translate a design-friendly model into a synthesis-efficient structure. Examples illustrate the approach.

Abstract
We construct functional encryption (FE) schemes for the orthogonality (OFE) relation where each ciphertext encrypts some vector (Formula Presented) and each decryption key, associated to some vector (Formula Presented), allows to determine if (Formula Presented) is orthogonal to (Formula Presented) or not. Motivated by compelling applications, we aim at schemes which are function hidding, i.e. (Formula Presented) is not leaked. Our main contribution are two such schemes, both rooted in existing constructions of FE for inner products (IPFE), i.e., where decryption keys reveal the inner product of (Formula Presented) and (Formula Presented). The first construction builds upon the very efficient IPFE by Kim et al. (SCN 2018) but just like the original scheme its security holds in the generic group model (GGM). The second scheme builds on recent developments in the construction of efficient IPFE schemes in the standard model and extends the work of Wee (TCC 2017) in leveraging these results for the construction of FE for Boolean functions. Conceptually, both our constructions can be seen as further evidence that shutting down leakage from inner product values to only a single bit for the orthogonality relation can be done with little overhead, not only in the GGM, but also in the standard model. We discuss potential applications of our constructions to secure databases and provide efficiency benchmarks. Our implementation shows that the first scheme is extremely fast and ready to be deployed in practical applications.

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