Abstract. We consider two systems of constructive modal logic which are computationally motivated. Their modalities admit several computational interpretations and are used to capture intensional features such as notions of computation, constraints, concurrency, etc. Both systems have so far been studied mainly from type-theoretic and category-theoretic perspectives, but Kripke models for similar systems were studied independently. Here we bring these threads together and prove duality results which show how to relate Kripke models to algebraic models and these in turn to the appropriate categorical models for these logics.

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Synchronous languages ensure deterministic concurrency, but at the price of heavy restrictions on what programs are considered valid, or constructive. Meanwhile, sequential languages such as C and Java offer an intuitive, familiar programming paradigm but provide no guarantees with regard to deterministic concurrency. The sequentially constructive model of computation (SC MoC) presented here harnesses the synchronous execution model to achieve deterministic concurrency while addressing concerns that synchronous languages are unnecessarily restrictive and difficult to adopt.In essence, the SC MoC extends the classical synchronous MoC by allowing variables to be read and written in any order as long as sequentiality expressed in the program provides sufficient scheduling information to rule out race conditions. This allows to use programming patterns familiar from sequential programming, such as testing and later setting the value of a variable, which are forbidden in the standard synchronous MoC. The SC MoC is a conservative extension in that programs considered constructive in the common synchronous MoC are also SC and retain the same semantics. In this paper, we identify classes of variable accesses, define sequential constructiveness based on the concept of SC-admissible scheduling, and present a priority-based scheduling algorithm for analyzing and compiling SC programs.

We classify gate level circuits with cycles based on their stabilization behavior. We define a formal class of combinational circuits, the constructive circuits, for which signals settle to a unique value in bounded time, for any input, under a simple conservative delay model, called the up-bounded non-inertial (UN) delay. Since circuits with combinational cycles can exhibit asynchronous behavior, such as non-determinism or metastability, it is crucial to ground their analysis in a formal delay model, which previous work in this area did not do.We prove that ternary simulation, such as the practical algorithm proposed by Malik, decides the class of constructive circuits. We prove that three-valued algebra is able to maintain correct and exact stabilization information under the UN-delay model, and thus provides an adequate electrical interpretation of Malik's algorithm, which has been missing in the literature. Previous work on combinational circuits used the upbounded inertial (UI) delay to justify ternary simulation. We show that the match is not exact and that stabilization under the UI-model, in general, cannot be decided by ternary simulation. We argue for the superiority of the UN-model for reasons of complexity, compositionality and electrical adequacy. The UN-model, in contrast to the UI-model, is consistent with the hypothesis that physical mechanisms cannot implement non-deterministic choice in bounded time.As the corner-stone of our main results we introduce UN-Logic, an axiomatic specification language for UN-delay circuits that mediates between the real-time behavior and its abstract simulation in the ternary domain. We present a symbolic simulation calculus for M. Mendler ( ) Faculty 284 Form Methods Syst Des (2012) 40:283-329 circuit theories expressed in UN-logic and prove it sound and complete for the UN-model. This provides, for the first time, a correctness and exactness result for the timing analysis of cyclic circuits. Our algorithm is a timed extension of Malik's pure ternary algorithm and closely related to the timed algorithm proposed by Riedel and Bruck, which however was not formally linked with real-time execution models.

The clinical symptoms of diabetic neuropathy (DN) manifest in a time dependent manner as a positive symptoms (i. e. pain, hypersensitivity, tingling, cramps, cold feet etc.) during its early stages and by a loss of function (i. e. loss of sensory perception, delayed wound healing etc.) predominating in the later stages. Elevated blood glucose alone cannot explain the development and progression of DN and the lowering of blood glucose is insufficient in preventing and/or reversing neuropathy in patients with type 2 diabetes. Recently it has been shown that the endogenous reactive metabolite methylglyoxal (MG) can contribute to the gain of function via post-translational modification in DN of neuronal ion channels involved in chemosensing and action potential generation in nociceptive nerve endings. Dicarbonyls, such as MG, that are elevated in diabetic patients, modify DNA as well as extra- and intracellular proteins, leading to the formation of advanced glycation endproducts (AGEs). Increased formation of AGEs leads to increased cellular stress, dysfunction and ultimately cell death. The interaction of AGE-modified proteins through cell surface receptors, such as RAGE, can lead to increased cellular activation and sustained inflammatory responses, which are the molecular hallmarks of the later, degenerative, stages of DN. The direct and indirect effects of dicarbonyls on nerves or neuronal microvascular network provides a unifying mechanism for the development and progression of DN. Targeting the accumulation of MG and/or prevention of RAGE interactions may therefore provide new, more effective, therapeutic approaches for the treatment of DN.