Modal Logic over Higher Dimensional Automata

Higher dimensional automata (HDA) are a model of concurrency [5, 8] that can express most of the traditional partial order models like Mazurkiewicz traces, pomsets, event structures, or Petri nets. Modal logics, interpreted over Kripke structures, are the logics for reasoning about sequential behavior and interleaved concurrency. Modal logic is a well behaved subset of first-order logic; many variants of modal logic are decidable. However, there are no modal-like logics for the more expressive HDA models. We introduce and develop a modal logic over HDAs which incorporates two modalities for reasoning about “during” and “after”. This general higher dimensional modal logic (HDML) is shown to be decidable and we provide a complete axiomatic system for it. With higher dimensional automata concurrent systems can be modeled at different levels of abstraction, not only as all possible interleavings of their concurrent actions. HDAs can model concurrent systems at any granularity level and make no assumptions about the durations of the actions. Moreover, HDAs are not constrained to only before-after modeling, but represent also the “during”. Work has been done on defining temporal logics over Mazurkiewicz traces [4], and strong results like decidability and expressive completeness are known [7]. For general partial orders, temporal logics usually become undecidable [1]. For the more expressive event structures there are fewer works on modal logics. There is hardly any work on logics for higher dimensional automata and, as far as we know, there is no work on modal logics for HDAs. In practice, modal logics, like temporal logics or dynamic logics, are desirable because they are generally decidable, as opposed to full first-order logic, which is undecidable. That is why we introduce and develop a logic in the style of standard modal logic which has HDAs as models, hence, the name higher dimensional modal logic (HDML). This is our language for talking about general models of concurrent systems. We prove that HDML is decidable using an extension of the filtration technique. We associate to HDML an axiomatic system which is proven to be sound and complete for HDAs. The proof of completeness is rather involved and is based on a constructive method of building the canonical model for a consistent formula. The construction starts with a minimal model and gradually enlarges it using two constructions (for lifting and enriching the model) that may be associated to the two different diamond operators of HDML.