A note on strictly positive logics and word rewriting systems

We establish a natural translation from word rewriting systems to strictly positive polymodal logics. Thereby, the latter can be considered as a generalization of the former. As a corollary we obtain examples of undecidable strictly positive normal modal logics. The translation has its counterpart on the level of proofs: we formulate a natural deep inference proof system for strictly positive logics generalizing derivations in word rewriting systems. We also formulate some open questions related to the theory of modal companions of superintuitionistic logics that was initiated by L.L. Maximova and V.V. Rybakov. In this note we study the fragment of polymodal logic consisting of implications of the form A → B, where A and B are formulas built-up from ⊤ and propositional variables using just ∧ and the diamond modalities. We call such formulas A and B strictly positive and will often omit the word ‘strictly.’ The interest towards such weak logics independently emerged within two different disciplines: provability logic and description logic (see [5, 3, 1]). In both cases, it was observed that the strictly positive language combines simplicity and efficiency while retaining a substantial amount of expressive power of modal logic. Thus, strictly positive fragments of many standard modal logics are polytime decidable. The positive fragment of the (Kripke incomplete) polymodal provability logic GLP is both polytime decidable and complete w.r.t. a natural class of finite Kripke frames [3]. The positive variable-free fragment of this logic gives rise to a natural ordinal notation system up to the ordinal ε0 and allows for a proof-theoretic analysis of Peano arithmetic [1]. In the present paper we study some general questions related to strictly positive logics. In particular, we establish a link between proof systems for strictly positive logics and the standard word rewriting (semi-Thue) systems. 1 Strictly positive logics Consider a modal language LΣ with propositional variables p, q,. . . , a constant ⊤, conjunction ∧, and a possibly infinite family of symbols Σ = {ai : i ∈ I} ∗Steklov Mathematical Institute, RAS; Moscow M.V. Lomonosov State University; National Research University Higher School of Economics; email: [email protected]. Supported by Russian Foundation for Basic Research project 15–01–09218a.