Computational complexity of the word problem in modal algebras

Приводится доказательство $\mathrm{PSPACE}$-полноты проблемы равенства слов в классе всех нуль-порождённых модальных алгебр, или, эквивалентно, проблемы равенства константных слов в классе всех модальных алгебр. Также рассматривается вопрос о сложности равенства слов в произвольном многообразии модальных алгебр. Доказывается, что уже проблема равенства константных слов в многообразии модальных алгебр может быть сколь угодно трудной (включая как классы сложности, так и степени неразрешимости). Показано, как построить соответствующие многообразия. The paper deals with the word problem for modal algebras. It is proved that, for the variety of all modal algebras, the word problem is $\mathrm{PSPACE}$-complete if only constant modal terms or only $0$-generated modal algebras are considered. We also consider the word problem for different varieties of modal algebras. It is proved that the word problem for a variety of modal algebras can be $C$-hard, for any complexity class or unsolvability degree $C$ containing a $C$-complete problem. It is shown how to construct such varieties.

Cut-free Sequent Calculus and Natural Deduction for the Tetravalent Modal Logic

The tetravalent modal logic ($${\mathcal {TML}}$$ TML ) is one of the two logics defined by Font and Rius (J Symb Log 65(2):481–518, 2000) (the other is the normal tetravalent modal logic$${{\mathcal {TML}}}^N$$ TML N ) in connection with Monteiro’s tetravalent modal algebras. These logics are expansions of the well-known Belnap–Dunn’s four-valued logic that combine a many-valued character (tetravalence) with a modal character. In fact, $${\mathcal {TML}}$$ TML is the logic that preserves degrees of truth with respect to tetravalent modal algebras. As Font and Rius observed, the connection between the logic $${\mathcal {TML}}$$ TML and the algebras is not so good as in $${{\mathcal {TML}}}^N$$ TML N , but, as a compensation, it has a better proof-theoretic behavior, since it has a strongly adequate Gentzen calculus (see Font and Rius in J Symb Log 65(2):481–518, 2000). In this work, we prove that the sequent calculus given by Font and Rius does not enjoy the cut-elimination property. Then, using a general method proposed by Avron et al. (Log Univ 1:41–69, 2006), we provide a sequent calculus for $${\mathcal {TML}}$$ TML with the cut-elimination property. Finally, inspired by the latter, we present a natural deduction system, sound and complete with respect to the tetravalent modal logic.

Ideal related algebras and their logics

We investigate modal algebras that generalize the unary discriminator into two directions related to an ideal of the algebra. It turns out that some classes lead to well-known logics, while others have not yet been explored.

COMPLETE ADDITIVITY AND MODAL INCOMPLETENESS

In this article, we tell a story about incompleteness in modal logic. The story weaves together an article of van Benthem (1979), “Syntactic aspects of modal incompleteness theorems,” and a longstanding open question: whether every normal modal logic can be characterized by a class ofcompletely additivemodal algebras, or as we call them,${\cal V}$-baos. Using a first-order reformulation of the property of complete additivity, we prove that the modal logic that starred in van Benthem’s article resolves the open question in the negative. In addition, for the case of bimodal logic, we show that there is a naturally occurring logic that is incomplete with respect to${\cal V}$-baos, namely the provability logic$GLB$(Japaridze, 1988; Boolos, 1993). We also show that even logics that are unsound with respect to such algebras do not have to be more complex than the classical propositional calculus. On the other hand, we observe that it is undecidable whether a syntactically defined logic is${\cal V}$-complete. After these results, we generalize the Blok Dichotomy (Blok, 1978) to degrees of${\cal V}$-incompleteness. In the end, we return to van Benthem’s theme of syntactic aspects of modal incompleteness.

VARIETIES OF POSITIVE MODAL ALGEBRAS AND STRUCTURAL COMPLETENESS

Positive modal algebras are the$$\left\langle { \wedge , \vee ,\diamondsuit ,\square,0,1} \right\rangle $$-subreducts of modal algebras. We prove that the variety of positive S4-algebras is not locally finite. On the other hand, the free one-generated positive S4-algebra is shown to be finite. Moreover, we describe the bottom part of the lattice of varieties of positive S4-algebras. Building on this, we characterize (passively, hereditarily) structurally complete varieties of positive K4-algebras.