A Dichotomy for Some Elementarily Generated Modal Logics

Abstract

In this paper we consider the normal modal logics of elementary classes defined by first-order formulas of the form ∀x0∃x1 . . . ∃xn xiR λ xj. We prove that many properties of these logics, such as finite axiomatisability, elementarity, axiomatisability by a set of canonical formulas or by a single generalised Sahlqvist formula, together with modal definability of the initial formula, either simultaneously hold or simultaneously do not hold. arXiv:1406.5700v2 [math.LO] 27 Feb 2015Briefly, we prove that for any class C in question, conditions (I-i) -(I-x) either simultaneously hold, or simultaneously do not hold, and this is determined by the existence in the corresponding diagram of an undirected cycle not passing through the universally quantified point, provided that the diagram is "minimal", i.e., none of its edges may be removed without affecting the corresponding formula, and "rooted", i.e., each of its points is reachable from x 0 via a directed path.We exclude from our list such algorithmical properties as decidability, finite model property and complexity, and do not deal with them in this paper, since an easy (but seemingly unpublished) argument shows that all logics in our class have f.m.p. and are PSPACE-complete regardless of the mentioned cycle. But we cannot help mentioning that the dichotomies in the complexity-theoretic setting have recently become known to the logical community. For example, in [13] the modal logics given by universal Horn sentences are classified into those that are in NP and those that are PSPACE-hard and this classification was further refined in [27]. The authors of [25] classified universal relational constraints with respect to the complexity of reasoning in the description logic EL.This work is in line with current research in theoretical modal logic. First, this result can be considered as a straighforward generalisation of Hughes' paper [16] about the reflexive-successor logic. The axiomatics of [16] was generalised in [1] to the case of first-order conditions of the form ∀x∃y(xR λ y ∧ φ(y)) where φ(y) is a generalised Kracht formula [18], and for some particular logics of this form finite axiomatisability, the finite model property and elementarity are studied there. The authors of [1] also conjectured that within their class there is a coincidence between finite axiomatisability and elementarity, and between ∆-elementarity and elementarity (cf. [2]).Another central problem of modal logic is: given an elementary class, i.e., a first-order formula, provide an explicit axiomatisation of the corresponding modal logic (this was done in [14]), and describe its properties, for example, in terms of (I-i)-(I-x) (cf. problems 6.6 and 6.8 ibid.) Since the product of two elementary classes is elementary [9], the school of many dimensional modal logic deals mainly with such problems (e.g., [23], [24], and [8] for older results). In general, the algorithmic problem 'given a firstorder formula, decide whether each of (I-i)-(I-x) holds' should be undecidable due to the undecidability ...

Non-finitely axiomatisable modal product logics with infinite canonical axiomatisations

“…, (u n , v j ), a contradiction, proving (30). (24). Similarly, by (28)-(29), for every 1 ≤ j ≤ ℓ there is some w j ∈ U such that w j ≠ u and M, (w j , v) ⊧b j , and so M, (u, v) ⊧ b j by (25)…”

“…So suppose (u, v) ∈ Z is neither a-type nor b-type. By ( 26)-( 27), for every 1 ≤ i ≤ k there is some z i ∈ V such that z i ≠ v and M, (u, z i ) ⊧ âi , and so M, (u, v) ⊧ a i by (24). Similarly, by ( 28)-( 29), for every 1 ≤ j ≤ ℓ there is some w j ∈ U such that w j ≠ u and M, (w j , v) ⊧ bj , and so M, (u, v) ⊧ b j by (25).…”

“…While any set of canonical formulas always axiomatises a canonical logic, Hodkinson and Venema [21] show that there are canonical logics that are barely canonical in the sense that every axiomatisation for such logics must contain infinitely many non-canonical axioms. Further examples of barely canonical elementarily generated logics are given in [11,3,24]. Kikot [24] also obtained the following general dichotomy result: If a class of Kripke frames is definable by first-order formulas of the form ∀x 0 ∃x 1 .…”

“…Further examples of barely canonical elementarily generated logics are given in [11,3,24]. Kikot [24] also obtained the following general dichotomy result: If a class of Kripke frames is definable by first-order formulas of the form ∀x 0 ∃x 1 . .…”

Canonical extensions and ultraproducts of polarities

“…The first examples of varieties with this property were given by Hodkinson and Venema [66], and include the variety RRA of representable relation algebras. Many more can be found in [51,8,77].…”