Two-variable Logic with Counting and a Linear Order

Abstract

Abstract. We study the finite satisfiability problem for the two-variable fragment of first-order logic extended with counting quantifiers (C 2 ) and interpreted over linearly ordered structures. We show that the problem is undecidable in the case of two linear orders (in the presence of two other binary symbols). In the case of one linear order it is NExpTime-complete, even in the presence of the successor relation. Surprisingly, the complexity of the problem explodes when we add one binary symbol more: C 2 with one linear order and in the presence of other binary predicate symbols is equivalent, under elementary reductions, to the emptiness problem for multicounter automata.

Order-Invariance of Two-Variable Logic is Decidable

“…Only the case of finite ordered structures with at least three successor relations remains open. For ESO 2 it is only known that it is decidable on the class of finite ordered (<, S)-structures [10,21,4] and NExpTime-complete for (S 1 , S 2 )structures [3]. We note that the latter result combined with the observation from above establishes the decidability of successor-invariance.…”

“…Previously, only the satisfiability problem for EMSO 2 was known to be NExpTime-complete on this class of structures. The extension of ESO 2 with counting quantifiers was recently shown to be decidable on such structures, but only with very high complexity [4].…”

Finite Model Reasoning in Expressive Fragments of First-Order Logic

“…Moreover, adding transitivity axioms allows one to write sentences that have only infinite models (e.g. replacing the last conjunct in the formula (5) by the transitivity axiom (8)). Hence, the question therefore arises as to whether transitivity (or related properties like orderings or equivalence relations) could be added at reasonable computational cost.…”

“…[29]). An example of this phenomenon is the extension of C 2 with one linear order and one successor of a linear order augmented with an additional binary relation studied in [8].…”

Two-variable First-Order Logic with Counting in Forests

“…In the context of extensions of C 2 , proving general satisfiability usually is not an easy task. In fact, most such extensions, including [4,3,1], do not touch this problem; an exception is [11]. Thus, the contributions of the present paper are: (a) we lift the earlier restriction to forests of bounded rank; and (b) we consider general satisfiability for C 2 [↓] (thus allowing interpretation over a single, infinite forest).…”