Order-Invariance of Two-Variable Logic is Decidable

Abstract

It is shown that order-invariance of two-variable first-logic is decidable in the finite. This is an immediate consequence of a decision procedure obtained for the finite satisfiability problem for existential second-order logic with two first-order variables (ESO 2 ) on structures with two linear orders and one induced successor. We also show that finite satisfiability is decidable on structures with two successors and one induced linear order. In both cases, so far only decidability for monadic ESO 2 has been known. In addition, the finite satisfiability problem for ESO 2 on structures with one linear order and its induced successor relation is shown to be decidable in non-deterministic exponential time.

Finite Satisfiability of the Two-Variable Guarded Fragment with Transitive Guards and Related Variants

“…undecidable [Kieroński et al 2014] 1 linear order ( * ) NEXPTIME-complete [Otto 2001] 2 linear orders ( * ) Sat: ? FinSat in 2-NEXPTIME [Zeume and Harwath 2016] 3 linear orders undecidable [Kieroński 2011] The upper bounds in the results marked with ( * ) are actually obtained for full FO 2 . The corresponding lower bounds, if present, hold for GF 2 .…”

“…undecidable [Kieroński et al 2014] 1 linear order ( * ) NEXPTIME-complete [Otto 2001] 2 linear orders ( * ) Sat: ? FinSat in 2-NEXPTIME [Zeume and Harwath 2016] 3 linear orders undecidable [Kieroński 2011]…”

“…Namely, the following fragments are undecidable for both finite and unrestricted satisfiability: GF 2 with two transitive relations [Kieroński 2005], GF 2 with one transitive relation and one equivalence relation [Kieroński et al 2014], GF 2 with three equivalence relations [Kieroński and Otto 2012], GF 2 with three linear orders [Kieroński 2011]. Decidability of the satisfiability and finite satisfiability problems for many of the remaining cases has been recently established involving several papers not mentioned above [Otto 2001;Schwentick and Zeume 2012;Manuel and Zeume 2013;Bojańczyk et al 2011;Kieroński et al 2014;Zeume and Harwath 2016;Kieroński et al 2017].…”

Finite satisfiability for two‐variable, first‐order logic with one transitive relation is decidable

“…Stronger complexity‐theoretic upper bounds are available when the distinguished predicates are required to be interpreted as linear orders: the satisfiability and finite satisfiability problems for together with one linear order are both NExpTime ‐complete ; the finite satisfiability problem for together with two linear orders is in 2 ‐NExpTime (falling to ExpSpace when all non‐navigational predicates are unary ); with three linear orders, satisfiability and finite satisfiability are both undecidable . Also somewhat related to is the propositional modal logic known as navigational XPATH , which features a signature of proposition letters interpreted over vertices of some finite, ordered tree, together with modal operators giving access to vertices standing in the relations of daughter and next‐sister , as well as their transitive closures.…”

Extending Two-Variable Logic on Trees

“…This lead to extensive studies of FO 2 over various classes of structures, where some distinguished relational symbols are interpreted in a special way, e.g., as equivalences or linear orders. The finite satisfiability problem for FO 2 remains decidable over structures where one [17] or two relation symbols [18] are interpreted as equivalence relations; where one [21] or two relations are interpreted as linear orders [25,27]; where two relations are interpreted as successors of two linear orders [19,11,8]; where one relation is interpreted as linear order, one as its successor and another one as equivalence [3]; where one relation is transitive [26]; where an equivalence closure can be applied to two binary predicates [16]; where deterministic transitive closure can be applied to one binary relation [6]. It is known that the finite satisfiability problem is undecidable for FO 2 with two transitive relations [15], with three equivalence relations [17], with one transitive and one equivalence relation [18], with three linear orders [14], with two linear orders and their two corresponding successors [19].…”