We call a first-order formula one-dimensional if its every maximal block of existential (universal) quantifiers leaves at most one variable free. We consider the one-dimensional restrictions of the guarded fragment, GF, and the tri-guarded fragment, TGF, the latter being a recent extension of GF in which quantification for subformulas with at most two free variables need not be guarded, and which thus may be seen as a unification of GF and the two-variable fragment, FO2. We denote the resulting formalisms, resp., GF1, and TGF1. We show that GF1 has an exponential model property and NExpTime-complete satisfiability problem (that is, it is easier than full GF). For TGF1 we show that it is decidable, has the finite model property, and its satisfiability problem is TwoExpTime-complete (NExpTime-complete in the absence of equality). All the above-mentioned results are obtained for signatures with no constants. We finally discuss the impact of their addition, observing that constants do not spoil the decidability but increase the complexity of the satisfiability problem.
Answering Boolean conjunctive queries over the guarded fragment is decidable, however, as yet no practical decision procedure exists. Meanwhile, ordered resolution, as a practically oriented algorithm, is widely used in state-of-art modern theorem provers. In this paper, we devise a resolution decision procedure, which not only proves decidability of querying of the guarded fragment, but is implementable in any `off-the-shelf' resolution theorem prover with modest effort. Further, we extend the procedure to querying the loosely guarded fragment. The difficulty in querying a knowledge base of (…
In previous work, Abramsky, Dawar and Wang (LiCS 2017) and Abramsky and Shah (CSL 2018) have shown how a range of model comparison games which play a central role in finite model theory, including Ehrenfeucht-Fraisse, pebbling, and bisimulation games, can be captured in terms of resource-indexed comonads on the category of relational structures. Moreover, the coalgebras for these comonads capture important combinatorial parameters such as tree-width and tree-depth. The present paper extends this analysis to quantifier-guarded fragments of first-order logic. We give a systematic account, coveri…
In this paper we prove that the uniform one-dimensional guarded fragment, which is a natural polyadic generalization of the guarded two-variable logic, has the Craig interpolation property. We will also prove that the satisfiability problem of uniform guarded fragment is NEXPTIME-complete.
We develop a doubly-exponential decision procedure for the satisfiability problem of guarded separation logic -- a novel fragment of separation logic featuring user-supplied inductive predicates, Boolean connectives, and separating connectives, including restricted (guarded) versions of negation, magic wand, and septraction. Moreover, we show that dropping the guards for any of the above connectives leads to an undecidable fragment. We further apply our decision procedure to reason about entailments in the popular symbolic heap fragment of separation logic. In particular, we obtain a doubly-ex…
In logics with the Craig interpolation property (CIP) the existence of an interpolant for an implication follows from the validity of the implication. In logics with the projective Beth definability property (PBDP), the existence of an explicit definition of a relation follows from the validity of a formula expressing its implicit definability. The two-variable fragment, FO2, and the guarded fragment, GF, of first-order logic both fail to have the CIP and the PBDP. We show that nevertheless in both fragments the existence of interpolants and explicit definitions is decidable. In GF, both probl…
An important class of decidable first-order logic fragments are those satisfying a guardedness condition, such as the guarded fragment (GF). Usually, decidability for these logics is closely linked to the tree-like model property - the fact that satisfying models can be taken to have tree-like form. Decision procedures for the guarded fragment based on the tree-like model property are difficult to implement. An alternative approach, based on restricting first-order resolution, has been proposed, and this shows more promise from the point of view of implementation. In this work, we connect the …
Guarded Kleene Algebra with Tests (GKAT) is an efficient fragment of KAT, as it allows for almost linear decidability of equivalence. In this paper, we study the (co)algebraic properties of GKAT. Our initial focus is on the fragment that can distinguish between unsuccessful programs performing different actions, by omitting the so-called early termination axiom. We develop an operational (coalgebraic) and denotational (algebraic) semantics and show that they coincide. We then characterize the behaviors of GKAT expressions in this semantics, leading to a coequation that captures the covariety o…
Institute of Computer Science University of Wrocław（弗罗茨瓦夫大学计算机科学研究所）；Technische Universität Dresden（德累斯顿工业大学）
The Triguarded Fragment (TGF) is among the most expressive decidable fragments of first-order logic, subsuming both its two-variable and guarded fragments without equality. We show that the TGF has the finite model property (providing a tight doubly exponential bound on the model size) and hence finite satisfiability coincides with satisfiability known to be N2ExpTime-complete. Using similar constructions, we also establish 2ExpTime-completeness for finite satisfiability of the constant-free (tri)guarded fragment with transitive guards.
