Guarded Fragment with Regular Guards (RGF)

• RGF is a first-order logic that extends the guarded fragment by allowing guards defined through regular path expressions from ICPDL, thereby unifying various constrained-guard extensions.
• It maintains the guarded quantification discipline while enabling the simulation of transitive, equivalence, and closure guards through regular programs such as Kleene plus and star.
• The satisfiability problem for RGF is 2EXPTIME-complete, with a maximal EXPSPACE fragment available, though query entailment remains undecidable, highlighting both its expressive power and limitations.

Searching arXiv for the specified paper to ground the article and citation. Guarded Fragment with Regular Guards (RGF) is a first-order logic that extends the classical Guarded Fragment (GF) by allowing guards to be defined not only by atomic relations, but by regular path expressions from Propositional Dynamic Logic with Intersection and Converse (ICPDL). In this sense, RGF combines the expressive power of GF with ICPDL, while preserving the guarded discipline. It uniformly generalizes many previously studied extensions of GF, including transitive guards, equivalence guards, transitive closure, equivalence closure, and conjunctions of such constraints. The central result is that the satisfiability problem for RGF is 2EXPTIME-complete, showing that the combination is not harder than ICPDL or GF, despite the increased expressivity (Bednarczyk et al., 11 Sep 2025).

1. Formal language and semantics

RGF is defined over a vocabulary partitioned as $\Sigma \cup R$, where $\Sigma$ is an equality- and constant-free relational signature with predicates of arbitrary arity, and $R$ is a set of binary “regular predicates.” A crucial restriction is that regular predicates appear only in guards. Variables are $x_1, x_2, \ldots$, with standard first-order semantics.

The guard language is based on ICPDL programs. Programs $\pi$ and $\rho$ are generated by

$$\pi,\rho \Coloneqq B \mid \bar{B} \mid \pi \circ \rho \mid \pi \cup \rho \mid \pi \cap \rho \mid \pi^{\ast} \mid \pi^{+} \mid \varphi{?},$$

where $B \in R$, $\bar{B}$ is converse, $\circ$ is concatenation, $\Sigma$0 is union, $\Sigma$1 is intersection, $\Sigma$2 is Kleene star, $\Sigma$3 is one-or-more, and $\Sigma$4 is a test on a unary $\Sigma$5-predicate or on a formula with one free variable. In a structure $\Sigma$6, each program is interpreted as a binary relation $\Sigma$7. Tests and atomic relations are interpreted by

$\Sigma$8

and by the given interpretation of $\Sigma$9. Converse, composition, union, intersection, star, and plus are interpreted by the standard relational clauses, with

$R$0

An RGF-guard $R$1 for a subformula $R$2 is either an atomic $R$3-atom or an expression of the form $R$4, where $R$5 is an RGF-program. The free variables of the guard must include all free variables of the guarded subformula. The grammar of RGF formulas is

$R$6

where $R$7 and $R$8 is an RGF-guard. Universal quantification and the remaining Boolean connectives are defined in the usual way.

The semantics follows first-order logic, except that guarded quantification is restricted to tuples satisfying the guard. Thus, a formula such as

$R$9

quantifies only over those $x_1, x_2, \ldots$0 that are reachable from the current $x_1, x_2, \ldots$1 via $x_1, x_2, \ldots$2, and the universal counterpart behaves dually. When the guard is atomic, this recovers the classical GF pattern. When the guard is $x_1, x_2, \ldots$3, quantification is controlled by a binary relation induced by a regular path expression. This is the defining shift from atomic guards to regular guards (Bednarczyk et al., 11 Sep 2025).

2. Relation to GF, PDL, and constrained-guard formalisms

RGF strictly extends GF by replacing atomic guards with relational guards induced by path expressions. This yields a uniform account of a range of previously studied semantically constrained guards.

Transitive guards can be simulated by introducing a fresh binary relation $x_1, x_2, \ldots$4 and using $x_1, x_2, \ldots$5 as the guard program, so that $x_1, x_2, \ldots$6. Equivalence guards can be simulated by

$x_1, x_2, \ldots$7

again for a fresh $x_1, x_2, \ldots$8. Transitive closure of a binary guard $x_1, x_2, \ldots$9 is expressed by choosing $\pi$0 to be $\pi$1 or $\pi$2. Conjunctions of transitive guards are captured by program intersection, for example

$\pi$3

The framework therefore uniformly captures guarded transitive and guarded equivalence closure constructions that had previously been studied in more restricted two-variable settings.

Compared to GF, the guarded nature of quantification is preserved: all quantification remains syntactically guarded. Compared to ICPDL, RGF works over arbitrary arity $\pi$4-predicates and has full first-order syntax, while using ICPDL only in the guard component. A notable consequence is that satisfiability remains 2EXPTIME-complete, the same upper complexity class as GF and ICPDL. The paper also notes that the finite model property is delicate and fails for certain forward fragments (Bednarczyk et al., 11 Sep 2025).

This suggests that the significance of RGF lies less in introducing a new complexity class than in unifying disparate constrained-guard extensions under a single guard mechanism.

3. Illustrative expressive patterns

The expressive gain over plain GF is visible already in simple formulas. A transitive closure guard can be used in

$\pi$5

This requires every $\pi$6-node to have a witness reachable by the transitive closure of $\pi$7. Plain GF cannot use $\pi$8 as a guard.

Intersection and converse can appear together in a single regular guard: $\pi$9 Here the witness $\rho$0 must be both reachable via $\rho$1 and backward-reachable along $\rho$2.

Equivalence closure is expressible by taking

$\rho$3

A guarded sentence of the form

$\rho$4

then enforces invariance of the unary property $\rho$5 on the $\rho$6-equivalence classes. The source explicitly notes that this is not definable in plain GF, which can only guard by atomic $\rho$7.

A further example of a property definable in RGF but not in GF is

$\rho$8

where the universal quantifier is guarded by $\rho$9. The examples make clear that the increase in expressivity is tied specifically to the ability to define admissible quantification domains by regular path relations, rather than by atomic predicates alone (Bednarczyk et al., 11 Sep 2025).

4. Satisfiability, normal forms, and upper-bound techniques

Any RGF sentence can be nondeterministically transformed in polynomial time into a guarded normal form that preserves the number of variables, forwardness, and the set of program operators. The normalized sentence has the shape

$$\pi,\rho \Coloneqq B \mid \bar{B} \mid \pi \circ \rho \mid \pi \cup \rho \mid \pi \cap \rho \mid \pi^{\ast} \mid \pi^{+} \mid \varphi{?},$$0

where each $$\pi,\rho \Coloneqq B \mid \bar{B} \mid \pi \circ \rho \mid \pi \cup \rho \mid \pi \cap \rho \mid \pi^{\ast} \mid \pi^{+} \mid \varphi{?},$$1 is of one of several schematic guarded forms, including existential witness clauses, universal local constraints, and explicit two-variable variants. This Scott-like normalization is a central preparatory step for the complexity analysis (Bednarczyk et al., 11 Sep 2025).

The main theorem states that $$\pi,\rho \Coloneqq B \mid \bar{B} \mid \pi \circ \rho \mid \pi \cup \rho \mid \pi \cap \rho \mid \pi^{\ast} \mid \pi^{+} \mid \varphi{?},$$2 is 2EXPTIME-complete. The lower bound follows from ICPDL and from $$\pi,\rho \Coloneqq B \mid \bar{B} \mid \pi \circ \rho \mid \pi \cup \rho \mid \pi \cap \rho \mid \pi^{\ast} \mid \pi^{+} \mid \varphi{?},$$3 with transitive guards. The upper bound proceeds in two stages.

For the two-variable fragment in normal form, the proof begins with a sparse model lemma: if $$\pi,\rho \Coloneqq B \mid \bar{B} \mid \pi \circ \rho \mid \pi \cup \rho \mid \pi \cap \rho \mid \pi^{\ast} \mid \pi^{+} \mid \varphi{?},$$4 is satisfiable, then it has a model $$\pi,\rho \Coloneqq B \mid \bar{B} \mid \pi \circ \rho \mid \pi \cup \rho \mid \pi \cap \rho \mid \pi^{\ast} \mid \pi^{+} \mid \varphi{?},$$5 realizing at most $$\pi,\rho \Coloneqq B \mid \bar{B} \mid \pi \circ \rho \mid \pi \cup \rho \mid \pi \cap \rho \mid \pi^{\ast} \mid \pi^{+} \mid \varphi{?},$$6 many FO 2-types, while preserving all 1-types and introducing no new 2-types. On this basis, the construction defines a “$$\pi,\rho \Coloneqq B \mid \bar{B} \mid \pi \circ \rho \mid \pi \cup \rho \mid \pi \cap \rho \mid \pi^{\ast} \mid \pi^{+} \mid \varphi{?},$$7-table” $$\pi,\rho \Coloneqq B \mid \bar{B} \mid \pi \circ \rho \mid \pi \cup \rho \mid \pi \cap \rho \mid \pi^{\ast} \mid \pi^{+} \mid \varphi{?},$$8 mapping triples $$\pi,\rho \Coloneqq B \mid \bar{B} \mid \pi \circ \rho \mid \pi \cup \rho \mid \pi \cap \rho \mid \pi^{\ast} \mid \pi^{+} \mid \varphi{?},$$9—two 1-types and a set $B \in R$0 of guards—to a set of feasible 2-types, under size, closure, compatibility, and guard-satisfaction constraints. Each model induces such a table.

From $B \in R$1, one constructs an ICPDL sentence

$B \in R$2

whose models are type-labeled graphs with predicates $B \in R$3 and $B \in R$4 for 1-types and 2-types. The clauses $B \in R$5–$B \in R$6 enforce uniqueness of types, existence of witnesses, prohibition of unsupported 2-types, and availability of required 2-types. The modality

$B \in R$7

ensures connectedness of the ICPDL model. Although $B \in R$8 is exponential in $B \in R$9, its intersection width is polynomial, bounded by $\bar{B}$0, and satisfiability of $\bar{B}$1 is therefore decidable in 2EXPTIME using the ICPDL algorithm. The correctness theorem states that $\bar{B}$2 is satisfiable if and only if there exists a $\bar{B}$3-table $\bar{B}$4 such that $\bar{B}$5 is satisfiable.

The general multi-variable case is reduced to the two-variable case by a fusion argument. The sentence is decomposed into a GF component $\bar{B}$6, which fixes chosen sets of realized 1-types and 2-types restricted to $\bar{B}$7-atoms, and a two-variable RGF component $\bar{B}$8, which enforces that 2-types realized under guards belong to the chosen set. Satisfiability of the original sentence is equivalent to the existence of FO-compatible sparse sets for which both components are satisfiable. The models are then fused by arranging copies in a two-dimensional grid and adding guarded witnesses by a circular scheme. The method relies critically on the fact that $\bar{B}$9-predicates occur only in guards. Enumerating the relevant type sets and invoking the 2EXPTIME procedures for GF and for the two-variable RGF fragment yields a 2EXPTIME algorithm overall (Bednarczyk et al., 11 Sep 2025).

5. Finite satisfiability, query entailment, and undecidability

Finite satisfiability for ICPDL is open, and the finite satisfiability result for RGF is therefore conditional: if $\circ$0 is decidable and 2EXPTIME-complete, then so is $\circ$1. The finite model property is not available in general. Several forward fragments are shown to lack it, namely

$\circ$2

The proofs use formulas that force infinite chains while forbidding the loops that would be needed for finite realization (Bednarczyk et al., 11 Sep 2025).

For query entailment, the paper considers knowledge bases $\circ$3 and conjunctive queries $\circ$4, asking whether $\circ$5 in the finite or unrestricted setting. For tree-shaped conjunctive queries, entailment reduces to satisfiability by rolling-up.

The main negative result is stronger: conjunctive-query entailment is undecidable, for both finite and unrestricted semantics, already for the fluted two-variable GF fragment with a single transitive guard. The source emphasizes that this strengthens earlier results of Gottlob–Pieris–Tendera in three ways: the logic is more restricted, the queries are pure CQs rather than UCQs, and the result also applies to finite entailment. The proof uses “snake” grid structures in triangular boustrophedon order, with a transitive guard $\circ$6 and binary grid relations $\circ$7 and $\circ$8. Propagation and consistency axioms are encoded in guarded form, and the octant tiling problem, both finite and infinite, is reduced to entailment via a conjunctive query that closes the grid by ensuring missing $\circ$9-edges. Homomorphism arguments are used to inject snakes into models of the knowledge base (Bednarczyk et al., 11 Sep 2025).

A plausible implication is that the guarded use of regular path constraints preserves decidability for satisfiability, but does not in itself protect query answering once unrestricted conjunctive-query homomorphisms are admitted.

6. Maximal EXPSPACE fragment, translations, and broader connections

The paper identifies a largest fragment of RGF below 2EXPTIME in a natural sense: the forward fragment

$\Sigma$00

Its satisfiability problem is EXPSPACE-complete. This fragment is forward index-normal, meaning that quantifier order matches variable order, and its guards are one-way, with no converse, while allowing plus, star, and tests. The upper bound is obtained by showing that satisfiable sentences admit proper regular tree-shaped models with short witnessing paths, using links, half-links, and unravelling, and then verifying such models in alternating exponential time, hence EXPSPACE. The matching maximality statement is that adding any further operator to Kleene plus—composition $\Sigma$01, union $\Sigma$02, or intersection $\Sigma$03—already yields 2EXPTIME-hardness (Bednarczyk et al., 11 Sep 2025).

Beyond complexity classification, RGF has explicit connections to several adjacent areas. In database theory and ontology-based data access, guarded first-order logic over higher-arity $\Sigma$04-predicates with regular path guards aligns with settings where tree-shaped conjunctive queries can be rolled up to RGF sentences in 2EXPTIME. In description logics, RGF uniformly encodes systems with regular roles, including $\Sigma$05, $\Sigma$06, and higher-arity $\Sigma$07, by standard translations, and it also embeds the guard component of ICPDL. The undecidability transfer yields that CQ entailment is undecidable for $\Sigma$08 extended with intersection existential restrictions $\Sigma$09, role inclusions $\Sigma$10, and a single transitivity axiom. In modal and dynamic logic, the significance is that guards are themselves ICPDL programs, which are standard objects in program verification.

The broader complexity-theoretic message is explicit: RGF combines GF and ICPDL without exceeding their shared 2EXPTIME satisfiability bound. This consolidates a collection of decidability results that had been known for more specialized two-variable constrained-guard logics, and lifts them to full guarded first-order logic without equality (Bednarczyk et al., 11 Sep 2025).

7. Limitations and open problems

Several limitations remain explicit. Finite satisfiability is only conditionally decidable because decidability of finite satisfiability for ICPDL, already for LoopPDL, has remained open for decades. Several forward fragments fail to have the finite model property, while the status of other fragments, including $\Sigma$11 and $\Sigma$12, is left open.

The boundary between the EXPSPACE fragment and the full 2EXPTIME behavior is only partially charted. Adding converse or composition to the EXPSPACE fragment raises complexity, but the identification of further robust fragments that retain lower complexity while preserving useful expressive power remains open. The source also isolates a number of model-theoretic questions, including axiomatizability, interpolation, and preservation theorems under regular guards, and proposes extending the guard formalism beyond binary relations to regular guards for higher-arity relations. For query answering, the undecidability results leave open the task of characterizing decidable subclasses for conjunctive-query entailment in the presence of regular guards (Bednarczyk et al., 11 Sep 2025).

Taken together, these open problems delineate RGF as a logic with a sharply analyzed satisfiability theory, strong negative results for query entailment, and a substantial but still incomplete map of its finite-model and model-theoretic behavior.