Monotone Precedence Formulas

• Monotone precedence formulas are defined as expressions whose semantics depends on a background order with precedence-like relations subject to a monotonicity discipline, often via interval-preservation.
• They bridge two research traditions: monadic first-order logic over linear orders and operator-precedence temporal logics like POTL and OPTL, each employing distinct structural approaches.
• These formalisms support advanced verification methods, including automata-based compilation and SMT-based model checking, while ensuring a three-variable collapse and robust expressiveness.

“Monotone Precedence Formulas” is not a standard formalism explicitly introduced under that name in the cited literature. The nearest established frameworks are, first, monadic first-order formulas over linear orders equipped with binary relations satisfying an order-monotonicity condition called interval-preservation, and, second, temporal logics interpreted over operator-precedence structures, notably POTL and OPTL, whose modalities navigate chain relations induced by an operator precedence matrix. As an Editor’s term, “monotone precedence formulas” can therefore denote formulas whose semantics depends on a background order together with precedence-like relations constrained by a monotonicity discipline, while the cited work also shows that “monotone” can mean something different—semantic monotonicity versus syntactic positivity—in modal logic (Fortin, 2019, Chiari et al., 2024, Chiari et al., 2018, Dvorkin, 2 Feb 2026).

1. Terminological scope and adjacent formalisms

Two distinct research traditions are closest to the topic.

In the first, the underlying semantic object is a linear order $(A,\le)$ enriched with unary predicates and designated binary relations $R\subseteq A\times A$. The relevant monotonicity notion is not called “precedence” in the source paper, but interval-preserving relation: for every interval $I\subseteq A$, the image $R(I)$ is an interval of $(R(A),\le)$, and for every interval $J\subseteq A$, the inverse image $R^{-1}(J)$ is an interval of $(R^{-1}(A),\le)$. This setting directly captures many precedence-like predicates whose behavior is compatible with the ambient order (Fortin, 2019).

In the second, the underlying objects are operator-precedence words. Here precedence is given by an operator precedence matrix

$$M : (\Sigma \cup \{\#\})^2 \to \{\lessdot, \doteq, \gtrdot\},$$

where $a\lessdot b$ means “yields precedence,” $R\subseteq A\times A$0 means “equal in precedence,” and $R\subseteq A\times A$1 means “takes precedence.” The cited work emphasizes that these are not order relations despite the notation; rather, they induce a hidden nesting structure through chains. The associated logical formalisms are POTL and OPTL, which are formulas interpreted over precedence-structured words rather than over plain linear orders (Chiari et al., 2024, Chiari et al., 2018).

A useful distinction follows. In the linear-order setting, “monotone precedence” concerns binary relations constrained by interval geometry. In the operator-precedence setting, “precedence formulas” are formulas whose modalities explicitly follow precedence-induced chains and summary or hierarchical paths. The literature therefore supports the topic through a family of closely related formalisms rather than a single named theory.

2. Linear orders with interval-preserving relations

The most direct order-theoretic framework studies structures of the form

$R\subseteq A\times A$2

where $R\subseteq A\times A$3 is a linear order, each $R\subseteq A\times A$4 is a designated binary relation, and each $R\subseteq A\times A$5 is a monadic predicate. The language is monadic first-order logic with equality, order, unary predicates, and the binary predicates $R\subseteq A\times A$6: $R\subseteq A\times A$7

The key semantic restriction is interval-preservation. Operationally, the first half of the definition can be read as a betweenness condition: if $R\subseteq A\times A$8 and $R\subseteq A\times A$9 with $I\subseteq A$0, and $I\subseteq A$1 lies between $I\subseteq A$2 and $I\subseteq A$3, then whenever $I\subseteq A$4 has some preimage under $I\subseteq A$5, it has a preimage inside $I\subseteq A$6. The converse condition is imposed symmetrically on $I\subseteq A$7. This is the precise order-monotonicity discipline under which precedence-like relations remain tame (Fortin, 2019).

Several examples in the source material are especially close to precedence formulas. If $I\subseteq A$8 is a partial function that is increasing or decreasing, then its graph

$I\subseteq A$9

is interval-preserving. Likewise, for a temporal structure $R(I)$0 and propositions $R(I)$1,

$R(I)$2

is interval-preserving. These examples show that monotone “jumps,” future witnesses constrained by all intermediate points, and order-compatible reachability relations all fall inside the framework.

The same paper also states the main limitation sharply. It does not define arbitrary precedence relations, and the positive theory depends critically on interval-preservation. The introduction contrasts this with failure phenomena for unrestricted binary structure: adding one unrestricted equivalence relation can kill the $R(I)$3-variable property, and linearly ordered graphs do not have the $R(I)$4-variable property for any $R(I)$5. A plausible implication is that “monotone precedence formulas” are mathematically robust only when their precedence predicates satisfy a strong order-compatibility condition.

3. Expressive collapse, star-free dynamic logic, and low-variable normal forms

The central expressiveness theorem for the linear-order setting is that monadic first-order logic has the three-variable property over linear orders with interval-preserving binary relations. The abstract states: “We show that over the class of linear orders with additional binary relations satisfying some monotonicity conditions, monadic first-order logic has the three-variable property,” and the formal statement is

$R(I)$6

The title abbreviates this as FO = FO$R(I)$7 over the class under consideration (Fortin, 2019).

The proof factors through a star-free variant of Propositional Dynamic Logic with converse. Its full syntax is: $R(I)$8 for state formulas, and

$R(I)$9

for path formulas. The interval-preserving fragment restricts path formulas to

$(R(A),\le)$0

where $(R(A),\le)$1 are structured complement-like cases replacing arbitrary complement. The paper proves that all path formulas in this restricted fragment remain interval-preserving.

The main translation theorem states that every $(R(A),\le)$2 formula with at least one free variable is equivalent to a positive boolean combination of formulas of the form

$(R(A),\le)$3

where $(R(A),\le)$4 and $(R(A),\le)$5 lies in the restricted PDL fragment. This yields two important normal-form consequences: one-free-variable formulas correspond to PDL state formulas, and two-free-variable formulas correspond to PDL path formulas. Since PDL formulas translate back into first-order formulas using at most three variable names, the collapse to $(R(A),\le)$6 follows.

The combinatorial engine is a Helly-type fact for intervals in a linear order: $(R(A),\le)$7 Applied to interval images of path formulas, this turns quantified witness conditions into pairwise relational constraints. The quantified normal form

$(R(A),\le)$8

is replaced by a positive boolean combination of pairwise formulas $(R(A),\le)$9. For precedence-oriented reasoning, this is a direct methodological template: represent atomic precedence predicates by interval-preserving relations, preserve interval-preservation under converse, composition, and intersection, translate into star-free path formulas, and conclude a three-variable collapse.

4. Operator-precedence temporal logics: OPTL and POTL

The operator-precedence tradition approaches precedence formulas through temporal logic rather than first-order collapse. An operator precedence automaton is

$J\subseteq A$0

where $J\subseteq A$1 is an operator precedence alphabet, $J\subseteq A$2 is a finite state set, $J\subseteq A$3 initial states, $J\subseteq A$4 final states, and $J\subseteq A$5 is partitioned into push, shift, and pop transitions. The induced structural object is the chain. A simple chain is written

$J\subseteq A$6

with

$J\subseteq A$7

and composed chains recursively nest chains inside chain bodies. The chain relation is the basis for the logical semantics (Chiari et al., 2024, Chiari et al., 2018).

OPTL is the temporal logic introduced for operator precedence languages. Its syntax includes atomic propositions, Boolean connectives, linear next/back operators, matching operators $J\subseteq A$8, OP-summary until/since operators parameterized by $J\subseteq A$9, and hierarchical until/since operators. The matching operators use maximal chains: $R^{-1}(J)$0

$R^{-1}(J)$1

OP-summary paths may either move linearly to the next position or jump over the body of a maximal chain, which makes the logic suitable for hidden syntactic structure rather than only visible call/return nesting (Chiari et al., 2018).

POTL is the precedence-oriented temporal logic used in later model checking work. The paper focuses on the future fragment $R^{-1}(J)$2 over finite words. A model is an operator-precedence word

$R^{-1}(J)$3

from which one obtains the chain relation

$R^{-1}(J)$4

POTL extends ordinary linear temporal logic with precedence-sensitive next, chain next, summary-path until/release, hierarchical next, and hierarchical until/release. The syntax given in the source is

$R^{-1}(J)$5

where $R^{-1}(J)$6. Particularly characteristic is chain next: $R^{-1}(J)$7 iff there exists $R^{-1}(J)$8 such that

$R^{-1}(J)$9

The summary and hierarchical path modalities use $(R^{-1}(A),\le)$0, allowing formulas to move along linear order while jumping through chain links or along positions sharing the same chain mate (Chiari et al., 2024).

The two logics occupy closely related but distinct places. OPTL is presented as a new temporal logic for operator precedence languages and proved at least as expressive as NWTL, while POTL is the precedence-based logic used for symbolic model checking. Neither paper defines a monotone fragment of these logics. A plausible implication is that, in the operator-precedence tradition, “monotone precedence formulas” would have to be extracted as a restricted fragment of POTL or OPTL rather than inherited from the original definitions.

5. Verification methods: automata compilation and SMT-based symbolic model checking

The verification theory of precedence formulas has developed along two lines.

For OPTL, the main method is automata-theoretic compilation. Given a formula $(R^{-1}(A),\le)$1, the paper constructs an operator precedence automaton

$(R^{-1}(A),\le)$2

whose states are atoms over the closure $(R^{-1}(A),\le)$3. The paper states that it designs a procedure that, given a formula in the logic, builds an automaton recognizing the sentences satisfying the formula, “whose size is at most exponential in the length of the formula.” It also proves that OPTL is at least as expressive as analogous logics defined for visible pushdown languages, and that hierarchical operators do not increase expressiveness because they can be translated into summary operators, albeit with exponential blowup in formula size (Chiari et al., 2018).

For POTL, the 2024 work replaces explicit-state automata constructions by symbolic reasoning. It gives “the first symbolic, SMT-based approach for model checking POTL properties.” Instead of constructing an automaton for both the POTL formula and the model of the program, the method encodes them into a sequence of SMT formulas, and the search for a violating trace is carried out by an SMT solver in a bounded-model-checking fashion. The encoding is based on a tree-shaped tableau for the future fragment of POTL on finite words, a next normal form $(R^{-1}(A),\le)$4, and a finite-sorted SMT signature containing predicates such as $(R^{-1}(A),\le)$5, $(R^{-1}(A),\le)$6, $(R^{-1}(A),\le)$7, $(R^{-1}(A),\le)$8, and $(R^{-1}(A),\le)$9. The soundness statements recorded in the paper are: if the tableau for $$M : (\Sigma \cup \{\#\})^2 \to \{\lessdot, \doteq, \gtrdot\},$$0 has an accepted branch, then $$M : (\Sigma \cup \{\#\})^2 \to \{\lessdot, \doteq, \gtrdot\},$$1 is satisfiable; and if $$M : (\Sigma \cup \{\#\})^2 \to \{\lessdot, \doteq, \gtrdot\},$$2 is satisfiable for some $$M : (\Sigma \cup \{\#\})^2 \to \{\lessdot, \doteq, \gtrdot\},$$3, then $$M : (\Sigma \cup \{\#\})^2 \to \{\lessdot, \doteq, \gtrdot\},$$4 is satisfiable (Chiari et al., 2024).

The experimental evaluation reported for the SMT-based method uses benchmarks including Quicksort, a Jensen-style banking application, and C++ stack implementations with exceptions, and compares against the explicit-state tool POMC. The paper reports that POMC often exhibits exponential blow-up, whereas the SMT-based approach scales much better, especially with larger bitwidths, arrays, and hierarchical operators. One reported Quicksort family increases POMC times from $$M : (\Sigma \cup \{\#\})^2 \to \{\lessdot, \doteq, \gtrdot\},$$5 seconds as bitwidth grows, while the SMT approach stays around $$M : (\Sigma \cup \{\#\})^2 \to \{\lessdot, \doteq, \gtrdot\},$$6–$$M : (\Sigma \cup \{\#\})^2 \to \{\lessdot, \doteq, \gtrdot\},$$7 seconds and solves larger instances under 23 seconds. The same paper also notes one family where POMC was consistently faster: a nearly deterministic Stack benchmark with little nondeterminism. These results concern verification methodology rather than the definition of monotone precedence formulas, but they show that precedence-based formula languages now support both automata compilation and symbolic SMT workflows (Chiari et al., 2024).

6. Monotonicity, positivity, and conceptual boundaries

A final source of ambiguity lies in the word monotone itself. In the linear-order paper, monotonicity refers to a semantic discipline on binary relations: interval-preservation. In the operator-precedence papers, precedence is structural, but no monotone fragment is isolated. In modal logic, by contrast, monotonicity means semantic preservation under increasing valuations of designated propositional variables, and the corresponding syntactic notion is positivity (Fortin, 2019, Chiari et al., 2024, Dvorkin, 2 Feb 2026).

This distinction matters because semantic monotonicity and syntactic positivity need not coincide automatically. The paper “Monotonicity versus positivity in modal logics” defines the Lyndon positivity property (LPP): every formula that is monotone in some variable(s) is equivalent over the logic to a formula positive in those variable(s). It proves LPP for all normal modal logics with the Lyndon interpolation property, proves that all logics between $$M : (\Sigma \cup \{\#\})^2 \to \{\lessdot, \doteq, \gtrdot\},$$8 and $$M : (\Sigma \cup \{\#\})^2 \to \{\lessdot, \doteq, \gtrdot\},$$9 do not have LPP, and proves that all canonical monotone modal logics preserved under bisimulation products have both LIP and LPP. In particular, it shows LIP and LPP for all logics axiomatizable over the minimal monotone logic EM by means of closed formulas and formulas of the form $a\lessdot b$0, where $a\lessdot b$1 is positive (Dvorkin, 2 Feb 2026).

For the topic at hand, this yields a precise caution. A formula language may be “precedence-based” without being monotone in the modal-semantic sense, and it may be “monotone” in the modal-semantic sense without involving order-compatible precedence relations. No cited paper proves that POTL or OPTL possesses a monotone fragment equivalent to a positive fragment, and the operator-precedence papers explicitly do not define “monotone precedence formulas.” Conversely, the linear-order paper does provide a clear semantic criterion—interval-preservation—for when precedence-like binary predicates support low-variable collapse and star-free dynamic-logic normal forms.

Taken together, the literature supports a restrained synthesis. “Monotone precedence formulas” is best understood as a cross-cutting label for formulas interpreted over precedence-structured models under an explicit monotonicity discipline. In the strongest current formalization, that discipline is interval-preservation on binary relations over a linear order, yielding the theorem $a\lessdot b$2 and a translation into a star-free PDL fragment (Fortin, 2019). In the operator-precedence tradition, the nearest concrete realizations are POTL and OPTL, which supply a rich precedence-sensitive modal vocabulary and effective model-checking procedures but do not yet identify a dedicated monotone fragment (Chiari et al., 2024, Chiari et al., 2018).