Manna–Pnueli Games: Synthesis and Complexity

• Manna–Pnueli games are infinite-duration synthesis games that extend finite-trace methods to ω-regular domains using a temporal hierarchy of safety, guarantee, recurrence, and persistence conditions.
• They translate LTLf⁺ and PPLTL⁺ formulas into game arenas via DFA products with state labeling and local memory, preserving the structure of local F, G, GF, and FG atoms.
• Empirical and complexity analyses show that native MP strategies can streamline synthesis, offering advantages over Emerson–Lei reductions, especially on mixed safety and guarantee benchmarks.

Manna–Pnueli games are infinite-duration synthesis games whose winning conditions are organized by the Manna–Pnueli temporal hierarchy. In the formulation used for LTLf$^+$ and PPLTL$^+$ synthesis, a Manna–Pnueli objective is a positive Boolean combination of local and recurring event predicates—$\mathsf{F}a$, $\mathsf{G}a$, $\mathsf{GF}a$, and $\mathsf{FG}a$—interpreted over runs of a game arena labeled by events (Hausmann et al., 20 Aug 2025). This native embedding of the hierarchy contrasts with reductions that force all objectives into Emerson–Lei form. A particularly important specialization is the obligation level, consisting of positive Boolean combinations of safety and guarantee properties; for that fragment, the game structure collapses to deterministic weak automata and weak Büchi games with markedly simpler symbolic algorithms (Giacomo et al., 20 Apr 2026). In the broader synthesis tradition, Manna–Pnueli games also denote the classical Church-style setting over $\omega$-words, which has been generalized from finite alphabets to the Baire space $\mathbb{N}^{\omega}$ using N-memory automata and transducers (Brütsch et al., 2016).

1. Hierarchical basis and semantic classes

The Manna–Pnueli hierarchy classifies $\omega$-regular temporal properties by lifting finite-trace properties to infinite traces via prefix quantification. If $R \subseteq (2^{AP})^*$ is a finite-trace property, then over infinite traces $^+$0 the hierarchy uses four canonical prefix modalities: safety $^+$1, guarantee or co-safety $^+$2, recurrence $^+$3, and persistence $^+$4 (Giacomo et al., 20 Apr 2026).

Class	Prefix characterization	MP atom form
Safety	every finite prefix is in $^+$5	$^+$6
Guarantee	some finite prefix is in $^+$7	$^+$8
Recurrence	infinitely many prefixes are in $^+$9	$\mathsf{F}a$0
Persistence	all but finitely many prefixes are in $\mathsf{F}a$1	$\mathsf{F}a$2

Safety properties are precisely those with a set of finite bad prefixes $\mathsf{F}a$3 such that a trace violates the property iff some prefix belongs to $\mathsf{F}a$4. Guarantee properties dually have a set of finite good prefixes $\mathsf{F}a$5 such that satisfaction holds iff some prefix in $\mathsf{F}a$6 occurs (Giacomo et al., 20 Apr 2026). In the event-based game presentation, an MP formula is a positive Boolean combination over atoms $\mathsf{F}a$7, $\mathsf{F}a$8, $\mathsf{F}a$9, and $\mathsf{G}a$0, with $\mathsf{G}a$1 ranging over a finite event set $\mathsf{G}a$2. For runs $\mathsf{G}a$3, the semantics are: $\mathsf{G}a$4

$\mathsf{G}a$5

$\mathsf{G}a$6

$\mathsf{G}a$7

An MP objective $\mathsf{G}a$8 consists of the event set, a state labeling $\mathsf{G}a$9, and an MP formula $\mathsf{GF}a$0; a run is winning iff the induced event word satisfies $\mathsf{GF}a$1 (Hausmann et al., 20 Aug 2025).

A central structural fact is that MP objectives are $\mathsf{GF}a$2-regular but not prefix-independent, because local $\mathsf{GF}a$3/$\mathsf{GF}a$4 atoms depend on whether an event has happened at least once or has held at all positions so far. This differentiates them from pure Emerson–Lei objectives, which use only $\mathsf{GF}a$5 and $\mathsf{GF}a$6 atoms (Hausmann et al., 20 Aug 2025).

2. Logical encodings: from finite-trace formulas to infinite-trace games

LTLf$\mathsf{GF}a$7 and PPLTL$\mathsf{GF}a$8 lift finite-trace formulas to infinite traces by prefix quantification. Their syntax is

$\mathsf{GF}a$9

where $\mathsf{FG}a$0 is an LTLf or PPLTL formula over $\mathsf{FG}a$1 (Hausmann et al., 20 Aug 2025). Writing $\mathsf{FG}a$2 for the finite words satisfying $\mathsf{FG}a$3, each quantified formula $\mathsf{FG}a$4 denotes the corresponding prefix-quantified language $\mathsf{FG}a$5. These logics are as expressive as full LTL over infinite traces; the obligation fragment, obtained by restricting $\mathsf{FG}a$6 to $\mathsf{FG}a$7 and $\mathsf{FG}a$8, defines exactly the Manna–Pnueli obligation properties (Giacomo et al., 20 Apr 2026).

The usual synthesis pipeline starts from a positive Boolean combination of atoms $\mathsf{FG}a$9. Each finite-trace $\omega$0 is compiled into a DFA $\omega$1, and the game arena is built from the product of these DFAs. The infinite-trace objective is then determined not by an $\omega$2-automaton determinization step, but by interpreting the Manna–Pnueli lifting over the product states (Hausmann et al., 20 Aug 2025). This is the key reason DFA-based finite-word technology can be reused in infinite-trace synthesis.

For native MP games, the mapping from quantified subformulas to event atoms is direct:

• $\omega$3 becomes $\omega$4,
• $\omega$5 becomes $\omega$6,
• $\omega$7 becomes $\omega$8,
• $\omega$9 becomes $\mathbb{N}^{\omega}$0.

If $\mathbb{N}^{\omega}$1, then replacing each atom $\mathbb{N}^{\omega}$2 by the corresponding temporal atom over event $\mathbb{N}^{\omega}$3 yields an MP objective equivalent to the original formula (Hausmann et al., 20 Aug 2025).

The obligation fragment has several simplifications. In particular,

$\mathbb{N}^{\omega}$4

and an LTLf formula $\mathbb{N}^{\omega}$5 is equi-realizable with the obligation formula $\mathbb{N}^{\omega}$6 (Giacomo et al., 20 Apr 2026). This suggests that, at the second level of the hierarchy, infinite-trace synthesis inherits much of the finite-prefix structure of LTLf.

3. Arena construction and native versus reduced objectives

A game arena is a finite directed graph $\mathbb{N}^{\omega}$7 in which system and environment alternate moves and every node has at least one successor. In the synthesis setting induced by finite-trace DFAs, the arena alternates the system choice of $\mathbb{N}^{\omega}$8 and the environment choice of $\mathbb{N}^{\omega}$9, while the DFA product updates deterministically according to the resulting letter (Giacomo et al., 20 Apr 2026).

The 2025 formulation distinguishes two routes from LTLf$\omega$0/PPLTL$\omega$1 to a solvable game (Hausmann et al., 20 Aug 2025). The first route reduces local atoms to Emerson–Lei objectives. For $\omega$2, the construction adds the initial state to $\omega$3 and turns every non-accepting state into a non-accepting sink; for $\omega$4, it removes the initial state from $\omega$5 and turns every accepting state into an accepting sink. This sink adjustment records whether a local condition has already been irrevocably decided. The resulting event objective then uses only $\omega$6 and $\omega$7 atoms.

The second route keeps the Manna–Pnueli objective native. If the arena does not already store local-event memory, it can be augmented to $\omega$8, where the memory component records which local $\omega$9 events have already occurred and which local $R \subseteq (2^{AP})^*$0 events have remained continuously true. The update is

$R \subseteq (2^{AP})^*$1

With labeling $R \subseteq (2^{AP})^*$2, local atoms can be partially evaluated per memory value $R \subseteq (2^{AP})^*$3, leaving only Emerson–Lei subobjectives on each memory layer (Hausmann et al., 20 Aug 2025).

This decomposition organizes the augmented arena into a DAG of subarenas $R \subseteq (2^{AP})^*$4, indexed by local-memory valuations $R \subseteq (2^{AP})^*$5. For each $R \subseteq (2^{AP})^*$6, the formula

$R \subseteq (2^{AP})^*$7

contains only $R \subseteq (2^{AP})^*$8/$R \subseteq (2^{AP})^*$9 atoms and therefore defines an Emerson–Lei subgame. Solving these subgames bottom-up along the DAG yields a global solution to the MP game (Hausmann et al., 20 Aug 2025).

A plausible implication is that the native MP construction preserves more of the original formula structure than an unconditional reduction to recurrence and persistence, which is exactly the motivation stated for introducing Manna–Pnueli games as a separate game model.

4. Solution methods and complexity for general Manna–Pnueli games

Emerson–Lei games are solved symbolically using Zielonka trees, which compile Boolean combinations of $^+$00/$^+$01 atoms into nested fixpoint equations evaluated with BDDs. For an EL game with $^+$02 nodes and $^+$03 events, the solver runs in time

$^+$04

and strategies require at most $^+$05 memory values (Hausmann et al., 20 Aug 2025).

For MP games, two reductions to EL games are given. The naïve reduction lifts all local atoms into the EL setting on the augmented arena, yielding time

$^+$06

where $^+$07 is the number of EL events and $^+$08 the number of local events. If the arena already carries local memory, this becomes

$^+$09

The compositional DAG reduction instead solves at most $^+$10 EL subgames, giving

$^+$11

with strategy memory at most $^+$12. If local memory is already in the arena, the bound improves to

$^+$13

and $^+$14 memory, matching the EL asymptotics (Hausmann et al., 20 Aug 2025).

The end-to-end synthesis complexity is worst-case optimal for the underlying logics. LTLf$^+$15 synthesis is 2EXPTIME-complete, because LTLf subformulas may require DFAs of size $^+$16; PPLTL$^+$17 synthesis is EXPTIME-complete, reflecting the single-exponential DFA blow-up for PPLTL (Hausmann et al., 20 Aug 2025).

The strategy model reflects the game decomposition. EL strategies use Zielonka-tree leaves as memory values. MP strategies additionally track the current local-memory valuation $^+$18, so the controller chooses moves according to the EL strategy $^+$19 attached to the current subgame and updates $^+$20 via the memory-update function (Hausmann et al., 20 Aug 2025).

An important limitation is empirical as well as structural: MP games are not universally faster than EL games. The native MP route often helps on formulas mixing local $^+$21 with $^+$22, but the EL route can be better on guarantee-dominated specifications because sink-adjusted DFAs sometimes reduce the nested fixpoint burden substantially (Hausmann et al., 20 Aug 2025).

5. Obligation games: deterministic weak automata and linear-time weak-game solving

Obligation properties are positive Boolean combinations of safety and guarantee properties and form the second level of the temporal hierarchy of Manna and Pnueli. In LTLf$^+$23, the obligation fragment restricts quantifiers to $^+$24 and $^+$25, and defines exactly these obligation properties (Giacomo et al., 20 Apr 2026).

The automata-theoretic simplification is unusually strong. Starting from a symbolic DFA $^+$26 for a finite-trace formula $^+$27, the construction defines absorbing variants

$^+$28

Then

$^+$29

For $^+$30, once an accepting DFA state is reached, it becomes absorbing; for $^+$31, once a rejecting state is reached, rejection becomes absorbing. The resulting $^+$32-automata are deterministic weak automata (DWAs), meaning every strongly connected component is either fully accepting or fully rejecting (Giacomo et al., 20 Apr 2026).

DWAs are closed under union, intersection, and complement. Products use the standard synchronous construction

$^+$33

Intersection marks $^+$34, union marks $^+$35, and complement flips acceptance. Any obligation formula $^+$36 can therefore be translated into an equivalent DWA of size

$^+$37

This size matches the DFA blow-up already present in LTLf compilation (Giacomo et al., 20 Apr 2026).

Once the DWA is built, the synthesis game becomes a weak Büchi game. The paper studies three symbolic solvers. Standard nested fixpoints for Büchi and co-Büchi objectives require $^+$38 symbolic operations. The SafeReach procedure alternates safety and reachability fixpoints with monotone region growth and has complexity

$^+$39

Most notably, an SCC-based bottom-up algorithm solves weak games with $^+$40 nodes in

$^+$41

symbolic operations, and yields memoryless strategies by composing per-SCC safety and reachability strategies (Giacomo et al., 20 Apr 2026).

This explains the claim that obligation synthesis over infinite traces can be performed with virtually the same effectiveness as LTLf synthesis. The overall worst-case complexity remains 2ExpTime because DWA construction is double exponential in formula size, but once the DWA is available, game solving is linear in the arena (Giacomo et al., 20 Apr 2026).

A representative obligation formula is

$^+$42

which states that along any infinite trace that stays within the domain, each goal eventually holds in some prefix. The corresponding game combines a safety monitor for $^+$43 with reachability objectives for the $^+$44, and memoryless winning strategies suffice (Giacomo et al., 20 Apr 2026).

6. Empirical behavior, misconceptions, and broader synthesis context

The implemented solvers confirm that Manna–Pnueli games are primarily a structural optimization, not a universal dominance result. In the 2025 experiments, MP games often offered significant advantages on mixed formulas, including counter-style benchmarks and implication patterns of the form

$^+$45

but EL games outperformed MP games on a pure guarantee pattern consisting only of $^+$46-atoms. The stated reason is that sink-loop adjustments in the EL pipeline can introduce shortcuts that substantially reduce nested fixpoints, whereas the MP pipeline still constructs and solves the local-memory DAG (Hausmann et al., 20 Aug 2025).

The obligation-specific experiments sharpen this picture. In the LydiaSyft$^+$47 prototype, synthesis for obligation LTLf$^+$48 formulas was practically on par with LTLf synthesis over finite traces. DWA construction typically dominated runtime, while game solving was fast and often negligible. Minimization was polynomial and sometimes helpful, although its benefit depended on the representation. The SCC algorithm remained theoretically linear, but in the prototype its performance was limited by monolithic transition encoding overhead, whereas the fixpoint-based solvers were robust (Giacomo et al., 20 Apr 2026).

A recurring misconception is that “Manna–Pnueli games” denote a single fixed acceptance condition. The cited work uses the term in at least two technically different ways. In the 2025 synthesis framework, MP games are native objectives over $^+$49, $^+$50, $^+$51, and $^+$52 atoms on finite arenas built from products of DFAs (Hausmann et al., 20 Aug 2025). In the 2026 obligation study, the same hierarchy is specialized to positive Boolean combinations of $^+$53/$^+$54 components, where the corresponding games reduce to weak Büchi games on DWAs (Giacomo et al., 20 Apr 2026). The two viewpoints are compatible, but the second isolates a fragment with strictly simpler automata and game-solving behavior.

The broader lineage includes classical Church synthesis and its extensions. Over finite alphabets, Manna–Pnueli and Church-style games ask whether there exists a causal strategy ensuring that the interleaved input/output $^+$55-word lies in a specification language. “Playing Games in the Baire Space” generalizes this from the Cantor space to the Baire space $^+$56, where plays are sequences of natural numbers and winning conditions are recognized by deterministic N-memory parity automata. The induced game arena has vertices $^+$57, is MSO-interpretable, and supports decidability of the winner and effective synthesis of an N-memory transducer implementing a winning strategy for Output (Brütsch et al., 2016).

Taken together, these results place Manna–Pnueli games at a precise point between finite-trace DFA techniques and full $^+$58-regular synthesis. The native game view exposes the hierarchy directly; the obligation fragment shows that the second level already admits DFA-like closure, $^+$59 minimization, and linear-time weak-game solving; and the broader synthesis literature shows that the same game-theoretic perspective extends beyond finite alphabets when the automata model is chosen carefully (Hausmann et al., 20 Aug 2025).