First-Order Action Synthesis (FAS)

• FAS is a formal process that synthesizes action models using first-order logic integrated with temporal constraints and structured representations.
• It applies to automated planning and multi-agent systems by leveraging decidable logic fragments, theorem-proving, and learning methods for complex scheduling and verification tasks.
• Recent research in FAS combines logical frameworks, neural architectures, and dynamic models to address challenges from robotics to theoretical physics.

First-order Action Synthesis (FAS) refers to the formal process of generating, generalizing, or recognizing action models, schemas, and plans within frameworks that support first-order representations—principally encompassing structured logic, temporal modalities, and variable-rich symbolic descriptions. FAS is critical in automated planning, multi-agent systems, and theoretical physics where action, time, and objects with interdependent properties must be systematically managed. Research in FAS spans logic-based formalisms, learning grounded models from data, synthesis in distributed systems under resource bounds, and the specification and solution of complex planning tasks with temporal and environmental constraints.

1. Temporal Logic Frameworks for First-order Action Synthesis

First-order temporal logics for actions underpin rigorous specification and synthesis in FAS, notably extending classical first-order logic by introducing modal operators qualified by action terms ("Dal" logic) (0705.1999). In these frameworks, actions are not merely transitions between states but are fully characterized by temporal parameters, durations, delayed preconditions, and complex effect profiles. The syntax typically allows formulas such as:

• $$at(t, x) \rightarrow [move(t, d, x, y)]\ at(t + d, y)$$
• $$T(t_1, x_1) \rightarrow [a(t, d, x_2)]\ p(t_2, x_3)$$
• Modal operators: $[a(t, d, x)] \varphi$

Crucial innovations include first-order modalities with unifiable and quantifiable terms, enabling tight linkage of temporal variables and object parameters across actions and state formulas. The explicit homomorphism $time: W \rightarrow T$ grounds each world (state) to a temporal axis, essential for generating action sequences whose executions satisfy duration and ordering constraints.

Dal recognizes that action effects may persist, overlap, or delay, and contains mechanisms for specifying nuanced causality—for instance, applying effects only if preconditions continue to hold until the action or event occurs (as seen in elevator or thrown-object scenarios). The existence of a decidable fragment allows systematic tableau-based synthesis and verification, guaranteeing termination in practical planning algorithms and supporting tool development for complex scheduling and hybrid temporal domains.

2. Synthesis under Assumptions and Domain Constraints

Formal planning and synthesis must respect assumptions about environment dynamics, resource availability, and fairness (Aminof et al., 2018). Assumptions are not arbitrary logical formulas but must be environment realizable, meaning there exists an environment strategy fulfilling the assumption regardless of agent actions. Formally, for an assumption $\omega$ and goal $\gamma$, agent strategies $\sigma_A$ must guarantee:

$$\forall \sigma_E \in Str_E(\omega):\ \pi_{(\sigma_A, \sigma_E)} \models \gamma$$

Domains are captured as tuples $D = (E, A, I, Pre, \Delta)$ mapping states, actions, preconditions, and nondeterministic transitions. The domain itself is compiled into a linear-time property $\omega_D$, constraining feasible traces to respect initial states and available transitions, expressed in LTL (Linear Temporal Logic) with extensions to automata-theoretic analysis (DPW, DFA).

Agent goals and environmental constraints are characterized in LTL, LTLf (finite traces), or LDLf formalisms. Synthesis reduces to solving parity or reachability games, with worst-case complexity in 2EXPTIME, but provides a uniform basis for FOND (Fully Observable Non-Deterministic) planning, fair planning, and multi-agent extensions. The framework is applicable to robotic planning, hybrid systems, and verification contexts, emphasizing soundness with respect to domain and stratified assumptions.

3. Parameterized Synthesis for First-order Logic over Data Words

Many distributed systems require synthesis of controllers and protocols for unbounded or parameterized sets of participants (Bérard et al., 2019, Grange et al., 22 Apr 2024). Executions are represented as data words—sequences of action-process pairs—with first-order logic specifications tailored to this setting. Undecidability is a central challenge if both the system and the environment can access an unrestricted number of processes. For FO logic without order, the synthesis problem is undecidable by reduction from the halting problem for 2-counter Minsky machines.

Decidability is recoverable by restricting environment access to a bounded number of processes, enabling reduction to parameterized vector games characterized by configuration mappings $C: L \rightarrow \mathbb{N}^T$ (with $L$ as the set of action counts up to threshold $B$ and $T$ as process types). Winning conditions, represented by counting formulas such as

$$\#_{B,\ell}(y) = \bigwedge_{a \in A,\ \ell(a) < B}\ \exists^{=\ell(a)} z.\ [y \sim z \wedge a(z)]\wedge\ldots$$

lead to cutoffs: finite bounds on process numbers sufficient to generalize correctness for all larger configurations.

Prefix First-order Logic (prefFO) further refines expressiveness by allowing reasoning about prefixes within each process's data word (Grange et al., 22 Apr 2024). The prefix order $x \lesssim y :\equiv x \sim y \wedge x < y$ constrains analysis to local process history, forming a finite, tree-structured type space. Synthesis then reduces to finite token games with double cutoffs, making it feasible for distributed system controllers under realistic assumptions.

4. Online and Grounded Action Model Learning

Synthesis is not merely theoretical: learning action representations from data—whether in abstract domains or from raw perceptual input—is essential for autonomous agents. Online Action Recognition proposes induction of first-order (STRIPS) action schemas directly from observed transitions (Suárez-Hernández et al., 2020). The core is Action Unification (AU), an optimization over injective partial mappings between parameters, rendered as a Weighted Partial MaxSAT (WPMS) instance:

$I^*(\Phi) = W_{big}\cdot N_{np} + N_{param}$

where $N_{np}$ is the count of non-preserved predicates and $N_{param}$ is the number of lifted parameters. The OARU algorithm incrementally generalizes action models, building a hierarchical library from trivial grounded actions observed and recognized.

Grounded learning integrates symbolic planning with perceptual grounding (e.g., learning action schemas directly tied to parsed image representations) (Liberman et al., 2022). Parsed images in an O2D language yield state descriptions via a learned abstraction $h$, matching constraints:

• (C1) Different O2D states produce different abstractions.
• (C2) Predicted transitions match observed transitions.

The combination of combinatorial structure search (favoring simple, human-like schemas) with image-based grounding allows new planning instances to be specified as image pairs—broadening applicability to vision-driven robotics and interpretable model creation.

5. Neural and Dynamical Models for Action Synthesis

Recognition and synthesis in human-inspired systems leverage dynamic (kinematic) information and advanced neural architectures (Gharaee et al., 2021). Hierarchical SOMs (Self Organizing Maps) coupled with supervised neural networks process joint positions and their first- and second-order dynamics (velocity, acceleration):

$v(t) = p(t) - p(t-1)$

$a(t) = v(t) - v(t-1)$

Preprocessing incorporates ego-centered transformations and scaling, making learned representations invariant to orientation and distance. Attention mechanisms focus on highly dynamic joints, aligning computational models with biologically inspired perception. Merged features improve action categorization—cluster formation in SOM layers validates similarity-based structure, and accuracy reaches up to 90% under optimal feature integration. Such dynamic models complement logical synthesis, providing robust recognition where temporal derivatives are proxies for forces and intentions.

6. First-order Action Principles in Gauge and Physical Theories

First-order action synthesis in theoretical physics takes a foundational role in unified field theories (Gallagher et al., 2023). The action functional is constructed as a 4-form:

$I = \int L,\quad L = L_G + L_M$

$$L_G = B^{ab} \wedge R_{ab},\quad B^{ab} = (i/2)\ \varphi^a \wedge \varphi^b$$

with $R^{ab}$ as curvature and $\varphi^a$ a pregeometric scalar field ("khronon" field). The metric is a derived quantity $g = \varphi^a \otimes \varphi_a$. Matter coupling and canonical Noether currents for energy-momentum and spin arise naturally via variational derivatives. The theory is symmetric under Lorentz, shift (translation of $\varphi^a$), and diffeomorphism transformations. Shift symmetry yields conserved "shadow" charges, interpreted as sources of effective dark matter.

The action principle is proposed as a candidate for unifying both gravitation and the Standard Model, enforcing consistent definition of energy-momentum sources and offering a route for embedding traditional second-order Yang-Mills into a robust first-order gauge framework.

7. Synthesis with Temporal and Agent Programming Logics

Complex agent behaviors are realized by synthesizing correct execution policies for programs in languages such as Golog, within situation calculus frameworks and under nondeterministic environment models (Hofmann et al., 1 Oct 2024). The synthesis process defines games between controllable and uncontrollable actions, with execution traces verified against temporal goals specified in LTLf (Linear Temporal Logic on finite traces):

Transformations such as "tail normal form" (TNF) and "next normal form" (XNF) facilitate tracking of temporal formula satisfaction. The game arena is composed of tuples integrating world types, residual programs, effects, and subformula progress. Winning policy extraction guarantees robust agent behavior under all environmental outcomes, supported experimentally in domains with unbounded objects and non-local effects. Scalability remains a challenge, particularly regarding symmetrical structuring and state-space explosion.

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In summary, First-order Action Synthesis spans logical, data-driven, neural, and physical paradigms. It interlinks expressiveness, decidability, planning under constraints, learning from perception, and unification of action models. Contemporary research provides rigorous frameworks and algorithmic tools for synthesizing plans and models in domains from robotics to foundational physics, leveraging modal temporal logics, automata-theoretic reductions, optimization-based learning, and hierarchical neural architectures. The persistent direction is towards more scalable, interpretable, and grounded synthesis, as well as extending the theoretical underpinnings to complex, multi-agent, and unified field settings.