First-Order Action Synthesis (FAS)

• First-Order Action Synthesis (FAS) is a framework for specifying and generating action models using first-order formalisms with rich temporal and logical features.
• It integrates methods like modal logics, environmental assumption-based planning, learning from observations, and symbol grounding to support robust reasoning and control.
• Research in FAS focuses on balancing expressivity with decidability using automata-theoretic, game-theoretic, and unification-based techniques across diverse domains.

First-Order Action Synthesis (FAS) refers to the process of specifying, generating, or learning action models in first-order formalisms—often with rich temporal, logical, or algebraic structure—to support reasoning, planning, or control in dynamic, interactive systems. In recent research, FAS encompasses a variety of approaches spanning modal logic with first-order modalities, synthesis from observations, symbol grounding on perceptual data, robust planning under environment assumptions, and, in theoretical physics, first-order action principles for fundamental interactions. The domain includes both the formal synthesis of system behaviors subject to temporal and causal constraints and, more abstractly, the mathematical specification of actions or interactions in first-order frameworks.

1. First-Order Modal Logics for Actions and Temporal Properties

One approach to FAS, exemplified by the Dal logic (0705.1999), is to directly encode actions, their parameters, and temporal features as first-order modalities within a multi-modal logic. Actions appear as modal operators parameterized by first-order terms, e.g., $[a(t_1, t_2, \ldots, t_n)]\, \varphi$, allowing the variables in the action term to be unified with variables in the embedded formula. Temporal reasoning is enabled by explicit modeling of time axes (discrete, dense, or continuous), with timed action terms such as $a(t,d,\ldots)$ encoding both the occurrence and duration of actions.

Dal expresses numerous temporally extended action relations, such as:

• Beginnings and endings of actions, and delayed preconditions/results
• State transitions governed by parametric action effects, e.g., $at(t,x,y) \to [move(t,d,x,y)]\, at(t+d,x,z)$
• Causal scenarios with temporal ordering (e.g., multiple agents affecting a shared object in temporally sensitive sequences)

While full first-order modal logic is undecidable, practically significant fragments (e.g., with grounded, quantifier-limited modalities) can be handled algorithmically using analytic tableaux rules, supporting the formal synthesis and verification of temporal action plans.

2. Synthesis under Environmental Assumptions

FAS in interactive, open-world settings often requires synthesizing agent plans that are robust against environmental non-determinism, constraints, or fairness assumptions (Aminof et al., 2018). Here, environmental assumptions are formalized as sets of strategies or as linear-time logic formulas (e.g., LTL/LTLf/LDLf), acting as constraints on admissible environment behaviors. Synthesizing a correct agent plan requires the agent's policy to guarantee its objective provided the environment conforms to the assumption, yielding challenges of consistency (ensuring the environment assumption is realizable) and semantic coupling between agent synthesis and domain constraints.

Algorithms in this setting leverage automata-theoretic constructions:

• Planning domains are encoded as automata (deterministic parity automata for infinite traces or DFAs for finite traces)
• These are combined (via intersection/conjunction) with automata for environment assumptions and temporal specifications
• Synthesis is solved via parity games or DFA search, producing agent strategies correct under all admissible environmental reactions

Consistency checking and environment realizability are prominent, requiring additional algorithmic steps (e.g., 2EXPTIME for infinite-trace environment realizability, PTIME for DFA/finite settings). Practical applications include robotics planning under physical trajectory constraints, safety-critical system design, and multi-agent rational synthesis.

3. Learning and Unification-based Synthesis of First-Order Action Models

A major branch of FAS focuses on the inductive synthesis of action models from observations in symbolic planning domains, particularly where the action and effect schemas are initially unknown. One methodology constructs grounded, logic-based action representations by processing observed state transitions (Suárez-Hernández et al., 2020). This is achieved by:

• Deriving a “Trivial Grounded Action” (TGA) for each observed transition, encapsulating the minimal preconditions and effects
• Merging and generalizing these TGAs using Action Unification (AU), an optimization-encoded unification closely related to logic variable unification but augmented by weighted partial MaxSAT formulations to balance preservation of predicate structure with generalization (“lifting” parameters to variables)
• Incremental library-building: As observations accumulate, similar actions are unified to produce a hierarchy of increasingly abstract action schemas

Performance in benchmark planning domains demonstrates real-time capabilities (processing times on the order of tens of milliseconds per observation) and high alignment with expert-specified action models. The approach naturally accommodates partial observability, incremental model refinement, and is provably NP-hard in the worst case but tractable empirically.

4. Symbol Grounding and Learning from Perceptual Input

Recent advances bridge symbolic FAS with perceptual data, offering methods to learn first-order planning domains directly from parsed images (Liberman et al., 2022). In this setting:

• Parsed images are encoded as “O2D states” (objects-in-2D), described by unary and binary predicates capturing visual/spatial relations (“left,” “above,” “shape”)
• The learning process discovers a grounded planning domain $D$ and an abstraction function $h$ that maps O2D descriptions to sets of planning predicates, preserving both distinguishability (no two images mapped to the same abstract state) and transition consistency (abstract transitions mirror those in the observed data)
• The action schemas learned are STRIPS-like with concise, interpretable preconditions and effects, and are optimized for simplicity using a combinatorial answer set program framework

Empirical results in grid, block-world, puzzle, and manipulation domains show that grounded first-order domains generalize to new instances with more objects, support off-the-shelf planning engines (e.g., Pyperplan), and remain computationally practical due to the compactness of the induced models.

5. Decidability and Abstraction in First-Order Synthesis for Distributed Systems

In distributed synthesis, FAS encounters challenges posed by unbounded numbers of processes and the need for specifications over data words (Grange et al., 22 Apr 2024). To achieve decidability:

• Executions are modeled as data words over a finite action alphabet and an infinite process identifier domain, with each process corresponding to a “class” within the word
• Specifications use a restricted fragment, prefix first-order logic on data words (\prefFOdw), which permits only limited ordered comparisons (within-process orderings) and requires quantification disciplined by prefixes
• Core to the methodology is the abstraction of executions into types (with all words of the same type indistinguishable to the logic up to quantifier depth $k$), with types organized into a finite tree via a partial order

Synthesis reduces to a token game played over process types, leveraging cut-off theorems to bound the number of tokens required for completeness. This enables finite-state synthesis algorithms for distributed protocols, albeit with the trade-off that the full power of arbitrary first-order order comparisons is lost to guarantee decidability.

6. FAS in Physical Theories: First-Order Actions in Gauge Theories

FAS also appears in the formal developments of gauge theories and gravity, where the action principle itself is formulated in first-order variables (Gallagher et al., 2023). In such approaches:

• The action $I = \int (L_G + L_M)$ is constructed from a gravitational part $L_G$ (built from a “khronon field” $\phi^a$ and a Lorentz-connection) and a matter part $L_M$ (incorporating minimally coupled Standard Model fields)
• The first-order structure is characterized by independent variation of the fields (no a priori relationship between metric and connection), polynomiality in fields, and action functionals involving wedge products, for example $L_G = B^{ab}\wedge R_{ab}$ with $B^{ab} = (i/2)\phi^a\wedge \phi^b$
• Symmetries including Lorentz invariance, diffeomorphism covariance, and internal “shift” symmetry in $\phi^a$ yield well-defined Noether currents and the possibility of “shadow charges,” which may carry quantum-mechanical significance

Such formulations offer pathways for unification of matter and gravity (potentially gravi-GUTs) and manifestly first-principle, first-order dynamics for fundamental interactions, resonating with the general spirit of FAS in constructing actions from first-order building blocks.

7. Comparative Perspective and Synthesis Frameworks

Across the diverse instances of FAS, common patterns emerge:

• Expressivity and decidability are frequently in tension, with syntactic or semantic restrictions (e.g., grounded modalities, prefix discipline, bounded quantification) often necessary to obtain algorithmic synthesis
• Symbol grounding, inductive bias toward succinctness or hand-like schemas, and integration with perception are prominent in learning-based FAS
• In formal verification and synthesis, automata-theoretic and game-theoretic methods (e.g., parity/DFA automata, tableau procedures, finite token games, and backward fixpoint strategies) are key to lifting first-order problems into finite object-level computations
• The diversity of domains—AI planning, distributed systems, real-time robotics, theoretical physics—highlights the broad applicability and importance of first-order action synthesis principles in modeling, learning, and controlling dynamic systems

The current research frontier explores richer temporal/durational phenomena, robust synthesis under uncertainty, symbol grounding for perception-driven agents, and foundational constructions for unified physical theories, all under the umbrella of first-order action synthesis.