Behavior Domain Definition Language (BDDL)

• BDDL is a formal specification language that precisely defines behavioral domains through structured syntax, logical predicates, and compositional rules.
• It facilitates rigorous modeling and verification in areas such as robotics, simulation, and game theory with modular and machine-interpretable designs.
• The language bridges high-level behavioral requirements with executable models, enhancing interoperability and automated reasoning in complex systems.

A Behavior Domain Definition Language (BDDL) is a class of formal specification languages developed to systematically define the domains, structures, or states relevant to agent behavior, planning tasks, simulated activities, or game models. BDDLs facilitate the precise, compositional, and often machine-interpretable description of behavioral domains, spanning symbolic state specifications, grid-based game configurations, and data domains for interoperability. Prominent variants and instantiations of BDDL have emerged in robotics, simulation, game theory, and informatics, each tailored to the specific formal requirements of the target domain.

1. Foundational Motivations and Definitions

The motivation for BDDL arises from the limitations of informal or ad hoc domain specification in agent behavior design, autonomous systems, and benchmarking. Classical approaches (e.g., unstructured state descriptions, untyped behavior trees, ad hoc game rule codifications) lead to ambiguity, lack of composability, and limited interoperability.

BDDLs, as exemplified in several research efforts, provide:

• A formal, often logic-based, language to specify domain elements (objects, states, resources).
• Structured representation of predicates, quantifiers, or compositional rules.
• Declarative separation between what constitutes a valid state/goal and how it is achieved or processed.

The definition of "domain" in this context is the set of all relevant possibilities (states, objects, facts, configurations) upon which information or agent policies operate, in accordance with information theory's definition: "information means selection from a set of possibilities (domain)" (Orthuber, 2018).

2. Syntax and Formal Semantics across Domains

BDDLs are instantiated for different purposes but share several recurring syntactic and semantic features:

Logic-Based Specification (Embodied AI, BEHAVIOR Benchmark) (Srivastava et al., 2021):

• Predicates encoding properties or relations (e.g., OnTopOf(apple, plate), Cooked(fish)).
• Variables and object constants, typically grounded in a hierarchical taxonomy (e.g., WordNet synsets).
• Standard and custom quantifiers (forall, exists, for_n, for_pairs, for_n_pairs).
• Logical connectives (and, or, not, implication).
• Activities characterized by initial and goal conditions as logical formulas over predicates:

$$\text{Activity}~\tau = \{ S_{\tau,0}, S_{\tau,g} \}$$

where $S_{\tau,0}$ is a set of permissible initial states and $S_{\tau,g}$ a set of admissible goal states.

Game Domain Specification (Grid Games) (Shaik et al., 2023):

• Symbolic parametrization over grid indices (?x, ?y), with relative addressing (e.g., ?x+1) for pattern specification.
• Domain files define actions with parameters, preconditions, and effects over board positions.
• Problem instances specify the board size, initial state, goals, and search depth.
• Semantics defined by transitions over board states, legal/illegal move handling, and recursive goal/strategy logic.

Atomic Behavior Specification via Linear Logic (Martens et al., 2018):

• Actions specified as linear logic formulas describing resource transformations, e.g.:

$set\_target: no\_target \multimap has\_target$

• Trees composed via typed combinators (sequence, selector), with resource-aware operational semantics.

Stateful Behavior Trees and DSLs (Serbinowska et al., 21 Nov 2024):

• Tree nodes parameterized with typed variables or blackboard states.
• Domain-specific languages (DSLs) specify variables, actions, checks, and temporal properties.
• Full integration with verification/model checking tools (e.g., nuXmv).

3. Expressiveness and Compositionality

A primary objective of BDDL design is compositional expressiveness, supporting:

• Modular construction of complex behavioral specifications from reusable components.
• Ability to capture unlimited and diverse instantiations of an activity or configuration (e.g., object-centric definitions permitting any number of objects, as in BEHAVIOR BDDL).
• Hierarchical and parameterized abstraction, using typed constructs or resource-based types.
• Reusability of domain elements or patterns (e.g., behavior "building blocks" (Martens et al., 2018)).

Linear logic-based frameworks explicitly encode resource flow, enabling compositional and sound reasoning about concurrent and sequential behaviors.

4. Applications and Instantiations

BDDL Variant	Domain of Use	Key Features
Predicate logic-based (BEHAVIOR)	Embodied AI, household task simulation	Object-centric, quantifiers, activity definition by state conditions
Symbolic/action-param (Grid Games)	2-player games (Tic-Tac-Toe, etc)	Symbolic positions, relative addressing, QBF encoding, linear scaling
Linear logic typed (BTL)	Agent programming, robotics	Resource-sensitive types, compositionality, soundness
DSL for SBTs (BehaVerify)	Verification, model checking, planning	State variables, temporal properties, scalable verification

Examples include:

• Simulation of diverse activities with scene-agnostic, declarative goal definitions in household benchmarks (Srivastava et al., 2021).
• Concise, lifted specification and efficient QBF encoding of grid-based games, with strategy extraction and validation (Shaik et al., 2023).
• Typed behavior specifications supporting abstraction, checking, and reuse in programming and verification (Martens et al., 2018, Serbinowska et al., 21 Nov 2024).

5. Formal Properties, Mechanization, and Verification

BDDLs, through their formal underpinnings, facilitate:

• Mechanizable translation to model checking (e.g., CTL/LTL specifications from SBT DSLs (Serbinowska et al., 21 Nov 2024)).
• Direct mapping of behavior specification to code (e.g., Haskell implementations of behavior primitives (Biggar et al., 26 Nov 2024)).
• Soundness and partial correctness via type systems (as proven for linear logic-based approaches (Martens et al., 2018)).
• Automation of requirements analysis, integration, and defect detection using similarity and equivalence metrics in mathematical frameworks for BTs (Ahmed et al., 2014).

This suggests BDDL frameworks are critical for rigorous software, agent, and game design workflows, where robustness and verifiability are required.

6. Relation to Domain Theory and Information Interoperability

Generalizing beyond agent and game domains, foundational work in informatics (Orthuber, 2018) formalizes domain specification through Domain Vectors (DV) and Domain Spaces (DS):

• DV: $(\mathrm{UL}, \text{sequence of numbers})$, tying a data item to an explicit, globally addressable domain definition (UL: uniform locator).
• DS: A metric space of DVs, enabling comparison, similarity search, and precise, language-independent interoperability.

A plausible implication is that future evolutions of BDDL may closely align with online, metric-embedded registries of domain definitions to support global interoperability, precision, and machine-synthesizable semantics in complex behavioral domains.

7. Significance, Challenges, and Future Directions

BDDLs address foundational challenges in autonomous systems, simulation, verification, and software engineering by:

• Enabling declarative, unambiguous specification of behavioral contexts and requirements.
• Supporting composability, scalability, and automated reasoning in diverse domains.
• Providing a bridge between high-level requirements and formal, executable models.

Remaining challenges include:

• Defining extensible, universally agreed taxonomies across domains.
• Standardizing the integration of explicit metrics (distance functions) for behavioral similarity.
• Automating the synthesis and refinement of BDDL definitions from natural language requirements.
• Ensuring that BDDL frameworks remain accessible and modifiable as domains evolve.

BDDLs continue to evolve as a core technology for formal behavioral specification, interoperability, and automated reasoning in AI, autonomous systems, games, and beyond.