Symbolic Markov Decision Process

• Symbolic Markov Decision Processes (SMDPs) are models where states, transitions, and rewards are represented symbolically using logic-based or algebraic encodings.
• They enable efficient manipulation of large state spaces through operations like Boolean set operations, image/preimage computation, and fixpoint iterations.
• SMDPs support advanced applications in formal verification, AI planning, and hybrid control by facilitating scalable, compact, and dynamic system modeling.

A Symbolic Markov Decision Process (SMDP) is a Markov Decision Process in which the state space, transition structure, and reward function are represented and manipulated symbolically rather than explicitly enumerated. This symbolic treatment is essential for handling large or structured models, such as those defined in terms of logic-based, factored, or parametric descriptions, or for models whose analysis requires sophisticated set operations, quantified reasoning, or compact encoding of transition relations. SMDPs provide a pivotal computational foundation for formal verification, planning, and synthesis in AI, model checking, and decision-theoretic control.

1. Formal Foundations and Symbolic Encoding

A classical finite MDP is a tuple $M = (S, A, P, R, \gamma)$, with state set $S$, action set $A$, transition kernel $P:S \times A \rightarrow \Delta(S)$, reward function $R:S \times A \rightarrow \mathbb{R}$, and discount factor $\gamma$. In an SMDP, each of these components is described and manipulated at the level of logical, algebraic, or decision-diagram representations that encode sets, functions, and relations succinctly:

• States and actions: Instead of explicit enumeration, symbolic MDPs use factored representations over variables or predicates, as in PDDL or RDDL, or compositionally as assignments to Boolean/numeric variables (Núñez-Molina et al., 2023).
• Transition relation: Transitions may be compactly represented by Boolean characteristic functions $\chi_E(x, x')$, algebraic decision diagrams (ADDs), or logical formulas specifying the effect of actions; e.g., the binary relation $T(x, x')$ holds iff $(x, x')$ are connected (1804.00206).
• Reward: Symbolic SMDPs encode rewards as symbolic expressions over state (and action) variables, possibly piecewise-defined (Sanner et al., 2012).

All set and function manipulations (e.g., set union, intersection, image under transitions, maximization) are implemented as symbolic operations, as opposed to looping over explicit tables. The fundamental toolkit consists of characteristic functions, Boolean formulas, BDDs/OBDDs, ADDs, XADDs, and related structures (1804.00206, Sanner et al., 2012, Núñez-Molina et al., 2023).

2. Core Symbolic Operations

Symbolic algorithms for SMDPs center on manipulating sets and functions via their Boolean or algebraic encodings:

• Boolean set operations: Union, intersection, and complement of state sets are performed as pointwise logical operations on their characteristic functions (e.g., $S \cap R = \{x : \chi_S(x) \land \chi_R(x)\}$) (1804.00206).
• Image/preimage: Symbolic image operators compute the set of one-step successors (or predecessors) of a set under the transition relation, e.g., $$\operatorname{Pre}(Z) = \{x \mid \exists x'.\, T(x, x') \wedge \chi_Z(x')\}$$.
• Controllable predecessor: In MDPs, the CPre operator discriminates between player and random states, combining existential and universal quantification over transitions in the symbolic domain.
• Fixpoint and attractor computations: Symbolically express $\mu$-calculus style iterations, e.g., repeatedly applying Pre or CPre to compute attractor sets or fixed-points necessary for reachability, safety, or Streett objectives.

Decision-diagram based structures (OBDD, ADD, MTBDD, XADD) enable efficient support for these operations, even as the explicit MDP would be intractably large (1804.00206, Sanner et al., 2012).

3. Symbolic Dynamic Programming and Strategy Synthesis

Symbolic dynamic programming (SDP) generalizes classical Bellman backups to settings where value functions and policies are represented as diagrams or symbolic expressions:

• Symbolic Bellman update: For a value function $V_k$ encoded as an ADD/XADD, the update

$$V_{k+1}(s) = \max_{a \in A} \left[ R(s, a) + \gamma \sum_{s'} P(s, a, s') V_k(s') \right]$$

is carried out by sequentially composing symbolic operations: addition, multiplication, existential quantification (sum/integration), and maximization, using recursive Apply and dynamic programming over the structure (Sanner et al., 2012, Núñez-Molina et al., 2023).

• Piecewise and continuous models: Symbolic DP can be applied to discrete, continuous, or hybrid models by using XADDs that support regression, integration, and case-based symbolic maximization (Sanner et al., 2012).
• Strategy iteration and symblicit methods: In mean-payoff or stochastic shortest path settings, symblicit algorithms combine symbolic manipulation (e.g., via pseudo-antichains) with explicit numerical solution on reduced quotients (from bisimulation lumping), enabling solution of extremely large monotonic MDPs (Bohy et al., 2014, Bohy et al., 2014).

4. Symbolic Model Checking and $\omega$-Regular Objectives

SMDPs are central to formal methods for verifying properties of reactive and stochastic systems specified by temporal logic or automata over infinite traces:

• Fairness and $\omega$-regular objectives: Symbolic algorithms address model checking of Streett, Büchi, and parity objectives by computing almost-sure (or positive) winning regions via fixpoint and attractor operations over OBDD/BDD encodings (1804.00206, Chatterjee et al., 2011).
• Sub-quadratic symbolic algorithms: Recent advances achieve $O(n \sqrt{m \log n})$ symbolic-step complexity for Streett-MDPs, employing techniques such as lock-step search, maintaining sets of "heads"/"tails" of changed edges, and switching between global and local SCC/MEC recomputation as needed (1804.00206).
• Incremental "win-lose" decomposition: Symbolic MDP algorithms can incrementally discover both winning and losing sets using symbolic SCC decomposition, with linear symbolic-step complexity in certain regimes (Chatterjee et al., 2011).

5. Symbolic Representation Languages and Logical Abstractions

Modern SMDP frameworks often leverage logic-based knowledge representation:

• Factored, relational encodings: PDDL, PPDDL, RDDL, pBC+, and other declarative languages describe SMDPs by first-order rules, effect axioms, and utility/reward constructs. State and action spaces are grounded from object sets and schema, transitioning via action-effect rules (Núñez-Molina et al., 2023, Wang et al., 2019).
• Succinctness and elaboration tolerance: Symbolic causality (e.g., in pBC+) prevents the blow-up found in naively grounded MDPs, as symbolic static/dynamic laws prune infeasible states and actions, and indirect/transitive effects are encoded once (Wang et al., 2019).

Table: Comparison of Symbolic MDP Representation Frameworks

Framework	Representation	Core symbolic structure
SPUDD	Boolean factored	ADD
SDP+XADD	Hybrid (discrete/cont.)	XADD
OBDD/BDD	Propositional, finite	BDD
pBC+	Logic programming, ASP	LPMLN, logical rules

6. Hybrid and Advanced Symbolic Control Algorithms

Mature SMDP methodologies employ hybrid and neurosymbolic approaches:

• Symbolic advice in planning: Integration of logical constraints via SAT or QBF solvers into MCTS, pruning actions or rollouts at planning time according to temporally extended properties (Busatto-Gaston et al., 2020).
• Neurosymbolic hybrids: Recent reviewed frameworks use symbolic SMDP abstractions as high-level guides for neural reinforcement learning, enabling subgoal-directed policies, parameter sharing, and generalization across problem instances (Núñez-Molina et al., 2023).
• Pseudo-antichain representations: For MDPs with partial-order state structure, pseudo-antichains efficiently encode state sets for reachability and value iteration, enabling orders-of-magnitude memory reduction compared with MTBDD-based approaches (Bohy et al., 2014, Bohy et al., 2014).

7. Empirical Performance and Applications

Experimental validation demonstrates that symbolic algorithms for MDPs achieve significant scalability and effectiveness:

• Model checking: Sub-quadratic symbolic algorithms for Streett and parity objectives scale to previously intractable VLTS benchmarks, consistently reducing symbolic steps and wall-clock time (1804.00206).
• Planning and control: Symbolic SDP with XADDs produces exact optimal policies for discrete-continuous MDPs with non-rectangular, highly structured value functions (Sanner et al., 2012).
• Automated planning/verification: Logic-based SMDP formalisms (pBC+, PDDL) support succinct, elaboration-tolerant problem specifications that facilitate compositional modeling and incremental domain extensions (Wang et al., 2019, Núñez-Molina et al., 2023).
• Hybrid symbolic/sampling-based control: Symbolic advice in MCTS yields large empirical gains and theoretical guarantees, demonstrating efficacy on high-dimensional domains (e.g., Pac-Man MDP state-space ~$10^{16-23}$) (Busatto-Gaston et al., 2020).

In aggregate, symbolic MDPs form a core computational abstraction for formal verification, AI planning, and synthesis in settings where raw enumeration is prohibitive, enabling algorithmic solutions across discrete, continuous, logical, and hybrid domains.