Symbolic Dynamic Programming (SDP)

• Symbolic Dynamic Programming (SDP) is a method that utilizes symbolic representations to efficiently solve optimization problems with structured state spaces.
• It leverages canonical data structures such as ADDs, XADDs, and BDDs to perform efficient operations like integration, maximization, and constraint pruning across hybrid domains.
• Advanced techniques like bounded-error XADD compression and constraint-based pruning help manage exponential growth, making SDP scalable for MDPs, synthesis, and combinatorial enumeration.

Symbolic Dynamic Programming (SDP) is a class of dynamic programming algorithms that exploit symbolic representations to encode, manipulate, and solve dynamic programming or optimization problems whose state spaces, dynamics, and objectives exhibit high regularity or structure. SDP methodologies leverage canonical data structures such as algebraic decision diagrams (ADDs), extended ADDs (XADDs), and ordered binary decision diagrams (OBDDs), enabling compact and exact manipulation of piecewise or combinatorial functions over discrete, continuous, or hybrid domains. SDP has been developed for a wide spectrum of contexts, including Markov Decision Processes (MDPs) with Boolean and continuous state variables, symbolic boolean realizability and synthesis, online planning with real-time interaction, and enumerative combinatorics via systems of generating functions.

1. Symbolic Foundations and Representational Structures

SDP advances classical numeric dynamic programming by lifting value functions, transition kernels, and system constraints to symbolic forms:

• ADDs/BDDs efficiently encode mappings $\{0,1\}^n \to \mathbb{R}$ or $\{0,1\}^n \to \{0,1\}$, and support operations such as pointwise addition, multiplication, maximization, and variable elimination, which are required for dynamic programming backups (Feng et al., 2012).
• XADDs generalize ADDs to support mixed Boolean and continuous variables with piecewise polynomial or linear functions in leaves, and decision nodes formed by arbitrary linear constraints (Sanner et al., 2012, Vianna et al., 2013).
• Symbolic Power Series Engines replace table-based DP with the manipulation of ordinary generating functions (OGFs) indexed by structured subproblem states, leading to systems of functional or algebraic equations in combinatorial enumeration (Ekhad et al., 2020).
• Project-Join Trees and Tree Decomposition guide bottom-up and top-down DP in symbolic Boolean synthesis, exploiting primal graph structure and providing guarantees on intermediate symbolic object sizes (Lin et al., 2024).

In all cases, these representations support lazy, cache-efficient, and structure-exploiting algorithms, delivering substantial savings in space and computational complexity when problem structure is amenable.

2. Symbolic Dynamic Programming for Hybrid and DC-MDPs

For hybrid and discrete-continuous MDPs, SDP formalizes the value backup using a joint symbolic treatment of Boolean and continuous variables. The central recursion for horizon-$H$ value functions $V_h(b, x)$ (with Boolean $b$ and continuous $x$) and quality functions $Q_h(b, x, a, y)$ (with parameterized actions $a(y)$) is:

$$Q_h(b,x,a,y) = \int_{b',x'} P(b',x'|b,x,a,y) [R(b,x,a,y,b',x') + \gamma V_{h-1}(b',x')] \, db' dx'$$

$$V_h(b,x) = \max_{a \in \mathcal{A}} \max_y Q_h(b,x,a,y)$$

This recursion is implemented symbolically, so that all intermediate functions (transition kernels, rewards, value functions) remain in XADD form (Vianna et al., 2013, Sanner et al., 2012).

XADDs enable:

• Efficient integration over Dirac-delta transition models via substitution.
• Symbolic maximization by introducing new partitioning tests.
• Marginalization over Boolean variables by table-driven summation with constraint pruning.

Constraint-based pruning can be invoked to cull infeasible case-paths via LP feasibility, substantially reducing graph size in practical domains (Sanner et al., 2012).

However, symbolic exact SDP in these domains typically engenders exponential growth in XADD size—especially as maximization or integration introduces new partitions—despite the substructure sharing in XADDs (Vianna et al., 2013).

3. Bounded-Error Approximate SDP via XADD Compression

To address intractable XADD growth, a bounded-error XADD compression procedure (XADDComp) enables scalable approximate SDP with quantifiable error bounds (Vianna et al., 2013). The method proceeds by:

• Merging pairs of leaf regions $d_1:f_1(x)$ and $d_2:f_2(x)$ into a single leaf $f^*(x) = c^T(x,1)$, minimizing the maximum absolute error over the union region $d_1 \cup d_2$.
• The error is defined as

$$\varepsilon(c) = \max_{i=1,2} \max_{x \in C(d_i)} |c^T(x,1) - C_i^T(x,1)|$$

where $C(d_i)$ are convex polytopes and $f_i(x) = C_i^T(x,1)$.

• This problem is cast as a constrained bilinear saddle point, then as a bilevel linear program where the maximum is achieved at a polytope vertex.

Constraint generation is employed to solve this LP to global optimality: at each iteration, constraints corresponding to the vertices with maximal error are added, and the master LP is resolved until all are satisfied or the error target is met.

Theoretical error guarantees state:

• The pointwise error introduced per merge does not exceed the per-merge bound $\varepsilon$.
• The total accumulated error across the full function is the maximum rather than the sum of per-merge errors.
• Embedding $\varepsilon$-compression into SDP yields, by induction, value function errors at most $$\Delta_h \leq \frac{1-\gamma^h}{1-\gamma}\varepsilon$$, and $\frac{\varepsilon}{1-\gamma}$ for infinite horizon.

Empirical results on benchmark domains (1D/2D Mars Rover, Inventory Control) demonstrate $10\times$-$20\times$ reductions in XADD node counts and total time, for $5$–$10\%$ per-backup errors, with realized value-function errors within $2\times$ the user-specified bound (Vianna et al., 2013).

4. SDP in Boolean Synthesis and Symbolic Model Checking

Symbolic DP extends to symbolic Boolean realizability and synthesis via BDD-driven algorithms (Lin et al., 2024). Given a CNF $\varphi(X,Y)$ with inputs $X$, outputs $Y$, and a tree decomposition with graded project-join trees:

• Bottom-up DP computes pre/post BDD-valuations at each node, projecting out internal variables as specified by the tree's graded partitioning.
• The realizability set $R_\varphi(X)$ is symbolically characterized as a BDD without explicit enumeration.
• Top-down DP then extracts witness functions for $Y$ by recursively substituting synthesized outputs into descendant BDD-valuations, ensuring that the synthesized outputs satisfy $\varphi$ for all $X$ where realizability holds.

This approach yields substantial time and memory benefits over heuristics-based BDD methods on a broad set of QBF and synthesis benchmarks, showing superior scalability for moderate treewidth (Lin et al., 2024).

5. SDP for Symbolic Generalization in Online Planning

In online planning and reinforcement learning, symbolic SDP generalizes the real-time dynamic programming paradigm (RTDP) to operate on aggregates of states sharing structure, as defined algorithmically by symbolic queries over ADDs/BDDs (Feng et al., 2012):

• Each on-line backup can update the value for a block $E$ of states defined by, e.g., value similarity or identical one-step reachability profiles.
• Updates exploit symbolic masking and existential abstraction in ADD space, efficiently propagating value updates through subspaces with shared structure.
• This delivers at least one order of magnitude reduction in required environment transitions and substantial wall-clock time improvements compared to classic, enumerative RTDP, particularly on factored MDP benchmarks with large discrete spaces.

6. Symbolic Dynamic Programming in Combinatorial Enumeration

SDP generalizes numeric DP recurrence solving in enumerative combinatorics by operating over systems of formal power series OGFs indexed by abstract state sets (Ekhad et al., 2020):

• The state-graph is a directed graph whose vertices encode distinguished combinatorial subproblems, with edges corresponding to symbolic recurrences.
• For pattern-restricted Dyck paths and subword avoidance problems, the SDP framework builds and solves closed systems of algebraic equations over the generating functions.
• Automation (e.g., in Maple) constructs the state-graph, encodes the enumerative recurrences, eliminates interior OGFs via Gröbner basis methods, and returns minimal defining algebraic equations.
• The approach yields closed-form algebraic (often D-finite) characterizations where the finite or finitely-parameterized state-graph closes, and enables direct derivation of $P$-recurrences and asymptotic results.

A major advantage is that the algorithmic engine is decoupled from individual forbidden-pattern specifications; once the SDP system is coded, applying to new pattern constraints is automatic (Ekhad et al., 2020).

7. Limitations, Complexity, and Prospective Directions

Despite the compactness and generality of symbolic representations, SDP faces inherent complexity challenges:

• Worst-case exponential growth in XADD/BDD size per horizon increment or per symbolic operation, albeit often mitigated in practice by structure and pruning (Sanner et al., 2012).
• Computational cost of symbolic maximization, integration, and constraint generation for large state or variable spaces.
• Constraint-based pruning in XADDs is only applicable for linear cases, with nonlinear pruning methods still an open problem (Sanner et al., 2012).
• Combinatorial explosion in the number of symbolic subproblems or SDP states in complicated combinatorial enumeration settings (Ekhad et al., 2020).
• Gröbner basis elimination may be intractable for large or highly interdependent symbolic equation systems (Ekhad et al., 2020).

Potential future developments include approximate symbolic compression (e.g., generalizing XADDComp and APRICODD), extension to general stochastic transition models (beyond Dirac-delta), integration with scalable heuristic and decomposition methods, and advances in canonical, tractable representations for nonlinear and high-dimensional partitions (Vianna et al., 2013, Sanner et al., 2012).

• "Bounded Approximate Symbolic Dynamic Programming for Hybrid MDPs" (Vianna et al., 2013)
• "Automatic Counting of Restricted Dyck Paths via (Numeric and Symbolic) Dynamic Programming" (Ekhad et al., 2020)
• "Dynamic Programming for Symbolic Boolean Realizability and Synthesis" (Lin et al., 2024)
• "Symbolic Generalization for On-line Planning" (Feng et al., 2012)
• "Symbolic Dynamic Programming for Discrete and Continuous State MDPs" (Sanner et al., 2012)