Logic-RL Framework and Relational LP

• Logic-RL Framework is an approach that unifies logical representations and reinforcement learning using relational LP to enable scalable and interpretable policy synthesis.
• It leverages symmetry-aware lifting to reduce the number of variables and constraints, thereby boosting computational efficiency in complex relational settings.
• Applications span Markov Decision Processes, Markov Logic Networks, and LP-based SVMs, demonstrating its impact on structured, dynamic decision-making tasks.

Logic-RL Framework

The Logic-RL framework encompasses a class of approaches that integrate formal logical representations with reinforcement learning (RL), aiming to enhance the expressiveness, efficiency, interpretability, and scalability of model specification, policy synthesis, and inference in relational and structured environments. A prominent instantiation of this paradigm is relational linear programming (RLP), which provides a declarative framework that combines logic programming constructs (objects, relations, quantified variables, rules) with the mathematical formalism of linear programs (LPs). This synthesis enables the relational specification of optimization tasks—automatically instantiated via logical knowledge bases—whose combinatorial explosion is mitigated by symmetry exploitation through lifted linear programming. The RLP framework is notably contrasted with classical LP template languages, such as AMPL, both in language design and computational workflow. Empirical evaluations demonstrate the effectiveness of RLP on tasks ranging from approximate inference in Markov logic networks and MDP solution to collective classification with LP-SVMs, underscoring its role as a foundational approach for logic-based reinforcement learning.

1. Relational Linear Programs: Foundations and Syntax

A relational linear program (RLP) is a declarative LP template where the objective function and constraints are specified in terms of logical objects, relations, and variables, rather than explicit index sets or fixed matrices. The essential components are:

• Relational Variables and Atoms: Variables in the LP are indexed by logical atoms (e.g., predicates such as edge(X,Y)), implicitly quantifying over all individuals defined in the logical knowledge base (LogKB).
• Quantification and Summation: Expressions like sum{edge(X,Y)} flow(X,Y) denote summation over all tuples (X,Y) satisfying the predicate edge, parallel to Prolog-style set-valued queries.
• Declarative Semantics: An RLP, together with a LogKB (logical facts and rules specifying objects and relations), induces a grounded LP by instantiating the template with all valid atoms, yielding the canonical LP triplet (A, b, c) (constraint matrix, right-hand side, cost vector).

This formulation allows for parameterized problem statements where the size and structure of the ground LP adapt automatically to the logical data, without explicit enumeration.

2. Distinction from Traditional LP Template Languages

Traditional LP systems, exemplified by AMPL, require propositional indexing constructs and imperative descriptions to generate ground instances, necessitating manual enumeration and model-instance coupling. RLPs depart from this paradigm through:

• Logical Template Structure: LPs are specified using logical predicates and clauses, with index sets implicit in logical queries, e.g., {edge(X,Y)} rather than {j in P}.
• Separation of Model and Data: The RLP acts as an invariant template, while the LogKB varies across problem instances, allowing for automatic reuse and minimal code adaptation.
• Intuitive Representation: Optimization over variable-sized, structured data is encoded in a compact, human-oriented language, facilitating modeling of systems with dynamic relational content.
• Automatic Grounding and Lifting: The workflow decouples declarative specification from instance-level instantiation and symmetric reduction, streamlining both comprehension and execution.

3. Applications: Inference and Learning over Structured Domains

RLPs are deployed for several prototypical AI and RL tasks:

• Markov Decision Processes (MDPs): The value function optimization is expressed relationally as constraints akin to Bellman inequalities:

$$\text{value}(S) \leq \text{reward}(S, A) + \gamma \sum_{\text{transProb}(S, T, A)} \text{transProb}(S, T, A) \cdot \text{value}(T)$$

Experiments on gridworld domains show that symmetry-aware (lifted) RLP models yield reduced LP size and faster solution times.

• Markov Logic Networks (MLNs): MAP inference is posed as an LP over marginal atoms, with objective and constraints specified relationally. Weighted clauses in the LogKB dictate template generation, and symmetry structure is leveraged in the grounded LP.
• LP-SVMs for Collective Inference: The LP-based SVM formulation is encoded using relational predicates for weights, hyperplanes, and constraints across training examples, with relational linkage (e.g., citations) enforced via additional logical constraints. Empirical results on networked datasets such as CORA demonstrate lower prediction error through collective regularization.

These applications illustrate that RLPs not only support generic template definition but also unlock computational gains via symmetry-driven lifting strategies.

4. Lifted Linear Programming and Symmetry Reduction

A central technical contribution is the lifted resolution of grounded LPs via detection and exploitation of symmetries:

• Symmetry Structure: After grounding, the LP may contain many equivalent variables/constraints due to relational isomorphisms. Symmetries are formalized as fractional automorphisms—pairs of doubly stochastic matrices (X, Y) satisfying $X_Q A = A X_P$, $c^T X_P = c^T$, $X_Q b = b$.
• Equitable Partitioning and Projection: The variable and constraint symmetries are grouped into equivalence classes (equitable partitions), and the LP is projected onto the fixed space defined by these partitions. If $x = \mathbb{B} y$, with $\mathbb{B}$ the characteristic matrix of the partition, the reduced LP reads:

$$\min_{y} c^T \mathbb{B} y \quad \text{s.t.} \quad A \mathbb{B} y \leq b$$

This dimensionality reduction can dramatically decrease the number of variables and constraints.

• Algorithmic Tools: Efficient color-passing algorithms detect symmetries; grounding and lifting are formalized in explicit pseudocode (Algorithm 1 and 2), automating the pipeline from template to solution.

Symmetry-based lifting is crucial for the tractable application of LP relaxations in large, relationally-structured domains.

5. Implications for Logic-Based Reinforcement Learning

The RLP framework is foundational for logic-based reinforcement learning (Logic-RL) in several respects:

• Declarative Model Specification: RL tasks in complex domains—where the number of states or relations is not known a priori—can be formulated relationally, enabling generalization across instances and structure-aware learning.
• Automated Instance Handling: Logic-driven template instantiation and instance grounding minimize manual effort and error when scaling to new environments, as is common in RL applications.
• Symmetry-Preserving Policy Synthesis: By exploiting inherent task symmetries, Logic-RL can achieve scalable policy evaluation or improvement—an essential advantage for high dimensional or relational state/action spaces.
• Compatibility with Classical RL Algorithms: The RLP approach provides a natural embedding of value function or policy optimization (including linear relaxations of dynamic programming equations) within a logical modeling framework, allowing seamless integration with off-the-shelf LP solvers.

This theoretically grounded and computationally effective integration positions RLP as a blueprint for Logic-RL systems combining the strengths of symbolic knowledge representation and mathematical optimization.

6. Limitations, Open Problems, and Future Directions

While RLP facilitates model expressivity and computational efficiency, several challenges persist:

• Scalability of Symmetry Detection: As instance sizes and the complexity of logical knowledge bases grow, the cost of symmetry detection and equitable partitioning may become significant.
• Quality of Lifting: Excessive symmetry reduction may cause loss of granularity, potentially undermining solution quality in settings requiring fine distinction between states or actions.
• Generalization to Nonlinear or Integer Domains: While RLP is well-aligned with linear relaxations, direct extensions to integer or nonlinear programming in the presence of logic-based symmetry remain less developed.
• Interfacing with Policy Representations: Translating lifted LP policies into actionable controllers for standard RL agents, especially in online or deep RL settings, may require further development.

Ongoing work seeks to expand RLP to broader classes of convex programs, improve automated detection of relevant symmetries, and develop frameworks for integrating statistical relational learning and deep function approximation with logical LP templates.

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In summary, the Logic-RL framework exemplified by relational linear programming enables the unification of logic programming and linear optimization for RL and AI applications. By declaratively specifying models over objects, relations, and variables and leveraging symmetry-aware solution strategies, it offers a principled mechanism for scalable, interpretable, and reusable policy and inference computation over structured domains. This approach underpins further advances in logic-based and relational RL, particularly where scalability and modularity are essential.