Playgol System in ILP

• Playgol System is an ILP framework that integrates an unsupervised play phase for autonomous predicate discovery with a supervised build phase for task-oriented synthesis.
• Its two-stage approach enables systematic decomposition of complex tasks by inventing reusable predicates, thereby reducing both hypothesis and sample complexity.
• Empirical evaluations in robot planning and string transformations demonstrate substantial performance improvements through the reuse of play-discovered concepts.

The Playgol System is an inductive logic programming (ILP) framework designed to improve program synthesis via a principled integration of unsupervised exploration (“play”) and supervised task solving (“build”). Playgol operationalizes the hypothesis that, analogous to the way children learn by playfully interacting with their environment before confronting externally imposed tasks, a program induction agent can benefit from an autonomous exploration phase that results in the discovery and reuse of general-purpose program fragments. This paradigm is aimed at reducing both hypothesis and sample complexity in program induction, particularly for domains characterized by combinatorial explosion of instance spaces and the need for decompositional, multi-step reasoning.

1. Conceptual Foundations and Motivation

The central idea of Playgol is to augment supervised program induction pipelines with an explicit unsupervised “playing” stage. In standard ILP (and more specifically in meta-interpretive learning, MIL), the learner attempts to synthesize programs (hypotheses) for provided tasks given a set of training examples and background knowledge (BK). This supervised approach is limited by the scope of the initial BK and the combinatorial space of possible programs.

Playgol introduces an initial phase where the learner autonomously samples tasks (play tasks) from the instance space, attempts to synthesize programs that solve these play tasks, and integrates successful solutions as new predicate symbols into the BK. In subsequent supervised learning (the building stage), solutions to user-supplied tasks can be constructed by composing these previously discovered predicates. This methodology supports the systematic decomposition of complex tasks, predicate invention, and the reduction of the syntactic and sample complexity required to express and learn the user’s target concepts.

2. Architecture and Learning Protocol

Playgol operates in two sequential modes:

2.1 Playing Stage (Unsupervised)

• Sampling: Play tasks are uniformly sampled from the instance space without user guidance or label constraints.
• Synthesis: For each play task, Playgol employs a core MIL engine (Metagol) to attempt program synthesis. Clause depth is incrementally increased until a solution is found or capacity is reached.
• Knowledge Integration: Each successful play solution, including invented sub-predicates, is incorporated into the BK for future reuse. Predicate invention, a key feature of MIL, is essential and all non-top-level invented predicates are retained (_NH3 variant), further increasing the richness of the induced BK.
• Properties: This process is unsupervised and knowledge growth is governed by the internal structure and instance space of the domain, yielding spontaneously generalizable concepts.

2.2 Building Stage (Supervised)

• Task Solving: The learner addresses the user-supplied build tasks, leveraging the enriched BK, including play-discovered predicates, for program induction.
• Reuse and Composition: Solutions synthesize target programs as compositions of both primitive and invented predicates, thus reducing program length and complexity compared to direct, flat construction.

This two-stage regime enables the system to discover and reuse intermediate reusable abstractions, analogous to learning subprocedures or macros in human programming.

3. Theoretical Analysis and Sample Complexity

The Playgol framework is mathematically grounded via formal propositions regarding hypothesis space and PAC-style sample complexity within MIL:

• Hypothesis Space: Let $p$ be the number of predicate symbols, $m$ the number of metarules, and $j+1$ the arity of rule bodies. The number of expressible programs with $n$ clauses is $(m \cdot p^{j+1})^n$ (Proposition in the paper).
• Sample Complexity: For error $\epsilon$ and confidence $1-\delta$, the number of training examples $s$ required is bounded as

$$s \geq \frac{1}{\epsilon} \left( n \ln m + (j+1) n \ln p + \ln \frac{1}{\delta} \right)$$

• Playgol Improvement Theorem: If unsupervised play leads to the discovery of $c$ new predicates and this enables a reduction in the number of clauses needed to represent solutions from $n$ to $n-k$, then Playgol reduces sample complexity when:

$n \ln p > (n-k) \ln (p+c)$

This quantifies the trade-off: although the number of predicate symbols increases, the decreased textual complexity of target programs (fewer clauses) can result in strictly lower sample complexity.

4. Empirical Evaluation and Outcomes

Playgol’s efficacy is demonstrated in two domains:

4.1 Robot Planning

• Environment: A grid-based robot (and ball) environment with atomic actions (up, down, grab, drop), with instance spaces up to $6^8$ transitions.
• Task Generation: Build tasks are drawn by sampling 1000 atoms; play tasks are varied from 0 to 2000.
• Results:
  • Baseline (no play): Metagol achieves $\sim$12% (5×5 grid) and $\sim$7% (6×6 grid) task success.
  • With play: Performance increases to nearly 100% and over 60%, respectively, demonstrating a major boost in problem-solving capacity due to reuse of play-discovered concepts.
  • Saving invented predicates (NH3 variant) provides statistically significant improvements, validating the importance of storing all structure discovered during play.

4.2 Real-World String Transformations

• Data: A suite of 94 real-world string transformation tasks (after FlashFill), with examples as $f(\text{input}, \text{output})$ pairs.
• Play Protocol: Play tasks involve creating random string examples and searching for programs consistent with them.
• Results: Predictive accuracy improves from $\sim$25% to nearly 37% when increasing play tasks from 0 to 2000.
• Case Study: Solution programs assembled during the build stage effectively compose multiple reusable, recursively defined predicates synthesized during play (e.g., “make uppercase”, “skip to first uppercase”).

5. Algorithmic Details and Meta-Interpretive Learning Mechanism

Playgol relies on meta-interpretive learning, instantiated as Metagol:

• Metarule-Driven Search: The induction process is governed by instantiation of metarules (e.g., chain, precon, postcon, tailrec). These templates define the inductive bias and permissible program structures.
• Predicate Invention: The system’s ability to invent and reuse auxiliary predicates underpins its power to reduce clause complexity and facilitate composition.
• Enumeration and Depth Control: In the playing stage, clause depth is increased incrementally to avoid combinatorial explosion and ensure only compact concepts are discovered.
• Integration Workflow: Algorithmic pseudocode illustrates a loop where tasks are sampled, solved, and successful solutions are folded into the BK for subsequent stages.

6. Implications, Significance, and Limitations

Several critical implications arise from the Playgol paradigm:

• Autonomous Knowledge Bootstrapping: Playgol enables the self-discovery of background knowledge, obviating the need for extensive hand-engineered BK.
• Program Decomposition: The system automatically constructs libraries of higher-order components, paralleling modularization in software engineering and human learning.
• Generalization and Reuse: Empirically, the ability to reuse both top-level and invented predicates leads to performance improvements, particularly on tasks with challenging combinatorial structure.
• Complexity Management: By leveraging play, Playgol implicitly manages hypothesis space and sample complexity in a data-driven, domain-agnostic fashion.

Limitations include the need for judicious sampling of play tasks—excessive or irrelevant play can potentially introduce spurious concepts, and the theory of optimal play task scheduling remains incomplete. Additionally, although gains are pronounced in tested domains, further empirical evaluation in domains such as graphics programming is identified as a necessary direction.

7. Outlook and Relation to Broader Research

Playgol represents a systematic advancement in bootstrapped program induction, distinct from conventional, exclusively supervised ILP systems. By interleaving unsupervised conceptual exploration (play) with classic ILP, the architecture achieves a functionally richer and theoretically grounded synthesis pipeline. The demonstrated reduction in sample complexity and the capacity for autonomous predicate discovery position Playgol as an influential methodology for scalable, self-improving program induction systems.

Proposed future work includes further theoretical refinement of play task sampling strategies, application to new target domains, and investigation of semisupervised play protocols aligned with user objectives. The general principle—integrating an autonomous, unsupervised phase to strengthen subsequent supervised learning—offers a template for related methodologies in the broader machine learning and program synthesis landscapes (Cropper, 2019).