Term Factorized Crashes Exploring Alternatives For Cyclic Term Handling In SWI-Prolog

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Introduction

This article delves into the intricacies of the term_factorized/3 predicate in SWI-Prolog, particularly focusing on the challenges posed by cyclic terms and potential solutions that circumvent the reliance on compare/3. The discussion stems from a specific test case that triggers a stack overflow error when using term_factorized/3 with cyclic terms. This article explores alternative approaches to handle cyclic terms within term_factorized/3, aiming to provide a robust and efficient solution for term factorization in SWI-Prolog.

The core issue arises when term_factorized/3 encounters cyclic terms, leading to infinite recursion within the underlying comparison mechanisms. Current implementations, which often rely on compare/3 for term ordering and lookup structures, struggle with these cyclic structures, resulting in stack overflow errors. This article proposes exploring alternative strategies that sidestep the complexities of compare/3 by employing techniques like hashing with depth limits and relying on (==)/2 for term equality checks. By doing so, we can potentially create a more resilient and performant term_factorized/3 predicate that effectively handles both standard and cyclic terms.

The Problem: Stack Overflow with Cyclic Terms

The initial problem was identified through a concise yet revealing test case. The query below, when executed in SWI-Prolog (version 9.3.25), results in a stack overflow error:

/* SWI-Prolog 9.3.25 */
?- X=s(s(X,Y),Z), Y=s(Y,X), term_factorized(X, _, _).
ERROR: Stack limit (1.0Gb) exceeded
ERROR:   Stack sizes: local: 0.3Gb, global: 0.3Gb, trail: 81.8Mb
ERROR:     [6,038,429] rbtrees:insert2(<compound red/4>,
<compound s/2>, 1, <compound black/4>, _76517588, _76517590)

This error indicates a fundamental limitation in the current implementation of term_factorized/3 when dealing with cyclic terms. The error message points to the rbtrees:insert2 predicate, suggesting that the issue lies within the data structure used for term lookup and comparison. The use of red-black trees, which rely on compare/3 for ordering, becomes problematic when cyclic terms introduce infinite recursion during comparison.

The root cause of the stack overflow is the recursive nature of cyclic terms combined with the comparison-based approach used in the red-black tree insertion. When term_factorized/3 encounters a cyclic term, it attempts to compare it with existing terms in the tree. However, the cyclic nature of the term leads to an infinite loop in the comparison process, eventually exhausting the stack space. This highlights the need for a more robust approach that can handle cyclic terms without resorting to potentially infinite comparisons.

Why compare/3 Fails with Cyclic Terms

The compare/3 predicate is a cornerstone of term ordering in Prolog. It determines the relative order of two terms, returning <, =, or > depending on whether the first term is less than, equal to, or greater than the second term. However, compare/3's reliance on structural comparison makes it vulnerable to cyclic terms. When comparing two cyclic terms, compare/3 can enter an infinite recursion as it attempts to traverse the infinitely nested structure.

Alternative predicates like compare_with_stack/3 and compare_extensive/3, while designed to address some limitations of compare/3, do not fully resolve the issue with cyclic terms. These alternatives may provide some mitigation, but they do not fundamentally alter the comparison-based approach that leads to stack overflows. Therefore, a more radical solution is required to effectively handle cyclic terms within term_factorized/3.

The fundamental problem lies in the transitive nature of the comparison required by tree-based data structures like red-black trees. When inserting a new term, the tree needs to maintain its ordering invariant, which necessitates comparing the new term with existing terms. With cyclic terms, this transitive comparison can lead to an infinite loop, regardless of the specific comparison predicate used. The key takeaway is that relying solely on comparison for term lookup and factorization is inherently problematic when dealing with cyclic structures.

A New Angle: Hashing and (==)/2

The proposed solution shifts the focus away from comparison-based approaches and towards a hashing-based strategy. The core idea is to implement term_factorized/3 without relying on compare/3 for its primary lookup mechanism. This involves using a hash lookup structure that employs a hash function specifically designed to handle cyclic terms effectively.

A hash function with a depth limit can provide a practical solution. By limiting the depth of term traversal during hash computation, we can prevent infinite recursion when encountering cyclic terms. This approach generates a hash value that is representative of the term's structure up to a certain depth, effectively truncating the infinite cycle. The resulting hash value can then be used for efficient lookup in a hash table or similar data structure.

The second key component of this strategy is the use of (==)/2 for equality checking. After a potential match is found in the hash table based on the hash value, (==)/2 can be used to verify that the terms are indeed identical. The (==)/2 predicate performs a syntactic equality check without attempting to order the terms, making it safe to use with cyclic terms. This combination of hashing with a depth limit and (==)/2 for equality provides a robust and efficient way to handle cyclic terms in term_factorized/3.

Implementing term_factorized/3 with Hashing

To implement this hashing-based approach, we need to design a suitable hash function and a hash lookup structure. The hash function should be efficient to compute and should distribute terms evenly across the hash table to minimize collisions. A depth-limited hash function, as described earlier, can effectively handle cyclic terms by preventing infinite recursion.

For the hash lookup structure, a simple hash table or a more sophisticated hash-based data structure like a cuckoo hash table can be used. The choice of data structure depends on the performance requirements and the expected number of terms to be factorized. A hash table provides fast average-case lookup times, while cuckoo hashing offers excellent worst-case performance.

The algorithm for term_factorized/3 using hashing would involve the following steps:

  1. Compute the depth-limited hash value of the input term.
  2. Lookup the hash value in the hash table.
  3. If a match is found, use (==)/2 to verify that the terms are syntactically equal.
  4. If the terms are equal, return the existing factorization.
  5. If no match is found or the terms are not equal, create a new factorization and insert it into the hash table.

This approach avoids the pitfalls of compare/3 and provides a more robust solution for handling cyclic terms in term_factorized/3.

Advantages of the Hashing Approach

The hashing-based approach offers several advantages over traditional comparison-based methods for handling cyclic terms in term_factorized/3:

  • Robustness: The depth-limited hash function and (==)/2 provide a safe and reliable way to handle cyclic terms without risking stack overflows.
  • Efficiency: Hashing provides fast average-case lookup times, making term factorization more efficient.
  • Simplicity: The algorithm is relatively straightforward to implement and maintain.
  • Scalability: Hash-based data structures can scale well to handle a large number of terms.

By sidestepping the complexities of compare/3, this approach offers a more practical and performant solution for term factorization in SWI-Prolog, especially when dealing with cyclic terms. The use of a depth-limited hash function ensures that the hashing process remains bounded, preventing infinite recursion and stack overflows. Furthermore, the reliance on (==)/2 for equality checks provides a safe and efficient way to verify term identity without resorting to potentially problematic comparison operations.

Fuzzy Testing: The Key to Discovery

It is worth noting that the problematic test case was discovered through fuzzy testing. Fuzzy testing is a technique that involves generating random or semi-random inputs to a program to uncover unexpected behavior or vulnerabilities. In this case, fuzzy testing proved invaluable in identifying the stack overflow issue with cyclic terms in term_factorized/3.

The success of fuzzy testing in this scenario highlights its importance in software development, particularly for uncovering edge cases and unexpected interactions. By systematically exploring a wide range of inputs, fuzzy testing can reveal potential problems that might be missed by traditional testing methods. This underscores the value of incorporating fuzzy testing into the development and maintenance process of Prolog systems.

Conclusion

The challenge of handling cyclic terms in term_factorized/3 highlights the limitations of comparison-based approaches when dealing with recursive data structures. The proposed solution, which leverages hashing with a depth limit and (==)/2 for equality checking, offers a promising alternative that avoids the pitfalls of compare/3. This approach provides a more robust, efficient, and scalable solution for term factorization in SWI-Prolog.

By shifting the focus from comparison to hashing, we can effectively handle cyclic terms without risking stack overflows. The combination of a depth-limited hash function and (==)/2 ensures that the factorization process remains bounded and efficient. Furthermore, the discovery of the problematic test case through fuzzy testing underscores the importance of this technique in uncovering edge cases and ensuring the robustness of Prolog systems. This exploration paves the way for a more resilient and performant term_factorized/3 predicate that can effectively handle both standard and cyclic terms, enhancing the overall capabilities of SWI-Prolog.