6.3 Strategies for efficient resolution (set of support, subsumption)

Resolution in first-order logic can be a real headache. It's like trying to find a needle in a haystack. But don't worry, there are some tricks to make it easier.

Two key strategies can help: set of support and subsumption. These methods narrow down the search space and get rid of unnecessary stuff, making resolution way more efficient.

Strategies for Efficient Resolution

Motivation and Goals

• Resolution-based theorem proving can be computationally expensive due to the large search space and potential for generating redundant clauses
• Strategies for efficient resolution aim to reduce the search space, eliminate redundant clauses, and guide the resolution process towards the desired goal
• Common strategies include set of support, subsumption, unit preference, input resolution, and ordered resolution
• The choice of strategy depends on the specific problem domain, the structure of the clauses, and the desired trade-off between completeness and efficiency

Implementation Considerations

• Implementing efficient resolution strategies requires careful consideration of data structures, clause representation, and heuristics for selecting clauses and literals
  • Efficient data structures (subsumption indexes, tries) can speed up subsumption checks and clause retrieval
  • Clause representation should facilitate quick comparisons and manipulations (efficient literal ordering, indexing)
  • Heuristics guide the selection of clauses and literals for resolution (unit preference, literal ordering)

Set of Support Strategy

Concept and Motivation

• The set of support strategy focuses the resolution process on a subset of clauses called the "set of support," typically derived from the negated goal clause
• Only clauses in the set of support or clauses derived from them are allowed to participate in resolution steps
• The set of support strategy maintains the refutation completeness of resolution while significantly reducing the search space

Construction and Maintenance

• The initial set of support is constructed by including the negated goal clause and any clauses that share literals with it
• As the resolution proceeds, newly derived clauses that contain literals from the set of support are added to the set
  • This ensures that the resolution process remains focused on clauses relevant to the goal
  • Clauses not connected to the set of support are excluded, reducing the search space
• The set of support strategy helps prevent the generation of irrelevant clauses and guides the resolution towards the goal

Subsumption for Optimization

Concept and Purpose

• Subsumption is a technique used to eliminate redundant clauses during the resolution process
• A clause $C_1$ subsumes another clause $C_2$ if all the literals in $C_1$ are present in $C_2$, making $C_2$ redundant
  • Example: The clause P(x) ∨ Q(x) subsumes the clause P(a) ∨ Q(a) ∨ R(a)
• Subsumption helps reduce the size of the clause set, preventing the generation and storage of redundant clauses

Application and Efficiency

• Subsumption checks are performed whenever a new clause is generated through resolution
• If a newly generated clause is subsumed by an existing clause, it can be discarded without affecting the completeness of the resolution
• Efficient data structures, such as subsumption indexes or tries, can be used to speed up subsumption checks
  • Subsumption indexes organize clauses based on their literals, allowing quick retrieval of potentially subsuming clauses
  • Tries represent clauses as paths in a tree-like structure, enabling efficient subsumption comparisons

Resolution Strategies: Comparison

Variety of Strategies

• Set of support and subsumption are two fundamental strategies for efficient resolution, but there are other techniques that can be employed
• Unit preference gives priority to resolving unit clauses (clauses with a single literal) to derive new unit clauses, as they often lead to quick contradictions
• Input resolution restricts resolution steps to involve at least one clause from the original input set, preventing the generation of irrelevant clauses
• Ordered resolution imposes an ordering on the literals within clauses and allows resolution only on the highest literals, reducing the number of possible resolutions

Combination and Evaluation

• Strategies can be combined to achieve better performance, such as using set of support with subsumption or unit preference
  • Example: Applying subsumption checks within the set of support strategy to further eliminate redundant clauses
• The effectiveness of different strategies depends on the structure and characteristics of the problem domain
  • Some strategies may work well for certain types of problems (e.g., unit preference for problems with many unit clauses)
  • Other strategies may be more general-purpose and applicable across different domains (e.g., set of support)
• Empirical evaluations and benchmarking are often conducted to compare the performance of different strategies on various problem instances
  • Metrics such as the number of generated clauses, resolution steps, and runtime are used to assess efficiency
  • Comparative studies help identify the strengths and weaknesses of different strategies in different scenarios