7.3 Heuristics and optimizations in ATP

Automated Theorem Proving (ATP) relies on smart search strategies to tackle complex problems. Heuristics, pruning, and caching are key tools that make ATP more efficient and powerful.

These techniques help ATP systems navigate vast solution spaces, avoid redundant work, and find proofs faster. By cleverly guiding the search and reusing previous results, ATP can solve tougher problems and scale to larger domains.

Heuristics for ATP Search

Heuristic Functions and Admissibility

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• Heuristics efficiently guide search through large problem spaces towards a solution without guaranteeing optimality
• Common heuristic functions estimate the "goodness" of a state by calculating the estimated distance or cost to the goal state
  • Admissible heuristics never overestimate the cost to reach the goal and guarantee an optimal solution will be found if one exists (straight-line distance in path planning)
  • Non-admissible heuristics may overestimate cost but tend to find solutions faster, though they may be suboptimal (using the average of multiple admissible heuristics)
• The choice of heuristic significantly impacts search efficiency and solution optimality
  • Stronger heuristics provide tighter bounds but may be more computationally expensive to calculate
  • Weaker heuristics are faster to compute but may expand more nodes during the search

Domain-Specific Heuristics and Pattern Databases

• Heuristics can be designed using domain knowledge to estimate the number of steps to the goal state (number of misplaced tiles in the 8-puzzle)
• Pattern databases are a type of heuristic that pre-compute exact solution costs for subproblems
  • The pre-computed costs are then used as admissible estimates for the overall problem
  • Pattern databases trade off increased memory usage for faster heuristic computation during search
• Examples of domain-specific heuristics:
  • Manhattan distance for sliding puzzle problems
  • Number of pieces threatened in chess
  • Sum of distances of each block from its goal position in block-stacking problems

Pruning Techniques for ATP

Eliminating Redundant and Suboptimal Branches

• Pruning removes branches from the search tree that are determined to be redundant, irrelevant, or suboptimal
  • Reduces the effective branching factor of the search space
  • Allows the search to focus on more promising paths
• Cycle checking detects when a state has been previously visited and eliminates cycles by not expanding those repeated states again
• Alpha-beta pruning tracks the current best options for the maximizing and minimizing agents in adversarial search
  • Stops evaluating a move when at least one possibility is found to be worse than a previously examined move
  • Alpha represents the best choice found so far for the maximizing player
  • Beta represents the best choice found so far for the minimizing player
  • The search updates alpha or beta values and prunes remaining branches when alpha becomes greater than or equal to beta

Heuristic-Based Pruning and Forward Pruning

• Killer moves and the history heuristic preferentially explore moves that were found to be good in other branches
  • Potentially eliminates less promising options without fully expanding them
  • Killer moves store a fixed number of previously successful moves at each search depth
  • The history heuristic maintains a global table of move scores based on how often they cause cutoffs
• Forward pruning techniques use shallow searches to estimate the bound of a position before doing a deeper search
  • Null-move pruning makes a "null" move, giving the opponent two consecutive moves, and searches to a reduced depth
    • If the null-move search result exceeds beta, the position is assumed to be a cut-node and is pruned
  • ProbCut uses a shallow search with a wide alpha-beta window to prove that a move is likely to be a cut-node, pruning the remaining branches

Caching and Memoization for ATP

Avoiding Redundant Computations with Caching

• Many search algorithms perform redundant computations, revisiting the same states multiple times
• Caching stores previously computed results to speed up subsequent calculations
  • Comes at the cost of increased memory usage
  • The optimal cache size depends on the available memory and the size of the search space
• Cache lookups incur some overhead, so the cached results must be sufficiently expensive to compute to make caching worthwhile
• Examples of caching in ATP:
  • Storing evaluated positions in chess engines
  • Caching heuristic values in A* search
  • Memoizing expensive function calls in game trees

Transposition Tables and Replacement Schemes

• Transposition tables are a type of cache that stores previously seen positions and their evaluation scores or best moves
  • Allows quickly looking up results if the same position is encountered again
  • Zobrist hashing efficiently computes a unique hash key for each state, enabling fast lookups
• Replacement schemes determine which cache entries to overwrite when the table is full
  • Always Replace: Overwrites the least recently used entry
  • Depth-Preferred: Prefers to keep entries from deeper searches
  • Two-Level Tables: Uses a small "hot" table for frequently accessed entries and a larger "cold" table for less frequent ones
• Transposition tables are widely used in chess engines and other game-playing programs to avoid redundant work

Impact of Optimizations on ATP Performance

Improved Efficiency and Scalability

• Heuristics and optimizations can dramatically improve the efficiency and scalability of ATP systems
  • Allows solving larger and more complex problems than brute-force search
  • Reduces the effective branching factor and search space size
• Admissible heuristics guarantee optimal solutions but may expand more nodes than non-admissible heuristics
  • The optimal heuristic depends on the goal of minimizing search time vs. solution cost
• Techniques like cycle checking, transposition tables, and pruning significantly reduce the number of nodes expanded
• Caching and memoization provide significant speedups for search algorithms that perform many redundant calculations
  • Increases memory overhead and code complexity

Performance Metrics and Tradeoffs

• Key metrics for evaluating the impact of heuristics and optimizations on search performance:
  • Execution time
  • Nodes expanded
  • Solution quality (path cost)
• Profiling tools can identify performance bottlenecks and guide optimization efforts to the most expensive and frequently called parts of a program
• Heuristics and optimizations often involve tradeoffs between computation time, memory usage, implementation complexity, and solution quality
  • The appropriate balance depends on the specific application and available resources
  • Examples of tradeoffs:
    • Admissible vs. non-admissible heuristics
    • Cache size vs. memory usage
    • Pruning aggressiveness vs. risk of missing optimal solutions