14.1 Automated Theorem Proving

Automated theorem proving uses computer programs to prove mathematical theorems. It employs formal logic, knowledge representation, and inference engines to derive new theorems from existing axioms and premises.

Key components include resolution and unification algorithms, which form the basis for proof search. However, limitations like undecidability and computational complexity pose challenges. Heuristics and strategies help navigate the vast search space efficiently.

Automated Theorem Proving

Principles of automated theorem proving

Top images from around the web for Principles of automated theorem proving

• 

  Formal verification of Matrix based MATLAB models using interactive theorem proving [PeerJ] View original

  Is this image relevant?

• 

  Symbolic computation | Sage supports symbolic computation us… | Flickr View original

  Is this image relevant?

• 

  ASIC-System on Chip-VLSI Design: Theorem proving View original

  Is this image relevant?

• 

  Formal verification of Matrix based MATLAB models using interactive theorem proving [PeerJ] View original

  Is this image relevant?

• 

  Symbolic computation | Sage supports symbolic computation us… | Flickr View original

  Is this image relevant?

1 of 3

Top images from around the web for Principles of automated theorem proving

• 

  Formal verification of Matrix based MATLAB models using interactive theorem proving [PeerJ] View original

  Is this image relevant?

• 

  Symbolic computation | Sage supports symbolic computation us… | Flickr View original

  Is this image relevant?

• 

  ASIC-System on Chip-VLSI Design: Theorem proving View original

  Is this image relevant?

• 

  Formal verification of Matrix based MATLAB models using interactive theorem proving [PeerJ] View original

  Is this image relevant?

• 

  Symbolic computation | Sage supports symbolic computation us… | Flickr View original

  Is this image relevant?

1 of 3

• Automated theorem proving uses computer programs to prove mathematical theorems
  • Employs formal logic and reasoning to derive new theorems from existing axioms and premises
• Key components of automated theorem proving systems include
  • Knowledge representation formalizes mathematical statements, axioms, and inference rules using languages like first-order logic, higher-order logic, and type theory
  • Inference engines apply inference rules to derive new theorems from existing knowledge using algorithms such as resolution, natural deduction, and term rewriting
  • Proof search strategies guide the exploration of the proof search space using heuristics and techniques like goal-oriented strategies, clause selection, and term ordering

Resolution and unification algorithms

• Resolution is a powerful inference rule for automated theorem proving that operates on the clausal form of logical statements (disjunction of literals) and resolves two clauses by finding a complementary pair of literals and generating a new clause
• Unification determines if two terms can be made identical by substituting variables with terms and finds the most general unifier (MGU) that makes terms identical while being as general as possible
• Resolution proof search systematically applies the resolution rule to derive the empty clause (contradiction) and is refutation complete, meaning if a set of clauses is unsatisfiable, resolution will derive the empty clause
• Efficient implementation techniques for resolution include indexing data structures for fast clause retrieval and subsumption and tautology elimination to prune redundant clauses

Limitations of automated proving

• Undecidability and computational complexity pose challenges as many interesting mathematical theories are undecidable (first-order logic) and theorem proving in expressive logics can be computationally expensive
• The explosive search space leads to the number of possible proof paths growing exponentially with the size of the problem, requiring efficient search space pruning and heuristics for scalability
• Automated provers lack human-like intuition and creativity, relying on systematic search and predefined inference rules, struggling with high-level reasoning, analogies, and domain-specific insights
• Formalizing informal mathematical statements is difficult as translating natural language theorems into formal logic requires expertise in both mathematics and formal logic

Heuristics in proof search

• Heuristics prioritize and select promising proof paths by estimating the likelihood of a clause leading to a successful proof using techniques like symbol counting, term weight, and literal ordering
• Strategies determine the overall proof search approach, including
  1. Goal-oriented strategies like set of support (focus on clauses derived from the negated goal) and unit preference (prioritize clauses with fewer literals)
  2. Saturation-based strategies such as the given clause algorithm (select the shortest clause for resolution) and discount (prefer clauses with symbols occurring in the goal)
• Machine learning techniques can learn heuristics and strategies from successful proofs and perform premise selection to identify relevant axioms and lemmas for a given conjecture
• Combining multiple strategies and heuristics through portfolio approaches (run different strategies in parallel) and strategy scheduling (switch between strategies based on progress and time allocation) can improve proof search effectiveness