Automated Theorem Proving in Logic Systems
Summary
Automated theorem proving (ATP) encompasses a range of computational techniques for establishing the validity of logical statements within formal systems. These systems employ proof calculi—such as resolution, superposition and sequent calculi—to derive conclusions from given axioms and hypotheses. Initially developed for first-order logic, ATP has matured to address higher-order logics that admit quantification over predicates and functions, thereby enabling the formalisation and verification of complex mathematical theories and software systems. Core mechanisms include saturation procedures, which systematically apply inference rules until no new conclusions can be drawn, and specialised decision algorithms that ensure termination by detecting loops or redundant inferences. Integration with interactive proof assistants allows seamless cooperation between human guidance and automated search, enhancing both usability and trustworthiness. Applications span hardware and software verification, security protocol analysis and the mechanisation of mathematical proofs. Recent trends feature the use of machine learning for premise selection, the formal verification of prover architectures and the extension of ATP to large formal libraries, all of which contribute to the growing impact of automated reasoning across science and engineering.
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Automated Theorem Proving in Logic Systems publication trend
The graph below shows the total number of articles in automated theorem proving in logic systems across all publications each year (not limited to Nature Index journals).
Technical terms
Automated theorem proving: Algorithmic methods for mechanically verifying the validity of logical formulae.
Saturation: The process of exhaustively applying inference rules until no further conclusions can be inferred or a contradiction is reached.
Superposition: An inference rule combining equality reasoning with resolution, commonly used in equational theorem proving.
Higher-order logic: A logic extending first-order by permitting quantification over predicates and functions.
Sequent calculus: A proof system that represents deductions as sequents, emphasising structural properties of proofs.
Unification: The operation of finding substitutions that make disparate logical expressions identical.
References
- Proof Theory and Decision Procedures for Deontic STIT Logics. Journal of Artificial Intelligence Research (2024).
- A Comprehensive Framework for Saturation Theorem Proving. Journal of Automated Reasoning (2022).
- Superposition with Lambdas. Journal of Automated Reasoning (2021).
- Efficient Full Higher-Order Unification. Logical Methods in Computer Science (2021).
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