An [[Automated Theorem Prover]], also known as an [[ATP]], is a computer program or system designed to automatically prove or disprove mathematical theorems. It uses formal logic and reasoning to analyze statements or formulas and determine their truth value. The main goal of an Automated Theorem Prover is to provide a reliable and efficient method for proving theorems without relying on human intervention. It is particularly useful in complex mathematical fields where manual proofs can be time-consuming, error-prone, or even impossible due to the sheer complexity of the problem. Automated Theorem Provers work by employing various algorithms and techniques from logic, mathematics, and computer science. They typically take a set of axioms, rules of inference, and a target statement as input. The program then attempts to derive a proof for the target statement based on the given axioms and inference rules. To accomplish this, ATPs may use techniques such as resolution-based methods, model checking, term rewriting systems, or even artificial intelligence algorithms like machine learning. These techniques allow ATPs to search for possible proof paths in a systematic manner while ensuring completeness (proving true statements) and soundness (not proving false statements). Automated Theorem Provers have numerous applications across various fields. In mathematics, they can help prove new theorems or verify existing ones. In computer science and software engineering, they are used for formal verification of software correctness or hardware designs. They are also utilized in artificial intelligence research for knowledge representation and reasoning. Despite their power and utility, Automated Theorem Provers still face certain limitations. Some problems may be undecidable or require excessive computational resources to solve effectively. Additionally, ATPs heavily rely on the quality of input axioms and inference rules; hence incorrect or incomplete information can lead to incorrect results. # Commonly Known ATPs Automated theorem provers are software tools or systems that use formal logic to automatically prove theorems or validate mathematical statements. They play a crucial role in fields such as mathematics, computer science, and artificial intelligence. Here are some commonly known implementations of automated theorem provers: 1. Prolog: Prolog is a popular programming language used for artificial intelligence and logic programming. It is often used as a platform for implementing automated theorem provers. Prolog-based systems typically use resolution-based inference methods to prove theorems. 2. Isabelle: Isabelle is an interactive theorem prover that supports various logics, including classical higher-order logic and constructive type theory. It provides a user-friendly environment for proving theorems using tactics and proof commands. 3. HOL (Higher Order Logic) systems: HOL is a family of theorem proving systems based on higher-order logic. HOL Light, HOL4, and Isabelle/HOL are some well-known implementations of HOL systems. 4. [[Coq]]: Coq is an interactive proof assistant based on dependent type theory. It allows users to construct formal proofs interactively, providing strong guarantees about their correctness. Coq has been widely used in both academia and industry for verifying software and proving mathematical theorems. 5. ACL2: ACL2 (A Computational Logic for Applicative Common Lisp) is a theorem prover based on first-order logic with equality. It uses rewriting techniques and induction to prove theorems about computer hardware, software, and mathematics. 6. [[Lean]]: Lean is an open-source interactive theorem prover developed at Microsoft Research. It supports dependent type theory and provides tools for constructing formal proofs using tactics. 7. E-Prover: E-Prover is an automated theorem prover based on first-order resolution calculus with equality reasoning capabilities. It has been successful in solving problems from various domains like mathematics, computer science, and hardware verification. 8. Vampire: Vampire is a powerful automated theorem prover that uses the superposition calculus, a technique combining resolution and paramodulation. It has been successful in solving many challenging problems from different areas of mathematics and computer science. These are just a few examples of commonly known implementations of automated theorem provers. There are numerous other systems and tools available, each with its own strengths and weaknesses, catering to different purposes and logics. # Robert Harper's Comment on ATP See [[@PrinciplesProgrammingLanguages2021#Math is a branch of Theoretical Computing Science]] ![[@PrinciplesProgrammingLanguages2021#Math is a branch of Theoretical Computing Science]] # Conclusion Overall, Automated Theorem Provers have revolutionized mathematical research and automated reasoning by providing an efficient way to prove complex statements and theorems, reducing the burden on human mathematicians and researchers. # References ```dataview Table title as Title, authors as Authors where contains(subject, "Automated Theorem Prover") or contains(subject, "ATP") or contains(subject, "Theorem Prover") or contains(title, "theorem prover") ```