Depending on the underlying logic, the problem of deciding the validity of a formula varies from trivial to impossible. For the frequent case of propositional logic, the problem is decidable but NP-complete, and hence only exponential-time algorithms are believed to exist for general proof tasks. For a first order predicate calculus, with no ("proper") axioms, Gödel's completeness theorem states that the theorems (provable statements) are exactly the logically valid well-formed formulas, so identifying valid formulas is recursively enumerable: given unbounded resources, any valid formula can eventually be proven.
However, invalid formulas (those that are not entailed by a given theory), cannot always be recognized. In addition, a consistent formal theory that contains the first-order theory of the natural numbers (thus having certain "proper axioms"), by Gödel's incompleteness theorem, contains a true statement which cannot be proven. In these cases, an automated theorem prover may fail to terminate while searching for a proof. Despite these theoretical limits, in practice, theorem provers can solve many hard problems, even in these undecidable logics.
A simpler, but related problem is proof verification, where an existing proof for a theorem is certified valid. For this, it is generally required that each individual proof step can be verified by a primitive recursive function or program, and hence the problem is always decidable.
Interactive theorem provers require a human user to give hints to the system. Depending on the degree of automation, the prover can essentially be reduced to a proof checker, with the user providing the proof in a formal way, or significant proof tasks can be performed automatically. Interactive provers are used for a variety of tasks, but even fully automatic systems have proven a number of interesting and hard theorems, including some that have eluded human mathematicians for a long time. However, these successes are sporadic, and work on hard problems usually requires a proficient user.
Another distinction is sometimes drawn between theorem proving and other techniques, where a process is considered to be theorem proving if it consists of a traditional proof, starting with axioms and producing new inference steps using rules of inference. Other techniques would include model checking, which is equivalent to brute-force enumeration of many possible states (although the actual implementation of model checkers requires much cleverness, and does not simply reduce to brute force).
There are hybrid theorem proving systems which use model checking as an inference rule. There are also programs which were written to prove a particular theorem, with a (usually informal) proof that if the program finishes with a certain result, then the theorem is true. A good example of this was the machine-aided proof of the four color theorem, which was very controversial as the first claimed mathematical proof which was essentially impossible to verify by humans due to the enormous size of the program's calculation (such proofs are called non-surveyable proofs). Another example would be the proof that the game Connect Four is a win for the first player.
Commercial use of automated theorem proving is mostly concentrated in integrated circuit design and verification. Since the Pentium FDIV bug, the complicated floating point units of modern microprocessors have been designed with extra scrutiny. In the latest processors from AMD, Intel, and others, automated theorem proving has been used to verify that division and other operations are correct.
First-order theorem proving is one of the most mature subfields of automated theorem proving. The logic is expressive enough to allow the specification of arbitrary problems, often in a reasonably natural and intuitive way. On the other hand, it is still semi-decidable, and a number of sound and complete calculi have been developed, enabling fully automated systems. More expressive logics, such as higher order and modal logics, allow the convenient expression of a wider range of problems than first order logic, but theorem proving for these logics is less well developed. The quality of implemented system has benefited from the existence of a large library of standard benchmark examples - the Thousands of Problems for Theorem Provers (TPTP) Problem Library - as well as from the CADE ATP System Competition (CASC), a yearly competition of first-order systems for many important classes of first-order problems.
Some important systems (all have won at least one CASC competition division) are listed below.
You can find information on some of these theorem provers and others at http://www.tptp.org/CASC/J2/SystemDescriptions.html . The TPTP library of test problems, suitable for testing first-order theorem provers, is available at http://www.tptp.org, and solutions from many of these provers for TPTP problems are in the TSTP solution library, available at http://www.tptp.org/TSTP.
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