Section: Scientific Foundations
Keywords : focused proof systems.
Focused proof systems
There is a great deal of non-determinism that is present in the search for proofs (in the sense of automated deduction). The non-determinism involved with generating lemmas is one extreme: when attempting to prove one formula it is possible to generate a potential lemma and then attempt to prove it and to use it to prove the original formula. In general, there are no clues as to what is a useful lemma to construct. The famous “cut-elimination” theorem say that it is possible to prove theorems without using lemmas (that is, by restricting to cut-free proofs). Of course, cut-free proofs are not appropriate for all domains of computation logic since they can be vastly larger than proofs containing cuts. Even when restricting to cut-free proofs make sense (as in logic programming, model checking, and some areas of automated reasoning), the construction of cut-free proofs still contains a great deal of non-determinism.
Structuring the non-deterministic choices within the search for cut-free proofs has received increasing attention in recent years with the development of focusing proofs systems . In such proof systems, there is a clear separation between non-deterministic choices for which no backtracking is required (“don't care non-determinism”) and choices were backtracking may be required (“don't know non-determinism”). Furthermore, when a backtrackable choice is required, that choice actually extends over a series of inference rules, representing a “focus” during the construction of a proof. One focusing-style proof systems was developed within the early work of providing a proof-theoretic foundations for logic programming via uniform proofs  . The first comprehensive analysis of focusing proofs was done in linear logic by Andreoli  . There it was shown that proofs are constructed in two alternating phases: a negative phase in which don't-care-non-determinism is done and a positive phase in which a focused sequence of don't-know-non-determinism choices is applied.
Since a great deal of automated deduction (in the sense of logic programming, type inference, and model checking) is done in intuitionistic and classical logic, there is a strong need to have comprehensive focusing results for these logics as well. In linear logic, the assignment of inference rules to the positive and negative phases is canonical (only the treatment of atomic formulas is left as a non-canonical choice). Within intuitionistic and classical logic, a number of inference rules do not have canonical treatments. Instead, several focusing-style proof systems have been developed one-by-one for these logics. A general scheme for putting all of these choices together has recently been developed within the team and will be described below.