## Section: Overall Objectives

### History

*Deduction modulo* is a formulation of predicate logic
where deduction is performed modulo an equivalence relation defined on
propositions. A typical example is the equivalence relation relating
propositions differing only by a re-arrangement of brackets around
additions, relating, for instance, the propositions $P\left(\right(x+y)+z)$
and $P(x+(y+z\left)\right)$. Reasoning modulo this equivalence relation
permits to drop the associativity axiom. Thus, in
Deduction modulo, a theory is formed with a set of axioms and an
equivalence relation. When the set of axioms is empty the theory is
called *purely computational*.

Deduction modulo was proposed at the end of the 20th century as a tool to simplify the completeness proof of equational resolution. Soon, it was noticed that this idea was also present in other areas of logic, such as Martin-Löf's type theory, where the equivalence relation is definitional equality, Prawitz' extended natural deduction, etc. More generally, Deduction modulo gives an account on the way reasoning and computation are articulated in a formal proof, a topic slightly neglected by logic, but of prime importance when proofs are computerized.

The early research on Deduction modulo focused on the design of general proof search methods—Resolution modulo, tableaux modulo, etc.—that could be applied to any theory formulated in Deduction modulo, to general proof normalization and cut elimination results, to the definitions of models taking the difference between reasoning and computation into account, and to the definition of specific theories—simple type theory, arithmetic, some versions of set theory, etc.—as purely computational theories.