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Eliminating definitions and Skolem functions in first-order logic
 

Summary: Eliminating definitions and Skolem functions
in first-order logic
JEREMY AVIGAD
Carnegie Mellon University
From proofs in any classical first-order theory that proves the existence of at least two elements,
one can eliminate definitions in polynomial time. From proofs in any classical first-order theory
strong enough to code finite functions, including sequential theories, one can also eliminate Skolem
functions in polynomial time.
Categories and Subject Descriptors: F.4.1 [Mathematical Logic and Formal Languages]:
Mathematical Logic--Proof Theory
General Terms: Algorithms, Theory
Additional Key Words and Phrases: definitions, Skolem functions, proof complexity, lengths of
proofs
1. INTRODUCTION
When working with a first-order theory, it is often convenient to use definitions.
That is, if (x) is a first-order formula with the free variables shown, one can
introduce a new relation symbol R to abbreviate , with defining axiom x (R(x)
(x)). Of course, this definition can later be eliminated from a proof, simply
by replacing every instance of R by . But suppose the proof involves nested
definitions, with a sequence of relation symbols R0, . . . , Rk abbreviating formulae

  

Source: Avigad, Jeremy - Departments of Mathematical Sciences & Philosophy, Carnegie Mellon University

 

Collections: Multidisciplinary Databases and Resources; Mathematics