We consider the Ideal Proof System (IPS) introduced by Grochow and Pitassi
and pose the question of which tautologies admit symmetric proofs, and of what
complexity. The symmetry requirement in proofs is inspired by recent work
establishing lower bounds in other symmetric models of computation. We link the
existence of symmetric IPS proofs to the expressive power of logics such as
fixed-point logic with counting and Choiceless Polynomial Time, specifically
regarding the graph isomorphism problem. We identify relationships and
tradeoffs between the symmetry of proofs and other parameters of IPS proofs
such as size, degree and linearity. We study these on a number of standard
families of tautologies from proof complexity and finite model theory such as
the pigeonhole principle, the subset sum problem and the Cai-F\”urer-Immerman
graphs, exhibiting non-trivial upper bounds on the size of symmetric IPS
proofs.

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