• Interleavings, by themselves, do not form a generating set for the symmetric group. However a generalisation of them, the set of allriffle shuffleson a deck of cards, does form such a set.
• Think of this as just merge sort in reverse. The decomposition is attained by sorting the inverse of a permutation.
Proposition
Every permutation can be decomposed into a product of riffle shuffles whose underlying circuit has logarithmic depth.
Corollary
There are quasipolynomial-size proofs permuting the inputs of threshold formulae, for any permutation.
Arbitrary permutations
• Interleavings, by themselves, do not form a generating set for the symmetric group. However a generalisation of them, the set of allriffle shuffleson a deck of cards, does form such a set.
• Think of this as just merge sort in reverse. The decomposition is attained by sorting the inverse of a permutation.
Proposition
Every permutation can be decomposed into a product of riffle shuffles whose underlying circuit has logarithmic depth.
Corollary
There are quasipolynomial-size proofs permuting the inputs of threshold formulae, for any permutation.
Arbitrary permutations
• Interleavings, by themselves, do not form a generating set for the symmetric group. However a generalisation of them, the set of allriffle shuffleson a deck of cards, does form such a set.
• Think of this as just merge sort in reverse. The decomposition is attained by sorting the inverse of a permutation.
Proposition
Every permutation can be decomposed into a product of riffle shuffles whose underlying circuit has logarithmic depth.
Corollary
There are quasipolynomial-size proofs permuting the inputs of threshold formulae, for any permutation.
Arbitrary permutations
• Interleavings, by themselves, do not form a generating set for the symmetric group. However a generalisation of them, the set of allriffle shuffleson a deck of cards, does form such a set.
• Think of this as just merge sort in reverse. The decomposition is attained by sorting the inverse of a permutation.
Proposition
Every permutation can be decomposed into a product of riffle shuffles whose underlying circuit has logarithmic depth.
Corollary
There are quasipolynomial-size proofs permuting the inputs of threshold formulae, for any permutation.
Conclusions
Summary
• We constructed quasipolynomial-size proofs ofPHPnin the fragment of Gentzen free of cuts between descendants of any structural rules.
• The normalisation procedures from deep inference appear crucial for controlling the complexity of these proofs.
Procedures internal to the Gentzen formalism seem inadequate.
• While the existence of small proofs ofPHPnis encouraging, the relative complexity of weak monotone systems is open.
• This problem seems fundamentally related to the relationship between divide-and-conquer induction and regular induction in the absence of negation, something which could be studied in an arithmetic setting.
• The Gentzen formulation of our proofs still contains some cuts. The complexity of (partially) eliminating these cuts is the topic of further study.
Summary
• We constructed quasipolynomial-size proofs ofPHPnin the fragment of Gentzen free of cuts between descendants of any structural rules.
• The normalisation procedures from deep inference appear crucial for controlling the complexity of these proofs.
Procedures internal to the Gentzen formalism seem inadequate.
• While the existence of small proofs ofPHPnis encouraging, the relative complexity of weak monotone systems is open.
• This problem seems fundamentally related to the relationship between divide-and-conquer induction and regular induction in the absence of negation, something which could be studied in an arithmetic setting.
• The Gentzen formulation of our proofs still contains some cuts. The complexity of (partially) eliminating these cuts is the topic of further study.
Summary
• We constructed quasipolynomial-size proofs ofPHPnin the fragment of Gentzen free of cuts between descendants of any structural rules.
• The normalisation procedures from deep inference appear crucial for controlling the complexity of these proofs.
Procedures internal to the Gentzen formalism seem inadequate.
• While the existence of small proofs ofPHPnis encouraging, the relative complexity of weak monotone systems is open.
• This problem seems fundamentally related to the relationship between divide-and-conquer induction and regular induction in the absence of negation, something which could be studied in an arithmetic setting.
• The Gentzen formulation of our proofs still contains some cuts. The complexity of (partially) eliminating these cuts is the topic of further study.
Summary
• We constructed quasipolynomial-size proofs ofPHPnin the fragment of Gentzen free of cuts between descendants of any structural rules.
• The normalisation procedures from deep inference appear crucial for controlling the complexity of these proofs.
Procedures internal to the Gentzen formalism seem inadequate.
• While the existence of small proofs ofPHPnis encouraging, the relative complexity of weak monotone systems is open.
• This problem seems fundamentally related to the relationship between divide-and-conquer induction and regular induction in the absence of negation, something which could be studied in an arithmetic setting.
• The Gentzen formulation of our proofs still contains some cuts. The complexity of (partially) eliminating these cuts is the topic of further study.
Summary
• We constructed quasipolynomial-size proofs ofPHPnin the fragment of Gentzen free of cuts between descendants of any structural rules.
• The normalisation procedures from deep inference appear crucial for controlling the complexity of these proofs.
Procedures internal to the Gentzen formalism seem inadequate.
• While the existence of small proofs ofPHPnis encouraging, the relative complexity of weak monotone systems is open.
• This problem seems fundamentally related to the relationship between divide-and-conquer induction and regular induction in the absence of negation, something which could be studied in an arithmetic setting.
• The Gentzen formulation of our proofs still contains some cuts.
The complexity of (partially) eliminating these cuts is the topic of further study.