13 research outputs found
Short Proofs for Slow Consistency
Let denote the finite
consistency statement "there are no proofs of contradiction in with
symbols". For a large class of natural theories , Pudl\'ak
has shown that the lengths of the shortest proofs of
in the theory
itself are bounded by a polynomial in . At the same time he conjectures that
does not have polynomial proofs of the finite consistency
statements . In contrast we show that Peano arithmetic
() has polynomial proofs of
,
where is the slow consistency statement for
Peano arithmetic, introduced by S.-D. Friedman, Rathjen and Weiermann. We also
obtain a new proof of the result that the usual consistency statement
is equivalent to iterations
of slow consistency. Our argument is proof-theoretic, while previous
investigations of slow consistency relied on non-standard models of arithmetic
Average-Case Hardness of Proving Tautologies and Theorems
We consolidate two widely believed conjectures about tautologies -- no
optimal proof system exists, and most require superpolynomial size proofs in
any system -- into a -isomorphism-invariant condition satisfied by all
paddable -complete languages or none. The condition is: for any
Turing machine (TM) accepting the language, -uniform input
families requiring superpolynomial time by exist (equivalent to the first
conjecture) and appear with positive upper density in an enumeration of input
families (implies the second). In that case, no such language is easy on
average (in ) for a distribution applying non-negligible weight
to the hard families.
The hardness of proving tautologies and theorems is likely related. Motivated
by the fact that arithmetic sentences encoding "string is Kolmogorov
random" are true but unprovable with positive density in a finitely axiomatized
theory (Calude and J{\"u}rgensen), we conjecture that any
propositional proof system requires superpolynomial size proofs for a dense set
of -uniform families of tautologies encoding "there is no
proof of size showing that string is Kolmogorov
random". This implies the above condition.
The conjecture suggests that there is no optimal proof system because
undecidable theories help prove tautologies and do so more efficiently as
axioms are added, and that constructing hard tautologies seems difficult
because it is impossible to construct Kolmogorov random strings. Similar
conjectures that computational blind spots are manifestations of
noncomputability would resolve other open problems
Upward Translation of Optimal and P-Optimal Proof Systems in the Boolean Hierarchy over NP
We study the existence of optimal and p-optimal proof systems for classes in
the Boolean hierarchy over . Our main results concern
, i.e., the second level of this hierarchy:
If all sets in have p-optimal proof systems, then all sets in
have p-optimal proof systems. The analogous implication for
optimal proof systems fails relative to an oracle.
As a consequence, we clarify such implications for all classes
and in the Boolean hierarchy over : either we can
prove the implication or show that it fails relative to an oracle. Furthermore,
we show that the sets and have p-optimal proof
systems, if and only if all sets in the Boolean hierarchy over
have p-optimal proof systems which is a new characterization of a conjecture
studied by Pudl\'ak