13 research outputs found
A parameterized halting problem, the linear time hierarchy, and the MRDP theorem
The complexity of the parameterized halting problem for nondeterministic Turing machines p-Halt is known to be related to the question of whether there are logics capturing various complexity classes [10]. Among others, if p-Halt is in para-AC0, the parameterized version of the circuit complexity class AC0, then AC0, or equivalently, (+, x)-invariant FO, has a logic. Although it is widely believed that p-Halt ∉. para-AC0, we show that the problem is hard to settle by establishing a connection to the question in classical complexity of whether NE ⊈ LINH. Here, LINH denotes the linear time hierarchy.
On the other hand, we suggest an approach toward proving NE ⊈ LINH using bounded arithmetic. More specifically, we demonstrate that if the much celebrated MRDP (for Matiyasevich-Robinson-Davis-Putnam) theorem can be proved in a certain fragment of arithmetic, then NE ⊈ LINH. Interestingly, central to this result is a para-AC0 lower bound for the parameterized model-checking problem for FO on arithmetical structures.Peer ReviewedPostprint (author's final draft
Some Lower Bounds in Parameterized AC^0
We demonstrate some lower bounds for parameterized problems via parameterized classes corresponding to the classical AC^0. Among others, we derive such a lower bound for all fpt-approximations of the parameterized clique problem and for a parameterized halting problem, which recently turned out to link problems of computational complexity, descriptive complexity, and proof theory. To show the first lower bound, we prove a strong AC^0 version of the planted clique conjecture: AC^0-circuits asymptotically almost surely can not distinguish between a random graph and this graph with a randomly planted clique of any size <= n^xi (where 0 <= xi < 1)
Speedup for Natural Problems and Noncomputability
A resource-bounded version of the statement "no algorithm recognizes all
non-halting Turing machines" is equivalent to an infinitely often (i.o.)
superpolynomial speedup for the time required to accept any coNP-complete
language and also equivalent to a superpolynomial speedup in proof length in
propositional proof systems for tautologies, each of which implies P!=NP. This
suggests a correspondence between the properties 'has no algorithm at all' and
'has no best algorithm' which seems relevant to open problems in computational
and proof complexity.Comment: 8 page
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
On p-optimal proof systems and logics for PTIME
"Vegeu el resum a l'inici del document del fitxer adjunt"
The parameterized space complexity of model-checking bounded variable first-order logic
The parameterized model-checking problem for a class of first-order sentences
(queries) asks to decide whether a given sentence from the class holds true in
a given relational structure (database); the parameter is the length of the
sentence. We study the parameterized space complexity of the model-checking
problem for queries with a bounded number of variables. For each bound on the
quantifier alternation rank the problem becomes complete for the corresponding
level of what we call the tree hierarchy, a hierarchy of parameterized
complexity classes defined via space bounded alternating machines between
parameterized logarithmic space and fixed-parameter tractable time. We observe
that a parameterized logarithmic space model-checker for existential bounded
variable queries would allow to improve Savitch's classical simulation of
nondeterministic logarithmic space in deterministic space .
Further, we define a highly space efficient model-checker for queries with a
bounded number of variables and bounded quantifier alternation rank. We study
its optimality under the assumption that Savitch's Theorem is optimal
On slicewise monotone parameterized problems and optimal proof systems for TAUT
"Vegeu el resum a l'inici del document del fitxer adjunt"
Optimal algorithms and proofs (Dagstuhl Seminar 14421)
This report documents the programme and the outcomes of the Dagstuhl Seminar 14421 "Optimal algorithms and proofs". The seminar brought together researchers working in computational and proof complexity, logic, and the theory of approximations. Each of these areas has its own, but connected notion of optimality; and the main aim of the seminar was to bring together researchers from these different areas, for an exchange of ideas, techniques, and open questions, thereby triggering new research collaborations across established research boundaries