7 research outputs found
Theories for TC0 and Other Small Complexity Classes
We present a general method for introducing finitely axiomatizable "minimal"
two-sorted theories for various subclasses of P (problems solvable in
polynomial time). The two sorts are natural numbers and finite sets of natural
numbers. The latter are essentially the finite binary strings, which provide a
natural domain for defining the functions and sets in small complexity classes.
We concentrate on the complexity class TC^0, whose problems are defined by
uniform polynomial-size families of bounded-depth Boolean circuits with
majority gates. We present an elegant theory VTC^0 in which the provably-total
functions are those associated with TC^0, and then prove that VTC^0 is
"isomorphic" to a different-looking single-sorted theory introduced by
Johannsen and Pollet. The most technical part of the isomorphism proof is
defining binary number multiplication in terms a bit-counting function, and
showing how to formalize the proofs of its algebraic properties.Comment: 40 pages, Logical Methods in Computer Scienc
On Rules and Parameter Free Systems in Bounded Arithmetic
We present model–theoretic techniques to obtain conservation
results for first order bounded arithmetic theories, based on a hierarchical
version of the well known notion of an existentially closed model.Ministerio de Educación y Ciencia MTM2005-0865
Existentially Closed Models and Conservation Results in Bounded Arithmetic
We develop model-theoretic techniques to obtain conservation results for first order Bounded Arithmetic theories, based on a hierarchical version of the well-known notion of an existentially closed model. We focus on the classical Buss' theories Si2 and Ti2 and prove that they are ∀Σbi conservative over their inference rule counterparts, and ∃∀Σbi conservative over their parameter-free versions. A similar analysis of the Σbi-replacement scheme is also developed. The proof method is essentially the same for all the schemes we deal with and shows that these conservation results between schemes and inference rules do not depend on the specific combinatorial or arithmetical content of those schemes. We show that similar conservation results can be derived, in a very general setting, for every scheme enjoying some syntactical (or logical) properties common to both the induction and replacement schemes. Hence, previous conservation results for induction and replacement can be also obtained as corollaries of these more general results.Ministerio de Educación y Ciencia MTM2005-08658Junta de Andalucía TIC-13
Recommended from our members
Mathematical Logic: Proof Theory, Constructive Mathematics (hybrid meeting)
The Workshop "Mathematical Logic: Proof Theory,
Constructive Mathematics" focused on
proofs both as formal derivations in deductive systems as well as on
the extraction of explicit computational content from
given proofs in core areas of ordinary mathematics using proof-theoretic
methods. The workshop contributed to the following research strands: interactions between foundations and applications; proof mining; constructivity in classical logic; modal logic and provability logic; proof theory and theoretical computer science; structural proof theory
Short Propositional Refutations for Dense Random 3CNF Formulas
Random 3CNF formulas constitute an important distribution for measuring the
average-case behavior of propositional proof systems. Lower bounds for random
3CNF refutations in many propositional proof systems are known. Most notably
are the exponential-size resolution refutation lower bounds for random 3CNF
formulas with clauses [Chvatal and Szemeredi
(1988), Ben-Sasson and Wigderson (2001)]. On the other hand, the only known
non-trivial upper bound on the size of random 3CNF refutations in a
non-abstract propositional proof system is for resolution with
clauses, shown by Beame et al. (2002). In this paper we
show that already standard propositional proof systems, within the hierarchy of
Frege proofs, admit short refutations for random 3CNF formulas, for
sufficiently large clause-to-variable ratio. Specifically, we demonstrate
polynomial-size propositional refutations whose lines are formulas
(i.e., -Frege proofs) for random 3CNF formulas with variables and clauses.
The idea is based on demonstrating efficient propositional correctness proofs
of the random 3CNF unsatisfiability witnesses given by Feige, Kim and Ofek
(2006). Since the soundness of these witnesses is verified using spectral
techniques, we develop an appropriate way to reason about eigenvectors in
propositional systems. To carry out the full argument we work inside weak
formal systems of arithmetic and use a general translation scheme to
propositional proofs.Comment: 62 pages; improved introduction and abstract, and a changed title.
Fixed some typo
Theories for TC0 and Other Small Complexity Classes
We present a general method for introducing finitely axiomatizable "minimal"two-sorted theories for various subclasses of P (problems solvable inpolynomial time). The two sorts are natural numbers and finite sets of naturalnumbers. The latter are essentially the finite binary strings, which provide anatural domain for defining the functions and sets in small complexity classes.We concentrate on the complexity class TC^0, whose problems are defined byuniform polynomial-size families of bounded-depth Boolean circuits withmajority gates. We present an elegant theory VTC^0 in which the provably-totalfunctions are those associated with TC^0, and then prove that VTC^0 is"isomorphic" to a different-looking single-sorted theory introduced byJohannsen and Pollet. The most technical part of the isomorphism proof isdefining binary number multiplication in terms a bit-counting function, andshowing how to formalize the proofs of its algebraic properties.Comment: 40 pages, Logical Methods in Computer Scienc
Theories for TC0 and Other Small Complexity Classes
We present a general method for introducing finitely axiomatizable "minimal"
two-sorted theories for various subclasses of P (problems solvable in
polynomial time). The two sorts are natural numbers and finite sets of natural
numbers. The latter are essentially the finite binary strings, which provide a
natural domain for defining the functions and sets in small complexity classes.
We concentrate on the complexity class TC^0, whose problems are defined by
uniform polynomial-size families of bounded-depth Boolean circuits with
majority gates. We present an elegant theory VTC^0 in which the provably-total
functions are those associated with TC^0, and then prove that VTC^0 is
"isomorphic" to a different-looking single-sorted theory introduced by
Johannsen and Pollet. The most technical part of the isomorphism proof is
defining binary number multiplication in terms a bit-counting function, and
showing how to formalize the proofs of its algebraic properties