Gödel sentences of bounded arithmetic

2000 ◽  
Vol 65 (3) ◽  
pp. 1338-1346 ◽  
Author(s):  
Gaisi Takeuti
Keyword(s):  
Polynomial Time ◽  
Special Kind ◽  
Interesting Case ◽  
Polynomial Hierarchy ◽  
Bounded Arithmetic ◽  
Original Language ◽  
Close Relationship ◽  
Np Problem ◽  
Good Contrast ◽  
Computable Functions

In [1], S. Buss introduced the systems of Bounded Arithmetic for (i = 0,1,2,…) which has a close relationship to classes in polynomial hierarchy.In [4], we defined a very special kind of proof-predicate Prfi for which gives detailed information on bounds of free variables used in the proof. There we also introduced infinitely many Gödel sentences for Prfi (k = 0, 1, 2, …) and showed that the properties of Prfi and are closely related to the P ≠ NP problem. Then we presented many conjectures on Prfi and which imply P ≠ NP.Now in [2], Feferman emphasized that the arithmetization of metamathematics must be carried out intensionally. Bounded Arithmetic is a very interesting case in this sense.In this paper, we also introduce the usual proof-predicate PRFi for and infinitely many Gödel sentences for PRFi(k= 0, 1, 2, …). Then we show that (Prfi, )and (PRFi, ) form a good contrast, this contrast is also closely related to the P ≠ NP problem, and present more conjectures which imply P ≠ NP.As in [4] we define to be the following extension of Buss' original .(1) We add finitely many function symbols which express polynomial time computable functions to Buss' original language of .(2) All basic axioms on function symbols and ≤ can be expressed by initial sequents without logical symbols.

scholarly journals POLYNOMIAL LOCAL SEARCH IN THE POLYNOMIAL HIERARCHY AND WITNESSING IN FRAGMENTS OF BOUNDED ARITHMETIC

2009 ◽  
Vol 09 (01) ◽  
pp. 103-138 ◽  
Author(s):  
ARNOLD BECKMANN ◽  
SAMUEL R. BUSS
Keyword(s):  
Local Search ◽  
Polynomial Time ◽  
Complexity Class ◽  
Polynomial Hierarchy ◽  
Bounded Arithmetic ◽  
Weak Base ◽  
Base Theory ◽  
Definable Functions ◽  
New Formula

The complexity class of [Formula: see text]-polynomial local search (PLS) problems is introduced and is used to give new witnessing theorems for fragments of bounded arithmetic. For 1 ≤ i ≤ k + 1, the [Formula: see text]-definable functions of [Formula: see text] are characterized in terms of [Formula: see text]-PLS problems. These [Formula: see text]-PLS problems can be defined in a weak base theory such as [Formula: see text], and proved to be total in [Formula: see text]. Furthermore, the [Formula: see text]-PLS definitions can be skolemized with simple polynomial time functions, and the witnessing theorem itself can be formalized, and skolemized, in a weak base theory. We introduce a new [Formula: see text]-principle that is conjectured to separate [Formula: see text] and [Formula: see text].

Forcing on bounded arithmetic II

1998 ◽  
Vol 63 (3) ◽  
pp. 860-868 ◽  
Author(s):  
Gaisi Takeuti ◽  
Masahiro Yasumoto
Keyword(s):  
Boolean Algebra ◽  
Polynomial Time ◽  
Bounded Arithmetic ◽  
Pigeon Hole Principle ◽  
Polynomial Size ◽  
Np Complete ◽  
Computable Functions

Forcing method on Bounded Arithmetic was first introduced by J. B. Paris and A. Wilkie in [10]. Then in [1], [2] and [3], M. Ajtai used the method to get excellent results on the pigeon hole principle and the modulo p counting principle. The forcing method on Bounded arithmetic was further developed by P. Beame, J. Krajíček and S. Riis in [4], [7], [6], [8], [5], [12], [11], [13]. It should be noted that J. Krajíček and P. Pudlák used an idea of Boolean valued in [9] and also Boolean valued notion is efficiently used for model theoretic constructions in [7], [6], [8], [5].In our previous paper [14], we developed a Boolean valued version of forcing on Bounded Arithmetic using Boolean algebra which is generated by polynomial size circuits from Boolean variables and discussed its relation with NP = co-NP problem and P = NP problem. Especially we proved the following theorem and related theorems as Theorems 2, 3 and 4 in Section 2.Theorem. If M[G] is not a model of S2, then NP ≠ co-NP and therefore P ≠ NP.However in the proof of the Theorem, we used a consequence of P = NP. More precisely we used the following as a consequence of NP = co-NP, though it is a consequence of P = NP but not a consequence of NP = co-NP.Suppose that NP = co-NP holds. Then there exists an NP-complete predicate ∃x ≤ t(a) A(x,a) with sharply bounded A(x, a) and a sharply bounded B(y, a) such that ∃x ≤ t(a) A(x,a) ↔ ∀y ≤ s(a)B(y, a). Then there exists polynomial time computable functions f and g such that the following two sequents hold.

scholarly journals FRAGMENTS OF APPROXIMATE COUNTING

2014 ◽  
Vol 79 (2) ◽  
pp. 496-525 ◽  
Author(s):  
SAMUEL R. BUSS ◽  
LESZEK ALEKSANDER KOŁODZIEJCZYK ◽  
NEIL THAPEN
Keyword(s):  
Polynomial Time ◽  
Open Problem ◽  
Bounded Arithmetic ◽  
Pigeonhole Principle ◽  
Approximate Counting ◽  
Image Position

AbstractWe study the long-standing open problem of giving $\forall {\rm{\Sigma }}_1^b$ separations for fragments of bounded arithmetic in the relativized setting. Rather than considering the usual fragments defined by the amount of induction they allow, we study Jeřábek’s theories for approximate counting and their subtheories. We show that the $\forall {\rm{\Sigma }}_1^b$ Herbrandized ordering principle is unprovable in a fragment of bounded arithmetic that includes the injective weak pigeonhole principle for polynomial time functions, and also in a fragment that includes the surjective weak pigeonhole principle for FPNP functions. We further give new propositional translations, in terms of random resolution refutations, for the consequences of $T_2^1$ augmented with the surjective weak pigeonhole principle for polynomial time functions.

Jolas, Eugene (1894–1952)

2018 ◽  
Author(s):  
Julian Hanna
Keyword(s):  
Experimental Writing ◽  
Major Focus ◽  
Avant Garde ◽  
Original Language ◽  
Berlin Alexanderplatz ◽  
European Culture ◽  
The World ◽  
English Translations ◽  
Close Relationship ◽  
Literary Journal

Eugene Jolas was a journalist, editor, translator, and poet who embodied the transatlantic character of modernism between the World Wars. The task of transition, the Paris-based literary journal he edited with his wife Maria Jolas and others between 1927 and 1938, was to translate European culture for Americans, and vice versa. transition’s list of contributors reads like a Who’s Who of the international avant-garde. Jolas’ wealth of contacts in the literary world arose from his previous job writing the column ‘Rambles Through Literary Paris’ for the Chicago Tribune Paris edition. The romantic, imagination-driven strain of modernism that Jolas promoted led to a close relationship with Expressionism and Surrealism. Publishing non-anglophone experimental writing in translation or (after 1933) in the original language was a major focus of transition. Jolas also provided English translations of key European modernist texts outside the magazine, including Alfred Döblin’s Berlin Alexanderplatz (1931).

A feasible theory for analysis

1994 ◽  
Vol 59 (3) ◽  
pp. 1001-1011 ◽  
Author(s):  
Fernando Ferreira
Keyword(s):  
Polynomial Time ◽  
Order Theory ◽  
Second Order ◽  
First Order ◽  
Weak König’S Lemma ◽  
König’S Lemma ◽  
Computable Functions

AbstractWe construct a weak second-order theory of arithmetic which includes Weak König's Lemma (WKL) for trees defined by bounded formulae. The provably total functions (with -graphs) of this theory are the polynomial time computable functions. It is shown that the first-order strength of this version of WKL is exactly that of the scheme of collection for bounded formulae.

scholarly journals Supraconvergence of a Finite Difference Scheme for Elliptic Boundary Value Problems of the Third Kind in Fractional Order Sobolev Spaces

2006 ◽  
Vol 6 (2) ◽  
pp. 154-177 ◽  
Author(s):  
E. Emmrich ◽  
R.D. Grigorieff
Keyword(s):  
Finite Element ◽  
Finite Difference ◽  
Difference Scheme ◽  
Finite Difference Scheme ◽  
Elliptic Boundary ◽  
Special Kind ◽  
Variable Coefficients ◽  
Second Order ◽  
Nonuniform Grids ◽  
Close Relationship

AbstractIn this paper, we study the convergence of the finite difference discretization of a second order elliptic equation with variable coefficients subject to general boundary conditions. We prove that the scheme exhibits the phenomenon of supraconvergence on nonuniform grids, i.e., although the truncation error is in general of the first order alone, one has second order convergence. All error estimates are strictly local. Another result of the paper is a close relationship between finite difference scheme and linear finite element methods combined with a special kind of quadrature. As a consequence, the results of the paper can be viewed as the introduction of a fully discrete finite element method for which the gradient is superclose. A numerical example is given.

A second order version of S2i and U21

1991 ◽  
Vol 56 (3) ◽  
pp. 1038-1063 ◽  
Author(s):  
Gaisi Takeuti
Keyword(s):  
Second Order ◽  
The Other ◽  
Polynomial Hierarchy ◽  
Bounded Arithmetic ◽  
Conservative Extension ◽  
First Order ◽  
Second Order Systems ◽  
Separation Problems

In [1] S. Buss introduced systems of bounded arithmetic , , , (i = 1, 2, 3, …). and are first order systems and and are second order systems. and are closely related to and respectively in the polynomial hierarchy, and and are closely related to PSPACE and EXPTIME respectively. One of the most important problems in bounded arithmetic is whether the hierarchy of bounded arithmetic collapses, i.e. whether = or = for some i, or whether = , or whether is a conservative extension of S2 = ⋃i. These problems are relevant to the problems whether the polynomial hierarchy PH collapses or whether PSPACE = PH or whether PSPACE = EXPTIME. It was shown in [4] that = implies and consequently the collapse of the polynomial hierarchy. We believe that the separation problems of bounded arithmetic and the separation problems of computational complexities are essentially the same problem, and the solution of one of them will lead to the solution of the other.

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