scholarly journals List Approximation for Increasing Kolmogorov Complexity

Axioms ◽  
2021 ◽  
Vol 10 (4) ◽  
pp. 334
Author(s):  
Marius Zimand
Keyword(s):  
Polynomial Time ◽  
Kolmogorov Complexity ◽  
Input String ◽  
Polynomial Size

It is impossible to effectively modify a string in order to increase its Kolmogorov complexity. However, is it possible to construct a few strings, no longer than the input string, so that most of them have larger complexity? We show that the answer is yes. We present an algorithm that takes as input a string x of length n and returns a list with O(n2) strings, all of length n, such that 99% of them are more complex than x, provided the complexity of x is less than n−loglogn−O(1). We also present an algorithm that obtains a list of quasi-polynomial size in which each element can be produced in polynomial time.

scholarly journals Computing the Expected Edit Distance from a String to a Probabilistic Finite-State Automaton

2017 ◽  
Vol 28 (05) ◽  
pp. 603-621 ◽  
Author(s):  
Jorge Calvo-Zaragoza ◽  
Jose Oncina ◽  
Colin de la Higuera
Keyword(s):  
Polynomial Time ◽  
Edit Distance ◽  
Input String ◽  
Finite State Automaton ◽  
Global View ◽  
Finite State ◽  
Median String

In a number of fields, it is necessary to compare a witness string with a distribution. One possibility is to compute the probability of the string for that distribution. Another, giving a more global view, is to compute the expected edit distance from a string randomly drawn to the witness string. This number is often used to measure the performance of a prediction, the goal then being to return the median string, or the string with smallest expected distance. To be able to measure this, computing the distance between a hypothesis and that distribution is necessary. This paper proposes two solutions for computing this value, when the distribution is defined with a probabilistic finite state automaton. The first is exact but has a cost which can be exponential in the length of the input string, whereas the second is a fully polynomial-time randomized schema.

scholarly journals Examining Fragments of the Quantified Propositional Calculus

2008 ◽  
Vol 73 (3) ◽  
pp. 1051-1080 ◽  
Author(s):  
Steven Perron
Keyword(s):  
Polynomial Time ◽  
Propositional Calculus ◽  
Proof System ◽  
Polynomial Size ◽  
Propositional Proof ◽  
Polynomial Time Hierarchy

AbstractWhen restricted to proving formulas, the quantified propositional proof system is closely related to the theorems of Buss's theory . Namely, has polynomial-size proofs of the translations of theorems of , and proves that is sound. However, little is known about when proving more complex formulas. In this paper, we prove a witnessing theorem for similar in style to the KPT witnessing theorem for . This witnessing theorem is then used to show that proves is sound with respect to formulas. Note that unless the polynomial-time hierarchy collapses is the weakest theory in the S2 hierarchy for which this is true. The witnessing theorem is also used to show that is p-equivalent to a quantified version of extended-Frege for prenex formulas. This is followed by a proof that Gi, p-simulates with respect to all quantified propositional formulas. We finish by proving that S2 can be axiomatized by plus axioms stating that the cut-free version of is sound. All together this shows that the connection between and does not extend to more complex formulas.

ON THE CIRCUIT-SIZE OF INVERSES

2011 ◽  
Vol 22 (08) ◽  
pp. 1925-1938 ◽  
Author(s):  
J. C. BIRGET
Keyword(s):  
Polynomial Time ◽  
Partial Function ◽  
Polynomial Size

We reprove a result of Boppana and Lagarias: If [Formula: see text] then there exists a partial function f that is computable by a polynomial-size family of circuits, but no inverse of f is computable by a polynomial-size family of circuits. We strengthen this result by showing, if [Formula: see text], that there exist length-preserving total functions that are one-way by circuit size and that are computable in uniform polynomial time. We also prove, if [Formula: see text], that there exist polynomially balanced total surjective functions that are one-way by circuit size; here non-uniformity is used.

scholarly journals The Computational Power of Discrete Hopfield Nets with Hidden Units

1996 ◽  
Vol 8 (2) ◽  
pp. 403-415 ◽  
Author(s):  
Pekka Orponen
Keyword(s):  
Polynomial Time ◽  
Boolean Functions ◽  
Polynomial Space ◽  
Turing Machines ◽  
Computational Power ◽  
Hopfield Networks ◽  
Polynomial Size ◽  
Class Of Functions

We prove that polynomial size discrete Hopfield networks with hidden units compute exactly the class of Boolean functions PSPACE/poly, i.e., the same functions as are computed by polynomial space-bounded nonuniform Turing machines. As a corollary to the construction, we observe also that networks with polynomially bounded interconnection weights compute exactly the class of functions P/poly, i.e., the class computed by polynomial time-bounded nonuniform Turing machines.

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 Improved Separations of Regular Resolution from Clause Learning Proof Systems

2014 ◽  
Vol 49 ◽  
pp. 669-703 ◽  
Author(s):  
M. L. Bonet ◽  
S. Buss ◽  
J. Johannsen
Keyword(s):  
Polynomial Time ◽  
Proof Systems ◽  
Polynomial Size ◽  
Logical Strength ◽  
Clause Learning ◽  
Unit Propagation ◽  
The Right ◽  
The Relationship

This paper studies the relationship between resolution and conflict driven clause learning (CDCL) without restarts, and refutes some conjectured possible separations. We prove that the guarded, xor-ified pebbling tautology clauses, which Urquhart proved are hard for regular resolution, as well as the guarded graph tautology clauses of Alekhnovich, Johannsen, Pitassi, and Urquhart have polynomial size pool resolution refutations that use only input lemmas as learned clauses. For the latter set of clauses, we extend this to prove that a CDCL search without restarts can refute these clauses in polynomial time, provided it makes the right choices for decision literals and clause learning. This holds even if the CDCL search is required to greedily process conflicts arising from unit propagation. This refutes the conjecture that the guarded graph tautology clauses or the guarded xor-ified pebbling tautology clauses can be used to separate CDCL without restarts from general resolution. Together with subsequent results by Buss and Kolodziejczyk, this means we lack any good conjectures about how to establish the exact logical strength of conflict-driven clause learning without restarts.

The 2-closure of a 32-transitive group in polynomial time

2018 ◽  
Vol 60 (2) ◽  
pp. 360-375
Author(s):  
A. V. Vasil'ev ◽  
D. V. Churikov
Keyword(s):  
Polynomial Time ◽  
Transitive Group

Learning Importance of Preferences

10.29007/v68w ◽  
2018 ◽  
Author(s):  
Ying Zhu ◽  
Mirek Truszczynski
Keyword(s):  
Polynomial Time ◽  
Polynomial Time Algorithm ◽  
Time Algorithm ◽  
Scoring Rules ◽  
Np Hard ◽  
Individual Preferences

We study the problem of learning the importance of preferences in preference profiles in two important cases: when individual preferences are aggregated by the ranked Pareto rule, and when they are aggregated by positional scoring rules. For the ranked Pareto rule, we provide a polynomial-time algorithm that finds a ranking of preferences such that the ranked profile correctly decides all the examples, whenever such a ranking exists. We also show that the problem to learn a ranking maximizing the number of correctly decided examples (also under the ranked Pareto rule) is NP-hard. We obtain similar results for the case of weighted profiles when positional scoring rules are used for aggregation.

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