Consistency Tests
This chapter explores various consistency tests for low-degree polynomials, focusing on methods to enhance their effectiveness. Key improvements include minimizing the number of variables involved, reducing the variable ranges, and lowering error probabilities. We detail specific tests such as Points-on-Line, Line-vs-Point, and Plane-vs-Plane, elucidating their representations and local tests for consistency within polynomial frameworks. The concepts of global and limited pluralism are examined, highlighting their significance in ensuring compatibility in polynomial assignments across multiple variables.
Consistency Tests
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Presentation Transcript
Consistency Tests for low degree polynomials
Introduction • In this chapter we examine consistency tests, and trying to improve their parameters: • reducing the number of variables accessed by the test. • reducing the variables’ range. • reducing error probability.
Introduction We present the tests: • Points-on-Line • Line-vs.-Point • Plane-vs.-Plane
Representation, Test, Consistency V from PCP[D, V, ) The Basic Terms: • Representation [.] • [.] isa set of variables, • To each variable a value is assigned, • The values are in the range 2v, • The values correspond to a single, polynomial ƒ: a f is of global degree r
local tests Representation, Test, Consistency D from PCP[D, V, ) • Test • A set of Boolean functions, • Each depends on at most D representation’s variables.
Representation, Test, Consistency • Consistency: • Measures an amount of conformation between the different values assigned to the representation variables. • We say that the values are consistent if they satisfy at least an -fraction of the local tests.
Geometry • Let us define some specific affine subspaces of: • lines()is the set of all lines (affine subspaces of dimension 2) of • planes()is the set of all planes (affine subspaces of dimension 3) of
Overview of the Tests • In each tests the variables in [.] represent some aspect of the given polynomial f, such as • f’s values on points of • f’s restriction to a line in • f’s restriction to a plane in • The local-tests check compatibility between the values of different variables in [.].
Simple Test: Points-on-Line Representation: • [.] has one variable [p] for each point p. • The variables are supposedly assigned the valueƒ(p). hence v = log ||
Points-on-Line: Test Test: • There’s one local-test for each linellines(). • Each test depends on all points ofl. • A testaccepts if and only if the values are consistent with a single degree-r univariate polynomial 2r
Points-on-Line: Consistency Alas, each local-test depends on a non constant number of variables (2r) Def: An assignment to is said to be globally consistentif values on most points agree with asingle, global degree-r polynomial. Thm[RuSu]: If a large (constant) fraction of the local-tests accept, then there is a polynomial ƒ (of degree-r) which agrees with the assigned values on most points.
Next Test: Line-vs.-Point Representation: • [.] has one variable [p] for each pointp, supposedly assignedƒ(p), • Plus, one variable [l] for each linellines(),supposedly assignedƒ’srestriction tol. Hence the range of [l] is all degree-r univariate poly’s
Line-vs.-Point: Test Test: • There’sone local-test for each pair of: • a line l lines(), and • a point p l . • A testacceptsif the value assigned to [p] equals the value of the polynomial assigned to [l]on the point p.
Global Consistency: Constant Error Thm [AS,ALMSS]: Probability of finding inconsistency, between value for [p] and value for line [l] on p, is high (constant) , unlessmost lines and most points agree with a single, global degree-rpolynomial. HereD = O(1) V = (r+1) log||& constant.
Can the Test Be Improved? Can error-probability be made smaller than constant (such as 1/log(n)), while keeping each local-test depending on constant number of representation variables?
What’s the problem? Adversary: randomly partition variables into k sets, each consistent with a distinct degree-r polynomialThis would cause the local-test’s success probability to be at least k-(D-1). (if all variables fall within the same set in the partition)
Consequently One therefore must further weaken the notion of global consistency sought after[ still, making sure it can be applied in order to deducePCPcharacterization ofNP].
Limited Pluralism Def: Given an assignment to ’s variables,a degree-r polynomial ƒ is said to be-permissible if it is consistent with at least a fraction of the values assigned. Global Consistency: assignment’s values consistent with any -permissible ƒ are acceptable.
Limited Pluralism - Cont. Formally: Def: A local test is said to err (with respect to ) if it accepts values that are NOTconsistent with any-permissible degree-rƒ’s.
Limited Pluralism - Cont. • Note that the adversary’s randomly partition does not trick the test this time: • If the test accepts when all the variables are from a set consistent with an r-degree polynomial, then the polynomial is really -permissible.
Plane-vs.-Plane: Representation Representation: • [.] has one variable [p] for each planepplanes(), • supposedlyassignedthe restriction of f to p. Hence the range of [p] is all degree-r two-variables poly’s
Plane-vs.-Plane: Test That is, a pair of plains intersecting by a line Test: • There’s one local-test for each line llines() and a pair of planes p1,p2planes() such that lp1 and lp2 • A testacceptsif and only if the value of[p1]restricted tol equals the value of[p2]restricted to l. HereD=O(1), v=2(r+1)log||.
Plane-vs.-Plane: Consistency Thm[RaSa]:As long as ³||-c for some constant 1 > c > 0, the tests err (w.r.t. ) with a very small probability, namely £-c’for some constant 1 > c’ > 0.
Plane-vs.-Plane: Consistency - Cont. The theorem states that, the plane-vs.-plane test, with very high probability (³ 1 - c’), either rejects, or accepts values of a -permissible polynomial .
Summary • We examined consistency tests, Points-on-Line,Line-vs.-Point and Plane-vs.-Plane. • By weakening to-permissible definition, we achieve an error probability which is below constant.
