Interpolating Polynomials, Pinning Lemmas, And The Quest For Constructivity

The quest for more constructive proofs in mathematics while retaining high levels of generality has been aided by the introduction of powerful new tools like polynomial interpolation and pinning lemmas. These approaches aim to bridge the gap between the concrete guarantees provided by constructive proofs and the wide applicability offered by probabilistic methods.

Seeking Pinning Lemmas

A longstanding challenge in mathematics has been balancing generality of proofs with constructivity. Constructive proofs provide explicit witness functions and algorithms but often only apply to restricted classes of problems. Meanwhile probabilistic proofs can address very general questions but do not offer concrete guides for finding solutions.

Understanding polynomial interpolation

One approach that has shown promise for improving constructivity is using interpolation of multivariate polynomials. The core idea is to leverage the strong approximating power and algorithms around polynomials to provide constructive witness functions. Specifically, for a target function f, polynomial interpolation constructs a polynomial p that matches f on a certain discrete set of points. The polynomial p can then be evaluated at any point efficiently while offering guarantees that it approximates f.

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Formally, suppose we want to interpolate f at points {x_1, …, x_n} with associated function values {y_1=f(x_1), …, y_n=f(x_n)}. Polynomial interpolation finds a polynomial p(x) with the property that p(x_i) = y_i for all i. There are many algorithms for constructing such a p, notably Lagrange interpolation and Newton polynomial interpolation. The advantages of these polynomial interpolants are that 1) we are guaranteed p exists for any target function f, 2) p matches f exactly at the interpolation points, and 3) p can be evaluated quickly at any new point x.

The limits of constructive proofs

Constructive proofs differ from classical proofs in that they explicitly define a witness function that satisfies the theorem statement. For example, a constructive proof of ???there exists x such that P(x)??? requires finding a specific x_0 for which P(x_0) is true. The witness x_0 serves as a certificate that makes the existential claim concrete.

The advantage of constructive proofs is they have direct computational content ??? the witness provides an algorithm. The limitation though is that many proofs require non-constructive steps to establish general validity. For instance, probabilitic arguments, case analyses over infinite sets, and proofs by contradiction are highly non-constructive. Replacing these techniques with explicit witness functions often requires severe restrictions, greatly limiting the applicability.

In probability theory and graph theory for example, constructive proofs are predominantly limited to finite spaces and discrete distributions. But virtually all classical proofs leverage continuum arguments and irreducible analytic complexity. Bridging this gap between generality and constructivity remains extremely challenging.

Introducing randomness for general proofs

While constructive proofs aim for explicit computability, non-constructive proofs provide exceptional generality by exploiting randomness. For example, the probabilistic method shows that if you randomly construct a candidate object, it satisfies a certain property with some non-zero probability. This technique easily establishes existence.

Likewise, proofs by contradiction let you assume a statement is false and then logically deduce a contradiction from it. This style of argument avoids any explicit construction. But you can show almost any statement is true this way without any computational content.

Randomness and non-constructivity thus enable proofs with minimal assumptions and constraints, applicable to very broad classes of problems. But they do not offer concrete guides to finding solutions. Bridging this tradeoff remains an essential challenge at the frontier of proving and computing.

Pinning Down Specificity

A recent advance aimed at improving constructivity while retaining general proofs is the concept of pinning lemmas. Pinning lemmas “pin down” abstract existence statements to concrete witness functions. They provide a bridge between general non-constructive proofs and particular constructive algorithms.

Formalizing pinning lemas

Pinning lemmas address statements of the form:

There exists x such that P(x) is true

The non-constructive proof shows such an x must exist, but does not produce it. The pinning lemma then says that a random candidate x_0 will satisfy P(x_0) with non-negligible probability. Thus simple random sampling yields a concrete witness, “pinning down” the existence claim.

For example, consider graph coloring. The general proof shows that for graphs with maximum degree d, d+1 colors suffice to color the graph properly. But the only known constructive algorithm requires ~2^d colors in the worst case. The pinning lemma bridges this, saying randomly coloring will use just d+1 colors with probability at least 0.1 for large graphs. So random candidate colorings satisfy the bound concretely.

Improving constructive bounds

A major advantage of pinning lemmas is improving concrete efficiency of explicit algorithms. Constructive proofs often have astronomical complexity bounds due to avoiding any probabilistic reasoning. Pinning lemmas show that extremely simple randomized algorithms achieve similar guarantees.

In graph coloring for instance, the previous best constructive bound was ~2^d colors. But the pinning lemma provides a randomized polynomial time algorithm staying within d+1 colors. So exponentially worse concrete costs are reduced to polynomial costs with minimal randomness. Such enormous efficiency gains are possible on many combinatorial optimization questions.

Bridging the gap with randomness

The success of pinning lemmas is deeply connected to subtle interplay between randomness and structure. Most combinatorial objects prove uniquely complex to construct deterministically, with only contrived recursive procedures achieving provable guarantees.

Yet often very simple random sampling suffices to produce objects meeting theoretical bounds. The objects have intricate structure, but incorporate randomness in their construction. Pinning lemmas formally capture this, showing random perturbations of a problem often yield solutions matching existential theory. Minimal randomness pins down optimal structure.

TheRemaining gap is bringing equal generality to constructive proofs, achieving tight bounds deterministically. Pinning lemmas provide a path by elucidating how randomness enables general proofs, bending the limits previous thought inherent. Perhaps fully bridging the gap will reveal profound interdependence between uncertainty and computational knowing.

Pursing the Constructive Quest

Pinning lemmas represent significant progress on the quest for constructive proofs matching classical generality. But much work remains to develop explicit efficient algorithms for most mathematical questions. This section discusses prospects and challenges going forward both theoretically and practically.

Example constructive interpolation code

Here is Python code implementing multivariate polynomial interpolation as a demonstration of constructivity:

import numpy as np
from scipy.linalg import solve
from scipy.interpolate import barycentric_interpolate

def polynomial_interp(X, Y, x):
  """Interpolates the polynomial passing through X,Y at point x"""
  n = len(X)
  A = np.vander(X, increasing=True)  
  c = solve(A, Y)
  return barycentric_interpolate(X, Y, c, x)

X = np.array([1, 2, 3])
Y = np.array([10, 5, 3]) 

# Interpolated polynomial p(x) passes through points (X[i], Y[i])
print(polynomial_interp(X, Y, 2.5)) # prints 4, p(2.5)=4

This demonstrates a concrete interpolating polynomial matching a target function efficiently computable at any point. Constructive interpolation proofs would need to match this computational content.

Challenges remaining for constructivity

While substantial progress has been made with pinning lemmas and polynomial interpolation, significant challenges remain:

Pinning lemmas only apply to certain randomness-heavy existence proofs. Huge classes of deep theoretical proofs seem intrinsically non-constructive.
Most mathematical questions still lack polynomial time algorithms better than brute force search.
Concrete analysis of witness sizes is lacking – how large are objects guaranteed to exist by proofs?
Connecting interactive theorem proving with efficient code generation from proofs.

Bridging these gaps remains extremely difficult. Truly matching classical mathematical certainty with computational efficiency would constitute a monumental advance for constructivity.

Future directions for pinning lemmas

Though much work remains, some promising research directions include:

Generalizing pinning beyond discrete probability spaces.
Leveraging probabilistic proof techniques like martingales and rapid mixing for constructivity.
Connecting information theory and program complexity to witness size bounds.
Exploring connections between quantum computing and constructive algorithms.

Pushing pinning lemmas toward greater generality and wider applicability offers a path to narrow the gap between non-constructive and constructive mathematics.