Undecidability in Mathematics and Physics

Abstract

Kurt Gödel’s seminal proof of his first Incompleteness Theorem in 1931 paved the way for Alonzo Church and Alan Turing to formally define the concept of computability and thus prove the undecidability of Hilbert’s Entscheidungsproblem in 1936. Turing’s method of getting to his undecidability result by reduction to the Halting Problem has become a general method in proofs of undecidability for a wide range of problems in Mathematics including the Domino Problem, which involves deciding whether a finite set of Wang tiles can tile the plane. Recently, Robinson’s proof of the undecidability of the Domino problem has come to play a vital role in Cubitt, Garcia and Wolf’s undecidability proof of the Spectral Gap problem in Physics. The undecidability result for deciding if a given quantum mechanical system has a Spectral Gap has wide ranging implications in Physics, including implications for the Yang–Mills mass gap conjecture which is a one of the millennium prize problems. In this talk, an attempt will be made to weave together all these strands back to the basic ideas involved in undecidability proofs including the ideas of recursive functions, Gödel numbering, diagonalization and most importantly the notion of computability and it’s limits within the general framework of Turing Machines.

Full text

Undecidability in Mathematics and Physics
Farhan Tanvir Chowdhury,
Aberystwyth University
TMT 2019, Greenwich
9 February 2019
Farhan Tanvir Chowdhury, Aberystwyth University Undecidability in Mathematics and Physics

Foundations
Parallel lines in Different Plane Geometries
End of 19th century (Frege and Peano): Efforts to axiomatize mathematics.
20th century (Russell and Whitehead): Formal theories framed in 1st Order Logic,
a method of stripping down propositions into precise formalised statements involving
only quantifiers (quantified variables) and their relations.
Farhan Tanvir Chowdhury, Aberystwyth University Undecidability in Mathematics and Physics

A Theory in 1st Order Logic
Peano Arithmetic (PA): Consists of 9 key axioms1, uses constant symbols with:
Function Symbols Relation symbols Logical symbols
+,⇥2^,_
0$,¬
S8,9,=
Example (For the Logic Enthusiasts)
Boolean expressions for De Morgan’s Laws in 1st order logic:
(¬P_Q)$(¬P)^(¬Q)
(¬P^Q)$(¬P)_(¬Q)
PA can contain arithmetic, an arithmetic statement like “2+2=4” in PA translates to:
S(S0)+S(S0)=S(S(S(S0)))
1the last axiom being the “axiom of induction”
Farhan Tanvir Chowdhury, Aberystwyth University Undecidability in Mathematics and Physics

The first general notion of Undecidability
PA formed basis for Prinicipia Mathematica, thought to be step towards complete
axiomatisation. But:
Theorem (1931: Gödel’s 1st Incompleteness Theorem)
Any “consistent” formal theory, strong enough to contain arithmetic (like PA ), is incomplete.
Definition (Axiomatic Independence)
A mathematical statement is undecidable in the Gödelian sense if neither it, nor its
negation, can be deduced using the rules of logic from the set of axioms being used.
Farhan Tanvir Chowdhury, Aberystwyth University Undecidability in Mathematics and Physics

The 2nd general notion of Undecidability
Methods in Gödel’s proof (like Gödel numbering and diagonalization) vital for Church
and Turing to show undecidability of the Entscheidungsproblem in 1936.
Leads to 2nd notion of undecidability:
Definition (Decision Problem)
A problem with a YES/NO answer is undecidable if there is no algorithm that terminates
with the correct answer for the problem.
Turing Machine: Mathematical model of computation involving a
machine manipulating symbols on a bi-infinite strip of tape accord-
ing to a table of rules.
Performs only one action at a time, stores no other information apart
from the state it is in and the symbols written on tape.
Universal Turing Machine (UTM): Turing machine that can simulate
any arbitrary Turing machine on any arbitrary input.
Diagram Source: http://www.cse.lehigh.edu/ glennb/um/1intro.pdf
Farhan Tanvir Chowdhury, Aberystwyth University Undecidability in Mathematics and Physics

The Halting Problem
All of Computer Science essentially boils down to Turing’s model of computation.
Algorithms: The class of recursive functions computable by a Turing machine.
If a decision problem is intractable by a Turing Machine, it is intractable in general!
Definition (The Halting Problem)
Does a UTM halt given an initial input? Or equivalently in modern computing terms:
Given a program P and some n2N, does P eventually halt when run on input n?
Theorem (1936 Turing)
The Halting Problem is undecidable.
Farhan Tanvir Chowdhury, Aberystwyth University Undecidability in Mathematics and Physics

Undecidability of the Halting Problem
Proof Sketch.
Consider an encoding of programs as natural numbers, and suppose there is an algorithm
for deciding when a program Phalts on an input n2N.
Write a new program Hsuch that
Hhalts on input n$program ndoes not halt on input n.
Taking n=H,His run on input Hto get contradiction:
Hhalts on input Hif and only if Hdoes not halt on input H.
Farhan Tanvir Chowdhury, Aberystwyth University Undecidability in Mathematics and Physics

The Domino Problem
Definition (Wang Tiles)
Unit squares in the plane with coloured edges which may be translated, but not
rotated or reflected. Right: A Wang Tile of 2 colours.
Definition (Tileset)
A finite collection of Wang Tiles, each of which can be copied infinitely
many times to tile the plane, an example on the left.
Definition (Domino Problem)
Is there a valid tiling of the entire plane using only translated copies of
the given tiles?
Farhan Tanvir Chowdhury, Aberystwyth University Undecidability in Mathematics and Physics

Undecidability of the Domino Problem
Theorem (1966 Berger)
The Domino Problem is undecidable.
Proven by reducing a decision problem for a tileset admitting only an aperiodic tiling of
the plane to the halting problem.
Berger’s tileset of 20426 tiles reduced greatly to a set of 6 tiles, and the overall proof
simplified, by Robinson in 1966.
The Robinson Tileset
Farhan Tanvir Chowdhury, Aberystwyth University Undecidability in Mathematics and Physics

What is the Spectral Gap?
Definition (Hamiltonian)
A self adjoint operator on a Hilbert Space corresponding to the total p.e and k.e of a system.
Definition (Spectrum)
The set of eigenvalues of an operator A:V!V, given V is a finite dimensional vector space.
Definition (Ground State Energy and Excitation Energies)
Ground State Energy (G.S.E): Smallest eigenvalue of given Hamiltonian H⇤(L), corresponding
eigenvectors are ground states.
Excitation Energies: Eigenvalues >G.S.E, corresponding eigenvectors are excited states.
Definition (Spectral Gap)
The spectral gap (H)of Hamiltonian H⇤(L)is defined as (H⇤(L)):=1(H⇤(L))0(H⇤(L)),
the difference between the smallest eigenvalues of H⇤(L).
Farhan Tanvir Chowdhury, Aberystwyth University Undecidability in Mathematics and Physics

Spectral Gap Problem for a 2D Lattice.
Consider a translationally invariant L⇥Llattice, with each site (vertex)
assigned a complex Hilbert space H(i)'Cdof local dimension d, and
to each pair of neighbouring sites we associate a Hamiltonian h(i,j)and
a further local Hamiltonian h1(i)representing some nearest neighbour
interaction between each site. Overall, this gives us the total Hamilto-
nian:
H⇤(L):= Xh(i,j)+Xh(i)
1
H⇤(L)is gapped if (H⇤(L))(+), and gapless if spectrum above ground state is continuous.
Definition (Spectral Gap Problem)
Given a general 2 body case H⇤(L)as described as input, is H⇤(L)gapped or gapless?
Farhan Tanvir Chowdhury, Aberystwyth University Undecidability in Mathematics and Physics

Undecidability of Spectral Gap: Proof Outline
Theorem (2015 Cubitt et al. . . )
9families of hamiltonians H⇤(L)parameterised by n 2N(essentially computable d 2⇥d2matrix
functions of parameter n) for which the Spectral Gap Problem is undecidable.
Represent structure of the Robinson tiles as classical Hamiltonian.
Add “quantum layer” on top of tiling Hamiltonian and choose suit-
able translationally-invariant coupling between layers.
Copies of a QTM encoded in a 1D history state Hamiltonian can be
effectively placed along one edge of all of the squares.
Ground state of the sandwiched Hamiltonian consists of the Robin-
son tiling configuration, with computational history states in the
quantum layer.
Each of these encodes the evolution of a quantum phase estimation
algorithm and UTM.
Diagram Source: Undecidability of the Spectral
Gap by Cubbit et al, Nature 2015.
Farhan Tanvir Chowdhury, Aberystwyth University Undecidability in Mathematics and Physics

Conclusion, Parting Thoughts and Future Directions
Analogue for Gödelian Incompleteness demonstrated in Condensed Matter Physics.
Undecidability for problems in Physics must be taken seriously, the Spectral Gap result
has been extended to even simpler 1D systems in 2018 by Bausch et al.
Yang Mills Mass Gap (a millennium prize problem) is an analogue of a Spectral Gap in
the context of QFT.
The above, along with the Haldane Conjecture may be undecidable.
Spectral Gap Undecidability has real world implications (future material applications).
Questions?
Farhan Tanvir Chowdhury, Aberystwyth University Undecidability in Mathematics and Physics