Algorithmic Thermodynamics

Algorithmic Thermodynamics
John C. Baez
Department of Mathematics, University of California
Riverside, California 92521, USA

Mike Stay
Computer Science Department, University of Auckland
and
Google, 1600 Amphitheatre Pkwy
Mountain View, California 94043, USA

email: baez@math.ucr.edu, stay@google.com

October 12, 2010

Abstract
Algorithmic entropy can be seen as a special case of entropy as studied in
statistical mechanics. This viewpoint allows us to apply many techniques
developed for use in thermodynamics to the subject of algorithmic infor-
mation theory. In particular, suppose we fix a universal prefix-free Turing
machine and let X be the set of programs that halt for this machine.
Then we can regard X as a set of ‘microstates’, and treat any function
on X as an ‘observable’. For any collection of observables, we can study
the Gibbs ensemble that maximizes entropy subject to constraints on ex-
pected values of these observables. We illustrate this by taking the log
runtime, length, and output of a program as observables analogous to the
energy E, volume V and number of molecules N in a container of gas.
The conjugate variables of these observables allow us to define quantities
which we call the ‘algorithmic temperature’ T , ‘algorithmic pressure’ P
and ‘algorithmic potential’ µ, since they are analogous to the temper-
ature, pressure and chemical potential. We derive an analogue of the
fundamental thermodynamic relation dE = T dS − P dV + µdN , and use
it to study thermodynamic cycles analogous to those for heat engines. We
also investigate the values of T, P and µ for which the partition function
converges. At some points on the boundary of this domain of convergence,
the partition function becomes uncomputable. Indeed, at these points the
partition function itself has nontrivial algorithmic entropy.

1

1 Introduction
Many authors [1, 6, 9, 12, 16, 24, 26, 28] have discussed the analogy between
algorithmic entropy and entropy as defined in statistical mechanics: that is,
the entropy of a probability measure p on a set X. It is perhaps insufficiently
appreciated that algorithmic entropy can be seen as a special case of the entropy
as defined in statistical mechanics. We describe how to do this in Section 3.
This allows all the basic techniques of thermodynamics to be imported to
algorithmic information theory. The key idea is to take X to be some version of
‘the set of all programs that eventually halt and output a natural number’, and
let p be a Gibbs ensemble on X. A Gibbs ensemble is a probability measure that
maximizes entropy subject to constraints on the mean values of some observables
— that is, real-valued functions on X.
In most traditional work on algorithmic entropy, the relevant observable
is the length of the program. However, much of the interesting structure of
thermodynamics only becomes visible when we consider several observables.
When X is the set of programs that halt and output a natural number, some
other important observables include the output of the program and logarithm
of its runtime. So, in Section 4 we illustrate how ideas from thermodynamics
can be applied to algorithmic information theory using these three observables.
To do this, we consider a Gibbs ensemble of programs which maximizes
entropy subject to constraints on:
• E, the expected value of the logarithm of the program’s runtime (which
we treat as analogous to the energy of a container of gas),
• V , the expected value of the length of the program (analogous to the
volume of the container), and
• N , the expected value of the program’s output (analogous to the number
of molecules in the gas).
This measure is of the form
1 −βE(x)−γV (x)−δN (x)
p= e
Z
for certain numbers β, γ, δ, where the normalizing factor

Z= e−βE(x)−γV (x)−δN (x)
x∈X

is called the ‘partition function’ of the ensemble. The partition function reduces
to Chaitin’s number Ω when β = 0, γ = ln 2 and δ = 0. This number is un-
computable [6]. However, we show that the partition function Z is computable
when β > 0, γ ≥ ln 2, and δ ≥ 0.
We derive an algorithmic analogue of the basic thermodynamic relation
dE = T dS − P dV + µdN.
Here:

2

• S is the entropy of the Gibbs emsemble,
• T = 1/β is the ‘algorithmic temperature’ (analogous to the temperature
of a container of gas). Roughly speaking, this counts how many times you
must double the runtime in order to double the number of programs in
the ensemble while holding their mean length and output fixed.
• P = γ/β is the ‘algorithmic pressure’ (analogous to pressure). This
measures the tradeoff between runtime and length. Roughly speaking,
it counts how much you need to decrease the mean length to increase the
mean log runtime by a specified amount, while holding the number of
programs in the ensemble and their mean output fixed.
• µ = −δ/β is the ‘algorithmic potential’ (analogous to chemical potential).
Roughly speaking, this counts how much the mean log runtime increases
when you increase the mean output while holding the number of programs
in the ensemble and their mean length fixed.
Starting from this relation, we derive analogues of Maxwell’s relations and
consider thermodynamic cycles such as the Carnot cycle or Stoddard cycle. For
this we must introduce concepts of ‘algorithmic heat’ and ‘algorithmic work’.

Charles Babbage described a computer powered by a steam engine; we de-
scribe a heat engine powered by programs! We admit that the significance of
this line of thinking remains a bit mysterious. However, we hope it points the
way toward a further synthesis of algorithmic information theory and thermo-
dynamics. We call this hoped-for synthesis ‘algorithmic thermodynamics’.

2 Related Work
Li and Vitńyi use the term ‘algorithmic thermodynamics’ for describing phys-
a
ical states using a universal prefix-free Turing machine U . They look at the

3

smallest program p that outputs a description x of a particular microstate to
some accuracy, and define the physical entropy to be

SA (x) = (k ln 2)(K(x) + Hx ),

where K(x) = |p| and Hx embodies the uncertainty in the actual state given
x. They summarize their own work and subsequent work by others in chapter
eight of their book [17]. Whereas they consider x = U (p) to be a microstate,
we consider p to be the microstate and x the value of the observable U . Then
their observables O(x) become observables of the form O(U (p)) in our model.
Tadaki [27] generalized Chaitin’s number Ω to a function ΩD and showed
that the value of this function is compressible by a factor of exactly D when
D is computable. Calude and Stay [5] pointed out that this generalization was
formally equivalent to the partition function of a statistical mechanical system
where temperature played the role of the compressibility factor, and studied
various observables of such a system. Tadaki [28] then explicitly constructed
a system with that partition function: given a total length E and number of
programs N, the entropy of the system is the log of the number of E-bit strings
in dom(U )N . The temperature is

1 ∆E
= .
T ∆S N

In a follow-up paper [29], Tadaki showed that various other quantities like the
free energy shared the same compressibility properties as ΩD . In this paper,
we consider multiple variables, which is necessary for thermodynamic cycles,
chemical reactions, and so forth.
Manin and Marcolli [20] derived similar results in a broader context and
studied phase transitions in those systems. Manin [18, 19] also outlined an
ambitious program to treat the infinite runtimes one finds in undecidable prob-
lems as singularities to be removed through the process of renormalization. In
a manner reminiscent of hunting for the proper definition of the “one-element
field” Fun , he collected ideas from many different places and considered how
they all touch on this central theme. While he mentioned a runtime cutoff as
being analogous to an energy cutoff, the renormalizations he presented are un-
computable. In this paper, we take the log of the runtime as being analogous
to the energy; the randomness described by Chaitin and Tadaki then arises as
the infinite-temperature limit.

3 Algorithmic Entropy
To see algorithmic entropy as a special case of the entropy of a probability
measure, it is useful to follow Solomonoff [24] and take a Bayesian viewpoint.
In Bayesian probability theory, we always start with a probability measure called
a ‘prior’, which describes our assumptions about the situation at hand before
we make any further observations. As we learn more, we may update this

4

prior. This approach suggests that we should define the entropy of a probability
measure relative to another probability measure — the prior.
A probability measure p on a finite set X is simply a function p : X → [0, 1]
whose values sum to 1, and its entropy is defined as follows:

S(p) = − p(x) ln p(x).
x∈X

But we can also define the entropy of p relative to another probability measure
q:
p(x)
S(p, q) = − p(x) ln .
q(x)
x∈X

This relative entropy has been extensively studied and goes by various other
names, including ‘Kullback–Leibler divergence’ [13] and ‘information gain’ [23].
The term ‘information gain’ is nicely descriptive. Suppose we initially as-
sume the outcome of an experiment is distributed according to the probability
measure q. Suppose we then repeatedly do the experiment and discover its out-
come is distributed according to the measure p. Then the information gained is
S(p, q).
Why? We can see this in terms of coding. Suppose X is a finite set of signals
which are randomly emitted by some source. Suppose we wish to encode these
signals as efficiently as possible in the form of bit strings. Suppose the source
emits the signal x with probability p(x), but we erroneously believe it is emitted
with probability q(x). Then S(p, q)/ ln 2 is the expected extra message-length
per signal that is required if we use a code that is optimal for the measure q
instead of a code that is optimal for the true measure, p.
The ordinary entropy S(p) is, up to a constant, just the relative entropy in
the special case where the prior assigns an equal probability to each outcome.
In other words:
S(p) = S(p, q0 ) + S(q0 )
when q0 is the so-called ‘uninformative prior’, with q0 (x) = 1/|X| for all x ∈ X.
We can also define relative entropy when the set X is countably infinite. As
before, a probability measure on X is a function p : X → [0, 1] whose values sum
to 1. And as before, if p and q are two probability measures on X, the entropy
of p relative to q is defined by

p(x)
S(p, q) = − p(x) ln . (1)
q(x)
x∈X

But now the role of the prior becomes more clear, because there is no probability
measure that assigns the same value to each outcome!
In what follows we will take X to be — roughly speaking — the set of
all programs that eventually halt and output a natural number. As we shall
see, while this set is countably infinite, there are still some natural probability
measures on it, which we may take as priors.

5

To make this precise, we recall the concept of a universal prefix-free Turing
machine. In what follows we use string to mean a bit string, that is, a finite,
possibly empty, list of 0’s and 1’s. If x and y are strings, let x||y be the con-
catenation of x and y. A prefix of a string z is a substring beginning with the
first letter, that is, a string x such that z = x||y for some y. A prefix-free
set of strings is one in which no element is a prefix of any other. The domain
dom(M ) of a Turing machine M is the set of strings that cause M to eventually
halt. We call the strings in dom(M ) programs. We assume that when the M
halts on the program x, it outputs a natural number M (x). Thus we may think
of the machine M as giving a function M : dom(M ) → N.
A prefix-free Turing machine is one whose halting programs form a
prefix-free set. A prefix-free machine U is universal if for any prefix-free Turing
machine M there exists a constant c such that for each string x, there exists a
string y with
U (y) = M (x) and |y| < |x| + c.
Let U be a universal prefix-free Turing machine. Then we can define some
probability measures on X = dom(U ) as follows. Let

| · |: X → N

be the function assigning to each bit string its length. Then there is for any
constant γ > ln 2 a probability measure p given by
1 −γ|x|
p(x) = e .
Z
Here the normalization constant Z is chosen to make the numbers p(x) sum to
1:
Z= e−γ|x| .
x∈X

It is worth noting that for computable real numbers γ ≥ ln 2, the normalization
constant Z is uncomputable [27]. Indeed, when γ = ln 2, Z is Chaitin’s famous
number Ω. We return to this issue in Section 4.5.
Let us assume that each program prints out some natural number as its
output. Thus we have a function

N: X →N

where N (x) equals i when program x prints out the number i. We may use this
function to ‘push forward’ p to a probability measure q on the set N. Explicitly:

q(i) = e−γ|x| .
x∈X:N (x)=i

In other words, if i is some natural number, q(i) is the probability that a program
randomly chosen according to the measure p will print out this number.

6

Given any natural number n, there is a probability measure δn on N that
assigns probability 1 to this number:

1 if m = n
δn (m) =
0 otherwise.

We can compute the entropy of δn relative to q:
δn (i)
S(δn , q) = − δn (i) ln
q(i)
i∈N
  (2)
= − ln  e−γ|x| + ln Z.
x∈X : N (x)=n

Since the quantity ln Z is independent of the number n, and uncomputable, it
makes sense to focus attention on the other part of the relative entropy:
 

− ln  e−γ|x|  .
x∈X : N (x)=n

If we take γ = ln 2, this is precisely the algorithmic entropy [7, 16] of the
number n. So, up to the additive constant ln Z, we have seen that algorithmic
entropy is a special case of relative entropy.
One way to think about entropy is as a measure of surprise: if you can
predict what comes next — that is, if you have a program that can compute it
for you — then you are not surprised. For example, the first 2000 bits of the
binary fraction for 1/3 can be produced with this short Python program:
print "01" * 1000
But if the number is complicated, if every bit is surprising and unpredictable,
then the shortest program to print the number does not do any computation at
all! It just looks something like
print "101000011001010010100101000101111101101101001010"
Levin’s coding theorem [15] says that the difference between the algorithmic
entropy of a number and its Kolmogorov complexity — the length of the
shortest program that outputs it — is bounded by a constant that only depends
on the programming language.
So, up to some error bounded by a constant, algorithmic information is infor-
mation gain. The algorithmic entropy is the information gained upon learning
a number, if our prior assumption was that this number is the output of a ran-
domly chosen program — randomly chosen according to the measure p where
γ = ln 2.
So, algorithmic entropy is not just analogous to entropy as defined in statis-
tical mechanics: it is a special case, as long as we take seriously the Bayesian

7

philosophy that entropy should be understood as relative entropy. This real-
ization opens up the possibility of taking many familiar concepts from thermo-
dynamics, expressed in the language of statistical mechanics, and finding their
counterparts in the realm of algorithmic information theory.
But to proceed, we must also understand more precisely the role of the
measure p. In the next section, we shall see that this type of measure is already
familiar in statistical mechanics: it is a Gibbs ensemble.

4 Algorithmic Thermodynamics
Suppose we have a countable set X, finite or infinite, and suppose
C1 , . . . , Cn : X → R is some collection of functions. Then we may seek a prob-
ability measure p that maximizes entropy subject to the constraints that the
mean value of each observable Ci is a given real number C i :

p(x) Ci (x) = C i .
x∈X

As nicely discussed by Jaynes [10, 11], the solution, if it exists, is the so-called
Gibbs ensemble:
1
p(x) = e−(s1 C1 (x)+···+sn Cn (x))
Z
for some numbers si ∈ R depending on the desired mean values C i . Here the
normalizing factor Z is called the partition function:

Z= e−(s1 C1 (x)+···+sn Cn (x)) .
x∈X

In thermodynamics, X represents the set of microstates of some physical
system. A probability measure on X is also known as an ensemble. Each func-
tion Ci : X → R is called an observable, and the corresponding quantity si is
called the conjugate variable of that observable. For example, the conjugate
of the energy E is the inverse of temperature T , in units where Boltzmann’s
constant equals 1. The conjugate of the volume V — of a piston full of gas, for
example — is the pressure P divided by the temperature. And in a gas contain-
ing molecules of various types, the conjugate of the number Ni of molecules of
the ith type is minus the ‘chemical potential’ µi , again divided by temperature.
For easy reference, we list these observables and their conjugate variables below.

8

THERMODYNAMICS

Observable Conjugate Variable

1
energy: E
T
P
volume: V
T
µi
number: Ni −
T
Now let us return to the case where X = dom(U ). Recalling that programs
are bit strings, one important observable for programs is the length:
| · | : X → N.

We have already seen the measure
1 −γ|x|
p(x) = e .
Z
Now its significance should be clear! This is the probability measure on programs
that maximizes entropy subject to the constraint that the mean length is some
constant ℓ:
p(x) |x| = ℓ.
x∈X

So, γ is the conjugate variable to program length.
There are, however, other important observables that can be defined for
programs, and each of these has a conjugate quantity. To make the analogy
to thermodynamics as vivid as possible, let us arbitrarily choose two more ob-
servables and treat them as analogues of energy and the number of some type
of molecule. Two of the most obvious observables are ‘output’ and ‘runtime’.
Since Levin’s computable complexity measure [14] uses the logarithm of runtime
as a kind of ‘cutoff’ reminiscent of an energy cutoff in renormalization, we shall
arbitrarily choose the log of the runtime to be analogous to the energy, and
denote it as
E : X → [0, ∞)
Following the chart above, we use 1/T to stand for the variable conjugate to E.
We arbitrarily treat the output of a program as analogous to the number of a
certain kind of molecule, and denote it as
N : X → N.

We use −µ/T to stand for the conjugate variable of N . Finally, as already
hinted, we denote program length as
V:X→N

9

so that in terms of our earlier notation, V (x) = |x|. We use P/T to stand for
the variable conjugate to V .

ALGORITHMS

Observable Conjugate Variable

1
log runtime: E
T
P
length: V
T
µ
output: N −
T
Before proceeding, we wish to emphasize that the analogies here were chosen
somewhat arbitrarily. They are merely meant to illustrate the application of
thermodynamics to the study of algorithms. There may or may not be a specific
‘best’ mapping between observables for programs and observables for a container
of gas! Indeed, Tadaki [28] has explored another analogy, where length rather
than log run time is treated as the analogue of energy. There is nothing wrong
with this. However, he did not introduce enough other observables to see the
whole structure of thermodynamics, as developed in Sections 4.1-4.2 below.
Having made our choice of observables, we define the partition function by
1
Z= e− T (E(x)+P V (x)−µN (x)) .
x∈X

When this sum converges, we can define a probability measure on X, the Gibbs
ensemble, by
1 1
p(x) = e− T (E(x)+P V (x)−µN (x)) .
Z
Both the partition function and the probability measure are functions of T, P
and µ. From these we can compute the mean values of the observables to which
these variables are conjugate:
E = p(x) E(x)
x∈X

V = p(x) V (x)
x∈X

N = p(x) N (x)
x∈X

In certain ranges, the map (T, P, µ) → (E, V , N ) will be invertible. This allows
us to alternatively think of Z and p as functions of E, V , and N . In this
situation it is typical to abuse language by omitting the overlines which denote
‘mean value’.

10

4.1 Elementary Relations
The entropy S of the Gibbs ensemble is given by

S=− p(x) ln p(x).
x∈X

We may think of this as a function of T, P and µ, or alternatively — as explained
above — as functions of the mean values E, V, and N . Then simple calculations,
familiar from statistical mechanics [22], show that

∂S 1
= (3)
∂E V,N T
∂S P
= (4)
∂V E,N T
∂S µ
=− . (5)
∂N E,V T
We may summarize all these by writing
1 P µ
dS = dE + dV − dN
T T T
or equivalently
dE = T dS − P dV + µdN. (6)
Starting from the latter equation we see:

∂E
=T (7)
∂S V,N

∂E
= −P (8)
∂V S,N

∂E
= µ. (9)
∂N S,V

With these deﬁnitions, we can start to get a feel for what the conjugate
variables are measuring. To build intuition, it is useful to think of the entropy
S as roughly the logarithm of the number of programs whose log runtimes,
length and output lie in small ranges E ± ∆E, V ± ∆V and N ± ∆N . This is at
best approximately true, but in ordinary thermodynamics this approximation
is commonly employed and yields spectacularly good results. That is why in
thermodynamics people often say the entropy is the logarithm of the number
of microstates for which the observables E, V and N lie within a small range of
their speciﬁed values [22].
If you allow programs to run longer, more of them will halt and give an
answer. The algorithmic temperature, T , is roughly the number of times

11

you have to double the runtime in order to double the number of ways to satisfy
the constraints on length and output.
The algorithmic pressure, P , measures the tradeoff between runtime and
length [4]: if you want to keep the number of ways to satisfy the constraints
constant, then the freedom gained by having longer runtimes has to be counter-
balanced by shortening the programs. This is analogous to the pressure of gas
in a piston: if you want to keep the number of microstates of the gas constant,
then the freedom gained by increasing its energy has to be counterbalanced by
decreasing its volume.
Finally, the algorithmic potential describes the relation between log run-
time and output: it is a quantitative measure of the principle that most large
outputs must be produced by long programs.

4.2 Thermodynamic Cycles
One of the first applications of thermodynamics was to the analysis of heat
engines. The underlying mathematics applies equally well to algorithmic ther-
modynamics. Suppose C is a loop in (T, P, µ) space. Assume we are in a region
that can also be coordinatized by the variables E, V, N . Then the change in
algorithmic heat around the loop C is defined to be

∆Q = T dS.
C

Suppose the loop C bounds a surface Σ. Then Stokes’ theorem implies that

∆Q = T dS = dT dS.
C Σ

However, Equation (6) implies that

dT dS = d(T dS) = d(dE + P dV − µdN ) = +dP dV − dµdN

since d2 = 0. So, we have

∆Q = (dP dV − dµdN )
Σ

or using Stokes’ theorem again

∆Q = (P dV − µdN ). (10)
C

In ordinary thermodynamics, N is constant for a heat engine using gas in a
sealed piston. In this situation we have

∆Q = P dV.
C

This equation says that the change in heat of the gas equals the work done on
the gas — or equivalently, minus the work done by the gas. So, in algorithmic

12

thermodynamics, let us define C P dV to be the algorithmic work done on our
ensemble of programs as we carry it around the loop C. Beware: this concept
is unrelated to ‘computational work’, meaning the amount of computation done
by a program as it runs.
To see an example of a cycle in algorithmic thermodynamics, consider the
analogue of the heat engine patented by Stoddard in 1919 [25]. Here we fix N
to a constant value and consider the following loop in the P V plane:

(P2 , V1 )

1 2
(P3 , V2 )
P (P1 , V1 )
3
4 (P4 , V2 )

V

We start with an ensemble with algorithmic pressure P1 and mean length V1 .
We then trace out a loop built from four parts:

1. Isometric. We increase the pressure from P1 to P2 while keeping the mean
length constant. No algorithmic work is done on the ensemble of programs
during this step.
2. Isentropic. We increase the length from V1 to V2 while keeping the number
of halting programs constant. High pressure means that we’re operating in
a range of runtimes where if we increase the length a little bit, many more
programs halt. In order to keep the number of halting programs constant,
we need to shorten the runtime significantly. As we gradually increase
the length and lower the runtime, the pressure drops to P3 . The total
difference in log runtime is the algorithmic work done on the ensemble
during this step.
3. Isometric. Now we decrease the pressure from P3 to P4 while keeping the
length constant. No algorithmic work is done during this step.
4. Isentropic. Finally, we decrease the length from V2 back to V1 while
keeping the number of halting programs constant. Since we’re at low
pressure, we need only increase the runtime a little. As we gradually
decrease the length and increase the runtime, the pressure rises slightly
back to P1 . The total increase in log runtime is minus the algorithmic
work done on the ensemble of programs during this step.

The total algorithmic work done on the ensemble per cycle is the difference in
log runtimes between steps 2 and 4.

13

4.3 Further Relations
From the elementary thermodynamic relations in Section 4.1, we can derive
various others. For example, the so-called ‘Maxwell relations’ are obtained by
computing the second derivatives of thermodynamic quantities in two diﬀerent
orders and then applying the basic derivative relations, Equations (7-9). While
trivial to prove, these relations say some things about algorithmic thermody-
namics which may not seem intuitively obvious.
We give just one example here. Since mixed partials commute, we have:
∂2E ∂2E
= .
∂V ∂S N ∂S∂V N

Using Equation (7), the left side can be computed as follows:
∂2E ∂ ∂E ∂T
= =
∂V ∂S N ∂V S,N ∂S V,N ∂V S,N

Similarly, we can compute the right side with the help of Equation (8):
∂2E ∂ ∂E ∂P
= =− .
∂S∂V N ∂S V,N ∂V S,N ∂S V,N

As a result, we obtain:
∂T ∂P
=− .
∂V S,N ∂S V,N
We can also derive interesting relations involving derivatives of the partition
function. These become more manageable if we rewrite the partition function
in terms of the conjugate variables of the observables E, V , and N :
1 P µ
β= , γ= , δ=− . (11)
T T T
Then we have
Z= e−βE(x)−γV (x)−δN (x)
x∈X

Simple calculations, standard in statistical mechanics [22], then allow us to
compute the mean values of observables as derivatives of the logarithm of Z
with respect to their conjugate variables. Here let us revert to using overlines
to denote mean values:
∂
E = p(x) E(x) =− ln Z
∂β
x∈X

∂
V = p(x) V (x) =− ln Z
∂γ
x∈X

∂
N = p(x) N (x) =− ln Z
∂δ
x∈X

14

We can go further and compute the variance of these observables using second
derivatives:
2 ∂2
(∆E)2 = p(x)(E(x)2 − E ) = ln Z
∂ 2β
x∈X

and similarly for V and N . Higher moments of E, V and N can be computed
by taking higher derivatives of ln Z.

4.4 Convergence
So far we have postponed the crucial question of convergence: for which values of
T, P and µ does the partition function Z converge? For this it is most convenient
to treat Z as a function of the variables β, γ and δ introduced in Equation (11).
For which values of β, γ and δ does the partition function converge?
First, when β = γ = δ = 0, the contribution of each program is 1. Since
there are infinitely many halting programs, Z(0, 0, 0) does not converge.
Second, when β = 0, γ = ln 2, and δ = 0, the partition function converges to
Chaitin’s number
Ω= 2−V (x) .
x∈X

To see that the partition function converges in this case, consider this mapping
of strings to segments of the unit interval:
empty
0 1
00 01 10 11
000 001 010 011 100 101 110 111
.
.
.

Each segment consists of all the real numbers whose binary expansion begins
with that string; for example, the set of real numbers whose binary expansion
begins 0.101 . . . is [0.101, 0.110) and has measure 2−|101| = 2−3 = 1/8. Since the
set of halting programs for our universal machine is prefix-free, we never count
any segment more than once, so the sum of all the segments corresponding to
halting programs is at most 1.
Third, Tadaki has shown [27] that the expression

e−γV (x)
x∈X

converges for γ ≥ ln 2 but diverges for γ < ln 2. It follows that Z(β, γ, δ) con-
verges whenever γ ≥ ln 2 and β, δ ≥ 0.
Fourth, when β > 0 and γ = δ = 0, convergence depends on the machine.
There are machines where infinitely many programs halt immediately. For these,
Z(β, 0, 0) does not converge. However, there are also machines where program

15

x takes at least V (x) steps to halt; for these machines Z(β, 0, 0) will converge
when β ≥ ln 2. Other machines take much longer to run. For these, Z(β, 0, 0)
will converge for even smaller values of β.
Fifth and finally, when β = γ = 0 and δ > 0, Z(β, γ, δ) fails to converge,
since there are infinitely many programs that halt and output 0.

4.5 Computability
Even when the partition function Z converges, it may not be computable. The
theory of computable real numbers was independently introduced by Church,
Post, and Turing, and later blossomed into the field of computable analysis [21].
We will only need the basic definition: a real number a is computable if there
is a recursive function that maps any natural number n > 0 to an integer f (n)
such that
f (n) f (n) + 1
≤a≤ .
n n
In other words, for any n > 0, we can compute a rational number that approx-
imates a with an error of at most 1/n. This definition can be formulated in
various other equivalent ways: for example, the computability of binary digits.
Chaitin [6] proved that the number

Ω = Z(0, ln 2, 0)

is uncomputable. In fact, he showed that for any universal machine, the values
of all but finitely many bits of Ω are not only uncomputable, but random:
knowing the value of some of them tells you nothing about the rest. They’re
independent, like separate flips of a fair coin.
More generally, for any computable number γ ≥ ln 2, Z(0, γ, 0) is ‘partially
random’ in the sense of Tadaki [3, 27]. This deserves a word of explanation. A
fixed formal system with finitely many axioms can only prove finitely many bits
of Z(0, γ, 0) have the values they do; after that, one has to add more axioms or
rules to the system to make any progress. The number Ω is completely random
in the following sense: for each bit of axiom or rule one adds, one can prove at
most one more bit of its binary expansion has the value it does. So, the most
efficient way to prove the values of these bits is simply to add them as axioms!
But for Z(0, γ, 0) with γ > ln 2, the ratio of bits of axiom per bits of sequence
is less than than 1. In fact, Tadaki showed that for any computable γ ≥ ln 2,
the ratio can be reduced to exactly (ln 2)/γ.
On the other hand, Z(β, γ, δ) is computable for all computable real numbers
β > 0, γ ≥ ln 2 and δ ≥ 0. The reason is that β > 0 exponentially suppresses
the contribution of machines with long runtimes, eliminating the problem posed
by the undecidability of the halting problem. The fundamental insight here is
due to Levin [14]. His idea was to ‘dovetail’ all programs: on turn n, run each
of the first n programs a single step and look to see which ones have halted. As
they halt, add their contribution to the running estimate of Z. For any k ≥ 0
and turn t ≥ 0, let kt be the location of the first zero bit after position k in the

16

estimation of Z. Then because −βE(x) is a monotonically decreasing function
of the runtime and decreases faster than kt , there will be a time step where the
total contribution of all the programs that have not halted yet is less than 2−kt .

5 Conclusions
There are many further directions to explore. Here we mention just three. First,
as already mentioned, the ‘Kolmogorov complexity’ [12] of a number n is the
number of bits in the shortest program that produces n as output. However,
a very short program that runs for a million years before giving an answer is
not very practical. To address this problem, the Levin complexity [15] of n
is defined using the program’s length plus the logarithm of its runtime, again
minimized over all programs that produce n as output. Unlike the Kolmogorov
complexity, the Levin complexity is computable. But like the Kolmogorov com-
plexity, the Levin complexity can be seen as a relative entropy—at least, up to
some error bounded by a constant. The only difference is that now we com-
pute this entropy relative to a different probability measure: instead of using
the Gibbs distribution at infinite algorithmic temperature, we drop the tem-
perature to ln 2. Indeed, the Kolmogorov and Levin complexities are just two
examples from a continuum of options. By adjusting the algorithmic pressure
and temperature, we get complexities involving other linear combinations of
length and log runtime. The same formalism works for complexities involving
other observables: for example, the maximum amount of memory the program
uses while running.
Second, instead of considering Turing machines that output a single natural
number, we can consider machines that output a finite list of natural numbers
(N1 , . . . , Nj ); we can treat these as populations of different “chemical species”
and define algorithmic potentials for each of them. Processes analogous to chem-
ical reactions are paths through this space that preserve certain invariants of the
lists. With chemical reactions we can consider things like internal combustion
cycles.
Finally, in ordinary thermodynamics the partition function Z is simply a
number after we fix values of the conjugate variables. The same is true in
algorithmic thermodynamics. However, in algorithmic thermodynamics, it is
natural to express this number in binary and inquire about the algorithmic
entropy of the first n bits. For example, we have seen that for suitable values
of temperature, pressure and chemical potential, Z is Chaitin’s number Ω. For
each universal machine there exists a constant c such that the first n bits of the
number Ω have at least n − c bits of algorithmic entropy with respect to that
machine. Tadaki [27] generalized this computation to other cases.
So, in algorithmic thermodynamics, the partition function itself has nontriv-
ial entropy. Tadaki has shown that the same is true for algorithmic pressure
(which in his analogy he calls ‘temperature’). This reflects the self-referential
nature of computation. It would be worthwhile to understand this more deeply.

17

Acknowledgements
We thank Leonid Levin and the denizens of the n-Category Caf´ for useful
e
comments. MS thanks Cristian Calude for many discussions of algorithmic in-
formation theory. JB thanks Bruce Smith for discussions on relative entropy.
He also thanks Mark Smith for conversations on physics and information the-
ory, as well as for giving him a copy of Reif’s Fundamentals of Statistical and
Thermal Physics.

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