Games, maps and randomness

George Barmpalias

Chinese Academy of Sciences

NanKai, October 2025

Overview

Games and randomness
- Allocations and Tree-embeddings
- Enumeration games
- Applications to algorithmic information
Inversions of functions
- Types of inversion and hardness
- Degrees of unsolvability of inversions
- Oneway functions and their strength
Progress and problems

Games with strings

The prefix $\preceq$ relation $\twomel$, $\omel$ gives them a tree structure.

$\twomel$ is the set of binary strings
$\omel$ is the set of strings natural numbers

The weight of a set $Q$ of strings is given by

$$\sum_{\sigma\in Q} 2^{-|\sigma|}.$$

A set of strings is prefix-free if no string in it is a prefix of another one.

$\Pone$ denotes player 1
$\Ptwo$ denotes player 2

Kraft game (binary)

Offline game

$\Pone$ picks a sequence $n_i, i < k $ of numbers with $\sum_{i< k} 2^{-n_i}\leq c$
$\Ptwo$ needs to pick an antichain $\sigma_i, i< k$ in $\twomel$ with $|\sigma_i|=n_i$

$\Ptwo$ has a winning-strategy iff $c\leq 1$.

Onine game. At each stage s:

$\Pone$ produces a number $n_s$ such that $\sum_{i\leq s} 2^{-n_i}\leq c$
$\Ptwo$ produces $\sigma_s$ with $|\sigma_s|=n_s$ and $\sigma_i, i\leq s$ is an antichain.

$\Ptwo$ has a winning-strategy iff $c\leq 1$.

Kraft Tree-games

Requests for strings have a tree-structure.

A splice-map is a map from $Q\subseteq\twomel$ to $S\subseteq\omel$ which is

length-preserving and order preserving.

In this case we say that $Q$ is a splice of $S$ and

the $\sigma\in Q$ that map to $\tau\in S$ are copies of $\tau$ in $Q$.

Weak embeddings of $S\subseteq \omel$ in $\twomel$ that preserve length.

Kraft Tree-game

Offline

$\Pone$ picks $S\subseteq\omega^{<\omega}$ of weight $\leq w$
$\Ptwo$ has to give a splice of $S$ in $\twomel$.

Online

Players enumerate $S\subseteq\omel$, $Q\subseteq\twomel$: at each stage s:

$\Pone$ enumerates $S\subseteq\omega^{<\omega}$ of weight $\leq w$.
$\Ptwo$ needs to maintain a splice $Q$ of $S$ in $\twomel$.

$\Ptwo$ wins $Q$ is a splice of $S$.

In the Kraft tree game, $\Ptwo$ has a winning strategy iff $w\leq 1$.

Information measures assign a complexity value to each string.

Kolmogorov complexity (universal but incomputable)
Time-bounded Kolmogorov complexity (computable).

If $I$ is a computable information measure, for every $x$ there is:

an algorithmically random real $z$ which computes $x$
the first $I_n(x)$ bits of $z$ determine the first $n$ bits of $x$.

If $I$ is computably enumerable $I_n(x)+\log n$ bits of $z$ suffice for $x\restr_n$.

Kraft Tree-game variations

dynamically restricted space for $\Ptwo$
dynamic shrinking of the strings of $\Pone$

These correspond to results for:

coding into algorithmically random reals
compressing down to the information content for (noncomputable) enumerable information measures

Disk game: collisions

tight-coding: $n$-bits to $n$-bits
every prefix of each string is coded
contaminated space but we allow $O(1)$ collisions
list-decoding

Enumeration games

In each round:

$\Pone$ picks $k$ numbers
$\Ptwo$ picks an even number between the min and max of the $k$ numbers.

Both choices are without repetition:

$\Pone$ cannot choose a number that he has chosen in previous stages
$\Ptwo$ cannot choose a number that he has chosen in previous stages.

Winning condition

$\Pone$ wins at a stage where $\Ptwo$ is unable to make a move
$\Ptwo$ wins if he has a legitimate move at each stage.

Who wins? This is open for $k>3$.

References for games

Allocation games

Game interpretation of Kolmogorov complexity. Muchnik et al.
The Kraft-Barmpalias-Lewis-Pye Lemma revisited. A. Shen
Compression of data streams to their information. Barmpalias & Lewis-Pye.
Optimal redundancy in computations from random oracles. Barmpalias & Lewis-Pye.
Compression of enumerations and gain. Barmpalias, Zhang, Zhan.

Martingale games

Granularity of wagers in games and the possibility of savings. Barmpalias, Fang.
Monotonous betting strategies in warped casinos. Barmpalias, Fang, Lewis-Pye.
Irreducibility of enumerable betting strategies. Barmpalias & Liu.

Hardness of inverting functions

The security of modern cryptographic protocols is based on the (assumed) hardness of certain computational problems.

Central in computational complexity is the notion of oneway functions:

finite maps that are easy to compute but hard to invert.

Oneway functions are used to make security protocols hard to break (hash maps).

Their existence is unknown: fundamental problem in computational complexity and modern cryptography.

Levin (2022) extended oneway maps to the reals and asked whether they exist.

Effective functions on the reals

This is a standard notion in computable analysis due to Turing (1936).

Computability of real functions is effective continuity:

computable real functions are continuous
every continuous real function is computable in some oracle

Probability and measure

Uniform measure on $2^{\omega}$ or Lebesgue measure on $[0,1]$.

A set of reals is positive if it has positive measure.

A function is positive is it maps every positive set to a positive set.

A property holds almost everywhere if it holds on a set of measure 1.

A a property $P$ of functions holds for almost all functions in a given class $\mathcal{C}$ if for almost all $w$ the property holds for all $g\leq_T w$ in $\mathcal{C}$.

Properties of interest: total, surjective, injective, collision-resistant.

Inversions and Collisions

We say that $g$ inverts $f$ on $y$ if $f(g(y))= y$.

If $I_g$ is the set of reals where $g$ inverts $f$ we say that:

$g$ fully inverts $f$ if $I_g= f(\twome)$
$g$ positively inverts $f$ if $I_g$ or $f\inv(I_g)$ is positive.

$f$ is effectively invertible on $y$ if $y$ computes a member of $f\inv(y)$.

We say that $g$ computes an $f$-collision on $x$ if $f(x)=f(g(z))$.

Complexity of inversion

There is no degree bound, even for total functions.

There is a total computable positive surjection $f$ which has no continuous full inversion.

Total computable injections have computable inverses.

If a total computable $f$ is injective on $R$ then $f:R\to 2^{\omega}$ is effectively invertible.

Inversion-hardness from non-injectivity and partiality (domain complexity).

Oneway functions and collision-resistance

Avoid: functions that map all positive sets to null sets do not qualify. A positive partial computable function $f$ is oneway if almost all functions fail to invert $f$ almost everywhere.

Collision-resistance is hardness of computing collisions
Avoid: functions that are injective on a positive set.

We say $f$ is collision-resistant if it is almost nowhere injective and almost all functions fail to compute $f$-collisions almost everywhere:

$$\textrm{$f(x)\neq f(h(x))$ for almost all $h, x$ with $h(x)\neq x$.}$$

Oneway and degrees of unsolvability

$f$ is oneway iff $f(g(y))\neq y$ for almost all functions $g$ and reals $y$. Zero-one law. If $f$ is not oneway then:

almost all oracles can invert $f$ on almost all $y$.
almost all oracles can compute a positive inversion of $f$.

Non-oneway maps are almost everywhere constant up to degree:

not oneway: $f(x)\oplus z \equiv_T x\oplus z$ for almost all $z$, $x$
not $0'$-oneway: $f(x)'\oplus z \equiv_T x'\oplus z$ for almost all $z, x$

Oneway and algorithmic randomness

A real is random if it is not a member of any null definable $G_\delta$ set.

The choice of definability determines the strength of the randomness notion.

Randomness with respect to $\Pi^0_2$ sets suffices for us.

Say $f$ is random-preserving if $f(x)$ is random for each random $x$. A partial computable $f$ is positive iff it is random-preserving.

Randomized computations

Randomized computations can occasionally be replaced by deterministic ones.

Probabilistic inversions and collisions

Given $f, g:\subseteq 2^{\omega}\to 2^{\omega}$ we say that $g$ is a probabilistic inversion of $f$ if

$$\hspace{0.1cm}\mu(\sqbrad{(y,r)}{f(g(y,r))=y})>0$$

A partial computable random-preserving function $f$ is

oneway if it has no effective probabilistic inversion.
collision-resistant if no $(x,z)$ with $f(x)=f(z)$ is probabilistically computable.

The following are equivalent:

almost all oracles compute a positive inversion of $f$
there is an effective probabilistic inversion of $f$.

Existence of oneway functions

A partial shuffle of the input bits based on an enumeration of $0'$ is oneway.

There is a total linear-time computable oneway surjection $f$ such that any probabilistic inversion of $f$ computes $0'$.

The unused bits in a shuffle make collisions easy to find, probabilistically.

There is a total poly-time computable collision-resistant oneway surjection.

Collision-resistance is achieved by effective perturbation of the output (hashing).

Ko and Friedman (1980s): Time complexity for real functions.

Strength of oneway - new results

Gauged by the oracles that probabilistically invert them.

If $f$ is a positive partial computable function then

$f$ has a positive inversion $g\leq_T 0''$ (not optimal)
if $f$ is total it has a positive inversion $g\leq_T 0'$ (optimal)

If $f$ is a positive partial computable function then

$f$ has a probabilistic inversion $g\leq_T 0'$ (optimal)
if $f$ is injective it has an effective positive inversion.

The maximum strength of total and partial oneway functions is $0'$.

Positive inversions harder than probabilistic

There is a partial computable $f:\subseteq\twome\to\twome$ which is

positive and nowhere effectively invertible
a.e. probabilistically invertible.

Oracle $0''$ can positively invert every positive effective $f$ but $0'$ cannot:

There is an effective positive $f$ which is probabilistically invertible but has no positive inversion $\leq_T 0'$: restrict $x\oplus z\mapsto z$ to a complex positive $\Pi^0_2$ class.

Classify the complexity of positive inversions: I think close to $0'$ (in-progress:).

Our oneway is a shuffle

A shuffle $f:\twome\to\twome$ is given by a computable injection $(a_i)\in\omega^{\omega}$ and $$ f(x)(i):=x(a_i). $$

Shuffle maps on the reals

A shuffle $f:\twome\to\twome$ is given by a computable injection $(a_i)\in\omega^{\omega}$ and $$ f(x)(i):=x(a_i). $$

Shuffle maps on the reals

A shuffle $f:\twome\to\twome$ is given by a computable injection $(a_i)\in\omega^{\omega}$ and $$ f(x)(i):=x(a_i). $$

Shuffle maps on the reals

A shuffle $f:\twome\to\twome$ is given by a computable injection $(a_i)\in\omega^{\omega}$ and $$ f(x)(i):=x(a_i). $$

Shuffle maps on the reals

A shuffle $f:\twome\to\twome$ is given by a computable injection $(a_i)\in\omega^{\omega}$ and $$ f(x)(i):=x(a_i). $$

Shuffle maps on the reals

Let $(a_i)$ be a computable enumeration of a noncomputable set.

The $(a_i)$-shuffle has the following properties:

it is a total computable positive surjection
strongly non-injective
- all inverse images are uncountable
- every injective restriction of it has null domain
every probabilistic inversion of it computes $\sqbrad{a_i}{i\in\omega}$.

Take $(a_i)$ be a computable enumeration of $0'$.