Lecture 18: PCF Böhm trees, Innocent strategies, Hyland-Ong Games

Lecture 18

$λ$ -calculus: Normal forms of the form

\begin{matrix} M ≝ λ x_{0}, \dots, x_{n} . W \\ W ≝ n ∣ case x^{A} M_{1} \dots M_{p} is [n_{1} \to W_{1} \dots n_{q} \to W_{q}] \\ where A = σ_{1} \to \dots \to σ_{p} \to n a t \end{matrix}

NB: this term is not allowed $λ f . f$ , where $f : n a t \to n a t$ in PCF Böhm trees, because of the form of $A$ above. Only way to fix it: $η$ -expand it: $λ f, x . f x$

Exercise: Show that any real PCF term reduces to a some (infinite) Böhm tree.

⟹ Infinite Böhm trees are $η$ -long normal forms.

σ ≝ n a t ∣ σ \to σ

NB: so $σ$ is of the form $σ_{1} \to \dots \to σ_{n} \to n a t$ .

Typing rules:

\frac{⊢ W : n a t}{⊢ λ x_{1}^{σ_{1}}, \dots, x_{n}^{σ_{n}} . W : σ_{1} \to \dots \to σ_{n} \to n a t}

\frac{}{⊢ n : n a t}

\frac{⊢ M_{1} : σ_{1} \dots M_{p} : σ_{p} ⊢ W_{1} : n a t \dots ⊢ W_{q} : n a t}{x : σ_{1} \to \dots \to σ_{p} \to n a t ⊢ case x \vec{M} is [\dots n_{i} \to W_{i} \dots] : n a t}

First goal: Compile PCF Btrees into “strategies” written using the following vocabulary:

$(n a t \to n a t) \to n a t \to n a t$ is turned into

The vocabulary will consist of moves of the form

?_{u} n_{u}

with the convention that

$?_{ε}$ : Opponent (O)
$n_{ε}$ : Player (P)

and

If $?_{v}$ is O for $σ_{i}$ , then $?_{i v}$ is a P for $A$
If $?_{v}$ is P for $σ_{i}$ , then $?_{i v}$ is a O for $A$

NB: this is reminiscient of values of $M \Rightarrow M^{'}$ being either values of $M^{'}$ or cells of $M$

+ an equipement of pointers from P moves back to O moves

cf. picture

Innocent Strategies

Innocent strategies: between memoryless ones and those retaining complete history.

Warmup Example:

l o r : n a t_{1} \to n a t_{2} \to n a t_{ε}

h ≝ λ f . case f (3) is [4 \to 7, 6 \to 9] : (n a t_{11} \to n a t_{1}) \to n a t_{ε}

Kierstead (student of Curry (who was working on higher-order recursive functions)) terms:

From now on,

case x \vec{M} is an abbrev. for case x \vec{M} of \underset{infinite list}{\underset{⏟}{[0 \to 0, 1 \to 1, \dots, 2 \to 2, \dots]}}

\begin{matrix} K i e r s t e a d : ((n a t \to n a t) \to n a t) \to n a t \\ K i e r s t e a d_{1} : λ f . case f (λ x . case f (λ y . case x)) \\ K i e r s t e a d_{2} : λ f . case f (λ x . case f (λ y . case y)) \end{matrix}

cf. picture

We will define a cartesian closed category of Hyland-Ong Games.

Morphisms = innocent strategies

Two equivalent forms:
- meager (set of views):
  - where Opponent moves point to their direct Player parent move
  - composition via abstract machines
- fat (set of plays)
  - composition is defined in one line
NB: it’s similar to what we had with sequential algorithms: meager (AM) vs. fat (abstract algorithms)

Example: cf. picture

Kierstead applied to $h$

Arenas (HO)

An arena is given by

(M, λ : M \to {O, P})

where

$M$ stands for the moves
$⊢\subseteq M \cup M \times M$
- if $⊢ m$ , then $m$ is an O
- if $m ⊢ n$ , then $m$ and $n$ are of opposite polarities

Cartesian Product

A = (M, λ, ⊢) A^{'} = (M^{'}, λ^{'}, ⊢^{'})

\begin{matrix} A \times A^{'} = (\underset{M^{″}}{\underset{⏟}{(M .1) \cup (M^{'} .2)}}, λ^{″}, ⊢^{″}) \end{matrix}

where

\begin{matrix} λ^{″} (m .1) = λ (m) \\ \frac{⊢ m}{⊢^{″} m .1} \\ \frac{⊢^{'} n}{⊢^{″} n .2} \\ \frac{m_{1} ⊢ m_{2}}{m_{1} .1 ⊢^{″} m_{2} .1} \end{matrix}

Function Space

A \Rightarrow A^{'} = (M^{″}, λ^{″}, ⊢^{″})

\begin{matrix} M^{″} = M .1 \cup M^{'} .2 \\ λ^{″} (m .1) = \overset{―}{λ (m)} (P becomes O and vice versa) \\ λ^{″} (n .2) = \overset{―}{λ^{'} (n)} \end{matrix}

And the rules are the same as the ones of the product, except the intial rules:

\frac{⊢^{'} n}{⊢^{″} n .2}

\frac{⊢ m ⊢^{'} n}{n .2 ⊢^{″} m .1}

\frac{m_{1} ⊢ m_{2}}{m_{1} .1 ⊢^{″} m_{2} .1}

\frac{n_{1} ⊢ n_{2}}{n_{1} .2 ⊢^{″} n_{2} .2}

Strategies (morphisms in our category)

A play is an alternating sequence of moves in $M$ starting with O, equipped with pointers from all occurrences (of non-initial moves) in the sequence in such a way that

if $m_{i}$ points to $m_{j}$ for $j < i$ , then $m_{j} ⊢ m_{i}$

(cf. picture)

A strategy:: is a set $σ$ of even-length plays closed under even prefixes

NB: definition by Russ Harmer. It is very practical for non-determinism.

A strategy $σ$ is deterministic:: if moreover: whenever $s m_{1}, s m_{2} \in σ$ , where the $m_{i}$ ’s point in $s$ , it comes that $m_{2} = m_{1}$

Innocent strategies

A strategy $σ$ of $A$ is called innocent (Hyland Ong (HO)):

if for all $s$ (ranging over plays of $A$ ):

s \in σ ⟺ \underset{P view of s}{\underset{⏟}{⌈ s ⌉}} \in σ

where the view of a play is defined as:

\begin{matrix} ⌈ ε ⌉ = ε \\ ⌈ s n ⌉ = ⌈ s ⌉ n if λ (n) = P \\ ⌈ s m ⌉ = m if λ (m) = O and ⊢ m \\ ⌈ s n s^{'} m ⌉ = ⌈ s n ⌉ m if m points inside n \end{matrix}

Exercise: When we had the interaction $h (f)$ , we spotted a play $s$ that belonged to $h$ . Compute $⌈ s ⌉$ .

A view:: is a sequence $s$ such that $⌈ s ⌉ = s$

Meager strategies

Given an innocent strategy $σ$ , we can define

m e a g e r (σ) ≝ {set of views s ∣ s \in σ}

Prop: $m e a g e r$ is injective.

Innocent strategies can be composed in two ways:

either as strategies (parallel composition + hiding)
or as meager strategies (abstract machines)

Parallel composition

Let

$σ$ be a strategy of $A \Rightarrow A^{'}$
$σ^{'}$ be a strategy of $A^{'} \Rightarrow A^{″}$

Then we define

σ^{'} \circ σ ≝ {v ∣ v \in p l a y s (A \Rightarrow A^{″}) and \exists \underset{sequence of moves taken in A, A^{'}, A^{″} together with pointers}{\underset{⏟}{u}} st v = u_{| A, A^{″}}, w ≝ u_{| A, A^{'}} \in σ, w^{'} ≝ u_{| A^{'}, A^{″}} \in σ^{'}}

NB:

Hiding: because the everything related to $A^{'}$ has been kept private (hidden) in the final result
Parallel composition: because $w$ and $w^{'}$ are obtained from the same $u$
This definition works even without determinism nor innocence, but the equivalence with meager strategies holds only when we have innocence

n a t = {{?} \cup ℕ ∣ {\begin{cases} λ (?) = O, λ (n) = P \\ ⊢ ?, ? ⊢ n \end{cases}}

Next time:

give a precise definition of the compilation from PCF Böhm trees to innocent strategies
Suggest: the characterisation of the image of this compilation (which is injective: two different Böhm tress give rise to different strategies)
Well-parenthesisation: exclude catch and or_tester
How to interpret some imperative features in the setting of HO games (imperative features ⟶ non-innocent strategies)

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