- I learnt this proof from Anuj Dawar's course in "logic and complexity, lent 2024".
- Reference handout

- We add a new quantifier for bounded quantification over integers
- We add new terms that are given by
`#φ`

which counts the number of elements that satisfy phi. - We add addition and multiplication operators on number terms.
- We continue to have the fixpoint operator to take a fixpont of a positive predicate of a relation symbol.

- Cops announce where they are moving to on the graph. Can move to arbitrary vertices. (cops have helicopters)
- The cops lack secure crypto, so the robber learns of their new planned locations as they form the plan, and decides on a new vertex to move based on this information.
- Robber has a motorbike, so all the robber needs is
*some*path from her current position to her new position that is free of cops (who are still their their current positions) - Robber moves to her new position and chills there.
- Police then move with their (slow) helicopters to their committed position, and play continues.
- Robber wins if robber can play the game infinitely without being caught.
- Cops win if they can trap the robber into a configuration such that

```
def do_cops_win (robber_cur_pos : Vec2,
cops_cur_pos : List[Vec2],
graph : Graph,
robber_stragy,
cop_strategy):
while True:
# cops make strategy.
cops_new_pos = cop_strategy (cops_cur_pos) # cops don't know where robber is?
# insecure channel, robber learns of this and plots strategy
robber_new_pos = robber_strategy(graph,
cops_cur_pos, # robber knows where the cops currently are
cops_new_pos, # robber also wknows where the cops plan to be (insecure channel)
robber_cur_pos)
# robber tries to go to new position. If an avoiding path does not exist,
# robber is trapped.
if not avoiding_path_exists(robber_cur_pos, cops_cur_pos, robber_new_pos):
return "cops win"
# robber successfully makes it.
robber_cur_pos = robber_new_pos
# cops slowly make their way to their new position.
cops_cur_pos = cops_new_pos
```

`n+1`

cops needed for `nxn`

grid. `3x3`

grid, cells denoted by `o`

:
```
XYZ
1|ooo
2|ooo
3|ooo
```

Cops will occupy the first column and the first cell of the first row:
```
XYZ
1|cco
2|coo
3|coo
```

This forces the robber to live in the rest of the grid, and crucially, cuts of access
to the top left position `1X`

. So in the next round, the cops will take the cop at `1X`

and move her to `2Y`

. The inaccessible cell is marked with `-`

, where the cops have a
guarantee that the robber cannot be:
```
XYZ
1|-co
2|cco
3|coo
```

Now note that the cell `2Y`

is similarly inaccessible, and thus also becomes `-`

in the next turn.
```
XYZ
1|-co
2|-co
3|cco
```

Now we see that `3X`

is unreachable, so we can take the cop at `3X`

and move her to `3Z`

to make `3Y`

inaccessible:
```
XYZ
1|-co
2|-co
3|-cc
```

Repeat:
```
XYZ
1|-co
2|-cc
3|--c
```

Victory for the cops:
```
XYZ
1|-cc
2|--c
3|--c
```

- In general the cops have a winning strategy if the can "sweep" the graph, and maintain some territory that they expand in each turn.
- Note that the for nxn grid, the
(NOT min-cut) is*max-cut*`n`

, and any smaller graph that we get after removing cut edges will have at most`n`

edges. So we can use`n`

cops to repeatedly cut of access. There's probably a`log(2n)`

number of rounds strategy by subdivision that cuts of access by halving each round, either horizontally or vertically.

- what you get if you cut a torus into squares.
- Create an
`nxn`

grid, and connect vertices`(i, j) → ((i+1) % n, j)`

and similarly connect`(i, j) → (i, (j+1)%n)`

. This gives a torus. - We claim that for the cops and robbers game, $k$ cops does not suffice to win.

- For simplicitly, once again, let $k = 4$.
- Suppose the cops are on a set $X$ of $k$ vertices.
- We want to show that $G - X$ ha a connected component with at least half the vertices of $G$.
- Let's consider one possible configuration where they are in a row:

```
WXYZ
1|oooo
2|CCCC
3|oooo
4|oooo
```

- See that this does not cut off access, since we can move from
`(-, 1)`

to`(-, 4)`

as the top and bottom edges are glued. So this is useless for the cops.

- Now let's try and consider a configuration where all the cops are on distinct columns:

```
WXYZ
1|Cooo
2|oCoo
3|ooCo
4|oooC
```

- In this case too, the graph is connected, and has a large connected component.
- The only way the cops can cut of a region is to encircle it:

```
WXYZ
1|oooo
2|oCCo
3|oCCo
4|oooo
```

- In this case, the outside has
`kˆ2 - k`

vertices, which is larger that`kˆ2/2`

:`kˆ2 - k = k(k -1) > k * k/2`

.

- We will claim that system of linear equations over F2 (integers mod 2) cannot be solved by FPC (first order fixpoint + counting).
- The problem can clearly be solved in polytime by gaussian elimination.

- consider structures over the domain
`x1, ..., xn`

and`e1...em`

. -
`xi`

are variables,`ei`

are equations. - unary relation
`E0(ei)`

if RHS of`ei`

is 0. - unary relation
`E1(ei)`

if RHS of`ei`

is 1. -
`M(xi, ej)`

if`xi in ej`

. - Note that this suffices, since any linear equation over F2 can be written as
`x1 + ... xn = 0/1`

. - See that this problem is equivalent to XOR-SAT.
- See that we can negate variables by writing
`not(xi) = 1 + xi`

. - Let
`Solv(F2)`

be the class of structures representing solvable systems.

- Let $G$ be a toroidal graph.
- Define $E_G$, the system of equations next, based on $G$.
- For each edge, we get two variables $x[e, 0], x[e, 1]$ for each edge $e$.
- For each vertex $v \in V$, let $e[v, 1], e[v, 2], e[v, 3], e[v, 4]$ be the edges incident on $v$.
- We make 16 equations: $x[e[v, 1], i] + x[e[v, 2], j] + x[e[v, 3], k] + x[e[v, 4], l] = (i + j + k + l)$ for $i, j, k, l \in \{ 0, 1\}$.
- Intuitively, we pick a bitstring $\vec b$, and we write the equation $\sum_j x[e[v, j], \vec b[j]] = \texttt{parity}(\vec b)$.
- First see that this system is satisfiable by setting $x[-, i]$ to $i$, since the RHS is given by addint $i + j + k + l$, so will the lhs!

$\begin{aligned}
&x[e[v, 1], i] + x[e[v, 2], j] + x[e[v, 3], k] + x[e[v, 4], l] =_? (i + j + k + l) \\
&= i + j + k + l =_? i + j + k + l
\end{aligned}$

- We make another version of the equation system, $\tilde E_G$, which is the same system, except for a fixed vertex $c$ (for contradition), we change the parity of the equation to be .
- See that we perturb $16$ equations, for all possible choices of $b$ for a fixed $c$.
- See that this set of equations is unsatisfiable. Add up all equations that only have variables of the form $x[e, 0]$.
- The sum of all left hand sides will be double counting each variable, since we make an equation for each vertex, but we have a variable per edge. So the LHS will be zero.
- However, see that due to our perturbation, the RHS must be 1, since we changed the parity (TODO: make this more rigorous / proof by example).
- Now we show that for each $k$, there is a large enough $G$ such that $E_G \equiv{C[k]} E_{\tilde G}$.
- This means that FPC cannot distinguish between the cases where the solution is true and false.
- Thus the problem
`Solv(Z2)`

cannot be solved by FPC.

- Suppose that $G$ is the above toroidal graph such that the robber has a winning strategy for the $2k$ cops and robbers game played on $G$.
- We can use this to construct a winning strategy for the duplication in the $k$ pebble bijection game on $E_G$ and $E_{\tilde{G}}$.