OpenLLM-France/wikimedia huggingface dataset:
# Finite Model Theory/FO BFP

Discrimination, Subtask: How to describe a set of structures independent from a logic? Easy for finitely describable structures: by a finite set of structures.
Difficult: Infinite set of structures. Approach: the notion of pattern, i.e. Partial Isomorphisms that can be enhanced by induction.
$\exists _{x}\exists _{y}xRy$, can be described by single structure, like:
$\exists _{x}\exists _{y}\neg xRy$, can be described by a set of structures of size 2, like:
$\neg \exists _{x}\exists _{y}xRy$, requires infinitely many structures, since ...
In the latter case it is important how the pattern is extended, thus we need a rule for it. This can be achieved by induction, i.e. a structure for which the pattern holds is extended by a single element s.t. the pattern applies for x and y (i.e. all unified variables) with this element.
So, finally we get a structure where the pattern is applied to each single combination of elements (for the unifier case) - i.e. for all elements - and this is where FO comes in here.
In the basics section we defined the notion of finite (partial?) isomorphism. Now one can ask for a reasonable definition of two structures being finitely isomorphic. We discuss miscellaneous approaches here and will see that - on finite relational structures - one is equivalent to the concept of elementary equivalence.

## Loosely Finite Isomorphic Structures
#### All or Any

i.e. there is any subset of A with a FI to B, what is trivial in most cases, since one can pick a single element from each A and B. But on the other hand requiring that for all subsets of A there is a FI would demand a total FI from A to B i.e. would include the notion of Isomorphism(?):

#### Total

So one can make the definition relative to the domain-size m of FI, that is:
(shallow)
(deep)
# Finite Model Theory/FO EFM

The method for employing Ehrenfeucht-Fraisse-Games for (in-)expressibility-proofs is given by the following:
Theorem
Let P be a property of finite σ-structures. Then the following are equivalent
* P is not expressible in FO
* for every k $\in \mathbb {N}$ there exist two finite σ-structures ${\mathfrak {A}}_{k}$ and ${\mathfrak {B}}_{k}$, such that the following are both satisfied
** ${\mathfrak {A}}_{k}\equiv _{k}{\mathfrak {B}}_{k}$
** ${\mathfrak {A}}_{k}$ has P and ${\mathfrak {B}}_{k}$ does not have P
Remarks
* Thus using the EFM works roughly as follows:
** choose a k
** construct two structures - one with the property, one without - that are big enough s.t. the duplicator wins the k-ary EFG
** show that this can be expanded with k
* So, a non-expressible property (i.e. the effort to check it) must be somehow 'expandable' with k
Examples
# Finite Model Theory/FO Discrimination
## The Property of Elementary Equivalence

FMT is about discrimination of structures. So the basic question is "Can I discriminate a structure from all different structures?", where different means not equal up to isomorphy. This can be done for finite structures (but not for infinite ones). E.g. ...

## Properties for a finite Number of Structures

So being of a certain structure can be seen as an elementary property, that discriminates a class of isomorphic structures from all others. Now we can think of properties that discriminate 2 different (nonisomorphic) structures from the rest. They can also be discriminated by simply connecting the properties, like e.g. ...
This can be extended to properties for a finite number of structures, like e.g. ...

## Properties for an infinite Number of Structures

Also properties that hold for an infinite number of structures can be thought of. But these are not discriminatable in the above way. E.g. ...
There are Logics that allow this sort of infinite disjunction, like ...
So, are there other ways to discriminate these structure in FO? Sometimes, like in the case of ...
So we need a decision tool (method) that decides if structures with a certain property can be discriminated or not, i.e. responds either "yes" or "no" for all possible properties, i.e. is sound and complete.
# Finite Model Theory/Logics and Structures

FMT studies logics on finite structures. An outline of the most important of these objects of study is given here.

## Logics

The logics defined here and used throughout the book are always relational, i.e. without function symbols, and finite, i.e. have a finite universe, without further notice.

### Fragments of FO

The subsequent restrictions can analogously be found in other logics like SO.
* MFO ...
* ESO and USO ...
* FO$^n$ ...

### Second Order Logic (SO)

Second-order logic extends first-order logic by adding variables and quantifiers that range over sets of individuals. For example, the second-order sentence $\forall S\,\forall x\,(x\in S\lor x\not \in S)$ says that for every set S of individuals and every individual x, either x is in S or it is not. That is, the rules are extended by:
* If X is a n-ary relation variable and t$_1$ ... t$_n$ are terms then X t$_1$ ... t$_n$ is a formula
* If φ is a formula and X a relation variable then $\exists _{X}\varphi$ is a formula

### Fragments of Second Order Logic

* The fragment monadic second-order logic (MSO) only unary relation variables ("set variables")are allowed.
* The existential fragment (ESO) is second-order logic without universal second-order quantifiers, and without negative occurrences of existential second-order quantifiers.
* USO ...

### Infinitary Logics
#### Notion

The intention is to extend FO by an infinite disjunction element over a set ψ of formulas (of infinitary logic)
$\bigvee \psi$
So the following infinitary logics can be defined
* $L_{\infty \varpi }$ where ψ is an arbitrary set of formulas, e.g. uncountable
* $L_{\varpi \varpi }$ where ψ is a countable set of formulas
* $L_{\infty \varpi }^{n}$ ...

#### Definition

syntax ...
semantics ...
# Finite Model Theory/Motivation

We give a simple example of applications of FMT first and then some typical examples.

## FMT everywhere

Some simple examples that show how elementary the issues are that FMT cares about:

## Databases

A typical application area of FMT is database theory:
* Consider a companies' database that contains all managers together with the 'is superordinate' relation amongst them. In a proper hierarchy the database should contain no circles, i.e. a manager can not be a superordinate of his superordinate. Querying this corresponds to checking a graph for cyclicity. As from above this cannot be done in First Order Logic (FO).
* Say two managers want to find out if one of them is more powerful than the other. They need to query the database if the number of their subordinates is equal, i.e. the cardinalities of the sets of subordinates (say, directly and indirectly) is equal. This can't be done in FO ("FO cannot count"). This is the reason SQL is extended by a counting function.
* Consider a database $F$ of airports and connection flights among them, where $F(x,y)$means there is a direct flight from airport $x$ to airport $y$. We can write an order-zero query of the direct reachability of airport $b$ from airport $a$ as
$q_{0}(a,b)=F(a,b)$.
Now in order to query connections with one change of plane we write
$q_{1}(a,b)=\exists _{c}F(a,c)\land F(c,b)$ and get $Q_{1}(a,b)=q_{0}(a,b)\lor q_{1}(a,b)$
for connections with zero or one change. Thus in order to extend this to reachability (of no matter how many changes) we have to write
$Q=\bigvee _{k\in \mathbb {N} }q_{k}$
which is not a FO expression. So we are fine with FO for a restricted reachability up to a certain k but not for reachibility as it appears in graph theory. In fact it can be shown that reachability cannot be queried in FO.
# Finite Model Theory/FO EFG
## Logic and Games

Connection between Games and logical Quantifiers...

## Simple Games

* S chooses first all n from one/both side(s), then D chooses all
* one match as in 1. for every i = 1 ... n where the i'th choice must include the i-1'th.

## Definition

Ehrenfeucht-Fraisse-Game (EFG)
Let ${\mathfrak {A}}$ and ${\mathfrak {B}}$ be finite relational structures, with the same set of relation symbols, and n be a fixed natural number.
A game is said to be an Ehrenfeucht-Fraisse-game iff it is constituted the following way
* 2 Players: Spoiler S and Duplicator D
* Game tree:
** Root: a pair of empty sequences
** Branch:
*** Move of S: S chooses exactly one element of either A or B that has not been chosen already (latter is not relevant in the first move)
*** Move of D: D chooses exactly one element of the remaining set that has not been chosen already, i.e. if S has chosen from A D has to choose from B and if S chose one from B D has to choose from A.
** Terminator: The game is finished iff α and β are of size n.
* Turn: Moves are carried out alternatingly, S is to move first.
* Payoff: D wins iff the final position defines a partial isomorphism by mapping the i-th elements of α and β, for all i. Else S wins
* Information: the players are assumed to have perfect information
# Finite Model Theory/Model Theory

Many important theorems of Model Theory do not hold when restricted to the finite case, like Gödel's completeness theorem or the compactness theorem:

## Failure of Compactness Theorem

Consider the following sentence σ3
$\exists _{x}\exists _{y}\exists _{z}(x\neq y\land y\neq z\land x\neq z)$
that says that there are at least 3 different elements in a universe. One can expand σ3 easily for n other than 3. So, let Σ = {σ1, σ2, σ3, ...} be the infinite set of all these sentences. Now Σ is obviously not satisfiable by a finite model, although every finite subset of Σ is. Ok, but why does that matter? One of the most useful tools in general Model theory is the Compactness theorem, stating: "Let Σ be a set of FO sentences. If every finite subset of Σ is satisfiable, then Σ is satisfiable." But as just shown this doesn't hold for the finite case, thus there is no Compactness theorem in Finite Model Theory!
# Foundations of Education and Instructional Assessment/Classroom Management

    What is the importance of keeping students engaged in learning?
    Is positive discipline all about being nice?
    How can brain research help teachers understand their students?
    When should students be rewarded or punished?
    Should students be praised?
    How can students be motivated both intrinsically and extrinsically?
# Calculus/Multivariable Calculus/Chain Rule
### Rules of taking Jacobians

If f : R$^m$ → R$^n$, and h(x) : R$^m$ → R are differentiable at 'p':
* $J_{\mathbf {p} }(\mathbf {f} +\mathbf {g} )=J_{\mathbf {p} }\mathbf {f} +J_{\mathbf {p} }\mathbf {g}$
* $J_{\mathbf {p} }(h\mathbf {f} )=hJ_{\mathbf {p} }\mathbf {f} +\mathbf {f} (\mathbf {p} )J_{\mathbf {p} }h$
* $J_{\mathbf {p} }(\mathbf {f} \cdot \mathbf {g} )=\mathbf {g} ^{T}J_{\mathbf {p} }\mathbf {f} +\mathbf {f} ^{T}J_{\mathbf {p} }\mathbf {g}$
Important: make sure the order is right - matrix multiplication is not commutative!

#### Chain rule

The chain rule for functions of several variables is as follows. For f : R$^m$ → R$^n$ and g : R$^n$ → R$^p$, and g o f differentiable at p, then the Jacobian is given by
    $\left(J_{\mathbf {f} (\mathbf {p} )}\mathbf {g} \right)\left(J_{\mathbf {p} }\mathbf {f} \right)$
Again, we have matrix multiplication, so one must preserve this exact order. Compositions in one order may be defined, but not necessarily in the other way.
# Calculus/Multivariable Calculus/Double Integrals
### Order of integration

In some cases the first integral of the entire iterated integral is difficult or impossible to solve, therefore, it can be to our advantage to change the order of integration.
$\int _{a}^{b}\,\int _{f(x)}^{g(x)}h(x,y)\,dxdy$
$\int _{c}^{d}\,\int _{e(y)}^{f(y)}h(x,y)\,dydx$
As of the writing of this, there is no set method to change an order of integration from dxdy to dydx or some other variable. Although, it is possible to change the order of integration in an x and y simple integration by simply switching the limits of integration around also, in non-simple x and y integrations the best method as of yet is to recreate the limits of the integration from the graph of the limits of integration.
In higher order integration that can't be graphed, the process can be very tedious. For example, dxdydz can be written into dzdydx, but first dxdydz must be switched to dydxdz and then to dydzdx and then to dzdydx (as 3-dimensional cases can be graphed, this method would lack parsimony).
# Calculus/Multivariable Calculus/Limits and Continuity
## Limits and continuity

Before we can look at derivatives of multivariate functions, we need to look at how limits work with functions of several variables first, just like in the single variable case.
If we have a function f : R$^m$ → R$^n$, we say that f(x) approaches b (in R$^n$) as x approaches a (in R$^m$) if, for all positive ε, there is a corresponding positive number δ, |f(x)-b| < ε whenever |x-a| < δ, with x ≠ a.
This means that by making the difference between x and a smaller, we can make the difference between f(x) and b as small as we want.
If the above is true, we say
* f(x) has limit b at a
* $\lim _{\mathbf {x} \rightarrow \mathbf {a} }\mathbf {f} (\mathbf {x} )=\mathbf {b}$
* f(x) approaches b as x approaches a
* f(x) → b as x → a
These four statements are all equivalent.

### Rules

Since this is an almost identical formulation of limits in the single variable case, many of the limit rules in the one variable case are the same as in the multivariate case.
For f and g, mapping R$^m$ to R$^n$, and h(x) a scalar function mapping R$^m$ to R, with
* f(x) → b as x → a
* g(x) → c as x → a
* h(x) → H as x → a
then:
* $\lim _{\mathbf {x} \rightarrow \mathbf {a} }(\mathbf {f} +\mathbf {g} )=\mathbf {b} +\mathbf {c}$
* $\lim _{\mathbf {x} \rightarrow \mathbf {a} }(h\mathbf {f} )=H\mathbf {b}$
and consequently
* $\lim _{\mathbf {x} \rightarrow \mathbf {a} }(\mathbf {f} \cdot \mathbf {g} )=\mathbf {b} \cdot \mathbf {c}$
* $\lim _{\mathbf {x} \rightarrow \mathbf {a} }(\mathbf {f} \times \mathbf {g} )=\mathbf {b} \times \mathbf {c}$
when H≠0
* $\lim _{\mathbf {x} \rightarrow \mathbf {a} }({\mathbf {f} \over h})={\mathbf {b} \over H}$

### Continuity

Again, we can use a similar definition to the one variable case to formulate a definition of continuity for multiple variables.
If f : R$^m$ → R$^n$, f is continuous at a point a in R$^m$ if f(a) is defined and
    $\lim _{\mathbf {x} \rightarrow \mathbf {a} }\mathbf {f} (\mathbf {x} )=\mathbf {f} (\mathbf {a} )$
Just as for functions of one dimension, if f, g are both continuous at p, f+g, λf (for a scalar λ), f·g, and f×g are continuous also. If φ : R$^m$ → R is continus at p, φf, f/φ are too if φ is never zero.
From these facts we also have that if A is some matrix which is n×m in size, with x in R$^m$, a function f(x)=A x is continuous in that the function can be expanded in the form x$_1$a$_1$+...+x$_m$a$_m$, which can be easily verified from the points above.
If f : R$^m$ → R$^n$ which is in the form f(x) = (f$_1$(x),...,f$_n$(x) is continuous if and only if each of its component functions are a polynomial or rational function, whenever they are defined.
Finally, if f is continuous at p, g is continuous at f(p), g(f(x)) is continuous at p.

### Special note about limits

It is important to note that we can approach a point in more than one direction, and thus, the direction that we approach that point counts in our evaluation of the limit. It may be the case that a limit may exist moving in one direction, but not in another.
# Calculus/More Differentiation Rules
## External Links

* Online interactive exercises on derivatives
* Visual Calculus - Interactive Tutorial on Derivatives, Differentiation, and Integration
Navigation:
Main Page · Precalculus · Limits · Differentiation · Integration · Parametric and Polar Equations · Sequences and Series · Multivariable Calculus · Extensions · References
# Grand Theft Auto: San Andreas/Missions/OG Loc
## Prerequisites

* Mission: Nines and AK's
* Location: Big Smoke's house, Idlewood, Los Santos

## Walkthrough

Your first mission for Smoke, you must pick up another friend of yours called "OG Loc"... or just Jeffrey. Watch the cutscene then drive over to Freddy's house, after the cutscene hop into the PCJ and follow Freddy. No need to shoot, conserve your ammo for later on in the mission. All you need to do is follow the bike until you reach the Idlewood Basketball court. There are 3 Vagos waiting, kill them with your Tec-9 or Mac-10. Then kill Freddy, after that, drive OG Loc to his new job at Burger Shot.

## In-depth walkthrough

This walkthrough exploits the obvious design of Freddy's route. He has greater speed but chooses the longest and hardest routes (hoping you will follow and collide with something and end up losing him). This in-depth walkthrough takes full advantage of this.
When you start the mission Freddy is down an alleyway to the right. Drive close enough that he says, "Oh chase me chase me!" and drives off, then high-tail it back to the left, turn right, and then go along the sloping walled footpath to your left. By now Freddy should be far behind you.
Now slow down (so he doesn't vanish off the radar) and continue down the hill to the immediate left, accelerating as much as necessary to keep him on the map but behind you.
Freddy will take (or be taking) the road immediately to the right, so take the next one after it. This is a shorter route, and, once again, will put you in front of him for sure if he's managed to get ahead of you once more.
When you both re-emerge onto the main road (pictured) he will be taking the overpass to the left, so do that also, even if you are ahead of him. Keep to the far right, even scraping against the wall if necessary, so oncoming cars won't (hopefully) end up hitting you.
The rest of the route is more about following him directly so give him a chance to get slightly ahead, but any time he goes down an alley (or anywhere else that you would possibly hit something and fall off your bike) take the shortest proper road available.

## Tips

* If you are having problems staying on the bike, try increasing your Bike Skill before starting the mission.
* If you should lose track of Freddy, he leaves a tire-trail behind him, so try looking at the ground to see which route he took.
* Freddy takes a predictable and inefficient route. If you second-guess his movements you can take the more direct route, and you will hopefully be ahead of him for most of the "chase"! Don't let his marker come too near to the edge of the radar though, as you can still lose him even though you are in front.

## Rewards

* Respect: +5

## New Missions Available

* Running Dog
* Life's a Beach
# Grand Theft Auto: San Andreas/Missions/Outrider
## Prerequisites

* Mission: Mike Toreno
* Location: CJ's garage, Doherty, San Fierro

## Walkthrough

After the phone call, get to the car. After a cutscene, you must drive to a van. After another cutscene, pick up the RPG and Sniper Rifle if you wish, then get on the bike. You now have to destroy four roadblocks before the van arrives at each one. At some of the roadblocks, there are snipers standing in the windows of buildings looking on to the road, so watch out for these. When you have cleared the final roadblock, follow the van to its destination to complete the mission. After the mission, you have a 3-star wanted level.

## Rewards

* Cash: $9,000
* Respect: +25

## New Missions Available

* Snail Trail
# Grand Theft Auto: San Andreas/Missions/Oysters

You will probably find your first oyster during the Da Nang Thing mission. There are fifty to collect after that.

## Locations

Spoilers
* At the stern of the Da Nang boat
* At the stern of the Export boat
* In the pool of the Tiki Motel, Pilgrim, Las Venturas
* In the Glen Park pool, Los Santos
Reward: $100,000

## External Links

See the full Oyster location map
There is another oyster the CIA guys dock, under the boat roof.
# Grand Theft Auto: San Andreas/Missions/Pier 69
## Prerequisites

* Mission: Ice Cold Killa
* Location: Pier 69, Esplanade North, San Fierro

## Walkthrough

Go up the stairs with Cesar, and after the cutscene, use a sniper rifle to take out the six targets as quickly as possible. After another cutscene, you have to kill T-Bone. After another cutscene, you have to chase Ryder in a boat, and kill him. When returning to the shore, be careful not to land at the Easter Bay naval base, or you'll get a 5-star wanted level.

## Tips

* After clearing up the roof, you can stay there and use the sniper rifle to take out some of the ground force too.
* You don't need to get wet with Ryder. Snipe him out while he is swimming to save a boat chase.

## Rewards

* Cash: $15,000
* Respect: +40

## New Missions Available

* Toreno's Last Flight
# Grand Theft Auto: San Andreas/Missions/Photo Opportunity
## Prerequisites

* Mission: Deconstruction
* Location: CJ's garage, Doherty, San Fierro

## Walkthrough

Be prepared for lots of driving. First, you must drive out to see Cesar near the Panopticon, then you must take Cesar to Angel Pine. Once you arrive, get on the roof, and get ready, because if you miss a photo opportunity, you have to start the mission all over again. There are a total of four photographs to take, and you must get the person's face in shot:
* Ryder, who arrives in his Picador truck in the parking lot across the street.
* T-Bone Mendez, who arrives in a ZR-350 in the same location.
* Mike Toreno, who arrives in a brown Washington from the street to the right.
* Jizzy, who arrives in a Broadway from the left, stopping next to the Washington.
If you get all the photos, Cesar drops you off in a gas station just south of Angel Pine, and you complete the mission.

## Rewards

* Respect: +15
* Item: Camera

## New Missions Available

* Jizzy
# Grand Theft Auto: San Andreas/Missions/Photo Ops

When you arrive in San Fierro you can start taking pictures of various local places of interest. Most of these snapshot locations look slightly unusual compared to the normal textures and constructions in the game but the only way to check a location is to view it through the camera viewfinder. A true snapshot location will have a small purple icon with a camera on it.
To take a snapshot you need to zoom in on the icon and try to centre it in the viewfinder.
The easiest way to start taking snapshots is at the entrance gate to the San Fiero airport where there is a plane above the gate.

## List of Photo Opportunities

Spoilers
* Airplane over the entrance to San Fiero airport
* Iron and glass roof over the platform in the Doherty station
* USS Numnutz submarine in the harbour
* Sign over the donut shop, Juniper Hill
* Statue in the square, Downtown
* Clock face, tower, North Esplande
* Observatory dome, City Hall
* Country Club tower
* The cinema, Hashbury
* Hashbury Joint Festival banner (there are two, this is the one that faces onto the park)
* FinalBuild board, construction lot, Garcia
* Construction crane, building site, Doherty
(incomplete)

## External Links

See the full Snapshots location map
See another Snapshots location map
Because of inaccuracies in the official strategy guide, BradyGames has released full-page updates of both the San Fierro Photo Ops List (PDF, 137 KB) and the San Fierro Photo Ops Location Shots (PDF, 13.5 MB).