Forking

In stability theory and its generalizations, forking is a way of making precise the notion of "generic extensions" of types and of independence. A formula which forks over a set of parameters $$C$$ is thought of as being "non-generic." If $$p$$ is a complete type over some set $$C$$, the non-forking extensions of $$p$$ are those extensions which contain no formula forking over $$C$$; these are thought of as the "generic" extensions of $$p$$.

Forking is also related to a notion of independence: one says that tuples $$a$$ and $$b$$ are independent over a set $$C$$ if $$\operatorname{tp}(a/bC)$$ does not fork over $$C$$. This turns out to be a useful notion in the setting of stable theories, and their generalizations. In stable and simple theories, one has in particular the machinery of the forking calculus.

Definitions
Fix some complete theory $$T$$, and let $$\mathbb{U}$$ be a monster model. If $$C$$ is a small set of parameters and $$\phi(x;b)$$ is a formula with parameters from $$\mathbb{U}$$, one says that $$\phi(x;b)$$ divides over $$C$$ if there is a $$C$$-indiscernible sequence $$\langle b_i \rangle_{i < \omega}$$ of realizations of $$\operatorname{tp}(b/C)$$ such that $$\bigwedge_{i < \omega} \phi(x;b_i)$$ is inconsistent. Alternatively, without mentioning indiscernible sequences, one can define dividing as follows: a formula $$\phi(x;b)$$ is said to $$k$$-divide over $$C$$, for $$k$$ a positive integer, if there is a sequence $$\langle b_i \rangle_{i < \omega}$$ of realizations of $$\operatorname{tp}(b/C)$$ such that $$\bigwedge_{i < \omega} \phi(x;b_i)$$ is $$k$$-inconsistent, that is, such that any $$k$$-element subset of $$\{\phi(x;b_i)\}$$ is inconsistent.

A formula $$\phi(x;b)$$ is said to fork over $$C$$ if it implies a finite disjunction of formulas (with parameters from $$\mathbb{U}$$) each of which divides over $$C$$. That is, there is some $$c_i \in \mathbb{U}$$ and $$\psi_i(x;y)$$ such that $$T \vdash \forall x : \phi(x;b) \rightarrow \bigvee_{i = 1}^n \psi_i(x;c_i)$$ Any formula which divides over $$C$$ forks over $$C$$, though there are examples of the converse failing.

A partial type $$\Sigma(x)$$ is said to fork over $$C$$ (resp. divide over $$C$$) if it implies some formula which forks (resp. divides) over $$C$$. If $$p$$ is a complete type over $$C$$, a non-forking extension of $$p$$ is an extension which does not fork over $$C$$.

If $$a$$ is a tuple and $$B$$ and $$C$$ are small sets, one says that $$a$$ is (forking) independent from $$B$$ over $$C$$, written $$a \downarrow_C B$$, if $$\operatorname{tp}(a/BC)$$ doesn't fork over $$C$$, i.e., $$\operatorname{tp}(a/BC)$$ is a non-forking extension of $$\operatorname{tp}(a/B)$$. If $$A$$, $$B$$, and $$C$$ are small sets, then $$A \downarrow_C B$$ means that $$a \downarrow_C B$$ for every finite tuple $$a$$ from $$A$$. In general, this notion is not symmetric: $$A \downarrow_C B$$ need not be equivalent to $$B \downarrow_C A$$. Simple theories are exactly the theories for which this symmetry always holds; they include stable theories.

The case of ACF
In the theory of algebraically closed fields, forking turns out to be related to algebraic independence. If $$L/K$$ is an extension of fields (seen as substructures of the monster) and $$a$$ is a finite tuple, then it turns out that $$a \downarrow_K L$$ if and only if $$tr.deg(a/K) = tr.deg(a/L)$$.

General properties of forking
Forking is defined to ensure that it has the following extension property: if $$C' \supset C$$, any partial type over $$C'$$ which does not fork over $$C$$ can be extended to a complete type over $$C'$$ which does not fork over $$C$$. Dividing does not have this property. In fact, a partial type $$\Sigma(x)$$ can be extended to a global complete type which does not divide over $$C$$ if and only if the original type $$\Sigma(x)$$ does not fork over $$C$$. A global type divides over $$C$$ if and only if it forks over $$C$$.

Dividing can be defined more directly, in terms of indiscernible sequences. Specifically, the following are equivalent:


 * $$\operatorname{tp}(A/bC)$$ doesn't divide over $$C$$
 * For every $$C$$-indiscernible sequence $$I$$ beginning with $$b$$, there is some $$I' \equiv_{bC} I$$ such that $$I'$$ is $$AC$$-indiscernible.

Forking has the following "left transitivity" property: if $$\operatorname{tp}(a/BC)$$ doesn't fork over $$C$$ and $$\operatorname{tp}(a'/BCa)$$ doesn't fork over $$Ca$$, then $$\operatorname{tp}(aa'/BC)$$ doesn't fork over $$C$$. In terms of independence, this means that $$a \downarrow_C B$$ and $$a' \downarrow_{Ca} B$$ together imply $$aa' \downarrow_C B$$. This is the mirror image of the more familiar kind of "right transitivity" which holds in simple theories: if $$\operatorname{tp}(a/BC)$$ doesn't fork over $$C$$ and $$\operatorname{tp}(a/BB'C)$$ doesn't fork over $$BC$$, then $$\operatorname{tp}(a/BB'C)$$ doesn't fork over $$C$$. (In symbols: $$a \downarrow_C B$$ and $$a \downarrow_{CB} B'$$ imply $$a \downarrow_C BB'$$.)

TODO: monotonicity, base monotonicity, finite character, normality, "rigidity."

Forking vs dividing
In general, any formula or type which divides over $$C$$ also forks over $$C$$. The converse need not hold. The standard example of this is the theory of a dense circular order. If one considers the structure with underlying set $$[0,1) \subset \mathbb{Q}$$ and with a ternary predicate $$C(x,y,z)$$ interpreted as saying that $$x < y < z$$ or $$y < z < x$$ or $$z < x < y$$, then the formula $$x = x$$ does not divide over $$\emptyset$$, but does fork over $$\emptyset$$. In complete detail, the formula $$x = x$$ implies the disjunction of the two formulas $$C(x,0,1/4)$$ and $$C(x,1/2,3/4)$$. All tuples $$(b,c)$$ with $$b \ne c$$ have the same type over $$\emptyset$$, and the sequence of formulas $$C(x,1/2,1/3), C(x,1/3,1/4), C(x,1/4,1/5), \ldots$$ is $$2$$-inconsistent. So forking is not the same as dividing in this theory.

On the other hand, it is known that forking and dividing agree in simple theories (including stable theories), o-minimal theories, and C-minimal theories (such as ACVF). Furthermore, in an NIP setting, it is proven by Chernikov and Kaplan that if $$C$$ is a model, then a formula forks over $$C$$ if and only if it divides over $$C$$. That is, forking and dividing agree over models. This holds more generally in NTP$${}_2$$ theories (again by Chernikov and Kaplan).

One useful consequence of forking=dividing (when it holds) is that no type forks over its base. That is, $$\operatorname{tp}(a/C)$$ never forks over $$C$$. This comes from the easily verifiable fact that no formula over $$C$$ can divide over $$C$$.

Forking in simple theories
In simple theories, one has the full machinery of the forking calculus. Forking satisfies symmetry: $$A \downarrow_C B$$ is equivalent to $$B \downarrow_C A$$. This allows one to make definitions like Lascar rank, weight, independence, domination, and so on. Forking also satisfies local character: there is a cardinal $$\kappa$$ such that for every finite tuple $$a$$ and set $$C$$, there is some subset $$C_0 \subset C$$ such that $$|C_0| < \kappa$$ and $$a \downarrow_{C_0} C$$. In other words, every type doesn't fork over some subtype of size bounded by $$\kappa$$.

Forking in NIP theories
In NIP theories, non-forking has an equivalent description in terms of Lascar or compact (KP) strong types. As in any theory, $$\operatorname{tp}(a/BC)$$ doesn't fork over $$C$$ if and only if $$\operatorname{tp}(a/BC)$$ has an extension to a global type $$p$$ which doesn't divide over $$C$$. In the NIP setting, however, it turns out that the following are equivalent for a global type $$p$$:


 * $$p$$ doesn't divide over $$C$$
 * $$p$$ is Lascar $$C$$-invariant. That is, for every formula $$\phi(x;y)$$ and every $$b$$ and $$b'$$, if $$b$$ and $$b'$$ have the same Lascar strong type over $$C$$, then $$\phi(x;b) \in p(x) \iff \phi(x;b') \in p(x)$$. Equivalently, $$p(x)$$ is fixed by the group of Lascar strong automorphisms over $$C$$.
 * $$p$$ is $$bdd(C)$$-invariant. Equivalently, if $$\phi(x;y)$$ is a formula and $$b$$ and $$b'$$ have the same compact strong type over $$C$$, then $$\phi(x;b) \in p(x) \iff \phi(x;b') \in p(x)$$.

For stable theories, Lascar strong type, compact strong type, and strong type all agree, so one recovers the characterization of non-forking over $$C$$ as having an $$\operatorname{acl}^{eq}(C)$$-invariant global extension.

Forking in stable theories
Main article: Forking in stable theories.