Bouldering 1.: Formalism

Introduction

These notes endeavour to sketch a formalism of bouldering or climbing, in general. The physics of climbing is well established. Numerous papers have reported on the advancement of the climbing robot design and movement planning. Here, a language, perhaps restricted in immediate applicability, but one of considerable flexibility is constructed. It will be developed in multiple steps. Each adds a new notion or details to the previous ones.

Motivation

Once the apt reader has finished digesting the paragraphs will readily conclude that very little novelty has been shared with them. There is a start and a desired end. If someone is skilled enough and have the power and energy they can link the two positions through a sequence of appropriate moves. The aim is to lay out a formalism which is not only descriptive but has the capability to be a framework of prescribing solutions of climbing problems.

Notation

This post leads us through terrains of set theory. We therefore equip ourselves with a few utilities to make our passage swifter.

• Sets are denoted by calligraphic fonts: $\mathcal{A}$.
• the $!$ symbol expresses requirement. For example, $! \mathcal{A} \neq \emptyset$ means we require $\mathcal{A}$ not to be an empty set.
• The cardinality of a set is denoted by the operator $ \vert \cdot \vert $. The cardinality measures how many or how much elements are in a set. For example, $\mathcal{A} = \lbrace x, y \rbrace $, then $ \vert \mathcal{A} \vert = 2$.
• The Cartesian product of two sets is defined and denoted as: $\mathcal{A} \times \mathcal{B} = { (a, b): a \in \mathcal{A}, b \in \mathcal{B} }$. For example, $\mathcal{A} = \lbrace x, \rbrace, \mathcal{B} = { v, w }$, then $\mathcal{A} \times \mathcal{B} = \lbrace (x, v), (x, w), (y, v), (y,w) \rbrace$.
• A function, $f$ assigns a single element from a set to each element of a set. As such, it is a subset of the Cartesian product of two sets. For example, $\mathcal{A} = \lbrace x, y \rbrace, \mathcal{B} = \lbrace v, w \rbrace$, then $! f = \lbrace (x, v), (y, w)\rbrace \subset \mathcal{A} \times \mathcal{B}$.
• The power set of a set is denoted by $P(\cdot)$. The power set is the collection of all subsets of a set. For example, $\mathcal{A} = \lbrace x, y \rbrace$, then $P(\mathcal{A}) = \lbrace \emptyset, \lbrace x \rbrace, \lbrace y \rbrace, \lbrace x, y \rbrace \rbrace$.

First move

Rough climb

Forget the harness, the chalk, the climbing mates, focus on the crags in front of us. The aim is to move from the starting location $\ell_{s}$ to the top $\ell_{f}$. Therefore a climb, $V$ is a function that links the two locations:

\[\begin{eqnarray} V & = & \ell_{s} \rightarrow \ell_{f} \\ V & = &(\ell_{s}, \ell_{f}) \end{eqnarray}\]

It is likely that the climber moves from $\ell_{s}$ to $\ell_{f}$ through multiple transitions on the wall. That is the climb is a sequence of location changes:

\[\begin{eqnarray} !n & \in & \mathbb{N}: n > 1 \quad & \text{(finite number of locations)}\\ % V & = & \{ \ell_{1}, \ell_{2}, ..., \ell_{n} \} \quad & \text{(sequence of positions)}\\ % \forall & i & \in [ 1, n - 1]: \ell_{i} \prec \ell_{i + 1} \quad & \text{(ordered in time)} \\ \ell_{1} & = & \ell_{s}, \ell_{n} = \ell_{f} \quad & \text{(by definition)} % \end{eqnarray}\]

where the $\prec$ symbol signifies precedence.

To summarise, a beginner’s definition of a climb has been established which captures the spatial aspects of the activity.

The reader is the most certainly alert to that there are two major and a minor issues with the definition above.

• It cannot represent climbs which are unfinished.
• Nor can it be used to denote climbs which did not start.
• The minor issue follows from the syntax. It is not clear whether two locations are different if they have different indices.

Second move

Locations

The locations, $\ell$ have been used with some tacit liberly so far. Let us tie those which the rocks offer in a set denoted by $\mathcal{L}$.

\[\begin{eqnarray} \mathcal{L} & \neq & \emptyset \quad & \text{(there are locations)} \\ \vert \mathcal{L} \vert & \leq & \vert \mathbb{N} \vert \quad & \text{(locations can be indexed)} \end{eqnarray}\]

A special location, $\ell_{0} \in \mathcal{L}$ exists, the terminating location. It represents the most sorrowsome scenario, that of the climb cannot be continued. It has the following properties:

• Once in $\ell_{0}$ no further movements are possible apart from those to $\ell_{0}$. The climb is over.
• It can be reached from any other locations.

Path

Now, that the two major shortcomings are seemingly resolved, the minor issues is being addressed.

Quietly, the meaning of the subscripts became conflated. Firstly, it marked locations as they were visited in time. Secondly, they were differentiated locations in space. To reconcile the two meanings an index set is introduced, $\mathcal{T}$ which selects the locations in the order as they are visited.

\[\begin{eqnarray} !n & \in & \mathbb{N}: n > 0 \quad & \text{(finite "steps")} \\ % \mathcal{T} & = & [1, n] \quad & \text{(time index)}\\ \end{eqnarray}\]

The path, $\mathcal{P}$ is a function that orders the visited positions. In other words, it assigns a location to each time step. \(\begin{eqnarray} \mathcal{P} = \mathcal{T} \rightarrow \mathcal{L} \end{eqnarray}\)

Two level indices make this relationship more clear:

\[\begin{eqnarray} ! i & \in & \mathcal{I} = [1, \vert \mathcal{L} \vert ] \quad & \text{(location indices)} \\ % R & = & \{ (t, \ell_{i_{t}}): t \in \mathcal{T}, \ell_{i_{t}} \in \mathcal{L} \} \quad & \text{(assign a location to a time)} % \end{eqnarray}\]

The notion of the path as the sequence of positions have been introduced. Still, there is more to climbing. Hold on!

Third move

Holds

From a geometrical perspective, the location refers to a point of no extent. This definition is in constrast with the basics of climbing where a location is composed of multiple sub-locations. These are customarily called holds. (In addition, each hold is of non-zero extent, otherwise there would be nothing to hold on to.)

For example, two holds for the legs, two for the hands, when bouldering, and one for the rope, when climbing (no free solo this time!) in the easiest case.

To clarify the relationship between the locations, $\mathcal{L}$ and yet undefined wall, $\mathcal{H}$ a small number of equations are laid out:

\[\begin{eqnarray} \mathcal{\mathcal{H}} & \neq & \emptyset \quad & \text{(there is something to climb)} \\ % \vert \mathcal{H} \vert & \leq & \mathbb{N} \quad & \text{(indexable)} \\ % h & \in & \mathcal{H} \quad & \text{(a hold)} \\ % \mathcal{L}& \ni & \ell \subset \mathcal{H} \quad & \text{(multiple holds make a location)} \\ \end{eqnarray}\]

A location is thus a subset of holds which are elements of the wall.

The wall is augmented with a special hold, $h_{0}$ which means no non-temporary attachment to the wall. The location containing only this is the terminating location. Together with this, an exact definition of the locations are constructed.

\[\begin{eqnarray} \ell_{0} & = & \{ h_{0} \} \\ % \mathcal{L} & = & \ell \subset \mathcal{H}: \quad & \text{} \\ % \ell & = &\{ h_{0} \} \quad & \text{(terminating)} \\ % & \land & \quad & \text{(or)} \\ % h_{0} & \notin & \ell \quad & \text{(non-terminating)} \end{eqnarray}\]

An other distinguished hold is introduced, the free hold, $h_{f} \in \mathcal{H}$. It represents a locality which is not part of the physical wall: a temporary lack of attachment.

The definition of path remains unchanged for it solely relied on locations. Only layer of resolution has been added to our terminology.

Fourth move

Move

A quick one forward.

A move, $m$ takes from one location to an other. As such, it is a function taking from a location back to the wall ro to a fall. The set of all moves is denoted by $\mathcal{M}$

\[\begin{eqnarray} \mathcal{M} & = & \mathcal{L} \rightarrow \mathcal{L} \\ \mathcal{M} & \subseteq & \mathcal{L} \times \mathcal{L} \quad & \text{(Cartesian product)} \end{eqnarray}\]

The definition above allows for

• moves that terminate the climb i.e. $(\ell_{i\neq 0}, \ell_{0}) \in \mathcal{M}$
• moves that leave the climber in the same location e.g. $(\ell_{i}, \ell_{i}) \in \mathcal{M}$

The second point prompts us thinking. What might be the purpose of staying on the same position? Having a rest? Waiting for the helicopter? Rearrange the limbs on the holds? Or, mostly likely, to change the body configuration?

Fifth move

Limbs

The body of the climber is denoted by $\mathcal{B}$. The limbs, or body parts, $b$ are the elements of the body.

\[\begin{eqnarray} ! \mathcal{B} & \neq & \emptyset \quad & \text{(body has limbs)} \\ % \vert \mathcal{B} \vert & \le & \vert \mathbb{N} \vert \quad & \text{(finite number of body parts)} \end{eqnarray}\]

Holdup

A move not necessarily results in changing the holds. This can happen in three ways.

1. Some limbs are shuffled between the holds e.g. leg switch.
2. The limbs stay on the same holds but their position with respect to them is altered.
3. The body parts which are not in contact with the holds change position e.g. drawing the hips closer to the wall.

Sixth move

Arrangement

Let us discuss the first scenario.

The arrangement of the limbs on the holds constitute the set $\mathcal{A}$. An element just simply lists which body part is on which hold.

\[\mathcal{A} = \{ \{ (b, h): b \in B , h \in \ell \}: B \subset \mathcal{B}, \ell \subset \mathcal{L} \}\]

Let 1, 2, 3, 4 refer to the right, left legs and right, left hands in that order. Please note that the indices of the holds are irrelevant in the following examples. An arrangement where the left hand does not touch the wall is represented as

\[a = \{ (b_{1}, h_{1}), (b_{2}, h_{2}), (b_{3}, h_{3}), (b_{4}, h_{0}) \} \, .\]

The terminating arrangement is one where no limbs are attached to the wall an they will not be so:

\[a = \{ (b_{1}, h_{0}), (b_{2}, h_{0}) (b_{3}, h_{0}), (b_{4}, h_{0}) \} \, .\]

Moves are can now be redefined through arrangments. For instance, a leg swap reads as

\[\{ (b_{1}, h_{1}), (b_{2}, h_{2}), (b_{3}, h_{3}), (b_{4}, h_{4}) \} \rightarrow \{ (b_{1}, h_{2}), (b_{2}, h_{1}), (b_{3}, h_{3}), (b_{4}, h_{4}) \} \, .\]

Grasping a new hold with the left hand is simply written by

\[\{ (b_{1}, h_{1}), (b_{2}, h_{2}), (b_{3}, h_{3}), (b_{4}, h_{4}) \} \rightarrow \{ (b_{1}, h_{2}), (b_{2}, h_{1}), (b_{3}, h_{3}), (b_{4}, h_{5}) \} \, .\]

Seventh move

Configuration

Body–hold configuration

The posture of the body parts with respect to the holds is called the body–hold configuration. It is represented by $\mathcal{C}^{BH}$. This set contains all concievable positions. Some of them are not physically possible for a given climber to achieve. The reason might be attributed to the body structure, skills or just to the lack of energy. The configuration of a limb depends on those of the others. However, all of them are an allowance of an arrangement. The actual arrangement selects subsets configurations, $\mathcal{C}^{AB}(a)$.

Let us consider the set of all conceivable configurations and an arrangement

\[\begin{eqnarray} ! \mathcal{C}^{BH} & \neq & \emptyset \quad & \text{(all configurations)} \\ ! a & \in & \mathcal{A} \quad & \text{(current arragement)} \end{eqnarray}\]

then

\[\begin{eqnarray} \mathcal{C}^{AB}(a) & = & \{ C^{AB}: C^{AB} \subset \mathcal{C}^{AB} \} \quad & \text{(possible configurations)} \\ % ! C^{AB} & \in & \mathcal{C}^{AB}(a): \vert C^{B} \vert \leq \vert \mathbb{N} \vert \quad & \text{(indexable)} \\ % ! C^{AB} \in \mathcal{C}^{AB}(a) & = & \{ c^{AB}_{i, j} : \forall (b_{i}, h_{j}) \in a \} \quad & \text{(a configuration of all limbs)} \, . \end{eqnarray}\]

The possible configurations are indexable: \(\mathcal{C}^{AB}(a) = \{ C^{AB}_{k}, k \in [1, K] \} \, .\)

An element of $C^{AB}_{k}$ represents a position of a limb on a hold: $c^{AB}_{i,j,k}$

Body configuration

The posture of body parts which do not interact with the holds directly can also influence the climb e.g. pulling oneself up whilst retaining the same grips (body–hold configuration). An constellation of these postures are called body configuration. Formalising them is a trivial affair following the previous paragraph. The entirety of all body configurations are collated in the set $\mathcal{C}^{B}$.

Again, an arrangement selects sets of physically achievable configurations.

\[\begin{eqnarray} ! \mathcal{C}^{B} & \neq & \emptyset \quad & \text{(all configurations)} \\ ! a & \in & \mathcal{A} \quad & \text{(current arragement)} \end{eqnarray}\]

Then an indexable family of sets of body configuration are written as:

\[\begin{eqnarray} \mathcal{C}^{B}(a) & = & \{ C^{B}: C^{B} \subset \mathcal{C}^{B} \} \quad & \text{(possible configurations)} \\ % ! C^{B} & \in & \mathcal{C}^{B}(a): \vert C^{B} \vert \leq \vert \mathbb{N} \vert \quad & \text{(indexable)} \\ % ! C \in \mathcal{C}^{B}(a) & = & \{ c^{B}_{i} : \forall b_{i} \in \mathcal{B}: \nexists h_{j} \in \mathcal{H}, (b_{i}, h_{j}) \in a\} \quad & \text{(exclude limbs on holds)} \, . \end{eqnarray}\]

Looking down

State

Our exercise so far has stacked these notions on to each other

1. location, $\ell \in \mathcal{L}$: what holds are the climber is in contact with
2. body and limbs, $b \in \mathcal{B}$: the physical entity of the climber
3. arrangement, $a \in \mathcal{A}$: which limbs are in contact with which holds
4. body–hold configuration, $C^{AB} \in \mathcal{C}^{AB}$: posture of the limbs which are in contact with the holds
5. body configuration, $C^{B} \in \mathcal{C}^{B}$ posture of the body parts which are not in contact with the holds

Collating items 1. and 3–5. yields the state of the climber. All states are contained in the set $S$. An element thereof is a quartet of the location, arrangement along with the body–hold and body configurations given the second compoment. Please note that this definition of a state is redundant. The two configurations are indexed by the arrangement which is the function of the location. Nevertheless, both are kept explicit in the definition for the ease of referring to them. Retaining them will also facilitate further abstract our formalism.

\[\begin{eqnarray} \mathcal{S} & \subset & \mathcal{L} \times \mathcal{A} \times P(\mathcal{C}^{AB}) \times P(\mathcal{C}^{B}) \quad & \text{(set of all states)}\\ % !s & \in & \mathcal{S} \quad & \text{(a state)}\\ % s & = & ( \ell \in \mathcal{L}, a \in \mathcal{A}, C^{AB} \in \mathcal{C}^{AB}(a) , C^{B} \in \mathcal{C}^{B}(a) ) \\ \end{eqnarray}\]

The null state, $s_{0} \in \mathcal{S}$ is introduced which is one that terminates the climb.

Eighth move

Move again

Having left behind all three holdups, we can now work with a tiered description of the state of the climber. The rock-lover alters their location, arragement, configuration in the hope that they progress towards the finish and eventually reach it. Any individual change or combinations thereof is a move.

\[\mathcal{M} \subset \mathcal{S} \times \mathcal{S}\]

The definition above however implies that there is one and only one move between two arbitrary states. This is not the case at all. But only superficially. If we only consider changing locations or arrangements as moves.

Let us call changing the location relocation. A relocation can be performed in multiple ways, each of them concatenating a sequence of movements. Since a movement is nothing else than modifying the body posture, each movement is a move. Even if it is performed as a part of a dyno when not holding on to the wall. (That is the reason beyond introducing the free hold, $h_{f}$). The definition of move therefore covers all possible actions on and from the wall.

Ninth move

Climb!

We are finally worked ourselves in a position where we can comfortably define what a climb is. It is nothing else but a chain of subsequent states.

\[\begin{eqnarray} !n & \in & \mathbb{N}: n > 0 \quad & \text{(finite "steps")} \\ % \mathcal{T} & = & [1, n] \quad & \text{(time index)}\\ % ! i & \in & \mathcal{I} = [1, \vert \mathcal{S} \vert ] \quad & \text{(state indices)} \\ % V & = & \{ (t, s_{i_{t}}): t \in \mathcal{T}, s_{i_{t}} \in \mathcal{S} \} \quad & \text{(assign a state to a time)} % \end{eqnarray}\]

Until now, we have been dissecting the act of climbing to identify the basic components of it. From now on, these elements will construct notions of simple and hopefully familiar meaning.

Tenth move

Relocation

The primary goal of the climber to reach the desired location, $\ell_{t_{n}}$ from the starting one, $\ell_{t_{1}}$. The change of locations is called relocation:

\[\mathcal{R} = \mathcal{L} \times \mathcal{L} .\]

It is clear that each relocation induces a move. The reverse does not hold true, however.

The relocation $e = (\ell_{t_{1}}, \ell_{t_{n}})$ is more often than not is achieved through multiple location changes. These constitute the path as defined earlier.

Links

Let us focus on the next set of holds ahead of us. We devise a set of movements (a.k.a. moves) that takes us there.

This chain of moves, or equivalently, states is called a link, $e \in \mathcal{E}$. These, indeed, can and likely be different across climbers.

\[\begin{eqnarray} ! \ell_{i}, \ell_{j} & \in & \mathcal{L} \quad & \text{(two locations to move between)} \\ % ! K & \in & \mathbb{N} \quad & \text{(number of moves)} \\ % \mathcal{T} & = & [1, K ] \quad & \text{(time)} \\ % e_{\ell_{i}, \ell_{j}} & = & \{(t_{k}, s_{k}), k \in [1, ..., K]\}: \quad & \text{(sequence of states (or induced moves) with time)} \\ % s_{1} & = & \{ \ell_{i}, a \in \mathcal{A}, C^{AB} \in \mathcal{C}^{AB}(a) , C^{B} \in \mathcal{C}^{B}(a) \} \quad & \text{(specified starting location)} \\ % \forall k, 1 < k < K: \ell_{k} & \subset & \ell_{i} \cup \ell_{j} \quad & \text{(no intermediate locations apart from those made of the start of final holds)} \\ % s_{K} & = & \{ \ell_{j}, a \in \mathcal{A}, C^{AB} \in \mathcal{C}^{AB}(a) , C^{B} \in \mathcal{C}^{B}(a) \} \quad & \text{(ends in the desired location)} \\ \end{eqnarray}\]

We recognise that a link is a climb between two fixed locations. The set of all links over the wall are denoted by $\mathcal{E}$. If $\ell_{j}$ is not reachable from $\ell_{j}$ then the link is ${s_{0}}$.

Eleventh move

Primitives

We cannot help but notice over several and separate hours on the rocks that some holds are akin. Likewise, locations oftentimes bear similarities. Moreover the types and arrangement of neighbouring locations recurr on and across walls. The repeated characteristics of these features are captured in the so-called primitives. They are distinguished from the actual elements with a bar (-) overscribed. All primitives are contained in an idealised wall, $\bar{\mathcal{H}}$:

\[\begin{eqnarray} \bar{h} & \in & \bar{\mathcal{H}} \quad & \text{(hold primitive)} \\ % \bar{\ell} & \in & \bar{\mathcal{L}} \subset P(\bar{\mathcal{H}}) \quad & \text{(location primitive)}\\ % \bar{r} & \in & \bar{\mathcal{R}} = \bar{\mathcal{L}} \times \bar{\mathcal{L}} \quad & \text{(relocation primitive)} \, . \end{eqnarray}\]

It immediately follows that all possible elements have a primitive. This primitive may only be a representation of that single element.

Attributes

The similarity is induced by one or multiple shared properties between the elements. Thef attributes are collected in the sets $\mathcal{A}^{\bar{\mathcal{H}}}, \mathcal{A}^{\bar{\mathcal{L}}}, \mathcal{A}^{\bar{\mathcal{R}}}$ for holds, locations and relocations, respectively. There is not separate sets of the properties of the primitives for a property is already abstracted. As such it applies to both the element and its primitive. In fact, the primities are defined by having a certain set of attributes.

They are attached to the object through a function

\[\begin{eqnarray} \mathcal{Z} & = & \{ \mathcal{H}, \bar{\mathcal{H}}, \mathcal{L}, \bar{\mathcal{L}}, \mathcal{R}, \bar{\mathcal{R}} \} \quad & \text{(shorthand for element kinds)} \\ % ! \mathcal{S} & \in & \mathcal{Z} \quad & \text{(select a kind)} \\ % f^{\mathcal{S}} & = & \mathcal{S} \times P(\mathcal{P}^{\bar{\mathcal{S}}}) \setminus \emptyset \quad & \text{(assign a subset of properties)} \, . \end{eqnarray}\]

Equivalence

Two elements of the same kind e.g two locations, are equivalent, $\equiv$ if they have the same properties:

\[\begin{eqnarray} ! \mathcal{S} & \in & \mathcal{Z} \quad & \\ % ! x, y &\in & \mathcal{S} \quad & \text{(elements of the same kind)} \\ % x \equiv y & \Leftrightarrow & f^{\mathcal{S}}(x) = f^{\mathcal{S}}(y) \quad & \text{(both have the same properties)} \, . % \end{eqnarray}\]

The primitive of an element is by definition equivalent to the element itself.

\[\begin{eqnarray} \forall \mathcal{S} & \in & \mathcal{Z} \quad & \text{(select a kind)} \\ % \forall x &\in & \mathcal{S}, \bar{x} \in \bar{S} \quad & \text{(element and primitive)} \\ % x & \equiv & \bar{x} \quad & \text{} % \end{eqnarray}\]

Similarity

Two elements are similar, $\sim$ if they have at least one shared property:

\[\begin{eqnarray} ! \mathcal{S} & \in & \mathcal{Z} \quad & \\ % ! x, y &\in & \mathcal{S} \quad & \text{(elements of the same kind)} \\ % x \sim y & \Leftrightarrow & f^{\mathcal{S}}(x) \cap f^{\mathcal{S}}(y) \neq \emptyset \quad & \text{(some properties in common)} \, . \end{eqnarray}\]

Twelfth move

Skill

The primitives help us establish whether we can move from one location to an other. If we can hold on to both types of locations and have the ability to perform that kind of movements that takes us from one to the other, we attempt it with reasonable confidence. So to speak we have the skills.

A skill, $\bar{e}_{\bar{\ell}_{i}, \bar{\ell}_{j}}$ in this formalism is a climb over relocation primitive. As such, it is a link primitive. For the sake of completeness, it is defined below:

\[\begin{eqnarray} ! \bar{\ell}_{i}, \bar{\ell}_{j} & \in & \bar{\mathcal{L}} \quad & \text{(two locations to move between)} \\ % ! K & \in & \mathbb{N} \quad & \text{(number of moves)} \\ % \mathcal{T} & = & [1, K ] \quad & \text{(time)} \\ % \bar{e}_{\bar{\ell}_{i}, \bar{\ell}_{j}} & = & \{(t_{k}, s_{k}), k \in [1, ..., K]\}: \quad & \text{(sequence of states (or induced moves) with time)} \\ % s_{1} & = & \{ \bar{\ell}_{i}, a \in \mathcal{A}, C^{AB} \in \mathcal{C}^{AB}(a) , C^{B} \in \mathcal{C}^{B}(a) \} \quad & \text{(specified starting location)} \\ % \forall k, 1 < k < K: \bar{\ell}_{k} & \subset & \bar{\ell}_{i} \cup \bar{\ell}_{j} \quad & \text{(no intermediate locations apart from those made of the start of final holds)} \\ % s_{K} & = & \{ \bar{\ell}_{j}, a \in \mathcal{A}, C^{AB} \in \mathcal{C}^{AB}(a) , C^{B} \in \mathcal{C}^{B}(a) \} \quad & \text{(ends in the desired location)} \\ \end{eqnarray}\]

The set of all skills is denoted by $\bar{\mathcal{E}}$. The idempotent relocations i.e. the start and the final locations are the same, induce the skill of holding on.

Thirteenth move

Effort

We were purposefully vague at defining what an attribute was. For a reason. The climber may have the skill to move from one location primitive to an other one. Sadly enough, they are just a tad too far apart. It requires a force that the climber will surely attain in a short time of training. Staying in some locations, too, may be harder than camping out in other ones. The physical cost of performing a move or maintaining a position is called the effort, $\gamma$. The effort function $f_{\Gamma}$ quantifies the phyisical exertion by mapping in on to the zero–one range.

\[\begin{eqnarray} ! \gamma & \in & \Gamma = [0, 1] \quad \text{(hard upper limit of effort)} \\ % ! f_{\Gamma} = \bar{\mathcal{E}} \times \Gamma \end{eqnarray}\]

Again, this function simply measures how much effort is required to maintain or change a position irrespective of the manifestations of the locations. The maximum attainable effort of a climber is symbolised by $\gamma_{\text{max}}$. The total usable effort if denoted by $\gamma_{\text{tot}}$.

Fourteenth move

Let us link together what we have achieved in this post. We started it with the sentence “ If someone is skilled enough and have the power they can link the two positions through a sequence of appropriate moves.” We will translate to the language of our formalism:

\[\begin{eqnarray} \exists \bar{E} & \in & \bar{\mathcal{E}} \quad & \text{(if someone is skilled ...)} \\ % ! \ell_{i}, \ell_{j} & \in & \mathcal{L} \quad & \text{(two positions)} \\ % \exists V & = & \{ (t_{k}, s_{k}), k \in [1, K] \} : \quad & \text{(link them through a sequence of movements)} \\ % s_{1} & = & \{ \ell_{i}, a \in \mathcal{A}, C^{AB} \in \mathcal{C}^{AB}(a) , C^{B} \in \mathcal{C}^{B}(a) \} \quad & \text{} \\ % s_{K} & = & \{ \ell_{j}, a \in \mathcal{A}, C^{AB} \in \mathcal{C}^{AB}(a) , C^{B} \in \mathcal{C}^{B}(a) \} \quad & \text{} \\ % \forall k & \in & [1, .., K - 1]: \bar{e}_{\bar{\ell}_{k}, \bar{\ell}_{k + 1}} \in \bar{E} \quad & \text{( ... enough)} \\ % & \land & \quad & \text{(and)} \\ % \forall k & \in & [1, .., K - 1]: f_{\Gamma}(\bar{e}_{\bar{\ell}_{k}, \bar{\ell}_{k + 1}}) \leq \gamma_{\text{max}} \quad & \text{(no too large single effort)} \\ % \sum\limits_{k=1}^{k = K - 1} && f_{\Gamma}(\bar{e}_{\bar{\ell}_{k}, \bar{\ell}_{k + 1}}) \leq \gamma_{\text{(tot)}} \quad & \text{(enough enery for the entire climb)} \, . \end{eqnarray}\]

The lines above may not be more legible or clearer that the English sentence. However, it certainly encompasses more than the ad hoc collection of written words. It provides us with a machinery in which climbs can be modelled. Rules can be derived, problems solved or their solvability decided.

Future moves

The formalism, not matter how gripping it was to develop, and how aesthetically enchanting it shines, it is only a descriptive one. In order to create a prescriptive algorithm that draws routes between locations, graph theoretical algorithms are required. A future post will lay out some paths to reach this goal.