A step forward: casting actions into convex relations

(1)

A step forward: casting actions

into convex relations

Dani¨el B. van de Pavert 11045418

Bachelor thesis Credits: 18 EC

Bachelor Opleiding Kunstmatige Intelligentie University of Amsterdam Faculty of Science Science Park 904 1098 XH Amsterdam Supervisor dr. M.A.F. Lewis

Institute for Logic, Language and Computation Faculty of Science University of Amsterdam Science Park 107 1098 XG Amsterdam January 31st, 2019 1

(2)

Abstract

We apply methods proposed by G¨ardenfors (2000) and Bolt et al. (2019) to model actions within conceptual spaces. To do this we use convex sets and convex relations. We define a conceptual space for nouns and another for sen-tences, and show how verbs can be defined as relations from the noun space to the sentence space in line with pregroup grammar. We give a demonstration of this using point-light walker data.

(3)

Acknowledgements

Foremost, I would like to express my sincere gratitude to Dr. Martha Lewis for her help during this project. Her guidance helped me in time of research and writing, and her knowledge of convex relations and conceptual spaces made this thesis possible.

I would also like to thank Dr. Troje, as he designed, created and provided the data used for this research. Without this data, the project would not have been possible.

(4)

1 Introduction

The need for programs to simulate actions is becoming more relevant in an era where programs need to support, execute and react to a wide variety of actions. These include actions the program has not encountered before and actions that vary in execution from source to source. At the time of writing this paper the most common method for modelling actions is by using a feed-forward neural network, this network processes a large amount of data to create a model of what defines the action. Feed-forward neural networks are well suited for controlled environments with limited variation but do not operate optimally in an environment with variations and new concepts. The recognizing of variations and the combining of concepts are essential in the development and deployment of explainable artificial intelligent systems.

Conceptual spaces theory is a framework created to represent concepts in a geo-metric structure. Gärdenfors and Warglien (2012) extend the theory of analysis of perceptual spaces with methods of analyzing actions and events. Bolt et al. (2019) expand on the modelling and interaction with actions using ConvexRel. In this paper, a method is formalized of interaction with an action in conceptual spaces. In section 2 we describe Gärdenfors’ theory of conceptual spaces and how actions can be represented. We also describe pregroup grammar as seen in Lambek (2001), the vector space model and formal semantics all of which are the background for convex relations as described by Bolt et al. (2019). This method builds on the work done by Bolt et. at and Gärdenfors and applies it in a practical manner, this leads to the research question:

Taking Gardenfors’s and Bolt’s account of actions, can actions be cast in convex relations?

(6)

2 Background

2.1 Actions in conceptual spaces

2.1.1 Conceptual spaces

Conceptual Spaces were introduced by G¨ardenfors (2000) to represent informa-tion and meaning at the conceptual level. These spaces are comprised of quality dimensions which represent features, like colour, shape or taste. Quality dimen-sions are composed of domains each expressing a facet of the feature represented by the quality dimension. For instance, a banana can be seen as a combination of the Taste, Colour and Texture domains and the domain Colour can be seen as a so-called colour spindle [1]. Concepts are convex subsets of a conceptual space e.g. dark red is a region on the lower half of the spindle on the red side.

Figure 1: colour spindle, (G¨ardenfors, 1996)

G¨ardenfors (2014) expands the focus on noun-noun composition with formal-izing verb spaces, adjectives and other linguistic structures. Extending this a systematical method of utilizing grammatical structures within conceptual spaces is created by Bolt et al. (2019).

2.1.2 Actions within Conceptual spaces

When G¨ardenfors first theorized Conceptual Spaces (G¨ardenfors, 1996) he dis-cussed several ways to model and analyze actions. He starts by looking at the methods proposed by Marr and Vaina (1982) and Vaina (1983) who use cylindri-cal models and differential equations to model movements of body parts. With this model actions can be seen as force patterns for motions with multiple body parts need multiple force vectors to express the action.

A second method discussed by G¨ardenfors is the Johansson (1973) experiment involving light bulbs on the joints of actors. The actors would be filmed doing various actions in a dark room leaving only the joint visible. The film would then be shown to other subjects who recognized the actions within tenths of a second. These experiments and the subsequent research from Runesson and

(7)

Frykholm (1981) are used to formulate in a later paper the principle of kinematic specification of dynamics (the KSD-principle) (Runesson, 1994). This principle states that the kinematics of a movement contain enough information to identify the underlying force pattern’s. For this research, we interpret this principle as that from limited data similar to the data used by Johansson (1973) a concept can be formed.

2.1.3 Modeling actions using point light data

The modelling of actions is not limited to conceptual spaces as seen in a series of papers by Troje (2002a,b, 2008). In this series, Troje collects, processes and analyzes biological motion to develop a framework that outputs a representation of said motion that allows for analysis using linear methods. Similar to the data collection as seen in Johansson (1973), Troje also uses point light data. Subjects were filmed with lights attached to their joints walking in a dark room. This data is then processed to create a mathematical representation of walking which is used to analyze various features of the subject e.g. gender, weight, happiness and nervousness. Crucially, the representation of different walking styles is points in a vector space. In this paper, we use the data gathered and processed by the series of Troje papers. We give a detailed description of the data in the Method section of this paper.

(8)

2.2 Composing Actions with nouns

2.2.1 Formal semantics

Formal semantics is one of the methods used to formalize meaning and sentence structure. The aim is to combine words grammatically, see if a sentence is true or false in a certain world and thus look at the linguistic meaning. Formal semantics distinguishes between several word types e.g. Nouns, Transitive Verbs and Intransitive Verbs. Proper nouns refer to a single entity within a certain world, Intransitive verbs are a set of nouns that can express that verb and transitive verbs are sets of ordered pairs of entities where the first element of the set can be a subject of the verb and the second element is an object of the verb. An example of a simple language can be specified as follows:

1. S → N V P 2. S → S conj S 3. V P → VtN

4. V P → Vi

5. N → Bob | Anna | J ames 6. Vi→ is boring | is hungry

7. Vt→ admires

With this syntax we can build sentences like:

Bob is boring and James admires Anna

To look at the semantics of a sentence meaning has to be given to the words. For this, two additional sets of rules are needed. The first set gives meaning to the individual words in the syntax proposed earlier with semantic rules using the notation [[w]] to denote the meaning of the word w:

• [[Bob]] = Bob (b) • [[Anna]] = Anna (a) • [[J ames]] = J ames (j)

• [[admires]] = {hx, yi|x admires y} = hj, ai, hb, ai, hj, bi • [[is boring]] = {x|x is boring} = {b, j}

The second set adds semantic composition rules to create a manner of evaluating the truth value of a given sentence:

• [[SNP VP]] = 1 ↔ [[N ]] ∈ [[V P ]]

• [[SS1 conj S2]] = [[conj]](h[[S1]], [[S2]]i)

• [[V PVtN ]] = {x|hx, [[N ]]i ∈ [[Vt]]}

(9)

The third set describes the rules for the conjunction and used in the example sentence.: [[and]] =     h1, 1i → 1 h1, 0i → 0 h0, 1i → 0 h0, 0i → 0    

Combining the syntax rules and the semantic rules tree notation can be used to evaluate the truth value of the sentence. An small example of the method used is when looking at the leftmost sentence S, when resolving a tree the direction of work is from bottom to top and from left to right. First the noun N is solved by inserting the meaning for the noun Bob leaving b under the sentence S. Secondly the verb phrase VP is solved by inserting the set corresponding to is boring from the semantic rules above leaving {b, j} under the sentence S. The last step of this example will check whether b is in the set {b, j} using the semantic rules, as this is true the left hand sentence evaluates to 1.

S S N Bob VP Vi is boring conj and S N James VP Vt admires N Anna 9

(10)

2.2.2 Pregroup grammar

To describe syntactic structure pregroup grammar can be used as seen in Lam-bek (2001). A pregroup grammar is built over a pregroup algebra. A pregroup algebra is a partially ordered set with a multiplication, where each element p has a left and right adjoint such that:

• p ∗ pr_{≤ 1 ≤ p}r_{∗ p}

• pl_{∗ p ≤ 1 ≤ p ∗ p}l

For the demonstration in this paper, the grammar is constructed from the al-phabet {n, s} of noun and sentence types. Using n and s several grammatical types can be built but this paper is limited to the use of the intransitive verb with type nr_{s. In the sentence above the conjunction and is used, in this case}

the type in pregroup grammar is sr_{s s}l_{. The sentence Bob is boring and James}

admires Anna is seen as n ∗ nr_{∗ s ∗ s}r_{∗ s ∗ s}l_{∗ n ∗ n}r_{∗ s ∗ n}l_{∗ n in pregroup}

grammar and is reduced as shown below:

n ∗ nr∗ s ∗ sr_{∗ s ∗ s}l_{∗ n ∗ n}r_{∗ s ∗ n}l_{∗ n = 1 ∗ s ∗ s}r_{∗ s ∗ s}l_{∗ n ∗ n}r_{∗ s ∗ n}l_{∗ n} = 1 ∗ s ∗ sr∗ s ∗ sl∗ n ∗ nr∗ s ∗ 1 = 1 ∗ s ∗ sr∗ s ∗ sl_{∗ 1 ∗ s ∗ 1} = 1 ∗ s ∗ sl∗ 1 ∗ s ∗ 1 = 1 ∗ s ∗ 1 = s (1) 2.2.3 Vector space model

Pregroup grammar can be mapped to vector spaces, basic types such as nouns and sentences are mapped to the noun space N and the sentence space S re-spectively. This mapping is done according to a set of rules as shown below:

• n → N • s → S

• p ∗ q → P ⊗ Q • pr_{, p}l_{→ P}

• reduction rules → tensor contraction

Nouns in pregroup grammar have type n, mapping to N a vector in the noun space. The pregroup type for a intransitive verbs is nr_{s thus mapping to N ⊗ S,}

a linear map N → S from the noun space to the sentence space. Using the vector space model allows for tensor operations between concepts. Tensor contractions work in a similar way as the reduction seen in pregroup grammar.

(11)

The sentence Bob is boring in pregroup grammar would be reduced like nnr_{s =}

s, the same sentence in the vector space model has type N ⊗ N ⊗ S and reduces to S. The operation of tensor contraction can be seen as an extension of matrix multiplication with subjects in higher dimensions e.g. the multiplication of cuboid subjects. For more information on tensor contractions see Coecke et al. (2013).

2.2.4 Convex relations

To define a convex relation the definition of a relation, the application of a relation and of a convex set are needed. A relation is defined as a subset of two sets such that R ⊆ A×B. A set A is convex if ∀a1, a2∈ A|p∗a1+(1−p)∗a2∈ A.

The application of a relation to an element a ∈ A creates a subset of B containing those elements b ∈ B such that a is related to B. A convex relation can now be defined as a relation that sends convex sets to convex sets. To compose a convex relation Bolt et al. (2019) provide a series of steps:

1. (a) Choose a compositional structure, such as a pregroup or combinatory categorial grammar.

(b) Interpret this structure as a category, the grammar category. 2. (a) Choose or craft appropriate meaning or concept spaces, such as vector

spaces or conceptual spaces.

(b) Organize these spaces into a category, the semantics category, with the same abstract structure as the grammar category.

3. Interpret the compositional structure of the grammar category in the se-mantics category via a functor preserving the necessary structure. 4. Bingo! This functor maps type reductions in the grammar category onto

algorithms for composing meanings in the semantics category.

In this paper we apply these steps using pregroup grammar for our grammar category and the conceptual space model for the semantics category. Further more Bolt et al. (2019) describe the ”cup” morphism denoted by :

: A × A → I :: {((a, a), ∗)|a ∈ A}

In pregroup grammar the -morphism is equivalent to pl_{p ≤ 1 and pp}r _{≤ 1.,}

i.e the inequalities needed for grammatical reductions as seen in example 1 and in the vector space model this is equivalent to the tensor contraction. A new mapping from the vector space model to ConvexRel can now be defined:

• S → S • P → P

• P ⊗ Q → P × Q

• tensor contraction → -morphism

(12)

In ConvexRel the -morphism picks out the elements of A × A such that when the element match it is a relation to the singleton set denoted by I, this is the same as applying a relation. The singleton set {∗} has a unique convex structure so it can be viewed as a simple conceptual space. A relation to the singleton set can be seen as True under this model and a lack of a relation to the singleton set can be seen as False. Consider the following sets using the nouns Bob and James and the verb is boring as seen in the formal semantics section 2.2.1 example:

• N = {Bob, J ames} • S = {∗}

• is boring = {< Bob, ∗ >, < J ames, ∗ >}

Following the method above we can now reduce the sentence Bob is boring : Bob is boring = ({Bob} × {< Bob, ∗ >, < J ames, ∗ >}) = {∗}

The relation to the singleton set {∗} means that the sentence Bob is boring is true and that Bob is in fact boring. Now consider this next group of sets, the noun set N and the sentence set S are similar in structure to the last example but the verb has type N × S is defined as V ⊆ N × S:

• N = {a1, a2}

• S = {s1, s2}

• N × S = {< a1, s1>, < a1, s2>, < a2, s1>, < a2, s2>}

• V ⊆ N × S = {< a1, s2>, < a2, s1>}

Using the -morphism we can now reduce a noun verb combination. A noun verb combination has type, and is subset of N × N × S. An example of a noun verb combination is:

noun × verb = {a1} × {< a1, s2>, < a2, s1>}

And the application of the verb to the noun using the -morphism leads to:

sentence = (noun × verb)

= ({a1} × {< a1, s2>, < a2, s1>})

(13)

3 Methods

3.1 Data

We firstly describe the mean walker provided by Troje consisting of 15 markers and the gait time. The 15 markers are located on the joints of the walker as seen in figure 2.

Figure 2: Mean walker, (Troje, 2002a)

The following description of the data is from Troje (2002a). Each marker is expressed in three dimensions meaning a single pose is a 15 ∗ 3 = 45 dimensional vector. The position of the 15 markers at time t describes a single posture of the walker. The representation of a single posture is thus a 45 dimensional vector p = (m1x, m1y, m1z, m2x, ...m15z)T. Troje transforms the data by decomposing

a series of postures from a single person j into a second order Fourier expansion:

pj(t) = pj,0+ pj,1sin(ωjt) + pj,2cos(ωjt) + pj,3sin(2ωjt) + pj,4cos(2ωjt) + errj

Following Troje (2002a) the residual term errj is discarded from all further

computations. The walk of an individual is therefore approximated the average posture pj,0, the four characteristic postures pj,1, pj,2, pj,3, and pj,4, and the

fundamental frequency ωj. A 45 dimensional vector comprised of the

funda-mental frequencies ω for each marker of the mean walker was also provided by Troje. For further information on the acquisition of the fundamental frequency see Troje (2002a,b).

With the movement of a single-dimensional aspect of a marker now expressed by 5 terms, a vector can be formed that completely characterizes a gait. This vector combines the 45-dimensional vector for a single posture with the 5-dimensional

(14)

vector expressing the movement of a single marker in one spacial dimension. The resulting vector of this combination has a dimensionality of 226 = 5 ∗ 45 + 1. The +1 is an addition to store the gait time of the walker.

The walker space is the 226-dimensional space containing the vectors charac-terizing gaits and proves useful for application in conceptual spaces as it has convex properties. This means that for every two points w1, w2 in the space all

points between them are also in the space.

The data received from Troje also contains 4 vectors each corresponding to the effect of a certain trait on the walker. These vectors also have a dimensionality of 226 = 5 ∗ 45 + 1. The 4 trait vectors provided are Gender, Weight, Happiness and Nervousness. The traits can be applied to the walker by addition with α being a scalar value, T being a trait vector and W being the walker:

Wnew= αT + Wold

For the sake of clarity and readability the α will be denoted as [αG, αH, αN, αW]

and the trait vectors will be denoted as TG, TH, TN, TW.

3.2 Modeling simple sentences in ConvexRel

To model a simple sentence a noun-verb combination is used as seen in Bolt et al. (2019). To do this a noun space N, a sentence space S and a representation of a verb are needed. In the sections below the composition of the spaces and the representation of verbs will be clarified.

3.2.1 Noun space

We define a noun space N based on the characteristics of the trait vectors, namely gender, happiness, nervousness and weight. The weightings α of each of the trait vectors range from -1 to 1, so the noun space N is the space [−1, 1]4. An individual i in the noun space N is a column vector of the four traits, for instance: i =     αG αH αN αW     i =     1 −0.5 0 −0.7    

Figure 3: These give a representation of an individual in the noun space N This point i describes a heavy slightly nervous yet happy man. Nouns within noun space N are convex regions and individuals are points. We can define heavy happy nervous men as:

heavy happy nervous men = {~α ∈ N |αG∈ [0, 1], αH ∈ [0, −1], αN ∈ [0, 1], αW ∈ [0, 1]}

(15)

Various combinations of traits can be seen as regions in the noun space N. As N is a 4-dimensional space an example is given in four 3-dimensional diagrams.

Figure 4: Four 3D diagrams to show the region heavy happy nervous men in the 4D noun space

3.2.2 Sentence space

We take the 226-dimensional walker space in which the gaits are represented to be the sentence space. Each sentence s can be represented as a subset in sen-tence space S. Each point in the sensen-tence space S represents a single individual expressing a single verb.

3.2.3 Representing verbs

A verb in pregroup grammar is represented like nrs as shown in the pregroup grammar section 2.2.2 of this paper. Transforming this representation into con-ceptual space model leads to the representation used in this paper: V ⊆ N × S. In the present work we concentrate on a single verb walks. We define walks as a subset of N × S. This is a relation telling us how to map from the noun space N to the sentence space S. We define walks as follows:

(16)

walks = {< ~α, ~w > | ~w = Wmean+ αG∗ TG+ αH∗ TH+ αN ∗ TN + αW ∗ TW}

The variables αG, αH, αN, αW denote a point in the noun space and TG, TH, TN, TW

denote the trait vectors as seen in the data section 3.1 of this paper. Working with a specific noun A ⊆ N note that to prove convexity of a verb we need to show that if ~w1, ~w2 ∈ (A × walks) then p ∗ ~w1+ (1 − p) ∗ ~w2 ∈ (A × walks).

Firstly suppose that A is convex i.e if ~α, ~β ∈ A then p ∗ ~α + (1 − p) ∗ ~β ∈ A. Defining a α and β and applying the -morphism leads to:

~ α =     αG αH αN αW     then ({~α} × walks ) = ~w1= Wmean+ αG∗ TG + αH∗ TH + αN ∗ TN + αW ∗ TW ~ β =     βG βH βN βW     then ({~β} × walks ) = ~w2= Wmean+ βG∗ TG + βH∗ TH + βN ∗ TN + βW ∗ TW

Now Suppose ~w3= p ∗ ~w1+ (1 − p) ∗ ~w2, this lead to:

~

w3= Wmean+ (p ∗ αG+ (1 − p)βG) ∗ TG

+ (p ∗ αH+ (1 − p)βH) ∗ TH

+ (p ∗ αN + (1 − p)βN) ∗ TN

+ (p ∗ αW + (1 − p)βW) ∗ TW

Reversing the process a noun can now be read from the scalar values in ~w3

~γ =     p ∗ αG+ (1 − p)βG p ∗ αH+ (1 − p)βH p ∗ αN+ (1 − p)βN p ∗ αW + (1 − p)βW    

And ~w3 = ({γ} × walks ) ∈ (A × walks)) since γ ∈ A since A is convex.

(17)

3.2.4 Composing nouns and verbs into sentences

Using the definitions for nouns, verbs and sentences we can now apply the methods from Bolt et al. (2019) to compose sentence given a noun and a verb. As we have seen in ConvexRel nouns have the type N , sentences have the type S and verbs have the type, and are the subset of N × S. Using this we can now represent noun verb combinations N × N × S and reduce this following the ConvexRel method. This can now be applied to a noun A ⊆ N to give a convex region B ⊆ S. Consider A to be the noun heavy happy nervous men from equation 2. For every John , Richard ∈ A the application of the convex relation walks returns a sentence John walks, Richard walks ∈ B. To illustrate this take John to be [0.1, −0.1, 0.1, 0.1]T _{and Richard to be [1, −1, 1, 1]}T_{, this}

lead to the following sentences s ∈ B:

John walks = ({John}× walks ) = Wmean+0.1∗TG+−0.1∗TH+0.1∗TN+0.1∗TW

Richard walks = ({Richard}× walks ) = Wmean+1∗TG+−1∗TH+1∗TN+1∗TW

Now take Eric to be between John and Richard thus Eric = p ∗ John + (1 − p) ∗ Richard, as A is convex Eric ∈ A. The application of walks to Eric leads to the following sentence:

Eric walks = ({Eric} × walks ) = Wmean+ (p ∗ 0.1 + (1 − p) ∗ 1) ∗ TG

+ (p ∗ −0.1 + (1 − p) ∗ −1) ∗ TH

+ (p ∗ 0.1 + (1 − p) ∗ 1) ∗ TN

+ (p ∗ 0.1 + (1 − p) ∗ 1) ∗ TW

As seen here Eric walks is in the same region as John walks and Richard walks i.e. Eric walks = p ∗ John walks + (1 − p) ∗ Richard walks and thus Eric walks ∈ B. Therefore the application of the verb walks to a noun is a convex relation from a convex region A ∈ N to a convex region B ∈ S

(18)

4 Results

Below are shown four frames, three different sentences in the sentence space and the mean walker. In this section, we show the result of the combinations of certain nouns with the verb walk. Nouns are instances of regions in the noun space. The use of certain nouns has a visible effect on the posture of the walker. As the values from the noun are applied to the mean walker as stated in section 3 (Methods) the mean walker transforms into a representation of the formed sentence.

Take the noun Beatrice as an instance of the region in the noun space associated with sad fat woman i.e. Beatrice = [−0.8, 1, 0, 0.6]T. We can now form the sentence Beatrice walks by applying the formula [2] seen in section 3 (Methods) of this paper:

Beatrice walks = (Beatrice × walks)

= (     −0.8 1 0 0.6     × {< ~α, ~w > | ~w = Wmean+ αG∗ TG+ αH∗ TH+ αN ∗ TN+ αW ∗ TW} = { ~w = Wmean− 0.8 ∗ TG+ 1 ∗ TH+ 0 ∗ TN + 0.6 ∗ TW}

The application of this formula on the mean walker can be seen in the figure 6 below with the mean walker [figure 5] as comparison. The human gait does not radically change so the gait may seem unchanged but on closer inspection, we can see that the hips have widened and the knees are further apart.

(19)

In order to better visualize the transformation from the mean walker we define two more nouns. Firstly Max = [1, 1, 1, 1]T _{an instance of the region on the}

noun space associated with sad nervous heavy men and secondly Felicia = [−1, −1, −1, −1] an instance of the region on the noun space associated with happy relaxed light women. With these nouns we can now form the sentences Max walks and Felicia walks:

Max walks = (Max × walks)

= (     1 1 1 1     × {< ~α, ~w > | ~w = Wmean+ αG∗ TG+ αH∗ TH+ αN ∗ TN + αW ∗ TW} = { ~w = Wmean+ 1 ∗ TG+ 1 ∗ TH+ 1 ∗ TN + 1 ∗ TW}

Felicia walks = (Felicia × walks)

= (     −1 −1 −1 −1     × {< ~α, ~w > | ~w = Wmean+ αG∗ TG+ αH∗ TH+ αN∗ TN+ αW ∗ TW} = { ~w = Wmean− 1 ∗ TG− 1 ∗ TH− 1 ∗ TN − 1 ∗ TW}

The application of this formula shows changes in the walkers’ postures as can be seen in the differences between Max walks [figure 8] and Felicia walks [figure 7].

Figure 7: Sentence: Felicia walks Figure 8: Sentence: Max walks

(20)

5 Conclusion

This paper, taking in to account the views of actions of Bolt et. al. and Gardenfors, demonstrated that actions can be cast into convex relations. To do this we introduced a new manner of combining the framework set by Bolt et al. (2019) and data shaped like the data provided by Troje. We take an algorithmic approach to demonstrate the application of work done by Bolt et al. (2019). We consider this model as a proof of concept of modelling actions within concep-tual spaces. Modelling actions within concepconcep-tual spaces is more often than not a theoretical aspect and not a practical example. With this paper, we aimed to provide a basis for future research in simulating action interactions in conceptual spaces.

A natural direction for future work would be the addition of other verbs or the addition of adjectives and adverbs. In this paper, nouns were represented in dimensions named after adjectives but no actual adjectives were used. An interesting future addition to this paper would be to formulate the nouns using actual adjectives. This would increase the meaningfulness of the used spaces as the nouns would be formulated more naturally. As regions are more compatible with the theory of conceptual spaces another clear further direction of research is the application of methods shown in this paper to regions instead of points. Lastly, a future direction would be to investigate how the methods used in this paper have an effect when used in multimodal NLP models.

References

Bolt, J., Coecke, B., Genovese, F., Lewis, M., Marsden, D., and Piedeleu, R. (2019). Interacting Conceptual Spaces I: Grammatical Composition of Con-cepts, pages 151–181. Springer International Publishing, Cham.

Coecke, B., Grefenstette, E., and Sadrzadeh, M. (2013). Lambek vs. lambek: Functorial vector space semantics and string diagrams for lambek calculus. Annals of Pure and Applied Logic, 164(11):1079 – 1100. Special issue on Seventh Workshop on Games for Logic and Programming Languages (GaLoP VII).

G¨ardenfors, P. (1996). Conceptual Spaces as a Basis for Cognitive Semantics. Philosophy and Cognitive Science: Categories, Consciousness, and Reason-ing, (1991):159–180.

G¨ardenfors, P. (2014). The geometry of meaning: Semantics based on conceptual spaces. MIT Press.

G¨ardenfors, P. (2000). Conceptual spaces : the geometry of thought. A Bradford book. MIT Press.

G¨ardenfors, P. and Warglien, M. (2012). Using conceptual spaces to model actions and events. Journal of Semantics, 29:487–519.

Johansson, G. (1973). Visual perception of biological motion and a model for its analysis. Brain Mapping: An Encyclopedic Reference, 3(2):125–130.

(21)

Lambek, J. (2001). Type grammars as pregroups. Grammars, 4(1):21–39. Marr and Vaina (1982). Representation and recognition of the movements of

shapes.

Runesson, S. (1994). Perception of biological motion: The ksd-principle and the implications of a distal versus proximal approach.

Runesson, S. and Frykholm, G. (1981). Visual perception of lifted weights. Journal of Experimental Psychology: Human Perception and Performance, 7. Troje, N. (2008). Retrieving information from human movement patterns.

Un-derstanding events: how humans see, represent, and act on events.

Troje, N. F. (2002a). Decomposing biological motion: A framework for analysis and synthesis of human gait patterns. Journal of Vision, 2(5):2–2.

Troje, N. F. (2002b). The little difference: Fourier based synthesis of gender-specific biological motion. Dynamic perception, pages 115–120.

Vaina (1983). From shapes and movements to objects and actions.

A step forward: casting actions into convex relations