Towards a computational- and algorithmic-level account of concept blending using analogies and amalgams

Figures & data

Figure 1. A schematic overview of HDTP's generalisation-based approach to analogy.

Table 1. Example formalisation of a stereotypical characterisation of a horse.

Sorts:

clade, entity, bodypart, ability

Entities:

mammal : clade horse : entity torso, legs : bodypart walk : ability

Predicates:

is_of_clade : $\mathit{\text{entity}}\times \mathit{\text{clade}}$ has_bodypart : $\mathit{\text{entity}}\times \mathit{\text{bodypart}}$

has_ability : $\mathit{\text{entity}}\times \mathit{\text{ability}}$

Laws of the horse characterisation:

${\mathit{\text{is}}}_{-}{\mathit{\text{of}}}_{-}\mathit{\text{clade}}$(horse, mammal) ${\mathit{\text{has}}}_{-}\mathit{\text{bodypart}}(\mathit{\text{horse}},\mathit{\text{legs}})$

${\mathit{\text{has}}}_{-}\mathit{\text{bodypart}}(\mathit{\text{horse}},\mathit{\text{torso}})$ ${\mathit{\text{has}}}_{-}\mathit{\text{ability}}(\mathit{\text{horse}},\mathit{\text{walk}})$

Figure 2. A reproduction of the examples originally given by Schwering et al. (2009) for the different types of higher-order anti-unifications applied in HDTP: a renaming (a), two different forms of fixation (b and c), an argument insertion (d), and a permutation (e).

Table 2. Example formalisation of a stereotypical characterisation of a dog.

Sorts:

clade, entity, bodypart, ability

Entities:

mammal : clade dog : entity tail : bodypart drool : ability

Predicates:

is_of_clade : $\mathit{\text{entity}}\times \mathit{\text{clade}}$ has_bodypart : $\mathit{\text{entity}}\times \mathit{\text{bodypart}}$

has_ability : $\mathit{\text{entity}}\times \mathit{\text{ability}}$

Laws of the dog characterisation:

${\mathit{\text{is}}}_{-}{\mathit{\text{of}}}_{-}\mathit{\text{clade}}$(dog, mammal) ${\mathit{\text{has}}}_{-}\mathit{\text{bodypart}}$(dog, tail) ${\mathit{\text{has}}}_{-}\mathit{\text{ability}}$(dog, drool)

Table 3. Shared generalisation of the “horse” and “dog” formalisations from Tables 1 and 2, respectively.

Sorts:

clade, entity, bodypart, ability

Entities:

mammal : clade E : entity B : bodypart A : ability

Predicates:

is_of_clade : $\mathit{\text{entity}}\times \mathit{\text{clade}}$ has_bodypart : $\mathit{\text{entity}}\times \mathit{\text{bodypart}}$

has_ability : $\mathit{\text{entity}}\times \mathit{\text{ability}}$

Laws of the shared characterisation:

${\mathit{\text{is}}}_{-}{\mathit{\text{of}}}_{-}\mathit{\text{clade}}(\mathit{\text{E}},\mathit{\text{mammal}})$ ${\mathit{\text{has}}}_{-}\mathit{\text{bodypart}}(\mathit{\text{E}},\mathit{\text{B}})$ ${\mathit{\text{has}}}_{-}\mathit{\text{ability}}(\mathit{\text{E}},\mathit{\text{A}})$

Figure 3. A diagram of an amalgam A from inputs $I_1$ and $I_2$ where $A={\overline{I}}_1\bigsqcup {\overline{I}}_2$.

Figure 4. A diagram that transfers content from source S to a target T via an asymmetric amalgam A.

Figure 5. Schematic overview of the houseboat blend as conceptualised by Goguen (2006): the conceptual theories for house and boat are generalised to a theory describing some object used by a person resting on some medium, and then combined to a houseboat theory featuring an object which is at the same time house and boat, resting on water, with residents living in it (who are at the same time passengers riding on it).

Figure 6. A conceptual overview of Goguen (2006)'s account of concept blending.

Figure 7. A conceptual overview of the COINVENT model of concept blending as described in Section 4.1: the shared generalisation G from S and T is computed with ${\phi }_S\left(G\right)=S_c$. The relation ${\phi }_S$ is subsequently re-used in the generalisation of S into $S^{\prime }$, which is then combined in an asymmetric amalgam with T into the proto-blend $T^{\prime }=S^{\prime }\bigsqcup T$ and finally, by application of ${\phi }_T$, completed into the blended output theory $T_B$. (⊆ indicates an element-wise subset relationship between sets of axioms and $⊑$ indicates subsumption between theories in the direction of the respective arrows.)

Figure 8. Blending schema for “Sign Forest” when inputs are typical concepts for “Sign” (traffic signpost) and “Forest” (forest of typical trees); the arrows indicate subsumption ($⊑$) as in Figure 3.

Table 4. Example formalisations of stereotypical characterisations for a bird S and a horse T.

Sorts:

clade, entity, bodypart, ability

Entities:

mammal, avialae : clade horse, bird : entity  torso, legs, wings : bodypart  walk, fly, lay_eggs : ability

Predicates:

is_of_clade : $\mathit{\text{entity}}\times \mathit{\text{clade}}$,  has_bodypart : $\mathit{\text{entity}}\times \mathit{\text{bodypart}}$,

has_ability : $\mathit{\text{entity}}\times \mathit{\text{ability}}$

Facts of the bird characterisation:

$\left({\alpha }_1\right)$ ${\mathit{\text{is}}}_{-}{\mathit{\text{of}}}_{-}\mathit{\text{clade}}(\mathit{\text{bird}},\mathit{\text{avialae}})$  $\left({\alpha }_2\right)$ ${\mathit{\text{has}}}_{-}\mathit{\text{bodypart}}(\mathit{\text{bird}},\mathit{\text{legs}})$

$\left({\alpha }_3\right)$ ${\mathit{\text{has}}}_{-}\mathit{\text{bodypart}}(\mathit{\text{bird}},\mathit{\text{torso}})$ $\left({\alpha }_4\right)$ ${\mathit{\text{has}}}_{-}\mathit{\text{bodypart}}(\mathit{\text{bird}},\mathit{\text{wings}})$

$\left({\alpha }_5\right)$ ${\mathit{\text{has}}}_{-}\mathit{\text{ability}}(\mathit{\text{bird}},\mathit{\text{walk}})$  $\left({\alpha }_6\right)$ ${\mathit{\text{has}}}_{-}\mathit{\text{ability}}(\mathit{\text{bird}},\mathit{\text{fly}})$

$\left({\alpha }_7\right)$ ${\mathit{\text{has}}}_{-}\mathit{\text{ability}}(\mathit{\text{bird}},{\mathit{\text{lay}}}_{-}\mathit{\text{eggs}})$

Facts of the horse characterisation:

$\left({\beta }_1\right)$ ${\mathit{\text{is}}}_{-}{\mathit{\text{of}}}_{-}\mathit{\text{clade}}(\mathit{\text{horse}},\mathit{\text{mammal}})$  $\left({\beta }_2\right)$ ${\mathit{\text{has}}}_{-}\mathit{\text{bodypart}}(\mathit{\text{horse}},\mathit{\text{legs}})$

$\left({\beta }_3\right)$ ${\mathit{\text{has}}}_{-}\mathit{\text{bodypart}}(\mathit{\text{horse}},\mathit{\text{torso}})$ $\left({\beta }_4\right)$ ${\mathit{\text{has}}}_{-}\mathit{\text{ability}}(\mathit{\text{horse}},\mathit{\text{walk}})$

Table 5. Abbreviated representation of the shared generalisation G based on the stereotypical characterisations for a horse and a bird, constituted by generalisations ${\alpha }_1={\phi }_S\left({\gamma }_1\right)$/${\beta }_1={\phi }_T\left({\gamma }_1\right)$, ${\alpha }_2={\phi }_S\left({\gamma }_2\right)$/${\beta }_2={\phi }_T\left({\gamma }_2\right)$, ${\alpha }_3={\phi }_S\left({\gamma }_3\right)$/${\beta }_3={\phi }_T\left({\gamma }_3\right)$, and ${\alpha }_5={\phi }_S\left({\gamma }_4\right)$/${\beta }_4={\phi }_T\left({\gamma }_4\right)$ (i.e. $S_c=\{{\alpha }_1,{\alpha }_2,{\alpha }_3,{\alpha }_5\}$ and $T_c=\{{\beta }_1,{\beta }_2,{\beta }_3,{\beta }_4\}$).

Entities:

C : $\mathit{\text{clade}}$, E : $\mathit{\text{entity}}$

Facts:

$\left({\gamma }_1\right)$ ${\mathit{\text{is}}}_{-}{\mathit{\text{of}}}_{-}\mathit{\text{clade}}(E,C)$  $\left({\gamma }_2\right)$ ${\mathit{\text{has}}}_{-}\mathit{\text{bodypart}}(E,\mathit{\text{legs}})$

$\left({\gamma }_3\right)$ ${\mathit{\text{has}}}_{-}\mathit{\text{bodypart}}(E,\mathit{\text{torso}})$  $\left({\gamma }_4\right)$ ${\mathit{\text{has}}}_{-}\mathit{\text{ability}}(E,\mathit{\text{walk}})$

Table 6. Abbreviated representation of the generalised source theory $S^{\prime }$ based on the stereotypical characterisations for a horse and a bird, including additional axioms ${\gamma }_5$, ${\gamma }_6$, and ${\gamma }_7$ obtained from generalising the remaining axioms from $S\setminus S_c=\{{\alpha }_4,{\alpha }_6,{\alpha }_7\}$.

Entities:

C : $\mathit{\text{clade}}$, E : $\mathit{\text{entity}}$

Facts:

$\left({\gamma }_1\right)$ ${\mathit{\text{is}}}_{-}{\mathit{\text{of}}}_{-}\mathit{\text{clade}}(E,C)$  $\left({\gamma }_2\right)$ ${\mathit{\text{has}}}_{-}\mathit{\text{bodypart}}(E,\mathit{\text{legs}})$

$\left({\gamma }_3\right)$ ${\mathit{\text{has}}}_{-}\mathit{\text{bodypart}}(E,\mathit{\text{torso}})$  $\left({\gamma }_4\right)$ ${\mathit{\text{has}}}_{-}\mathit{\text{ability}}(E,\mathit{\text{walk}})$

$\left({\gamma }_5\right)$ ${\mathit{\text{has}}}_{-}\mathit{\text{bodypart}}(E,\mathit{\text{wings}})$  $\left({\gamma }_6\right)$ ${\mathit{\text{has}}}_{-}\mathit{\text{ability}}(E,\mathit{\text{fly}})$

$\left({\gamma }_7\right)$ ${\mathit{\text{has}}}_{-}\mathit{\text{ability}}(E,{\mathit{\text{lay}}}_{-}\mathit{\text{eggs}})$

Table 7. Abbreviated representation of the proto-blend $T^{\prime }$ obtained from computing the asymmetric amalgam between $S^{\prime }$ and T.

Entities:

E : $\mathit{\text{entity}}$

Facts:

$\left({\delta }_1\right)$ ${\mathit{\text{is}}}_{-}{\mathit{\text{of}}}_{-}\mathit{\text{clade}}(\mathit{\text{horse}},\mathit{\text{mammal}})$  $\left({\delta }_2\right)$ ${\mathit{\text{has}}}_{-}\mathit{\text{bodypart}}(\mathit{\text{horse}},\mathit{\text{legs}})$

$\left({\delta }_3\right)$ ${\mathit{\text{has}}}_{-}\mathit{\text{bodypart}}(\mathit{\text{horse}},\mathit{\text{torso}})$  $\left({\delta }_4\right)$ ${\mathit{\text{has}}}_{-}\mathit{\text{ability}}(\mathit{\text{horse}},\mathit{\text{walk}})$

$\left({\delta }_5\right)$ ${\mathit{\text{has}}}_{-}\mathit{\text{bodypart}}(E,\mathit{\text{wings}})$ $\left({\delta }_6\right)$ ${\mathit{\text{has}}}_{-}\mathit{\text{ability}}(E,\mathit{\text{fly}})$

$\left({\delta }_7\right)$ ${\mathit{\text{has}}}_{-}\mathit{\text{ability}}(E,{\mathit{\text{lay}}}_{-}\mathit{\text{eggs}})$

Table 8. Abbreviated representation of $T_B={\phi }_T\left(T^{\prime }\right)$.

Facts:

$\left({\delta }_1\right)$ ${\mathit{\text{is}}}_{-}{\mathit{\text{of}}}_{-}\mathit{\text{clade}}(\mathit{\text{horse}},\mathit{\text{mammal}})$  $\left({\delta }_2\right)$ ${\mathit{\text{has}}}_{-}\mathit{\text{bodypart}}(\mathit{\text{horse}},\mathit{\text{legs}})$

$\left({\delta }_3\right)$ ${\mathit{\text{has}}}_{-}\mathit{\text{bodypart}}(\mathit{\text{horse}},\mathit{\text{torso}})$  $\left({\delta }_4\right)$ ${\mathit{\text{has}}}_{-}\mathit{\text{ability}}(\mathit{\text{horse}},\mathit{\text{walk}})$

$\left({\delta }_5\right)$ ${\mathit{\text{has}}}_{-}\mathit{\text{bodypart}}(\mathit{\text{horse}},\mathit{\text{wings}})$  $\left({\delta }_6\right)$ ${\mathit{\text{has}}}_{-}\mathit{\text{ability}}(\mathit{\text{horse}},\mathit{\text{fly}})$

$\left({\delta }_7\right)$ ${\mathit{\text{has}}}_{-}\mathit{\text{ability}}(\mathit{\text{horse}},{\mathit{\text{lay}}}_{-}\mathit{\text{eggs}})$

Table 9. Abbreviated representation of the final blended theory $T_B$ giving a characterisation of Pegasus after inconsistency check and repair (i.e. based on $S_{clash}=S\setminus \left\{{\alpha }_7\right\}$).

Facts:

$\left({\delta }_1\right)$ ${\mathit{\text{is}}}_{-}{\mathit{\text{of}}}_{-}\mathit{\text{clade}}(\mathit{\text{horse}},\mathit{\text{mammal}})$  $\left({\delta }_2\right)$ ${\mathit{\text{has}}}_{-}\mathit{\text{bodypart}}(\mathit{\text{horse}},\mathit{\text{legs}})$

$\left({\delta }_3\right)$ ${\mathit{\text{has}}}_{-}\mathit{\text{bodypart}}(\mathit{\text{horse}},\mathit{\text{torso}})$  $\left({\delta }_4\right)$ ${\mathit{\text{has}}}_{-}\mathit{\text{ability}}(\mathit{\text{horse}},\mathit{\text{walk}})$

$\left({\delta }_5\right)$ ${\mathit{\text{has}}}_{-}\mathit{\text{bodypart}}(\mathit{\text{horse}},\mathit{\text{wings}})$  $\left({\delta }_6\right)$ ${\mathit{\text{has}}}_{-}\mathit{\text{ability}}(\mathit{\text{horse}},\mathit{\text{fly}})$

Figure 9. A folding toothbrush like the one from the example in Section 4.5, characteristically featuring a hinge allowing the brush head to be folded back into the handle.

Table 10. Example formalisations of stereotypical characterisations for a pocketknife S and a toothbrush T.

Sorts:

entity, part, functionality

Entities:

toothbrush, pocketknife : entity  handle, brush_head, blade, hinge : part

brush, cut, fold : functionality

Predicates:

has_part : $\mathit{\text{entity}}\times \mathit{\text{part}}$,  has_functionality : $\mathit{\text{entity}}\times \mathit{\text{functionality}}$

Facts of the pocketknife characterisation:

$\left({\alpha }_1\right)$ ${\mathit{\text{has}}}_{-}\mathit{\text{part}}(\mathit{\text{pocketknife}},\mathit{\text{handle}})$   $\left({\alpha }_2\right)$ ${\mathit{\text{has}}}_{-}\mathit{\text{part}}(\mathit{\text{pocketknife}},\mathit{\text{blade}})$

$\left({\alpha }_3\right)$ ${\mathit{\text{has}}}_{-}\mathit{\text{functionality}}(\mathit{\text{pocketknife}},\mathit{\text{cut}})$   $\left({\alpha }_4\right)$ ${\mathit{\text{has}}}_{-}\mathit{\text{part}}(\mathit{\text{pocketknife}},\mathit{\text{hinge}})$

$\left({\alpha }_5\right)$ ${\mathit{\text{has}}}_{-}\mathit{\text{functionality}}(\mathit{\text{pocketknife}},\mathit{\text{fold}})$

Facts of the toothbrush characterisation:

$\left({\beta }_1\right)$ ${\mathit{\text{has}}}_{-}\mathit{\text{part}}(\mathit{\text{toothbrush}},\mathit{\text{handle}})$   $\left({\beta }_2\right)$ ${\mathit{\text{has}}}_{-}\mathit{\text{part}}(\mathit{\text{toothbrush}},{\mathit{\text{brush}}}_{-}\mathit{\text{head}})$

$\left({\beta }_3\right)$ ${\mathit{\text{has}}}_{-}\mathit{\text{functionality}}(\mathit{\text{toothbrush}},\mathit{\text{brush}})$

Table 11. Abbreviated representation of the shared generalisation G based on the stereotypical characterisations for a pocketknife and a toothbrush, constituted by generalisations ${\alpha }_1={\phi }_S\left({\gamma }_1\right)$/${\beta }_1={\phi }_T\left({\gamma }_1\right)$, ${\alpha }_2={\phi }_S\left({\gamma }_2\right)$/${\beta }_2={\phi }_T\left({\gamma }_2\right)$, and ${\alpha }_3={\phi }_S\left({\gamma }_3\right)$/${\beta }_3={\phi }_T\left({\gamma }_3\right)$ (i.e. $S_c=\{{\alpha }_1,{\alpha }_2,{\alpha }_3\}$ and $T_c=\{{\beta }_1,{\beta }_2,{\beta }_3,\}$).

Entities:

E : $\mathit{\text{entity}}$, P : $\mathit{\text{part}}$, F : $\mathit{\text{functionality}}$

Facts:

$\left({\gamma }_1\right)$ ${\mathit{\text{has}}}_{-}\mathit{\text{part}}(E,\mathit{\text{handle}})$  $\left({\gamma }_2\right)$ ${\mathit{\text{has}}}_{-}\mathit{\text{part}}(E,P)$  $\left({\gamma }_3\right)$ ${\mathit{\text{has}}}_{-}\mathit{\text{functionality}}(E,F)$

Table 12. Abbreviated representation of the generalised source theory $S^{\prime }$ based on the stereotypical characterisations for a toothbrush and a pocketknife, including additional axioms ${\gamma }_4$ and ${\gamma }_5$ obtained from generalising the remaining axioms from $S\setminus S_c=\{{\alpha }_4,{\alpha }_5\}$.

Entities:

E : $\mathit{\text{entity}}$, P : $\mathit{\text{part}}$, F : $\mathit{\text{functionality}}$

Facts:

$\left({\gamma }_1\right)$ ${\mathit{\text{has}}}_{-}\mathit{\text{part}}(E,\mathit{\text{handle}})$  $\left({\gamma }_2\right)$ ${\mathit{\text{has}}}_{-}\mathit{\text{part}}(E,P)$

$\left({\gamma }_3\right)$ ${\mathit{\text{has}}}_{-}\mathit{\text{functionality}}(E,F)$  $\left({\gamma }_4\right)$ ${\mathit{\text{has}}}_{-}\mathit{\text{part}}(E,\mathit{\text{hinge}})$

$\left({\gamma }_5\right)$ ${\mathit{\text{has}}}_{-}\mathit{\text{functionality}}(E,\mathit{\text{fold}})$

Table 13. Abbreviated representation of the proto-blend $T^{\prime }$ obtained from computing the asymmetric amalgam between $S^{\prime }$ and T.

Entities:

E : $\mathit{\text{entity}}$

Facts:

$\left({\delta }_1\right)$ ${\mathit{\text{has}}}_{-}\mathit{\text{part}}(\mathit{\text{toothbrush}},\mathit{\text{handle}})$  $\left({\delta }_2\right)$ ${\mathit{\text{has}}}_{-}\mathit{\text{part}}(\mathit{\text{toothbrush}},{\mathit{\text{brush}}}_{-}\mathit{\text{head}})$

$\left({\delta }_3\right)$ ${\mathit{\text{has}}}_{-}\mathit{\text{functionality}}(\mathit{\text{toothbrush}},\mathit{\text{brush}})$  $\left({\delta }_4\right)$ ${\mathit{\text{has}}}_{-}\mathit{\text{part}}(E,\mathit{\text{hinge}})$

$\left({\delta }_5\right)$ ${\mathit{\text{has}}}_{-}\mathit{\text{functionality}}(E,\mathit{\text{fold}})$

Table 14. Abbreviated representation of $T_B={\phi }_T\left(T^{\prime }\right)$.

Facts:

$\left({\delta }_1\right)$ ${\mathit{\text{has}}}_{-}\mathit{\text{part}}(\mathit{\text{toothbrush}},\mathit{\text{handle}})$  $\left({\delta }_2\right)$ ${\mathit{\text{has}}}_{-}\mathit{\text{part}}(\mathit{\text{toothbrush}},{\mathit{\text{brush}}}_{-}\mathit{\text{head}})$

$\left({\delta }_3\right)$ ${\mathit{\text{has}}}_{-}\mathit{\text{functionality}}(\mathit{\text{toothbrush}},\mathit{\text{brush}})$  $\left({\delta }_4\right)$ ${\mathit{\text{has}}}_{-}\mathit{\text{part}}(\mathit{\text{toothbrush}},\mathit{\text{hinge}})$

$\left({\delta }_5\right)$ ${\mathit{\text{has}}}_{-}\mathit{\text{functionality}}(\mathit{\text{toothbrush}},\mathit{\text{fold}})$

Schwering, A., Krumnack, U., Kühnberger, K.-U., & Gust, H. (2009). Syntactic principles of heuristic-driven theory projection. Journal of Cognitive Systems Research, 10(3), 251–269. doi: 10.1016/j.cogsys.2008.09.002

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Goguen, J. (2006). Mathematical models of cognitive space and time. Reasoning and cognition; proceedings of the interdisciplinary conference on reasoning and cognition (pp. 125–128). Tokyo: Keio University Press.

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