A Hybrid Reasoning Model for “Whole and Part” Cardinal Direction Relations

Table 2

(a) Composition of binary relations in the 𝑦-direction for the hybrid model (composed relations of {WN} and {WN,H}). (b) Composition of binary relations in the 𝑦-direction for the hybrid model (composed relations of {H} and {WN,H}). (c) Composition of binary relations in the 𝑦-direction for the hybrid model (composed relations of {S} and {WN,H}). (d) Composition of binary relations in the 𝑦-direction for the hybrid model (composed relations of {WN,H,S} and {WS}).

(a)  

Composed relations of  {WN} and {WN,H}		WN(c,b )		H(c,b)
		NTPPy(c,WN(b))	TPPy(c,WN(b)); ∧ ECy(c,H(b))	TPPy(c,H(b)); ∧ ECy(c,WN(b))	TPPy(c,H(b)); ∧ ECy(c,WS(b))	NTPPy(c,H(b))	EQy(c,H(b))
	NTPPy(b,WN(a))	NTPPy(c,WN(a))	NTPPy(c,WN(a))	NTPPy(c,WN(a))	NTPPy(c,WN(a))	NTPPy(c,WN(a))	NTPPy(c,WN(c))
WN(𝑏,𝑎)	TPPy(b,WN(a)); ∧ ECy(b,H(a))	NTPPy(c,WN(a))	NTPPy(c,WN(a))	NTPPy(c,WN(a))	TPPy(c,WN(a)); ∧ ECy(c,H(a))	NTPPy(c,WN(a))	TPPy(c,WN(a)); ∧ ECy(c,H(a))
	WN(b,a)	NTPPy(c,WN(a))		WN(c,a)

Note: the data in bold are results of the composition of relations.

(b)  

Composed relations of  {H} and {WN,H}		WN(c,b)		H(c,b)
		NTPPy(c,WN(b))	TPPy(c,WN(b)); ∧ ECy(c,H(b))	TPPy(c,H(b)); ∧ ECy(c,WN(b))	TPPy(c,H(b)); ∧ ECy(c,WS(b))	NTPPy(c,H(b))	EQy(c,H(b))
H(b,a)	TPPy(b,H(a)); ∧ ECy(b,WN(a))	NTPPy(c,WN(a))	TPPy(c,WN(a)); ∧ ECy(c,H(a))	TPPy(c,H(a)); ∧ ECy(c,WN(a))	NTPPy(c,H(a))	NTPPy(c,H(a))	TPPy(c,H(a)); ∧ ECy(c,WN(a))
	TPPy(b,H(a)); ∧ ECy(b,WS(a))	[ NTPPy(c,H(a)) ]  ∨; [ TPPy(c,H(a)); ∧ ECy(c,WN(a)) ]  ∨; [ TPPy(c,WN(a)); ∧ ECy(c,H(a)) ]  ∨; [ NTPPy(c,WN(a)) ]  ∨; [ POy(c,WN(a)); ∧ POy(c,H(a)); ∧ DCy(c,WS(a)) ]	[ NTPPy(c,H(a)) ]  ∨; [ PPy(c,H(a)); ∧ ECy(c,WN(a)) ]  ∨; [ POy(c,WN(a)); ∧ POy(c,H(a)); ∧ DCy(c,WS(a)) ]	NTPPy(c,H(a))	TPPy(c,H(a)); ∧ ECy(c,WN(a))	NTPPy(c,H(a))	TPPy(c,H(a)); ∧ ECy(c,WN(a))
	NTPPy(b,H(a))	[ NTPPy(c,H(a)) ]  ∨; [ TPPy(c,H(a)); ∧ ECy(c,WN(a)) ]  ∨; [ TPPy(c,WN(a)); ∧ ECy(c,H(a)) ]  ∨; [ NTPPy(c,WN(a)) ]  ∨; [ POy(c,WN(a)); ∧ POy(c,H(a)); ∧ DCy(c,WS(a)) ]	[ NTPPy(c,H(a)) ]  ∨; [ TPPy(c,H(a)); ∧ ECy(c,WN(a)) ]  ∨; [ POy(c,WN(a)); ∧ POy(c,H(a)); ∧ DCy(c,WS(a)) ]	NTPPy(c,H(a))	NTPPy(c,H(a))	NTPPy(c,H(a))	NTPPy(c,H(a))
	EQy(b,H(a))	NTPPy(c,WN(a))	TPPy(c,WN(a)); ∧ ECy(c,H(a))	TPPy(c,H(a)); ∧ ECy(c,WN(a))	TPPy(c,H(a)); ∧ ECy(c,WS(a))	NTPPy(c,H(a))	EQy(c,H(a))
	H(b,a)	WN(c,a) ∨ H(c,a)		H(c,a)

Note: the data in bold are the results of the composition of relations.

(c)  

Composed relations of  {S} and {WN,H}		WN(c,b)		H(c,b)
		NTPPy(c,WN(b))	TPPy(c,WN(b)); ∧ ECy(c,H(b))	TPPy(c,H(b)); ∧ ECy(c,WN(b))	TPPy(c,H(b)); ∧ ECy(c,WS(b))	NTPPy(c,H(b))	EQy(c,H(b))
S(b,a)	NTPPy(b,WS(𝑎))	U-13 relations	[ NTPPy(c,WS(a)) ]  ∨; [ TPPy(c,WS(a)); ∧ ECy(c,H(a)) ]  ∨; [ POy(c,WS(a)); ∧ POy(c,H(a)); ∧ DCy(c,WN(a)) ]  ∨; [ POy(c,WS(a)); ∧ POy(c,H(a)); ∧ ECy(c,WN(a)) ]  ∨; [ POy(c,WS(a)); ∧ POy(c,WN(a)); ∧ NTPPiy(c,H(a)) ]	NTPPy(c,WS(a))	NTPPy(c,WS(a))	NTPPy(c,WS(a))	NTPPy(c,WS(a))
	TPPy(b,WS(a)); ∧ ECy(b,H(a))	[ NTPPy(c,H(a)) ]  ∨; [ TPPy(c,H(a)); ∧ ECy(c,WN(a)) ]  ∨; [ TPPy(c,WN(a)); ∧ ECy(c,H(a)) ]  ∨; [ NTPPy(c,WN(a))]; ∨  [ POy(c,WN(a)); ∧ POy(c,H(a)); ∧ DCy(c,WS(a)) ]	[ TPPy(c,H(a)); ∧ ECy(c,WS(a)) ]  ∨; [ EQy(c,H(a)) ]  ∨; [ POy(c,WN(a)); ∧ POy(c,H(a)); ∧ ECy(c,WS(a)) ]	TPPy(c,WS(a)); ∧ ECy(c,H(a))	NTPPy(c,WS(a))	NTPPy(c,WS(a))	TPPy(c,WS(a)); ∧ ECy(c,H(a))
	S(b,a)	WN(c,a) ∨ H(c,a) ∨ WS(c,a)		WS(c,a)

Note: the data in bold are the results of the composition of relations.

(d)  

Composed relations of  {WN,H,S} and {WS}		WS(c,b)
WN(b,a)	NTPPy(b,WN(a))	NTPPy(c,WS(b))	TPPy(c,WS(b)) ∧ ECy(c,H(b))
	TPPy(b,WN(a)) ∧ ECy(b,H(a))	U-13 relations	[ NTPPy(c,WN(a)) ]  ∨; [ TPPy(c,WN(a)) ∧ ECy(c,H(a)) ]  ∨; [ POy(c,WN(a)) ∧ POy(c,H(a)) ∧ DCy(c,WS(a)) ]  ∨; [ POy(c,WN(a)) ∧ POy(c,H(a)) ∧ ECy(c,WS(a)) ]  ∨; [ POy(c,WN(a)) ∧ POy(c,WS(a)) ∧ NTPPiy(c,H(a)) ]
	WN(b,a)	WN(c,a) ∨ H(c,a) ∨ WS(c,a)
H(b,a)	TPPy(b,H(a)) ∧ ECy(b,WN(a))	TPPy(c,H(a)) ∧ ECy(c,WN(a))	[ NTPPy(c,H(a)) ]  ∨; [ TPPy(c,H(a)) ∧ ECy(c,WS(a)) ]  ∨; [ TPPy(c,WS(a)) ∧ ECy(c,H(a)) ]  ∨; [ NTPPy(c,WS(a)) ]  ∨; [ POy(c,WS(a)) ∧ POy(c,H(a)) ∧ DCy(c,WN(a)) ]
	TPPy(b,H(a)) ∧ ECy(b,WS(a))	TPPy(c,H(a)) ∧ ECy(c,WN(a))	NTPPy(c,WS(a))
	NTPPy(b,H(a))	[ NTPPy(c,H(a))] ∨; [ TPPy(c,H(a)) ∧ ECy(c,WS(a)) ]  ∨; [ TPPy(c,WS(a)) ∧ ECy(c,H(a)) ]  ∨; [ NTPPy(c,WS(a)) ]  ∨; [ POy(c,WS(a)) ∧ POy(c,H(a)) ∧ DCy(c,WN(a)) ]	[ NTPPy(c,H(a)) ]  ∨; [ TPPy(c,H(a)) ∧ ECy(c,WS(a)) ]  ∨; [ POy(c,WS(a)) ∧ POy(c,H(a)) ∧ DCy(c,WN(a)) ]
	EQy(b,H(a))	NTPPy(c,WS(a))	TPPy(c,WS(a)) ∧ ECy(c,H(a))
	H(b,a)	H(c,a) ∨ WS(c,a)
S(b,a)	NTPPy(b,WS(a))	NTPPy(c,WS(a))	NTPPy(c,WS(a))
	TPPy(b,WS(a)) ∧ ECy(b,H(a))	NTPPy(c,WS(a))	NTPPy(c,WS(a))
	S(b,a)	NTPPy(c,WS(a))

Note: the data in bold are the results of the composition of relations.