A (Hybrid) Logical Approach to Frame...
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A (Hybrid) Logical Approach to Frame Semantics
Sylvain Pogodalla1
(joint work with Laura Kallmeyer2 and Rainer Osswald2)
1LORIA–INRIA, Nancy, France2Heinrich Heine Universtitat, Dusseldorf, Germany
October 9–10, 2019, Gothenburg
S. Pogodalla (with L. Kallmeyer and R. Osswald) A (Hybrid) Logical Approach to Frame Semantics Oct. 9–10, 2019 1 / 30
Outline
1 Frame SemanticsThe Frame HypothesisFrames and Logic
2 Hybrid LogicReminder on Modal LogicHybrid Logic
3 Compositional Frame SemanticsQuantificationFrame Decomposition
S. Pogodalla (with L. Kallmeyer and R. Osswald) A (Hybrid) Logical Approach to Frame Semantics Oct. 9–10, 2019 2 / 30
Frame Semantics The Frame Hypothesis
Frame Semantics
Frame
A (cognitive or linguistic) structure that represents a situation.
A frame is a data-structure representing a stereotyped situation (Minsky 1974, p.1)
I thought of each case frame as characterizing a small abstract ‘scene’ or ‘situation’,so that to understand the semantic structure of the verb it was necessary tounderstand the properties of such schematized scenes (Fillmore 1982, p.115)
I propose that frames provide the fundamental representation of knowledge inhuman cognition (Barsalou 1992)
S. Pogodalla (with L. Kallmeyer and R. Osswald) A (Hybrid) Logical Approach to Frame Semantics Oct. 9–10, 2019 3 / 30
Frame Semantics The Frame Hypothesis
Frame Semantics
Frame
A (cognitive or linguistic) structure that represents a situation.
A frame is a data-structure representing a stereotyped situation (Minsky 1974, p.1)
I thought of each case frame as characterizing a small abstract ‘scene’ or ‘situation’,so that to understand the semantic structure of the verb it was necessary tounderstand the properties of such schematized scenes (Fillmore 1982, p.115)
I propose that frames provide the fundamental representation of knowledge inhuman cognition (Barsalou 1992)
S. Pogodalla (with L. Kallmeyer and R. Osswald) A (Hybrid) Logical Approach to Frame Semantics Oct. 9–10, 2019 3 / 30
Frame Semantics The Frame Hypothesis
Frame Semantics
Frame
A (cognitive or linguistic) structure that represents a situation.
A frame is a data-structure representing a stereotyped situation (Minsky 1974, p.1)
I thought of each case frame as characterizing a small abstract ‘scene’ or ‘situation’,so that to understand the semantic structure of the verb it was necessary tounderstand the properties of such schematized scenes (Fillmore 1982, p.115)
I propose that frames provide the fundamental representation of knowledge inhuman cognition (Barsalou 1992)
S. Pogodalla (with L. Kallmeyer and R. Osswald) A (Hybrid) Logical Approach to Frame Semantics Oct. 9–10, 2019 3 / 30
Frame Semantics The Frame Hypothesis
Frame Semantics
Frame
A (cognitive or linguistic) structure that represents a situation.
A frame is a data-structure representing a stereotyped situation (Minsky 1974, p.1)
I thought of each case frame as characterizing a small abstract ‘scene’ or ‘situation’,so that to understand the semantic structure of the verb it was necessary tounderstand the properties of such schematized scenes (Fillmore 1982, p.115)
I propose that frames provide the fundamental representation of knowledge inhuman cognition (Barsalou 1992)
S. Pogodalla (with L. Kallmeyer and R. Osswald) A (Hybrid) Logical Approach to Frame Semantics Oct. 9–10, 2019 3 / 30
Frame Semantics The Frame Hypothesis
The Frame Hypothesis (Lobner 2014)
H1 The human cognitive system operates with a single general format ofrepresentations
H2 If the human cognitive system operates with one general format ofrepresentations, this format is essentially Barsalou’s frames
Representation as frames in the human mind of:§ lexical linguistic expressions and their meanings§ complex linguistic expressions and their meanings
Verb meaning frames go beyond “case frames”. For instance§ Aspectual characteristics of the situation§ Structured relations between semantic arguments
Decompositional approach to meaning (Osswald and Van Valin 2014)
S. Pogodalla (with L. Kallmeyer and R. Osswald) A (Hybrid) Logical Approach to Frame Semantics Oct. 9–10, 2019 4 / 30
Frame Semantics The Frame Hypothesis
The Frame Hypothesis (Lobner 2014)
H1 The human cognitive system operates with a single general format ofrepresentations
H2 If the human cognitive system operates with one general format ofrepresentations, this format is essentially Barsalou’s frames
Representation as frames in the human mind of:§ lexical linguistic expressions and their meanings§ complex linguistic expressions and their meanings
Verb meaning frames go beyond “case frames”. For instance§ Aspectual characteristics of the situation§ Structured relations between semantic arguments
Decompositional approach to meaning (Osswald and Van Valin 2014)
S. Pogodalla (with L. Kallmeyer and R. Osswald) A (Hybrid) Logical Approach to Frame Semantics Oct. 9–10, 2019 4 / 30
Frame Semantics The Frame Hypothesis
The Frame Hypothesis (Lobner 2014)
H1 The human cognitive system operates with a single general format ofrepresentations
H2 If the human cognitive system operates with one general format ofrepresentations, this format is essentially Barsalou’s frames
Representation as frames in the human mind of:§ lexical linguistic expressions and their meanings§ complex linguistic expressions and their meanings
Verb meaning frames go beyond “case frames”. For instance§ Aspectual characteristics of the situation§ Structured relations between semantic arguments
Decompositional approach to meaning (Osswald and Van Valin 2014)
S. Pogodalla (with L. Kallmeyer and R. Osswald) A (Hybrid) Logical Approach to Frame Semantics Oct. 9–10, 2019 4 / 30
Frame Semantics The Frame Hypothesis
The Frame Hypothesis (Lobner 2014)
H1 The human cognitive system operates with a single general format ofrepresentations
H2 If the human cognitive system operates with one general format ofrepresentations, this format is essentially Barsalou’s frames
Representation as frames in the human mind of:§ lexical linguistic expressions and their meanings§ complex linguistic expressions and their meanings
Verb meaning frames go beyond “case frames”. For instance§ Aspectual characteristics of the situation§ Structured relations between semantic arguments
Decompositional approach to meaning (Osswald and Van Valin 2014)
S. Pogodalla (with L. Kallmeyer and R. Osswald) A (Hybrid) Logical Approach to Frame Semantics Oct. 9–10, 2019 4 / 30
Frame Semantics The Frame Hypothesis
Frame ExampleThe man walked into the house
l0
motionl1
man
path
walking
l2
house
agent
mover
pathmanner
endp at-regionpart-of
goal
S. Pogodalla (with L. Kallmeyer and R. Osswald) A (Hybrid) Logical Approach to Frame Semantics Oct. 9–10, 2019 5 / 30
Frame Semantics Frames and Logic
Formal Representation of FramesFrames as base-labelled feature structures with types and relations (Kallmeyer and Osswald 2013)
l0
motionl1
man
path
walking
l2
house
agent
mover
pathmanner
endp at-regionpart-of
goal
»
—
—
—
—
—
—
—
—
—
—
—
—
—
—
—
–
motion
agent 1
”
manı
mover 1
goal 2
«
house
at-region w
ff
path
«
path
endp v
ff
manner”
walkingı
fi
ffi
ffi
ffi
ffi
ffi
ffi
ffi
ffi
ffi
ffi
ffi
ffi
ffi
ffi
ffi
fl
part-of ( v , w )
Labelled attribute-value description (LAVD) language
0 : motion ^0 ¨ agent fi 1 ^0 ¨ agent “ 0 ¨ mover ^0 ¨ goal fi 2 ^0 ¨ path : path^0 ¨ manner : walking ^ 1 : man2 : house ^x 0 ¨ path ¨ endp, 2 ¨ at-regiony : part-of
S. Pogodalla (with L. Kallmeyer and R. Osswald) A (Hybrid) Logical Approach to Frame Semantics Oct. 9–10, 2019 6 / 30
Frame Semantics Frames and Logic
Formal Representation of FramesFrames as base-labelled feature structures with types and relations (Kallmeyer and Osswald 2013)
l0
motionl1
man
path
walking
l2
house
agent
mover
pathmanner
endp at-regionpart-of
goal
»
—
—
—
—
—
—
—
—
—
—
—
—
—
—
—
–
motion
agent 1
”
manı
mover 1
goal 2
«
house
at-region w
ff
path
«
path
endp v
ff
manner”
walkingı
fi
ffi
ffi
ffi
ffi
ffi
ffi
ffi
ffi
ffi
ffi
ffi
ffi
ffi
ffi
ffi
fl
part-of ( v , w )
Labelled attribute-value description (LAVD) language
0 : motion ^0 ¨ agent fi 1 ^0 ¨ agent “ 0 ¨ mover ^0 ¨ goal fi 2 ^0 ¨ path : path^0 ¨ manner : walking ^ 1 : man2 : house ^x 0 ¨ path ¨ endp, 2 ¨ at-regiony : part-of
S. Pogodalla (with L. Kallmeyer and R. Osswald) A (Hybrid) Logical Approach to Frame Semantics Oct. 9–10, 2019 6 / 30
Frame Semantics Frames and Logic
Formal Representation of FramesFrames as base-labelled feature structures with types and relations (Kallmeyer and Osswald 2013)
l0
motionl1
man
path
walking
l2
house
agent
mover
pathmanner
endp at-regionpart-of
goal
»
—
—
—
—
—
—
—
—
—
—
—
—
—
—
—
–
motion
agent 1
”
manı
mover 1
goal 2
«
house
at-region w
ff
path
«
path
endp v
ff
manner”
walkingı
fi
ffi
ffi
ffi
ffi
ffi
ffi
ffi
ffi
ffi
ffi
ffi
ffi
ffi
ffi
ffi
fl
part-of ( v , w )
Labelled attribute-value description (LAVD) language
0 : motion ^0 ¨ agent fi 1 ^0 ¨ agent “ 0 ¨ mover ^0 ¨ goal fi 2 ^0 ¨ path : path^0 ¨ manner : walking ^ 1 : man2 : house ^x 0 ¨ path ¨ endp, 2 ¨ at-regiony : part-of
S. Pogodalla (with L. Kallmeyer and R. Osswald) A (Hybrid) Logical Approach to Frame Semantics Oct. 9–10, 2019 6 / 30
Frame Semantics Frames and Logic
Base-Labelled Feature Structures and LAVD LanguageProperties:
A frame is represented as a base-labelled feature structure
It is specified using the LAVD language
It is the most general base-labelled feature structures that satisfy the given LAVDformula (minimal first-order model)
Suitable to frame decomposition and composition
Variable free
We take the semantic structures associated with the syntactic structures as genuinesemantic representations, not as some kind of yet to be interpreted logicalexpressions (Kallmeyer and Osswald 2013)
However:
No means for explicit quantification
Referential entities are treated as definites (reflected by the naming of nodes)
Similar to other “framish” formalisms, e.g., AMR, dependency structures. . .
Proposal
Frames are considered as relational models Ñ modal logic
Feature structures specification require the hybrid logic language extension
Use hybrid logic binders to quantify
S. Pogodalla (with L. Kallmeyer and R. Osswald) A (Hybrid) Logical Approach to Frame Semantics Oct. 9–10, 2019 7 / 30
Frame Semantics Frames and Logic
Base-Labelled Feature Structures and LAVD LanguageProperties:
A frame is represented as a base-labelled feature structure
It is specified using the LAVD language
It is the most general base-labelled feature structures that satisfy the given LAVDformula (minimal first-order model)
Suitable to frame decomposition and composition
Variable free
We take the semantic structures associated with the syntactic structures as genuinesemantic representations, not as some kind of yet to be interpreted logicalexpressions (Kallmeyer and Osswald 2013)
However:
No means for explicit quantification
Referential entities are treated as definites (reflected by the naming of nodes)
Similar to other “framish” formalisms, e.g., AMR, dependency structures. . .
Proposal
Frames are considered as relational models Ñ modal logic
Feature structures specification require the hybrid logic language extension
Use hybrid logic binders to quantify
S. Pogodalla (with L. Kallmeyer and R. Osswald) A (Hybrid) Logical Approach to Frame Semantics Oct. 9–10, 2019 7 / 30
Frame Semantics Frames and Logic
Base-Labelled Feature Structures and LAVD LanguageProperties:
A frame is represented as a base-labelled feature structure
It is specified using the LAVD language
It is the most general base-labelled feature structures that satisfy the given LAVDformula (minimal first-order model)
Suitable to frame decomposition and composition
Variable free
We take the semantic structures associated with the syntactic structures as genuinesemantic representations, not as some kind of yet to be interpreted logicalexpressions (Kallmeyer and Osswald 2013)
However:
No means for explicit quantification
Referential entities are treated as definites (reflected by the naming of nodes)
Similar to other “framish” formalisms, e.g., AMR, dependency structures. . .
Proposal
Frames are considered as relational models Ñ modal logic
Feature structures specification require the hybrid logic language extension
Use hybrid logic binders to quantify
S. Pogodalla (with L. Kallmeyer and R. Osswald) A (Hybrid) Logical Approach to Frame Semantics Oct. 9–10, 2019 7 / 30
Hybrid Logic Reminder on Modal Logic
Modal Logic (Blackburn 1993; Areces and ten Cate 2007)Formulas
Rel “ FuncY PropRel is a set of functional and non-functional relational symbols,
Prop a set of propositional variables
Nom a set of nominals (node names),
Svar a set of state variables
Stat “ NomY Svar.
Definition (Formulas)
The language of formulas Forms is defined as:
Forms ::“ J | p | φ | φ1 ^ φ2 | xRyφ | s | @sφ | Óx .φ | Eφ | Dx .φ
where p P Prop, φ, φ1, φ2 P Forms, R P Rel, s P Stat, x P Svar.
Moreover, we define:
φñ ψ ” φ_ ψ
rRsφ ” xRy φ
A
φ ” E φ
S. Pogodalla (with L. Kallmeyer and R. Osswald) A (Hybrid) Logical Approach to Frame Semantics Oct. 9–10, 2019 8 / 30
Hybrid Logic Reminder on Modal Logic
Modal Logic (Blackburn 1993; Areces and ten Cate 2007)Formulas
Rel “ FuncY PropRel is a set of functional and non-functional relational symbols,
Prop a set of propositional variables
Nom a set of nominals (node names),
Svar a set of state variables
Stat “ NomY Svar.
Definition (Formulas)
The language of formulas Forms is defined as:
Forms ::“ J | p | φ | φ1 ^ φ2 | xRyφ | s | @sφ | Óx .φ | Eφ | Dx .φ
where p P Prop, φ, φ1, φ2 P Forms, R P Rel, s P Stat, x P Svar.
Moreover, we define:
φñ ψ ” φ_ ψ
rRsφ ” xRy φ
A
φ ” E φ
S. Pogodalla (with L. Kallmeyer and R. Osswald) A (Hybrid) Logical Approach to Frame Semantics Oct. 9–10, 2019 8 / 30
Hybrid Logic Reminder on Modal Logic
Modal Logic (Blackburn 1993; Areces and ten Cate 2007)Formulas
Rel “ FuncY PropRel is a set of functional and non-functional relational symbols,
Prop a set of propositional variables
Nom a set of nominals (node names),
Svar a set of state variables
Stat “ NomY Svar.
Definition (Formulas)
The language of formulas Forms is defined as:
Forms ::“ J | p | φ | φ1 ^ φ2 | xRyφ | s | @sφ | Óx .φ | Eφ | Dx .φ
where p P Prop, φ, φ1, φ2 P Forms, R P Rel, s P Stat, x P Svar.
Moreover, we define:
φñ ψ ” φ_ ψ
rRsφ ” xRy φ
A
φ ” E φ
S. Pogodalla (with L. Kallmeyer and R. Osswald) A (Hybrid) Logical Approach to Frame Semantics Oct. 9–10, 2019 8 / 30
Hybrid Logic Reminder on Modal Logic
Modal Logic (Blackburn 1993; Areces and ten Cate 2007)Formulas
Rel “ FuncY PropRel is a set of functional and non-functional relational symbols,
Prop a set of propositional variables
Nom a set of nominals (node names),
Svar a set of state variables
Stat “ NomY Svar.
Definition (Formulas)
The language of formulas Forms is defined as:
Forms ::“ J | p | φ | φ1 ^ φ2 | xRyφ | s | @sφ | Óx .φ | Eφ | Dx .φ
where p P Prop, φ, φ1, φ2 P Forms, R P Rel, s P Stat, x P Svar.
Moreover, we define:
φñ ψ ” φ_ ψ
rRsφ ” xRy φ
A
φ ” E φ
S. Pogodalla (with L. Kallmeyer and R. Osswald) A (Hybrid) Logical Approach to Frame Semantics Oct. 9–10, 2019 8 / 30
Hybrid Logic Reminder on Modal Logic
Modal Logic (Blackburn 1993; Areces and ten Cate 2007)Formulas
Rel “ FuncY PropRel is a set of functional and non-functional relational symbols,
Prop a set of propositional variables
Nom a set of nominals (node names),
Svar a set of state variables
Stat “ NomY Svar.
Definition (Formulas)
The language of formulas Forms is defined as:
Forms ::“ J
| p | φ | φ1 ^ φ2 | xRyφ | s | @sφ | Óx .φ | Eφ | Dx .φ
where p P Prop, φ, φ1, φ2 P Forms, R P Rel, s P Stat, x P Svar.
Moreover, we define:
φñ ψ ” φ_ ψ
rRsφ ” xRy φ
A
φ ” E φ
S. Pogodalla (with L. Kallmeyer and R. Osswald) A (Hybrid) Logical Approach to Frame Semantics Oct. 9–10, 2019 8 / 30
Hybrid Logic Reminder on Modal Logic
Modal Logic (Blackburn 1993; Areces and ten Cate 2007)Formulas
Rel “ FuncY PropRel is a set of functional and non-functional relational symbols,
Prop a set of propositional variables
Nom a set of nominals (node names),
Svar a set of state variables
Stat “ NomY Svar.
Definition (Formulas)
The language of formulas Forms is defined as:
Forms ::“ J | p
| φ | φ1 ^ φ2 | xRyφ | s | @sφ | Óx .φ | Eφ | Dx .φ
where p P Prop
, φ, φ1, φ2 P Forms, R P Rel, s P Stat, x P Svar.
Moreover, we define:
φñ ψ ” φ_ ψ
rRsφ ” xRy φ
A
φ ” E φ
S. Pogodalla (with L. Kallmeyer and R. Osswald) A (Hybrid) Logical Approach to Frame Semantics Oct. 9–10, 2019 8 / 30
Hybrid Logic Reminder on Modal Logic
Modal Logic (Blackburn 1993; Areces and ten Cate 2007)Formulas
Rel “ FuncY PropRel is a set of functional and non-functional relational symbols,
Prop a set of propositional variables
Nom a set of nominals (node names),
Svar a set of state variables
Stat “ NomY Svar.
Definition (Formulas)
The language of formulas Forms is defined as:
Forms ::“ J | p | φ | φ1 ^ φ2
| xRyφ | s | @sφ | Óx .φ | Eφ | Dx .φ
where p P Prop, φ, φ1, φ2 P Forms
, R P Rel, s P Stat, x P Svar.
Moreover, we define:
φñ ψ ” φ_ ψ
rRsφ ” xRy φ
A
φ ” E φ
S. Pogodalla (with L. Kallmeyer and R. Osswald) A (Hybrid) Logical Approach to Frame Semantics Oct. 9–10, 2019 8 / 30
Hybrid Logic Reminder on Modal Logic
Modal Logic (Blackburn 1993; Areces and ten Cate 2007)Formulas
Rel “ FuncY PropRel is a set of functional and non-functional relational symbols,
Prop a set of propositional variables
Nom a set of nominals (node names),
Svar a set of state variables
Stat “ NomY Svar.
Definition (Formulas)
The language of formulas Forms is defined as:
Forms ::“ J | p | φ | φ1 ^ φ2 | xRyφ
| s | @sφ | Óx .φ | Eφ | Dx .φ
where p P Prop, φ, φ1, φ2 P Forms, R P Rel
, s P Stat, x P Svar.
Moreover, we define:
φñ ψ ” φ_ ψ
rRsφ ” xRy φ
A
φ ” E φ
S. Pogodalla (with L. Kallmeyer and R. Osswald) A (Hybrid) Logical Approach to Frame Semantics Oct. 9–10, 2019 8 / 30
Hybrid Logic Reminder on Modal Logic
Modal Logic (Blackburn 1993; Areces and ten Cate 2007)Formulas
Rel “ FuncY PropRel is a set of functional and non-functional relational symbols,
Prop a set of propositional variables
Nom a set of nominals (node names),
Svar a set of state variables
Stat “ NomY Svar.
Definition (Formulas)
The language of formulas Forms is defined as:
Forms ::“ J | p | φ | φ1 ^ φ2 | xRyφ
| s | @sφ | Óx .φ | Eφ | Dx .φ
where p P Prop, φ, φ1, φ2 P Forms, R P Rel
, s P Stat, x P Svar.
Moreover, we define:
φñ ψ ” φ_ ψ
rRsφ ” xRy φ
A
φ ” E φ
S. Pogodalla (with L. Kallmeyer and R. Osswald) A (Hybrid) Logical Approach to Frame Semantics Oct. 9–10, 2019 8 / 30
Hybrid Logic Reminder on Modal Logic
Modal Logic (Blackburn 1993; Areces and ten Cate 2007)Formulas
Rel “ FuncY PropRel is a set of functional and non-functional relational symbols,
Prop a set of propositional variables
Nom a set of nominals (node names),
Svar a set of state variables
Stat “ NomY Svar.
Definition (Formulas)
The language of formulas Forms is defined as:
Forms ::“ J | p | φ | φ1 ^ φ2 | xRyφ
| s | @sφ | Óx .φ | Eφ | Dx .φ
where p P Prop, φ, φ1, φ2 P Forms, R P Rel
, s P Stat, x P Svar.
Moreover, we define:
φñ ψ ” φ_ ψ
rRsφ ” xRy φ
A
φ ” E φ
S. Pogodalla (with L. Kallmeyer and R. Osswald) A (Hybrid) Logical Approach to Frame Semantics Oct. 9–10, 2019 8 / 30
Hybrid Logic Reminder on Modal Logic
Modal Logic (Blackburn 1993; Areces and ten Cate 2007)Models
Definition (Model)
A model M is a triple xM, pRMqRPRel,V y such that:
M is a non-empty set,
each RM is a binary relation on M,
V : Prop
Y Nom
ÝÑ ℘pMq is a valuation
such that if i P Nom then V piq is asingleton.
An assignment g is a mapping g : Svar ÝÑ M.g xm is an assignment that differs from g at most on x and such that g x
mpxq “ m.For s P Stat, we also define rssM,g to be the only m such that V psq “ tmu if s P Nomand rssM,g
“ gpsq if s P Svar.
S. Pogodalla (with L. Kallmeyer and R. Osswald) A (Hybrid) Logical Approach to Frame Semantics Oct. 9–10, 2019 9 / 30
Hybrid Logic Reminder on Modal Logic
Modal Logic (Blackburn 1993; Areces and ten Cate 2007)Models
Definition (Model)
A model M is a triple xM, pRMqRPRel,V y such that:
M is a non-empty set,
each RM is a binary relation on M,
V : Prop
Y Nom
ÝÑ ℘pMq is a valuation
such that if i P Nom then V piq is asingleton.
An assignment g is a mapping g : Svar ÝÑ M.g xm is an assignment that differs from g at most on x and such that g x
mpxq “ m.For s P Stat, we also define rssM,g to be the only m such that V psq “ tmu if s P Nomand rssM,g
“ gpsq if s P Svar.
S. Pogodalla (with L. Kallmeyer and R. Osswald) A (Hybrid) Logical Approach to Frame Semantics Oct. 9–10, 2019 9 / 30
Hybrid Logic Reminder on Modal Logic
Modal Logic (Blackburn 1993; Areces and ten Cate 2007)Models
Definition (Model)
A model M is a triple xM, pRMqRPRel,V y such that:
M is a non-empty set,
each RM is a binary relation on M,
V : Prop
Y Nom
ÝÑ ℘pMq is a valuation
such that if i P Nom then V piq is asingleton.
An assignment g is a mapping g : Svar ÝÑ M.g xm is an assignment that differs from g at most on x and such that g x
mpxq “ m.For s P Stat, we also define rssM,g to be the only m such that V psq “ tmu if s P Nomand rssM,g
“ gpsq if s P Svar.
S. Pogodalla (with L. Kallmeyer and R. Osswald) A (Hybrid) Logical Approach to Frame Semantics Oct. 9–10, 2019 9 / 30
Hybrid Logic Reminder on Modal Logic
Modal Logic (Blackburn 1993; Areces and ten Cate 2007)Models
Definition (Model)
A model M is a triple xM, pRMqRPRel,V y such that:
M is a non-empty set,
each RM is a binary relation on M,
V : Prop
Y Nom
ÝÑ ℘pMq is a valuation
such that if i P Nom then V piq is asingleton.
An assignment g is a mapping g : Svar ÝÑ M.g xm is an assignment that differs from g at most on x and such that g x
mpxq “ m.For s P Stat, we also define rssM,g to be the only m such that V psq “ tmu if s P Nomand rssM,g
“ gpsq if s P Svar.
S. Pogodalla (with L. Kallmeyer and R. Osswald) A (Hybrid) Logical Approach to Frame Semantics Oct. 9–10, 2019 9 / 30
Hybrid Logic Reminder on Modal Logic
Modal Logic (Blackburn 1993; Areces and ten Cate 2007)Satisfaction Relation
Definition (Satisfaction relation)
Let M be a model, w P M
, and g an assignment for M.
The satisfaction relation ( is defined as follows:
M,
g ,
w ( J
M,
g ,
w ( p iff w P V ppq for p P PropM,
g ,
w ( φ iff M,
g ,
w * φM,
g ,
w ( φ1 ^ φ2 iff M,
g ,
w ( φ1 and M,
g ,
w ( φ2
M,
g ,
w ( xRyφ iff there is a w 1P M such that RM
pw ,w 1q and M,
g ,
w 1( φ
M, g ,w ( s iff w “ rssM,g for s P StatM, g ,w ( @sφ iff M, g , rssM,g
( φ for s P StatM, g ,w (Óx .φ iff M, g x
w ,w ( φM, g ,w ( Eφ iff there is a w 1
P M such that M, g ,w 1( φ
M, g ,w ( Dx .φ iff there is a w 1P M such that M, g x
w 1 ,w ( φ
S. Pogodalla (with L. Kallmeyer and R. Osswald) A (Hybrid) Logical Approach to Frame Semantics Oct. 9–10, 2019 10 / 30
Hybrid Logic Reminder on Modal Logic
Modal Logic (Blackburn 1993; Areces and ten Cate 2007)Satisfaction Relation
Definition (Satisfaction relation)
Let M be a model, w P M
, and g an assignment for M.
The satisfaction relation ( is defined as follows:
M,
g ,
w ( JM,
g ,
w ( p iff w P V ppq for p P Prop
M,
g ,
w ( φ iff M,
g ,
w * φM,
g ,
w ( φ1 ^ φ2 iff M,
g ,
w ( φ1 and M,
g ,
w ( φ2
M,
g ,
w ( xRyφ iff there is a w 1P M such that RM
pw ,w 1q and M,
g ,
w 1( φ
M, g ,w ( s iff w “ rssM,g for s P StatM, g ,w ( @sφ iff M, g , rssM,g
( φ for s P StatM, g ,w (Óx .φ iff M, g x
w ,w ( φM, g ,w ( Eφ iff there is a w 1
P M such that M, g ,w 1( φ
M, g ,w ( Dx .φ iff there is a w 1P M such that M, g x
w 1 ,w ( φ
S. Pogodalla (with L. Kallmeyer and R. Osswald) A (Hybrid) Logical Approach to Frame Semantics Oct. 9–10, 2019 10 / 30
Hybrid Logic Reminder on Modal Logic
Modal Logic (Blackburn 1993; Areces and ten Cate 2007)Satisfaction Relation
Definition (Satisfaction relation)
Let M be a model, w P M
, and g an assignment for M.
The satisfaction relation ( is defined as follows:
M,
g ,
w ( JM,
g ,
w ( p iff w P V ppq for p P PropM,
g ,
w ( φ iff M,
g ,
w * φ
M,
g ,
w ( φ1 ^ φ2 iff M,
g ,
w ( φ1 and M,
g ,
w ( φ2
M,
g ,
w ( xRyφ iff there is a w 1P M such that RM
pw ,w 1q and M,
g ,
w 1( φ
M, g ,w ( s iff w “ rssM,g for s P StatM, g ,w ( @sφ iff M, g , rssM,g
( φ for s P StatM, g ,w (Óx .φ iff M, g x
w ,w ( φM, g ,w ( Eφ iff there is a w 1
P M such that M, g ,w 1( φ
M, g ,w ( Dx .φ iff there is a w 1P M such that M, g x
w 1 ,w ( φ
S. Pogodalla (with L. Kallmeyer and R. Osswald) A (Hybrid) Logical Approach to Frame Semantics Oct. 9–10, 2019 10 / 30
Hybrid Logic Reminder on Modal Logic
Modal Logic (Blackburn 1993; Areces and ten Cate 2007)Satisfaction Relation
Definition (Satisfaction relation)
Let M be a model, w P M
, and g an assignment for M.
The satisfaction relation ( is defined as follows:
M,
g ,
w ( JM,
g ,
w ( p iff w P V ppq for p P PropM,
g ,
w ( φ iff M,
g ,
w * φM,
g ,
w ( φ1 ^ φ2 iff M,
g ,
w ( φ1 and M,
g ,
w ( φ2
M,
g ,
w ( xRyφ iff there is a w 1P M such that RM
pw ,w 1q and M,
g ,
w 1( φ
M, g ,w ( s iff w “ rssM,g for s P StatM, g ,w ( @sφ iff M, g , rssM,g
( φ for s P StatM, g ,w (Óx .φ iff M, g x
w ,w ( φM, g ,w ( Eφ iff there is a w 1
P M such that M, g ,w 1( φ
M, g ,w ( Dx .φ iff there is a w 1P M such that M, g x
w 1 ,w ( φ
S. Pogodalla (with L. Kallmeyer and R. Osswald) A (Hybrid) Logical Approach to Frame Semantics Oct. 9–10, 2019 10 / 30
Hybrid Logic Reminder on Modal Logic
Modal Logic (Blackburn 1993; Areces and ten Cate 2007)Satisfaction Relation
Definition (Satisfaction relation)
Let M be a model, w P M
, and g an assignment for M.
The satisfaction relation ( is defined as follows:
M,
g ,
w ( JM,
g ,
w ( p iff w P V ppq for p P PropM,
g ,
w ( φ iff M,
g ,
w * φM,
g ,
w ( φ1 ^ φ2 iff M,
g ,
w ( φ1 and M,
g ,
w ( φ2
M,
g ,
w ( xRyφ iff there is a w 1P M such that RM
pw ,w 1q and M,
g ,
w 1( φ
M, g ,w ( s iff w “ rssM,g for s P StatM, g ,w ( @sφ iff M, g , rssM,g
( φ for s P StatM, g ,w (Óx .φ iff M, g x
w ,w ( φM, g ,w ( Eφ iff there is a w 1
P M such that M, g ,w 1( φ
M, g ,w ( Dx .φ iff there is a w 1P M such that M, g x
w 1 ,w ( φ
S. Pogodalla (with L. Kallmeyer and R. Osswald) A (Hybrid) Logical Approach to Frame Semantics Oct. 9–10, 2019 10 / 30
Hybrid Logic Hybrid Logic
Modal Logic (Blackburn 1993; Areces and ten Cate 2007)Formulas
Rel “ FuncY PropRel is a set of functional and non-functional relational symbols,
Prop a set of propositional variables
Nom a set of nominals (node names),
Svar a set of state variables
Stat “ NomY Svar.
Definition (Formulas)
The language of formulas Forms is defined as:
Forms ::“ J | p | φ | φ1 ^ φ2 | xRyφ | s | @sφ | Óx .φ | Eφ | Dx .φ
where p P Prop, φ, φ1, φ2 P Forms, R P Rel, s P Stat, x P Svar.
Moreover, we define:
φñ ψ ” φ_ ψ
rRsφ ” xRy φ
A
φ ” E φ
S. Pogodalla (with L. Kallmeyer and R. Osswald) A (Hybrid) Logical Approach to Frame Semantics Oct. 9–10, 2019 11 / 30
Hybrid Logic Hybrid Logic
Modal Logic (Blackburn 1993; Areces and ten Cate 2007)Formulas
Rel “ FuncY PropRel is a set of functional and non-functional relational symbols,
Prop a set of propositional variables
Nom a set of nominals (node names),
Svar a set of state variables
Stat “ NomY Svar.
Definition (Formulas)
The language of formulas Forms is defined as:
Forms ::“ J | p | φ | φ1 ^ φ2 | xRyφ
| s | @sφ | Óx .φ | Eφ | Dx .φ
where p P Prop, φ, φ1, φ2 P Forms, R P Rel
, s P Stat, x P Svar.
Moreover, we define:
φñ ψ ” φ_ ψ
rRsφ ” xRy φ
A
φ ” E φ
S. Pogodalla (with L. Kallmeyer and R. Osswald) A (Hybrid) Logical Approach to Frame Semantics Oct. 9–10, 2019 11 / 30
Hybrid Logic Hybrid Logic
Hybrid Logic (Blackburn 1993; Areces and ten Cate 2007)Formulas
Rel “ FuncY PropRel is a set of functional and non-functional relational symbols,
Prop a set of propositional variables
Nom a set of nominals (node names),
Svar a set of state variables
Stat “ NomY Svar.
Definition (Formulas)
The language of formulas Forms is defined as:
Forms ::“ J | p | φ | φ1 ^ φ2 | xRyφ
| s | @sφ | Óx .φ | Eφ | Dx .φ
where p P Prop, φ, φ1, φ2 P Forms, R P Rel
, s P Stat, x P Svar.
Moreover, we define:
φñ ψ ” φ_ ψ
rRsφ ” xRy φ
A
φ ” E φ
S. Pogodalla (with L. Kallmeyer and R. Osswald) A (Hybrid) Logical Approach to Frame Semantics Oct. 9–10, 2019 11 / 30
Hybrid Logic Hybrid Logic
Hybrid Logic (Blackburn 1993; Areces and ten Cate 2007)Formulas
Rel “ FuncY PropRel is a set of functional and non-functional relational symbols,
Prop a set of propositional variables
Nom a set of nominals (node names),
Svar a set of state variables
Stat “ NomY Svar.
Definition (Formulas)
The language of formulas Forms is defined as:
Forms ::“ J | p | φ | φ1 ^ φ2 | xRyφ
| s | @sφ | Óx .φ | Eφ | Dx .φ
where p P Prop, φ, φ1, φ2 P Forms, R P Rel
, s P Stat, x P Svar.
Moreover, we define:
φñ ψ ” φ_ ψ
rRsφ ” xRy φ
A
φ ” E φ
S. Pogodalla (with L. Kallmeyer and R. Osswald) A (Hybrid) Logical Approach to Frame Semantics Oct. 9–10, 2019 11 / 30
Hybrid Logic Hybrid Logic
Hybrid Logic (Blackburn 1993; Areces and ten Cate 2007)Formulas
Rel “ FuncY PropRel is a set of functional and non-functional relational symbols,
Prop a set of propositional variables
Nom a set of nominals (node names),
Svar a set of state variables
Stat “ NomY Svar.
Definition (Formulas)
The language of formulas Forms is defined as:
Forms ::“ J | p | φ | φ1 ^ φ2 | xRyφ
| s | @sφ | Óx .φ | Eφ | Dx .φ
where p P Prop, φ, φ1, φ2 P Forms, R P Rel
, s P Stat, x P Svar.
Moreover, we define:
φñ ψ ” φ_ ψ
rRsφ ” xRy φ
A
φ ” E φ
S. Pogodalla (with L. Kallmeyer and R. Osswald) A (Hybrid) Logical Approach to Frame Semantics Oct. 9–10, 2019 11 / 30
Hybrid Logic Hybrid Logic
Hybrid Logic (Blackburn 1993; Areces and ten Cate 2007)Formulas
Rel “ FuncY PropRel is a set of functional and non-functional relational symbols,
Prop a set of propositional variables
Nom a set of nominals (node names),
Svar a set of state variables
Stat “ NomY Svar.
Definition (Formulas)
The language of formulas Forms is defined as:
Forms ::“ J | p | φ | φ1 ^ φ2 | xRyφ | s
| @sφ | Óx .φ | Eφ | Dx .φ
where p P Prop, φ, φ1, φ2 P Forms, R P Rel, s P Stat
, x P Svar.
Moreover, we define:
φñ ψ ” φ_ ψ
rRsφ ” xRy φ
A
φ ” E φ
S. Pogodalla (with L. Kallmeyer and R. Osswald) A (Hybrid) Logical Approach to Frame Semantics Oct. 9–10, 2019 11 / 30
Hybrid Logic Hybrid Logic
Hybrid Logic (Blackburn 1993; Areces and ten Cate 2007)Formulas
Rel “ FuncY PropRel is a set of functional and non-functional relational symbols,
Prop a set of propositional variables
Nom a set of nominals (node names),
Svar a set of state variables
Stat “ NomY Svar.
Definition (Formulas)
The language of formulas Forms is defined as:
Forms ::“ J | p | φ | φ1 ^ φ2 | xRyφ | s | @sφ
| Óx .φ | Eφ | Dx .φ
where p P Prop, φ, φ1, φ2 P Forms, R P Rel, s P Stat
, x P Svar.
Moreover, we define:
φñ ψ ” φ_ ψ
rRsφ ” xRy φ
A
φ ” E φ
S. Pogodalla (with L. Kallmeyer and R. Osswald) A (Hybrid) Logical Approach to Frame Semantics Oct. 9–10, 2019 11 / 30
Hybrid Logic Hybrid Logic
Hybrid Logic (Blackburn 1993; Areces and ten Cate 2007)Formulas
Rel “ FuncY PropRel is a set of functional and non-functional relational symbols,
Prop a set of propositional variables
Nom a set of nominals (node names),
Svar a set of state variables
Stat “ NomY Svar.
Definition (Formulas)
The language of formulas Forms is defined as:
Forms ::“ J | p | φ | φ1 ^ φ2 | xRyφ | s | @sφ | Óx .φ
| Eφ | Dx .φ
where p P Prop, φ, φ1, φ2 P Forms, R P Rel, s P Stat, x P Svar.
Moreover, we define:
φñ ψ ” φ_ ψ
rRsφ ” xRy φ
A
φ ” E φ
S. Pogodalla (with L. Kallmeyer and R. Osswald) A (Hybrid) Logical Approach to Frame Semantics Oct. 9–10, 2019 11 / 30
Hybrid Logic Hybrid Logic
Hybrid Logic (Blackburn 1993; Areces and ten Cate 2007)Formulas
Rel “ FuncY PropRel is a set of functional and non-functional relational symbols,
Prop a set of propositional variables
Nom a set of nominals (node names),
Svar a set of state variables
Stat “ NomY Svar.
Definition (Formulas)
The language of formulas Forms is defined as:
Forms ::“ J | p | φ | φ1 ^ φ2 | xRyφ | s | @sφ | Óx .φ | Eφ
| Dx .φ
where p P Prop, φ, φ1, φ2 P Forms, R P Rel, s P Stat, x P Svar.
Moreover, we define:
φñ ψ ” φ_ ψ
rRsφ ” xRy φ
A
φ ” E φ
S. Pogodalla (with L. Kallmeyer and R. Osswald) A (Hybrid) Logical Approach to Frame Semantics Oct. 9–10, 2019 11 / 30
Hybrid Logic Hybrid Logic
Hybrid Logic (Blackburn 1993; Areces and ten Cate 2007)Formulas
Rel “ FuncY PropRel is a set of functional and non-functional relational symbols,
Prop a set of propositional variables
Nom a set of nominals (node names),
Svar a set of state variables
Stat “ NomY Svar.
Definition (Formulas)
The language of formulas Forms is defined as:
Forms ::“ J | p | φ | φ1 ^ φ2 | xRyφ | s | @sφ | Óx .φ | Eφ | Dx .φ
where p P Prop, φ, φ1, φ2 P Forms, R P Rel, s P Stat, x P Svar.
Moreover, we define:
φñ ψ ” φ_ ψ
rRsφ ” xRy φ
A
φ ” E φ
S. Pogodalla (with L. Kallmeyer and R. Osswald) A (Hybrid) Logical Approach to Frame Semantics Oct. 9–10, 2019 11 / 30
Hybrid Logic Hybrid Logic
Hybrid Logic (Blackburn 1993; Areces and ten Cate 2007)Formulas
Rel “ FuncY PropRel is a set of functional and non-functional relational symbols,
Prop a set of propositional variables
Nom a set of nominals (node names),
Svar a set of state variables
Stat “ NomY Svar.
Definition (Formulas)
The language of formulas Forms is defined as:
Forms ::“ J | p | φ | φ1 ^ φ2 | xRyφ | s | @sφ | Óx .φ | Eφ | Dx .φ
where p P Prop, φ, φ1, φ2 P Forms, R P Rel, s P Stat, x P Svar.
Moreover, we define:
φñ ψ ” φ_ ψ
rRsφ ” xRy φ
A
φ ” E φ
S. Pogodalla (with L. Kallmeyer and R. Osswald) A (Hybrid) Logical Approach to Frame Semantics Oct. 9–10, 2019 11 / 30
Hybrid Logic Hybrid Logic
Modal Logic (Blackburn 1993; Areces and ten Cate 2007)Models
Definition (Model)
A model M is a triple xM, pRMqRPRel,V y such that:
M is a non-empty set,
each RM is a binary relation on M,
V : Prop
Y Nom
ÝÑ ℘pMq is a valuation
such that if i P Nom then V piq is asingleton.
An assignment g is a mapping g : Svar ÝÑ M.g xm is an assignment that differs from g at most on x and such that g x
mpxq “ m.For s P Stat, we also define rssM,g to be the only m such that V psq “ tmu if s P Nomand rssM,g
“ gpsq if s P Svar.
S. Pogodalla (with L. Kallmeyer and R. Osswald) A (Hybrid) Logical Approach to Frame Semantics Oct. 9–10, 2019 12 / 30
Hybrid Logic Hybrid Logic
Hybrid Logic (Blackburn 1993; Areces and ten Cate 2007)Models
Definition (Model)
A model M is a triple xM, pRMqRPRel,V y such that:
M is a non-empty set,
each RM is a binary relation on M,
V : PropY Nom ÝÑ ℘pMq is a valuation
such that if i P Nom then V piq is asingleton.
An assignment g is a mapping g : Svar ÝÑ M.g xm is an assignment that differs from g at most on x and such that g x
mpxq “ m.For s P Stat, we also define rssM,g to be the only m such that V psq “ tmu if s P Nomand rssM,g
“ gpsq if s P Svar.
S. Pogodalla (with L. Kallmeyer and R. Osswald) A (Hybrid) Logical Approach to Frame Semantics Oct. 9–10, 2019 12 / 30
Hybrid Logic Hybrid Logic
Hybrid Logic (Blackburn 1993; Areces and ten Cate 2007)Models
Definition (Model)
A model M is a triple xM, pRMqRPRel,V y such that:
M is a non-empty set,
each RM is a binary relation on M,
V : PropY Nom ÝÑ ℘pMq is a valuation such that if i P Nom then V piq is asingleton.
An assignment g is a mapping g : Svar ÝÑ M.g xm is an assignment that differs from g at most on x and such that g x
mpxq “ m.For s P Stat, we also define rssM,g to be the only m such that V psq “ tmu if s P Nomand rssM,g
“ gpsq if s P Svar.
S. Pogodalla (with L. Kallmeyer and R. Osswald) A (Hybrid) Logical Approach to Frame Semantics Oct. 9–10, 2019 12 / 30
Hybrid Logic Hybrid Logic
Hybrid Logic (Blackburn 1993; Areces and ten Cate 2007)Models
Definition (Model)
A model M is a triple xM, pRMqRPRel,V y such that:
M is a non-empty set,
each RM is a binary relation on M,
V : PropY Nom ÝÑ ℘pMq is a valuation such that if i P Nom then V piq is asingleton.
An assignment g is a mapping g : Svar ÝÑ M.
g xm is an assignment that differs from g at most on x and such that g x
mpxq “ m.For s P Stat, we also define rssM,g to be the only m such that V psq “ tmu if s P Nomand rssM,g
“ gpsq if s P Svar.
S. Pogodalla (with L. Kallmeyer and R. Osswald) A (Hybrid) Logical Approach to Frame Semantics Oct. 9–10, 2019 12 / 30
Hybrid Logic Hybrid Logic
Hybrid Logic (Blackburn 1993; Areces and ten Cate 2007)Models
Definition (Model)
A model M is a triple xM, pRMqRPRel,V y such that:
M is a non-empty set,
each RM is a binary relation on M,
V : PropY Nom ÝÑ ℘pMq is a valuation such that if i P Nom then V piq is asingleton.
An assignment g is a mapping g : Svar ÝÑ M.g xm is an assignment that differs from g at most on x and such that g x
mpxq “ m.
For s P Stat, we also define rssM,g to be the only m such that V psq “ tmu if s P Nomand rssM,g
“ gpsq if s P Svar.
S. Pogodalla (with L. Kallmeyer and R. Osswald) A (Hybrid) Logical Approach to Frame Semantics Oct. 9–10, 2019 12 / 30
Hybrid Logic Hybrid Logic
Hybrid Logic (Blackburn 1993; Areces and ten Cate 2007)Models
Definition (Model)
A model M is a triple xM, pRMqRPRel,V y such that:
M is a non-empty set,
each RM is a binary relation on M,
V : PropY Nom ÝÑ ℘pMq is a valuation such that if i P Nom then V piq is asingleton.
An assignment g is a mapping g : Svar ÝÑ M.g xm is an assignment that differs from g at most on x and such that g x
mpxq “ m.For s P Stat, we also define rssM,g to be the only m such that V psq “ tmu if s P Nomand rssM,g
“ gpsq if s P Svar.
S. Pogodalla (with L. Kallmeyer and R. Osswald) A (Hybrid) Logical Approach to Frame Semantics Oct. 9–10, 2019 12 / 30
Hybrid Logic Hybrid Logic
Modal Logic (Blackburn 1993; Areces and ten Cate 2007)Satisfaction Relation
Definition (Satisfaction relation)
Let M be a model, w P M
, and g an assignment for M.
The satisfaction relation ( is defined as follows:
M,
g ,
w ( JM,
g ,
w ( p iff w P V ppq for p P PropM,
g ,
w ( φ iff M,
g ,
w * φM,
g ,
w ( φ1 ^ φ2 iff M,
g ,
w ( φ1 and M,
g ,
w ( φ2
M,
g ,
w ( xRyφ iff there is a w 1P M such that RM
pw ,w 1q and M,
g ,
w 1( φ
M, g ,w ( s iff w “ rssM,g for s P StatM, g ,w ( @sφ iff M, g , rssM,g
( φ for s P StatM, g ,w (Óx .φ iff M, g x
w ,w ( φM, g ,w ( Eφ iff there is a w 1
P M such that M, g ,w 1( φ
M, g ,w ( Dx .φ iff there is a w 1P M such that M, g x
w 1 ,w ( φ
S. Pogodalla (with L. Kallmeyer and R. Osswald) A (Hybrid) Logical Approach to Frame Semantics Oct. 9–10, 2019 13 / 30
Hybrid Logic Hybrid Logic
Hybrid Logic (Blackburn 1993; Areces and ten Cate 2007)Satisfaction Relation
Definition (Satisfaction relation)
Let M be a model, w P M, and g an assignment for M.The satisfaction relation ( is defined as follows:
M,
g ,
w ( JM,
g ,
w ( p iff w P V ppq for p P PropM,
g ,
w ( φ iff M,
g ,
w * φM,
g ,
w ( φ1 ^ φ2 iff M,
g ,
w ( φ1 and M,
g ,
w ( φ2
M,
g ,
w ( xRyφ iff there is a w 1P M such that RM
pw ,w 1q and M,
g ,
w 1( φ
M, g ,w ( s iff w “ rssM,g for s P StatM, g ,w ( @sφ iff M, g , rssM,g
( φ for s P StatM, g ,w (Óx .φ iff M, g x
w ,w ( φM, g ,w ( Eφ iff there is a w 1
P M such that M, g ,w 1( φ
M, g ,w ( Dx .φ iff there is a w 1P M such that M, g x
w 1 ,w ( φ
S. Pogodalla (with L. Kallmeyer and R. Osswald) A (Hybrid) Logical Approach to Frame Semantics Oct. 9–10, 2019 13 / 30
Hybrid Logic Hybrid Logic
Hybrid Logic (Blackburn 1993; Areces and ten Cate 2007)Satisfaction Relation
Definition (Satisfaction relation)
Let M be a model, w P M, and g an assignment for M.The satisfaction relation ( is defined as follows:
M, g ,w ( JM, g ,w ( p iff w P V ppq for p P PropM, g ,w ( φ iff M, g ,w * φM, g ,w ( φ1 ^ φ2 iff M, g ,w ( φ1 and M, g ,w ( φ2
M, g ,w ( xRyφ iff there is a w 1P M such that RM
pw ,w 1q and M, g ,w 1
( φ
M, g ,w ( s iff w “ rssM,g for s P StatM, g ,w ( @sφ iff M, g , rssM,g
( φ for s P StatM, g ,w (Óx .φ iff M, g x
w ,w ( φM, g ,w ( Eφ iff there is a w 1
P M such that M, g ,w 1( φ
M, g ,w ( Dx .φ iff there is a w 1P M such that M, g x
w 1 ,w ( φ
S. Pogodalla (with L. Kallmeyer and R. Osswald) A (Hybrid) Logical Approach to Frame Semantics Oct. 9–10, 2019 13 / 30
Hybrid Logic Hybrid Logic
Hybrid Logic (Blackburn 1993; Areces and ten Cate 2007)Satisfaction Relation
Definition (Satisfaction relation)
Let M be a model, w P M, and g an assignment for M.The satisfaction relation ( is defined as follows:
M, g ,w ( JM, g ,w ( p iff w P V ppq for p P PropM, g ,w ( φ iff M, g ,w * φM, g ,w ( φ1 ^ φ2 iff M, g ,w ( φ1 and M, g ,w ( φ2
M, g ,w ( xRyφ iff there is a w 1P M such that RM
pw ,w 1q and M, g ,w 1
( φM, g ,w ( s iff w “ rssM,g for s P Stat
M, g ,w ( @sφ iff M, g , rssM,g( φ for s P Stat
M, g ,w (Óx .φ iff M, g xw ,w ( φ
M, g ,w ( Eφ iff there is a w 1P M such that M, g ,w 1
( φM, g ,w ( Dx .φ iff there is a w 1
P M such that M, g xw 1 ,w ( φ
S. Pogodalla (with L. Kallmeyer and R. Osswald) A (Hybrid) Logical Approach to Frame Semantics Oct. 9–10, 2019 13 / 30
Hybrid Logic Hybrid Logic
Hybrid Logic (Blackburn 1993; Areces and ten Cate 2007)Satisfaction Relation
Definition (Satisfaction relation)
Let M be a model, w P M, and g an assignment for M.The satisfaction relation ( is defined as follows:
M, g ,w ( JM, g ,w ( p iff w P V ppq for p P PropM, g ,w ( φ iff M, g ,w * φM, g ,w ( φ1 ^ φ2 iff M, g ,w ( φ1 and M, g ,w ( φ2
M, g ,w ( xRyφ iff there is a w 1P M such that RM
pw ,w 1q and M, g ,w 1
( φM, g ,w ( s iff w “ rssM,g for s P StatM, g ,w ( @sφ iff M, g , rssM,g
( φ for s P Stat
M, g ,w (Óx .φ iff M, g xw ,w ( φ
M, g ,w ( Eφ iff there is a w 1P M such that M, g ,w 1
( φM, g ,w ( Dx .φ iff there is a w 1
P M such that M, g xw 1 ,w ( φ
S. Pogodalla (with L. Kallmeyer and R. Osswald) A (Hybrid) Logical Approach to Frame Semantics Oct. 9–10, 2019 13 / 30
Hybrid Logic Hybrid Logic
Hybrid Logic (Blackburn 1993; Areces and ten Cate 2007)Satisfaction Relation
Definition (Satisfaction relation)
Let M be a model, w P M, and g an assignment for M.The satisfaction relation ( is defined as follows:
M, g ,w ( JM, g ,w ( p iff w P V ppq for p P PropM, g ,w ( φ iff M, g ,w * φM, g ,w ( φ1 ^ φ2 iff M, g ,w ( φ1 and M, g ,w ( φ2
M, g ,w ( xRyφ iff there is a w 1P M such that RM
pw ,w 1q and M, g ,w 1
( φM, g ,w ( s iff w “ rssM,g for s P StatM, g ,w ( @sφ iff M, g , rssM,g
( φ for s P StatM, g ,w (Óx .φ iff M, g x
w ,w ( φ
M, g ,w ( Eφ iff there is a w 1P M such that M, g ,w 1
( φM, g ,w ( Dx .φ iff there is a w 1
P M such that M, g xw 1 ,w ( φ
S. Pogodalla (with L. Kallmeyer and R. Osswald) A (Hybrid) Logical Approach to Frame Semantics Oct. 9–10, 2019 13 / 30
Hybrid Logic Hybrid Logic
Hybrid Logic (Blackburn 1993; Areces and ten Cate 2007)Satisfaction Relation
Definition (Satisfaction relation)
Let M be a model, w P M, and g an assignment for M.The satisfaction relation ( is defined as follows:
M, g ,w ( JM, g ,w ( p iff w P V ppq for p P PropM, g ,w ( φ iff M, g ,w * φM, g ,w ( φ1 ^ φ2 iff M, g ,w ( φ1 and M, g ,w ( φ2
M, g ,w ( xRyφ iff there is a w 1P M such that RM
pw ,w 1q and M, g ,w 1
( φM, g ,w ( s iff w “ rssM,g for s P StatM, g ,w ( @sφ iff M, g , rssM,g
( φ for s P StatM, g ,w (Óx .φ iff M, g x
w ,w ( φM, g ,w ( Eφ iff there is a w 1
P M such that M, g ,w 1( φ
M, g ,w ( Dx .φ iff there is a w 1P M such that M, g x
w 1 ,w ( φ
S. Pogodalla (with L. Kallmeyer and R. Osswald) A (Hybrid) Logical Approach to Frame Semantics Oct. 9–10, 2019 13 / 30
Hybrid Logic Hybrid Logic
Hybrid Logic (Blackburn 1993; Areces and ten Cate 2007)Satisfaction Relation
Definition (Satisfaction relation)
Let M be a model, w P M, and g an assignment for M.The satisfaction relation ( is defined as follows:
M, g ,w ( JM, g ,w ( p iff w P V ppq for p P PropM, g ,w ( φ iff M, g ,w * φM, g ,w ( φ1 ^ φ2 iff M, g ,w ( φ1 and M, g ,w ( φ2
M, g ,w ( xRyφ iff there is a w 1P M such that RM
pw ,w 1q and M, g ,w 1
( φM, g ,w ( s iff w “ rssM,g for s P StatM, g ,w ( @sφ iff M, g , rssM,g
( φ for s P StatM, g ,w (Óx .φ iff M, g x
w ,w ( φM, g ,w ( Eφ iff there is a w 1
P M such that M, g ,w 1( φ
M, g ,w ( Dx .φ iff there is a w 1P M such that M, g x
w 1 ,w ( φ
S. Pogodalla (with L. Kallmeyer and R. Osswald) A (Hybrid) Logical Approach to Frame Semantics Oct. 9–10, 2019 13 / 30
Hybrid Logic Hybrid Logic
Example
l0
motionl1
man
path
walking
l2
house
agent
mover
pathmanner
endp at-regionpart-of
goal
l0 ^motion ^ xagentypl1 ^manq ^ xmoveryl1 ^ xgoalypl2 ^ houseq^
xmannerywalking ^ pDv w .xpathyppath^ xendpyvq^
@l2pxat-regionywq ^ @v pxpart-of ywqq
S. Pogodalla (with L. Kallmeyer and R. Osswald) A (Hybrid) Logical Approach to Frame Semantics Oct. 9–10, 2019 14 / 30
Hybrid Logic Hybrid Logic
Example
l0
motionl1
man
path
walking
l2
house
agent
mover
pathmanner
endp at-regionpart-of
goal
l0 ^motion ^ xagentypl1 ^manq ^ xmoveryl1 ^ xgoalypl2 ^ houseq^
xmannerywalking ^ pDv w .xpathyppath^ xendpyvq^
@l2pxat-regionywq ^ @v pxpart-of ywqq
S. Pogodalla (with L. Kallmeyer and R. Osswald) A (Hybrid) Logical Approach to Frame Semantics Oct. 9–10, 2019 14 / 30
Compositional Frame Semantics Quantification
Quantification
agent
name
theme name
agent nametheme
agenttheme theme
kissing
man
John
womanMary
kissing man Peter
kissing woman
Sue
kissing
man
Paul
agent
name
name
S. Pogodalla (with L. Kallmeyer and R. Osswald) A (Hybrid) Logical Approach to Frame Semantics Oct. 9–10, 2019 15 / 30
Compositional Frame Semantics Quantification
Quantification
agent
name
theme name
agent nametheme
agenttheme theme
kissing
man
John
womanMary
kissing man Peter
kissing woman
Sue
kissing
man
Paul
agent
name
name
(1) a. John kisses Mary
S. Pogodalla (with L. Kallmeyer and R. Osswald) A (Hybrid) Logical Approach to Frame Semantics Oct. 9–10, 2019 15 / 30
Compositional Frame Semantics Quantification
Quantification
agent
name
theme name
agent nametheme
agenttheme theme
kissing
man
John
womanMary
kissing man Peter
kissing woman
Sue
kissing
man
Paul
agent
name
name
(1) a. John kisses Mary
b. Epkissing^ xagentypxnameyJohnq ^ xthemeypxnameyMaryqq
S. Pogodalla (with L. Kallmeyer and R. Osswald) A (Hybrid) Logical Approach to Frame Semantics Oct. 9–10, 2019 15 / 30
Compositional Frame Semantics Quantification
Quantification
agent
name
theme name
agent nametheme
agenttheme theme
kissing
man
John
womanMary
kissing man Peter
kissing woman
Sue
kissing
man
Paul
agent
name
name
(2) a. Every man kisses Mary
S. Pogodalla (with L. Kallmeyer and R. Osswald) A (Hybrid) Logical Approach to Frame Semantics Oct. 9–10, 2019 15 / 30
Compositional Frame Semantics Quantification
Quantification
agent
name
theme name
agent nametheme
agenttheme theme
kissing
man
John
womanMary
kissing man Peter
kissing woman
Sue
kissing
man
Paul
agent
name
name
(2) a. Every man kisses Mary
b.
A
pÓ i .man ñ Epkissing^ xagentyi ^ xthemeypxnameyMaryqqq
S. Pogodalla (with L. Kallmeyer and R. Osswald) A (Hybrid) Logical Approach to Frame Semantics Oct. 9–10, 2019 15 / 30
Compositional Frame Semantics Quantification
Quantification
agent
name
theme name
agent nametheme
agenttheme theme
kissing
man
John
womanMary
kissing man Peter
kissing woman
Sue
kissing
man
Paul
agent
name
name
(3) a. Every man kisses some woman
S. Pogodalla (with L. Kallmeyer and R. Osswald) A (Hybrid) Logical Approach to Frame Semantics Oct. 9–10, 2019 15 / 30
Compositional Frame Semantics Quantification
Quantification
agent
name
theme name
agent nametheme
agenttheme theme
kissing
man
John
womanMary
kissing man Peter
kissing woman
Sue
kissing
man
Paul
agent
name
name
(3) a. Every man kisses some woman
b.
A
pÓ i .man ñ EpÓ i 1.woman^ Epkissing^ xagentyi ^ xthemeyi 1qqq
S. Pogodalla (with L. Kallmeyer and R. Osswald) A (Hybrid) Logical Approach to Frame Semantics Oct. 9–10, 2019 15 / 30
Compositional Frame Semantics Quantification
Quantification
agent
name
theme name
agent nametheme
agenttheme theme
kissing
man
John
womanMary
kissing man Peter
kissing woman
Sue
kissing
man
Paul
agent
name
name
(3) a. Every man kisses some woman
b.
A
pÓ i .man ñ EpÓ i 1.woman^ Epkissing^ xagentyi ^ xthemeyi 1qqq
c. EpÓ i .woman^
A
pÓ i 1.man ñ Epkissing^ xagentyi 1^ xthemeyiqqq
S. Pogodalla (with L. Kallmeyer and R. Osswald) A (Hybrid) Logical Approach to Frame Semantics Oct. 9–10, 2019 15 / 30
Compositional Frame Semantics Quantification
Example
endp
path
mannerat-region
goal
agent
movername
pathmannermover
agent
goal
name
motion
path v1
walking
housew
manJohn
motion
v2
pathwalking
man Peter
part
-of part-of
endp
S. Pogodalla (with L. Kallmeyer and R. Osswald) A (Hybrid) Logical Approach to Frame Semantics Oct. 9–10, 2019 18 / 30
Compositional Frame Semantics Quantification
Example
endp
path
mannerat-region
goal
agent
movername
pathmannermover
agent
goal
name
motion
path v1
walking
housew
manJohn
motion
v2
pathwalking
man Peter
part
-of part-of
endp
(4) a. Every man walked to some house
S. Pogodalla (with L. Kallmeyer and R. Osswald) A (Hybrid) Logical Approach to Frame Semantics Oct. 9–10, 2019 18 / 30
Compositional Frame Semantics Quantification
Example
endp
path
mannerat-region
goal
agent
movername
pathmannermover
agent
goal
name
motion
path v1
walking
housew
manJohn
motion
v2
pathwalking
man Peter
part
-of part-of
endp
(4) a. Every man walked to some house
b.
A
pÓ i .man ñ p EpÓ i 1.house^ pDa g . Epmotion^ xagentya^ xmoverya^xgoalyg ^ xpathypath^ xmannerywalking^ @ai ^ pDr v w .event^xpathyppath^ xendpyvq ^ @r pxat-regionywq ^ @v pxpart-of ywq ^ @r pg ^ i 1
qqqqqqq
S. Pogodalla (with L. Kallmeyer and R. Osswald) A (Hybrid) Logical Approach to Frame Semantics Oct. 9–10, 2019 18 / 30
Compositional Frame Semantics Quantification
Example
endp
path
mannerat-region
goal
agent
movername
pathmannermover
agent
goal
name
motion
path v1
walking
housew
manJohn
motion
v2
pathwalking
man Peter
part
-of part-of
endp
(4) a. Every man walked to some house
b. EpÓ i 1.house^ p
A
pÓ i .man ñ pDa g . Epmotion^ xagentya^ xmoverya^xgoalyg ^ xpathypath^ xmannerywalking^ @ai ^ pDr v w .event^xpathyppath^ xendpyvq ^ @r pxat-regionywq ^ @v pxpart-of ywq ^ @r pg ^ i 1
qqqqqqq
S. Pogodalla (with L. Kallmeyer and R. Osswald) A (Hybrid) Logical Approach to Frame Semantics Oct. 9–10, 2019 18 / 30
Compositional Frame Semantics Quantification
Compositional Frame Semantics
Semantic types and constants
Types: e, s, t
Constants:
event, kissing,motion, John, . . . : t # : s Ñ txagenty, xthemey, xmovery, xpart-of y, . . . : t Ñ t @ : s Ñ t Ñ t^,ñ : t Ñ t Ñ t E,
A
: t Ñ t Ó, D : ps Ñ tq Ñ t
Lexical Semantics
JJohnK “ John JsomeK “ λP Q. EpÓ i .P ^ pQ p# iqqqJMaryK “ Mary JeveryK “ λP Q.
A
pÓ i .P ñ pQ p# iqqqJmanK “ man JkissesK “ λo s. Epkissing^ xagentys ^ xthemeyoqJwomanK “ woman
S. Pogodalla (with L. Kallmeyer and R. Osswald) A (Hybrid) Logical Approach to Frame Semantics Oct. 9–10, 2019 20 / 30
Compositional Frame Semantics Quantification
Compositional Frame Semantics
Semantic types and constants
Types: e, s, t
Constants:
event, kissing,motion, John, . . . : t
# : s Ñ txagenty, xthemey, xmovery, xpart-of y, . . . : t Ñ t @ : s Ñ t Ñ t^,ñ : t Ñ t Ñ t E,
A
: t Ñ t Ó, D : ps Ñ tq Ñ t
Lexical Semantics
JJohnK “ John JsomeK “ λP Q. EpÓ i .P ^ pQ p# iqqqJMaryK “ Mary JeveryK “ λP Q.
A
pÓ i .P ñ pQ p# iqqqJmanK “ man JkissesK “ λo s. Epkissing^ xagentys ^ xthemeyoqJwomanK “ woman
S. Pogodalla (with L. Kallmeyer and R. Osswald) A (Hybrid) Logical Approach to Frame Semantics Oct. 9–10, 2019 20 / 30
Compositional Frame Semantics Quantification
Compositional Frame Semantics
Semantic types and constants
Types: e, s, t
Constants:
event, kissing,motion, John, . . . : t # : s Ñ t
xagenty, xthemey, xmovery, xpart-of y, . . . : t Ñ t @ : s Ñ t Ñ t^,ñ : t Ñ t Ñ t E,
A
: t Ñ t Ó, D : ps Ñ tq Ñ t
Lexical Semantics
JJohnK “ John JsomeK “ λP Q. EpÓ i .P ^ pQ p# iqqqJMaryK “ Mary JeveryK “ λP Q.
A
pÓ i .P ñ pQ p# iqqqJmanK “ man JkissesK “ λo s. Epkissing^ xagentys ^ xthemeyoqJwomanK “ woman
S. Pogodalla (with L. Kallmeyer and R. Osswald) A (Hybrid) Logical Approach to Frame Semantics Oct. 9–10, 2019 20 / 30
Compositional Frame Semantics Quantification
Compositional Frame Semantics
Semantic types and constants
Types: e, s, t
Constants:
event, kissing,motion, John, . . . : t # : s Ñ txagenty, xthemey, xmovery, xpart-of y, . . . : t Ñ t
@ : s Ñ t Ñ t^,ñ : t Ñ t Ñ t E,
A
: t Ñ t Ó, D : ps Ñ tq Ñ t
Lexical Semantics
JJohnK “ John JsomeK “ λP Q. EpÓ i .P ^ pQ p# iqqqJMaryK “ Mary JeveryK “ λP Q.
A
pÓ i .P ñ pQ p# iqqqJmanK “ man JkissesK “ λo s. Epkissing^ xagentys ^ xthemeyoqJwomanK “ woman
S. Pogodalla (with L. Kallmeyer and R. Osswald) A (Hybrid) Logical Approach to Frame Semantics Oct. 9–10, 2019 20 / 30
Compositional Frame Semantics Quantification
Compositional Frame Semantics
Semantic types and constants
Types: e, s, t
Constants:
event, kissing,motion, John, . . . : t # : s Ñ txagenty, xthemey, xmovery, xpart-of y, . . . : t Ñ t @ : s Ñ t Ñ t
^,ñ : t Ñ t Ñ t E,
A
: t Ñ t Ó, D : ps Ñ tq Ñ t
Lexical Semantics
JJohnK “ John JsomeK “ λP Q. EpÓ i .P ^ pQ p# iqqqJMaryK “ Mary JeveryK “ λP Q.
A
pÓ i .P ñ pQ p# iqqqJmanK “ man JkissesK “ λo s. Epkissing^ xagentys ^ xthemeyoqJwomanK “ woman
S. Pogodalla (with L. Kallmeyer and R. Osswald) A (Hybrid) Logical Approach to Frame Semantics Oct. 9–10, 2019 20 / 30
Compositional Frame Semantics Quantification
Compositional Frame Semantics
Semantic types and constants
Types: e, s, t
Constants:
event, kissing,motion, John, . . . : t # : s Ñ txagenty, xthemey, xmovery, xpart-of y, . . . : t Ñ t @ : s Ñ t Ñ t^,ñ : t Ñ t Ñ t
E,
A
: t Ñ t Ó, D : ps Ñ tq Ñ t
Lexical Semantics
JJohnK “ John JsomeK “ λP Q. EpÓ i .P ^ pQ p# iqqqJMaryK “ Mary JeveryK “ λP Q.
A
pÓ i .P ñ pQ p# iqqqJmanK “ man JkissesK “ λo s. Epkissing^ xagentys ^ xthemeyoqJwomanK “ woman
S. Pogodalla (with L. Kallmeyer and R. Osswald) A (Hybrid) Logical Approach to Frame Semantics Oct. 9–10, 2019 20 / 30
Compositional Frame Semantics Quantification
Compositional Frame Semantics
Semantic types and constants
Types: e, s, t
Constants:
event, kissing,motion, John, . . . : t # : s Ñ txagenty, xthemey, xmovery, xpart-of y, . . . : t Ñ t @ : s Ñ t Ñ t^,ñ : t Ñ t Ñ t E,
A
: t Ñ t
Ó, D : ps Ñ tq Ñ t
Lexical Semantics
JJohnK “ John JsomeK “ λP Q. EpÓ i .P ^ pQ p# iqqqJMaryK “ Mary JeveryK “ λP Q.
A
pÓ i .P ñ pQ p# iqqqJmanK “ man JkissesK “ λo s. Epkissing^ xagentys ^ xthemeyoqJwomanK “ woman
S. Pogodalla (with L. Kallmeyer and R. Osswald) A (Hybrid) Logical Approach to Frame Semantics Oct. 9–10, 2019 20 / 30
Compositional Frame Semantics Quantification
Compositional Frame Semantics
Semantic types and constants
Types: e, s, t
Constants:
event, kissing,motion, John, . . . : t # : s Ñ txagenty, xthemey, xmovery, xpart-of y, . . . : t Ñ t @ : s Ñ t Ñ t^,ñ : t Ñ t Ñ t E,
A
: t Ñ t Ó, D : ps Ñ tq Ñ t
Lexical Semantics
JJohnK “ John JsomeK “ λP Q. EpÓ i .P ^ pQ p# iqqqJMaryK “ Mary JeveryK “ λP Q.
A
pÓ i .P ñ pQ p# iqqqJmanK “ man JkissesK “ λo s. Epkissing^ xagentys ^ xthemeyoqJwomanK “ woman
S. Pogodalla (with L. Kallmeyer and R. Osswald) A (Hybrid) Logical Approach to Frame Semantics Oct. 9–10, 2019 20 / 30
Compositional Frame Semantics Quantification
Compositional Frame Semantics
Semantic types and constants
Types: e, s, t
Constants:
event, kissing,motion, John, . . . : t # : s Ñ txagenty, xthemey, xmovery, xpart-of y, . . . : t Ñ t @ : s Ñ t Ñ t^,ñ : t Ñ t Ñ t E,
A
: t Ñ t Ó, D : ps Ñ tq Ñ t
Lexical Semantics
JJohnK “ John
JsomeK “ λP Q. EpÓ i .P ^ pQ p# iqqq
JMaryK “ Mary
JeveryK “ λP Q.
A
pÓ i .P ñ pQ p# iqqq
JmanK “ man
JkissesK “ λo s. Epkissing^ xagentys ^ xthemeyoq
JwomanK “ woman
S. Pogodalla (with L. Kallmeyer and R. Osswald) A (Hybrid) Logical Approach to Frame Semantics Oct. 9–10, 2019 20 / 30
Compositional Frame Semantics Quantification
Compositional Frame Semantics
Semantic types and constants
Types: e, s, t
Constants:
event, kissing,motion, John, . . . : t # : s Ñ txagenty, xthemey, xmovery, xpart-of y, . . . : t Ñ t @ : s Ñ t Ñ t^,ñ : t Ñ t Ñ t E,
A
: t Ñ t Ó, D : ps Ñ tq Ñ t
Lexical Semantics
JJohnK “ John JsomeK “ λP Q. EpÓ i .P ^ pQ p# iqqqJMaryK “ Mary JeveryK “ λP Q.
A
pÓ i .P ñ pQ p# iqqqJmanK “ man
JkissesK “ λo s. Epkissing^ xagentys ^ xthemeyoq
JwomanK “ woman
S. Pogodalla (with L. Kallmeyer and R. Osswald) A (Hybrid) Logical Approach to Frame Semantics Oct. 9–10, 2019 20 / 30
Compositional Frame Semantics Quantification
Compositional Frame Semantics
Semantic types and constants
Types: e, s, t
Constants:
event, kissing,motion, John, . . . : t # : s Ñ txagenty, xthemey, xmovery, xpart-of y, . . . : t Ñ t @ : s Ñ t Ñ t^,ñ : t Ñ t Ñ t E,
A
: t Ñ t Ó, D : ps Ñ tq Ñ t
Lexical Semantics
JJohnK “ John JsomeK “ λP Q. EpÓ i .P ^ pQ p# iqqqJMaryK “ Mary JeveryK “ λP Q.
A
pÓ i .P ñ pQ p# iqqqJmanK “ man JkissesK “ λo s. Epkissing^ xagentys ^ xthemeyoqJwomanK “ woman
S. Pogodalla (with L. Kallmeyer and R. Osswald) A (Hybrid) Logical Approach to Frame Semantics Oct. 9–10, 2019 20 / 30
Compositional Frame Semantics Quantification
Compositional Frame Semantics
Semantic types and constants
Types: e, s, t
Constants:
event, kissing,motion, John, . . . : t # : s Ñ txagenty, xthemey, xmovery, xpart-of y, . . . : t Ñ t @ : s Ñ t Ñ t^,ñ : t Ñ t Ñ t E,
A
: t Ñ t Ó, D : ps Ñ tq Ñ t
Lexical Semantics
JJohnK “ John JsomeK “ λP Q. EpÓ i .P ^ pQ p# iqqqJMaryK “ Mary JeveryK “ λP Q.
A
pÓ i .P ñ pQ p# iqqqJmanK “ man JkissesK “ λo s. Epkissing^ xagentys ^ xthemeyoqJwomanK “ woman
JkissesK JMaryK JJohnK “Epkissing^ xagentypxnameyJohnq ^ xthemeypxnameyMaryqq
S. Pogodalla (with L. Kallmeyer and R. Osswald) A (Hybrid) Logical Approach to Frame Semantics Oct. 9–10, 2019 20 / 30
Compositional Frame Semantics Quantification
Compositional Frame Semantics
Semantic types and constants
Types: e, s, t
Constants:
event, kissing,motion, John, . . . : t # : s Ñ txagenty, xthemey, xmovery, xpart-of y, . . . : t Ñ t @ : s Ñ t Ñ t^,ñ : t Ñ t Ñ t E,
A
: t Ñ t Ó, D : ps Ñ tq Ñ t
Lexical Semantics
JJohnK “ John JsomeK “ λP Q. EpÓ i .P ^ pQ p# iqqqJMaryK “ Mary JeveryK “ λP Q.
A
pÓ i .P ñ pQ p# iqqqJmanK “ man JkissesK “ λo s. Epkissing^ xagentys ^ xthemeyoqJwomanK “ woman
pJeveryK JmanKq pλx .JkissesK JMaryK xq “
A
pÓ i .man ñ Epkissing^ xagentyi ^ xthemeypxnameyMaryqqq
S. Pogodalla (with L. Kallmeyer and R. Osswald) A (Hybrid) Logical Approach to Frame Semantics Oct. 9–10, 2019 20 / 30
Compositional Frame Semantics Quantification
Compositional Frame Semantics
Semantic types and constants
Types: e, s, t
Constants:
event, kissing,motion, John, . . . : t # : s Ñ txagenty, xthemey, xmovery, xpart-of y, . . . : t Ñ t @ : s Ñ t Ñ t^,ñ : t Ñ t Ñ t E,
A
: t Ñ t Ó, D : ps Ñ tq Ñ t
Lexical Semantics
JJohnK “ John JsomeK “ λP Q. EpÓ i .P ^ pQ p# iqqqJMaryK “ Mary JeveryK “ λP Q.
A
pÓ i .P ñ pQ p# iqqqJmanK “ man JkissesK “ λo s. Epkissing^ xagentys ^ xthemeyoqJwomanK “ woman
pJeveryK JmanKq pλx .pJsomeK JwomanKq pλy .JkissesK y xqq “
A
pÓ i .man ñ EpÓ i 1.woman^ Epkissing^ xagentyi ^ xthemeyi 1qqq
S. Pogodalla (with L. Kallmeyer and R. Osswald) A (Hybrid) Logical Approach to Frame Semantics Oct. 9–10, 2019 20 / 30
Compositional Frame Semantics Quantification
Compositional Frame Semantics
Semantic types and constants
Types: e, s, t
Constants:
event, kissing,motion, John, . . . : t # : s Ñ txagenty, xthemey, xmovery, xpart-of y, . . . : t Ñ t @ : s Ñ t Ñ t^,ñ : t Ñ t Ñ t E,
A
: t Ñ t Ó, D : ps Ñ tq Ñ t
Lexical Semantics
JJohnK “ John JsomeK “ λP Q. EpÓ i .P ^ pQ p# iqqqJMaryK “ Mary JeveryK “ λP Q.
A
pÓ i .P ñ pQ p# iqqqJmanK “ man JkissesK “ λo s. Epkissing^ xagentys ^ xthemeyoqJwomanK “ woman
pJsomeK JwomanKq pλy .pJeveryK JmanKq pλx .JkissesK y xqq “
EpÓ i .woman^
A
pÓ i 1.man ñ Epkissing^ xagentyi 1^ xthemeyiqqq
S. Pogodalla (with L. Kallmeyer and R. Osswald) A (Hybrid) Logical Approach to Frame Semantics Oct. 9–10, 2019 20 / 30
Compositional Frame Semantics Quantification
Compositional Frame Semantics (cont’d)
Lexical Semantics
walked “ λpp s.Da g . Epmotion^ xagentyp# aq ^ xmoveryp# aq^xgoalyp# gq ^ xpathypath^ xmannerywalking^ @as ^ ppp p# gqqq
to “ λn g .Dr v w .event^ xpathyppath^ xendpyp# vqq^@rxat-regionyp#wq ^ @vxpart-of yp#wq ^ @r pg ^ nq
S. Pogodalla (with L. Kallmeyer and R. Osswald) A (Hybrid) Logical Approach to Frame Semantics Oct. 9–10, 2019 24 / 30
Compositional Frame Semantics Quantification
Compositional Frame Semantics (cont’d)
Lexical Semantics
walked “ λpp s.Da g . Epmotion^ xagentyp# aq ^ xmoveryp# aq^xgoalyp# gq ^ xpathypath^ xmannerywalking^ @as ^ ppp p# gqqq
to “ λn g .Dr v w .event^ xpathyppath^ xendpyp# vqq^@rxat-regionyp#wq ^ @vxpart-of yp#wq ^ @r pg ^ nq
S. Pogodalla (with L. Kallmeyer and R. Osswald) A (Hybrid) Logical Approach to Frame Semantics Oct. 9–10, 2019 24 / 30
Compositional Frame Semantics Quantification
Compositional Frame Semantics (cont’d)
Lexical Semantics
walked “ λpp s.Da g . Epmotion^ xagentyp# aq ^ xmoveryp# aq^xgoalyp# gq ^ xpathypath^ xmannerywalking^ @as ^ ppp p# gqqq
to “ λn g .Dr v w .event^ xpathyppath^ xendpyp# vqq^@rxat-regionyp#wq ^ @vxpart-of yp#wq ^ @r pg ^ nq
S. Pogodalla (with L. Kallmeyer and R. Osswald) A (Hybrid) Logical Approach to Frame Semantics Oct. 9–10, 2019 24 / 30
Compositional Frame Semantics Quantification
Compositional Frame Semantics (cont’d)
Lexical Semantics
walked “ λpp s.Da g . Epmotion^ xagentyp# aq ^ xmoveryp# aq^xgoalyp# gq ^ xpathypath^ xmannerywalking^ @as ^ ppp p# gqqq
to “ λn g .Dr v w .event^ xpathyppath^ xendpyp# vqq^@rxat-regionyp#wq ^ @vxpart-of yp#wq ^ @r pg ^ nq
Jpevery manq pλx .psome houseqpλy .walked pto yq xqqK
“
A
pÓ i .man ñ p EpÓ i 1.house^ pDa g . Epmotion^ xagentya^ xmoverya^
xgoalyg ^ xpathypath^ xmannerywalking^ @ai ^ pDr v w .event^
xpathyppath^ xendpyvq ^ @r pxat-regionywq^
@v pxpart-of ywq ^ @r pg ^ i 1qqqqqqq
S. Pogodalla (with L. Kallmeyer and R. Osswald) A (Hybrid) Logical Approach to Frame Semantics Oct. 9–10, 2019 24 / 30
Compositional Frame Semantics Quantification
Compositional Frame Semantics (cont’d)
Lexical Semantics
walked “ λpp s.Da g . Epmotion^ xagentyp# aq ^ xmoveryp# aq^xgoalyp# gq ^ xpathypath^ xmannerywalking^ @as ^ ppp p# gqqq
to “ λn g .Dr v w .event^ xpathyppath^ xendpyp# vqq^@rxat-regionyp#wq ^ @vxpart-of yp#wq ^ @r pg ^ nq
Jpsome houseqpλy .pevery manq pλx .walked pto yq xqqK
“ EpÓ i 1.house^ p
A
pÓ i .man ñ pDa g . Epmotion^ xagentya^ xmoverya^
xgoalyg ^ xpathypath^ xmannerywalking^ @ai ^ pDr v w .event^
xpathyppath^ xendpyvq ^ @r pxat-regionywq^
@v pxpart-of ywq ^ @r pg ^ i 1qqqqqqq
S. Pogodalla (with L. Kallmeyer and R. Osswald) A (Hybrid) Logical Approach to Frame Semantics Oct. 9–10, 2019 24 / 30
Compositional Frame Semantics Frame Decomposition
Frame Decomposition and FrameNet (Osswald and Van Valin 2014)
Definitions of FN 1.5 frames of drying
Being dry: An [Item] is in a state of dryness (dehydrated, desiccated, dry, . . . )
Becoming dry: An [Entity] loses moisture with the outcome of being in a dry state(dehydrate, dry up, dry, . . . )
Cause to be dry: an [Agent] causes a [Dryee] (. . . ) to become dry (anhydrate, dehumidify,dehydrate, . . . )
«
Dry state
patient 2
ff
ÐÝ
»
—
–
Inchoation
result
«
Dry state
patient 2
ff
fi
ffi
fl
ÐÝ
»
—
—
—
—
—
—
—
—
—
–
Causation
cause
«
Activity
effector 1
ff
effect
»
—
–
Inchoation
result
«
Dry state
patient 2
ff
fi
ffi
fl
fi
ffi
ffi
ffi
ffi
ffi
ffi
ffi
ffi
ffi
fl
S. Pogodalla (with L. Kallmeyer and R. Osswald) A (Hybrid) Logical Approach to Frame Semantics Oct. 9–10, 2019 26 / 30
Compositional Frame Semantics Frame Decomposition
Frame Decomposition and FrameNet (Osswald and Van Valin 2014)
Definitions of FN 1.5 frames of drying
Being dry: An [Item] is in a state of dryness (dehydrated, desiccated, dry, . . . )
Becoming dry: An [Entity] loses moisture with the outcome of being in a dry state(dehydrate, dry up, dry, . . . )
Cause to be dry: an [Agent] causes a [Dryee] (. . . ) to become dry (anhydrate, dehumidify,dehydrate, . . . )
«
Dry state
patient 2
ff
ÐÝ
»
—
–
Inchoation
result
«
Dry state
patient 2
ff
fi
ffi
fl
ÐÝ
»
—
—
—
—
—
—
—
—
—
–
Causation
cause
«
Activity
effector 1
ff
effect
»
—
–
Inchoation
result
«
Dry state
patient 2
ff
fi
ffi
fl
fi
ffi
ffi
ffi
ffi
ffi
ffi
ffi
ffi
ffi
fl
S. Pogodalla (with L. Kallmeyer and R. Osswald) A (Hybrid) Logical Approach to Frame Semantics Oct. 9–10, 2019 26 / 30
Compositional Frame Semantics Frame Decomposition
Frame Decomposition and FrameNet (Osswald and Van Valin 2014)
Definitions of FN 1.5 frames of drying
Being dry: An [Item] is in a state of dryness (dehydrated, desiccated, dry, . . . )
Becoming dry: An [Entity] loses moisture with the outcome of being in a dry state(dehydrate, dry up, dry, . . . )
Cause to be dry: an [Agent] causes a [Dryee] (. . . ) to become dry (anhydrate, dehumidify,dehydrate, . . . )
«
Dry state
patient 2
ff
ÐÝ
»
—
–
Inchoation
result
«
Dry state
patient 2
ff
fi
ffi
fl
ÐÝ
»
—
—
—
—
—
—
—
—
—
–
Causation
cause
«
Activity
effector 1
ff
effect
»
—
–
Inchoation
result
«
Dry state
patient 2
ff
fi
ffi
fl
fi
ffi
ffi
ffi
ffi
ffi
ffi
ffi
ffi
ffi
fl
S. Pogodalla (with L. Kallmeyer and R. Osswald) A (Hybrid) Logical Approach to Frame Semantics Oct. 9–10, 2019 26 / 30
Compositional Frame Semantics Frame Decomposition
Frame Decomposition and FrameNet (Osswald and Van Valin 2014)
Definitions of FN 1.5 frames of drying
Being dry: An [Item] is in a state of dryness (dehydrated, desiccated, dry, . . . )
Becoming dry: An [Entity] loses moisture with the outcome of being in a dry state(dehydrate, dry up, dry, . . . )
Cause to be dry: an [Agent] causes a [Dryee] (. . . ) to become dry (anhydrate, dehumidify,dehydrate, . . . )
«
Dry state
patient 2
ff
ÐÝ
»
—
–
Inchoation
result
«
Dry state
patient 2
ff
fi
ffi
fl
ÐÝ
»
—
—
—
—
—
—
—
—
—
–
Causation
cause
«
Activity
effector 1
ff
effect
»
—
–
Inchoation
result
«
Dry state
patient 2
ff
fi
ffi
fl
fi
ffi
ffi
ffi
ffi
ffi
ffi
ffi
ffi
ffi
fl
S. Pogodalla (with L. Kallmeyer and R. Osswald) A (Hybrid) Logical Approach to Frame Semantics Oct. 9–10, 2019 26 / 30
Compositional Frame Semantics Frame Decomposition
Frame Decomposition and FrameNet (Osswald and Van Valin 2014)
Definitions of FN 1.5 frames of drying
Being dry: An [Item] is in a state of dryness (dehydrated, desiccated, dry, . . . )
Becoming dry: An [Entity] loses moisture with the outcome of being in a dry state(dehydrate, dry up, dry, . . . )
Cause to be dry: an [Agent] causes a [Dryee] (. . . ) to become dry (anhydrate, dehumidify,dehydrate, . . . )
«
Dry state
patient 2
ff
ÐÝ
»
—
–
Inchoation
result
«
Dry state
patient 2
ff
fi
ffi
fl
ÐÝ
»
—
—
—
—
—
—
—
—
—
–
Causation
cause
«
Activity
effector 1
ff
effect
»
—
–
Inchoation
result
«
Dry state
patient 2
ff
fi
ffi
fl
fi
ffi
ffi
ffi
ffi
ffi
ffi
ffi
ffi
ffi
fl
S. Pogodalla (with L. Kallmeyer and R. Osswald) A (Hybrid) Logical Approach to Frame Semantics Oct. 9–10, 2019 26 / 30
Compositional Frame Semantics Frame Decomposition
Frame Decomposition and FrameNet (Osswald and Van Valin 2014)
Definitions of FN 1.5 frames of drying
Being dry: An [Item] is in a state of dryness (dehydrated, desiccated, dry, . . . )
Becoming dry: An [Entity] loses moisture with the outcome of being in a dry state(dehydrate, dry up, dry, . . . )
Cause to be dry: an [Agent] causes a [Dryee] (. . . ) to become dry (anhydrate, dehumidify,dehydrate, . . . )
«
Dry state
patient 2
ff
ÐÝ
»
—
–
Inchoation
result
«
Dry state
patient 2
ff
fi
ffi
fl
ÐÝ
»
—
—
—
—
—
—
—
—
—
–
Causation
cause
«
Activity
effector 1
ff
effect
»
—
–
Inchoation
result
«
Dry state
patient 2
ff
fi
ffi
fl
fi
ffi
ffi
ffi
ffi
ffi
ffi
ffi
ffi
ffi
fl
S. Pogodalla (with L. Kallmeyer and R. Osswald) A (Hybrid) Logical Approach to Frame Semantics Oct. 9–10, 2019 26 / 30
Conclusion
Conclusion
Frame as a data-structure representing a situation
The semantic structure associated to a syntactic structure is a hybrid logicalformula that need to be further interpreted
Quantification does not belong to the frame language (contrary to Baldridge andKruijff 2002 or Kallmeyer and Richter 2014)
Inference
Link between dependency structures and logical representation (AMR,ECL—I. A. Mel’cuk, Clas, and Polguere 1995; I. Mel’cuk and Polguere 2018—, etc.)
Not variable free. . . But discourse referents are now made available!
S. Pogodalla (with L. Kallmeyer and R. Osswald) A (Hybrid) Logical Approach to Frame Semantics Oct. 9–10, 2019 27 / 30
Conclusion
Bibliographie I
Areces, Carlos and Balder ten Cate (2007). “Hybrid logics”. In: Handbook ofModal Logic. Ed. by Patrick Blackburn, Johan Van Benthem, and Frank Wolter.Vol. 3. Studies in Logic and Practical Reasoning. Elsevier. Chap. 14, pp. 821–868.doi: 10.1016/S1570-2464(07)80017-6.
Baldridge, Jason and Geert-Jan Kruijff (2002). “Coupling CCG and Hybrid LogicDependency Semantics”. In: Proceedings of 40th Annual Meeting of theAssociation for Computational Linguistics (ACL 2002). Philadelphia, Pennsylvania,USA: Association for Computational Linguistics, pp. 319–326. doi:10.3115/1073083.1073137. ACL anthology: P02-1041.
Barsalou, Lawrence (1992). “Frames, concepts, and conceptual fields”. In:Frames, fields, and contrasts: New essays in semantic and lexical organization.Ed. by Adrienne Lehrer and Eva Feder Kittey. Hillsdale: Lawrence ErlbaumAssociates, pp. 21–74.
Blackburn, Patrick (1993). “Modal Logic and Attribute Value Structures”. In:Diamonds and Defaults. Ed. by Maarten de Rijke. Vol. 229. Synthese Library.Springer Netherlands, pp. 19–65. isbn: 978-90-481-4286-6. doi:10.1007/978-94-015-8242-1_2.
Fillmore, Charles J. (1982). “Frame semantics”. In: Linguistics in the MorningCalm. Seoul, South Korea: Hanshin Publishing Co., pp. 111–137.
S. Pogodalla (with L. Kallmeyer and R. Osswald) A (Hybrid) Logical Approach to Frame Semantics Oct. 9–10, 2019 28 / 30
Conclusion
Bibliographie II
Gamerschlag, Thomas et al., eds. (2014). Frames and Concept Types. Vol. 94.Studies in Linguistics and Philosophy. Springer International Publishing. doi:10.1007/978-3-319-01541-5.
Kallmeyer, Laura and Rainer Osswald (2013). “Syntax-Driven Semantic FrameComposition in Lexicalized Tree Adjoining Grammars”. In: Journal of LanguageModelling 1.2, pp. 267–330. doi: 10.15398/jlm.v1i2.61.
Kallmeyer, Laura and Frank Richter (2014). “Quantifiers in Frame Semantics”. In:Formal Grammar. Ed. by Glyn Morrill et al. Vol. 8612. Lecture Notes in ComputerScience. Springer Berlin Heidelberg, pp. 69–85. doi:10.1007/978-3-662-44121-3_5.
Lobner, Sebastian (2014). “Evidence for frames from human language”. In:Frames and Concept Types. Applications in Language and Philosophy. Ed. byThomas Gamerschlag et al. Vol. 94. Studies in Linguistics and Philosophy 94.Dordrecht: Springer International Publishing. Chap. 2, pp. 23–67. isbn:978-3-319-01540-8. doi: 10.1007/978-3-319-01541-5_2.
Mel’cuk, Igor A., Andre Clas, and Alain Polguere (1995). Introduction a la
lexicologie explicative et combinatoire. Louvain-la-Neuve: Editions Duculot.
S. Pogodalla (with L. Kallmeyer and R. Osswald) A (Hybrid) Logical Approach to Frame Semantics Oct. 9–10, 2019 29 / 30
Conclusion
Bibliographie III
Mel’cuk, Igor and Alain Polguere (2018). “Theory and Practice of LexicographicDefinition”. In: Journal of Cognitive Science 19.4, pp. 417–470. hal open archive:halshs-02089593. url:http://cogsci.snu.ac.kr/jcs/index.php/issues/?uid=263&mod=document.
Minsky, Marvin (1974). A Framework for Representing Knowledge. Tech. rep. 306.MIT, AI Laboratory Memo. url: http://hdl.handle.net/1721.1/6089.
Osswald, Rainer and Robert D. Van Valin Jr. (2014). “FrameNet, FrameStructure, and the Syntax-Semantics Interface”. In: Frames and Concept Types.Ed. by Thomas Gamerschlag et al. Vol. 94. Studies in Linguistics and Philosophy.Springer International Publishing. Chap. 6, pp. 125–156. isbn: 978-3-319-01540-8.doi: 10.1007/978-3-319-01541-5_6.
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