Types from =1
Deterministic systems & languages
Types, tokens & examples
Varying alphabets & signatures
Conclusion
Types from frames as finite automata Tim Fernando Formal Grammar 2015, Barcelona
animate agent
smash
smash agent theme
animate concrete
theme
concrete
{smash, agent animate, theme concrete}
1 / 24
Types? Petersen (sortal) λe (smash(e) ∧ animate(agent(e)) ∧ concrete(theme(e)))
{smash, agent animate, theme concrete}
Or Muskens λyx λf ∃e [smash e ◦ agent ex ◦ theme ey ]f
[[L]] :=
T
s∈L
domain([[s]])
domain([[smash]]) ∩ domain([[agent]]; [[animate]]) ∩ domain([[theme]]; [[concrete]])
[[]] := λx.x [[sa]] := [[s]]; [[a]] = λx.[[a]]([[s]](x))
Record types?
agent theme
= =
b c
:
agent theme
: :
animate concrete
iff b : animate and c : concrete P λe (smash(e) ∧ animate(agent(e)) ∧ concrete(theme(e))) Cooper’s meaning function (λr : bg ) ϕ with agent : Ind - background bg = (presuppositions) theme : Ind p1 : smash(r ) dependent on - type ϕ = p2 : animate(r .agent) r : bg p3 : concrete(r .theme) bg ≈ signature/state q = {agent, theme, } ϕ ≈ language {smash, agent animate, theme concrete} ∪ q
Finite-state calculus Minimal DFA (Myhill-Nerode) via Brzozowski derivatives La La := {s | as ∈ L} Identity as indiscernibility (Leibniz) wrt Hennessy-Milner 1985 (Blackburn 1993)
L =
X
aLa + o(L) (Taylor series – Conway)
a∈Σ
Open-endedness of signatures (institution, Goguen & Burstall) Z Sign = Q (Grothendieck construction)
Link frames with timelines as strings (runs of automata)
Open-endedness (1) Jones did it slowly, deliberately, in the bathroom, with a knife, at midnight. (Davidson 1967) a q0 ;i qi (3) a1 q1 (2) agent jones . . slow . how deliberate ak qk h i where bathroom h i when midnight h i with-what knife - set Σ of labels a a
- relations ; ⊆ Q × Q for a ∈ Σ
Deterministic systems & languages
Types, tokens & examples
1
Deterministic systems & languages
2
Types, tokens & examples
3
Varying alphabets & signatures
4
Conclusion
Varying alphabets & signatures
Conclusion
6 / 24
Hennessy-Milner & traces Σ-deterministic system δ : Q × Σ + Q
a
q → δ(q, a)
trace δ (q) := domain(δq ) ⊆ Σ∗ where δq : Σ∗ + Q, 7→ q, sa 7→ δ(δq (s), a) (ΦΣ )
↓ a0 δq (aa0 )
ϕ ::= > | haiϕ | ϕ ∧ ϕ0 | ¬ϕ
(a ∈ Σ)
q |= haiϕ iff (q, a) ∈ domain(δ) and δ(q, a) |= ϕ iff a ∈ trace δ (q) and δq (a) |= ϕ
hiϕ := ϕ hasiϕ := haihsiϕ q |= hsiϕ iff s ∈ trace δ (q) and δq (s) |= ϕ
Identity of indiscernibles (Leibniz) trace δ (q) = {s ∈ Σ∗ | q |= hsi>} Does |= depend on more than trace δ (q)? ΦΣ (q) := {ϕ ∈ ΦΣ | q |= ϕ} trace δ (q) = {s ∈ Σ∗ | hsi> ∈ ΦΣ (q)}
Fact.
ΦΣ (q) = ΦΣ (q 0 )
iff trace δ (q) = trace δ (q 0 )
Holds also with ΦΣ closed under 3 where
q |= 3ϕ iff (∃s ∈ trace δ (q)) δq (s) |= ϕ _ 3ϕ ≈ hsiϕ s∈traceδ (q)
Components as derivatives (Brzozowski) Ls := {s 0 | ss 0 ∈ L}
L = L Lsa = (Ls )a
L = {s | ∈ Ls } aa0 a00 ∈ L iff a0 a00 ∈ La iff a00 ∈ Laa0 iff ∈ Laa0 a00 a0
a
a00
L → La → Laa0 → Laa0 a00
Minimal DFA & finality For all s, s 0 ∈ Σ∗ and L ⊆ Σ∗ , Ls = Ls 0 iff (∀w ∈ Σ∗ ) (sw ∈ L iff s 0 w ∈ L) so that the Myhill-Nerode Theorem says: L is regular
iff {Ls | s ∈ Σ∗ } is finite.
Finality: given a relation ; ⊆ Q × Σ × Q and q ∈ Q, let a
a
a
1 2 n L := {a1 · · · an ∈ Σ∗ | q ∈ domain(;; ;; · · · ; ;)}
for a unique morphism to {Ls | s ∈ L} [ a1 a 2 a {(q 0 , La1 ···an ) | q ;; ;; · · · ; ;n q 0 } a1 ···an ∈Σ∗
from the subset of Q accessible from q via ;.
Prefix-closed languages & coderivatives Fact. For L ⊆ Σ∗ , L =
X
aLa + o(L)
where o(L) :=
a∈Σ
and (i) (ii) (iii)
∅
if ∈ L otherwise
the following are equivalent L = trace δ (q) for some δ : Q × Σ + Q and q ∈ Q L is prefix-closed (s ∈ L whenever sa ∈ L) and non-empty ∈ L = pref(L) where pref(L) := {s | Ls 6= ∅}.
a-coderivative of L Fact.
aL
:= {s | sa ∈ L}
For any L ⊆ Σ∗ and a 6∈ Σ, L =
Deterministic systems & languages
a pref(La).
Types, tokens & examples
1
Deterministic systems & languages
2
Types, tokens & examples
3
Varying alphabets & signatures
4
Conclusion
Varying alphabets & signatures
Conclusion
12 / 24
Types as formulas Encode a type t as wff (t) — e.g., hat i> a particular a as singleton type {a} subtypeΣ (t, t 0 ) := {¬hsi(wff (t) ∧ ¬wff (t 0 )) | s ∈ Σ∗ } ≡ ¬3(wff (t) ∧ ¬wff (t 0 )) ≡ 2(wff (t) ⊃ wff (t 0 )) subtype as entailment inΣ (a, t) := subtypeΣ ({a}, t) ∪ nominalΣ (wff ({a})) {z } | sortal presupposition in Hybrid Logic nominalΣ (ϕ) := {¬(hs 0 i(ϕ ∧ hsi>) ∧ hs 00 i(ϕ ∧ ¬hsi>)) | s, s 0 , s 00 ∈ Σ∗ } ≡ {3(ϕ ∧ ψ) ⊃ 2(ϕ ⊃ ψ) | ψ ∈ ΦΣ }
Singletons, terminals & record labels For L ⊆ Σ∗ with aL 6∈ Σ singletonΣ (L) := {2(haL i> ⊃ hsi>) | s ∈ L} ∪ {2(haL i> ⊃ ¬hsi>) | s ∈ Σ∗ − L} L 7→ L + aL
terminalΣ (a) := {¬hsabi> | s ∈ Σ∗ and b ∈ Σ} ^ ≡ ¬3habi> b∈Σ
hsmashi> ∧ hagentihanimatei> ∧ hthemeihconcretei> L 7→
V
s∈L hsi>
smash agent theme
animate concrete
Record types from relations
x loc e
x L( loc e
Real Loc temp(loc,x)
: : : : : :
Real ) Loc temp(loc,x)
[[temp(loc,x)]]r = √ {(c, ) | [[temp]]([[loc]]r , [[x]]r , c)}
=
x L(Real) + loc L(Loc) + e L(temp(loc,x)) +
with T ∈ L(T ) and
x loc L( e l
: : : :
for T ∈ {Real, Loc, temp(loc,x)}
Real Loc ) temp(loc,x) R
Deterministic systems & languages
=
Types, tokens & examples
1
Deterministic systems & languages
2
Types, tokens & examples
3
Varying alphabets & signatures
4
Conclusion
x L(Real) + loc L(Loc) + e L(temp(loc,x)) + l L(R) +
Varying alphabets & signatures
Conclusion
16 / 24
A monster A-deterministic system δˆ Fin(A) := {Σ ⊆ A | Σ is finite} For X ∈ Fin(A) ∪ {A}, an X -state is a non-empty prefix-closed subset q of X ∗ δˆ = {(q, a, qa ) | q is an A-state and a ∈ q ∩ A} making δˆq = {(s, qs ) | s ∈ q} (sen(Σ))
ϕ ::= > | haiϕ | ϕ ∧ ϕ0 | ¬ϕ | 3Y ϕ
(a ∈ Σ, Y ⊆ Σ)
q |= 3Y ϕ iff (∃s ∈ q ∩ Y ∗ ) qs |= ϕ
Shorten 3Σ to 3
Σ-reducts for satisfaction For Σ ⊆ Σ0 ∈ Fin(A) and Σ0 -state q, sen(Σ) ⊆ sen(Σ0 ) q ∩ Σ∗ is a Σ-state qs is a Σ0 -state,
for s ∈ q
Fact. For every Σ ∈ Fin(A), ϕ ∈ sen(Σ) and A-state q, q |= ϕ iff
q ∩ Σ∗ |= ϕ
and if, moreover, s ∈ q ∩ Σ∗ , then q |= hsiϕ iff (q ∩ Σ∗ )s |= ϕ.
The functor Q : Fin(A)op → Cat For Σ ∈ Fin(A), Q(Σ) is the category with object non-empty prefix-closed q ⊆ Σ∗ morphisms (q, s) from q to qs , for q ∈ |Q(Σ)| and s ∈ q (q, s); (qs , s 0 ) = (q, ss 0 ) with identities (q, ) Q(Σ0 , Σ) : Q(Σ0 ) → Q(Σ)
for Σ ⊆ Σ0 ∈ Fin(A)
q 7→ q ∩ Σ∗ (q, s) 7→ (q ∩ Σ∗ , πΣ (s)) where πΣ (s) is the longest prefix of s in Σ∗ πΣ () := a πΣ (s) πΣ (as) :=
R
if a ∈ Σ otherwise.
Q (Grothendieck) & institutions (Goguen & Burstall) Signop =
R
Q
- objects (Σ, q) where Σ ∈ Fin(A) and q ∈ |Q(Σ)| - morphisms from (Σ0 , q 0 ) to (Σ, q) are pairs ((Σ0 , Σ), (q 00 , s)) of Fin(A)op -morphisms (Σ0 , Σ) and Q(Σ)-morphisms (q 00 , s) s.t. q 00 = q 0 ∩ Σ∗ and q = qs00 sen : Sign → Set - sen(Σ, q) := sen(Σ) - sen((Σ0 , Σ), (q 00 , s)) : ϕ 7→ hsiϕ Mod : Signop → Cat - |Mod(Σ, q)| := {q 0 ∈ |Q(Σ)| : q ⊆ q 0 } - Mod((Σ0 , Σ), (q 00 , s)) : qˆ 7→ (ˆ q ∩ Σ∗ )s
Deterministic systems & languages
Types, tokens & examples
1
Deterministic systems & languages
2
Types, tokens & examples
3
Varying alphabets & signatures
4
Conclusion
Varying alphabets & signatures
Conclusion
21 / 24
Back to smash animate agent
P
theme
concrete
λe (smash(e) ∧ animate(agent(e)) ∧ concrete(theme(e)))
C
smash
λr :
agent : theme : | {z bg
bg ≈ signature
p Ind 1 p2 Ind p3 } |
: : :
smash(r ) animate(r .agent) concrete(r .theme) {z } ϕ
Σ = {agent, theme, smash, animate, concrete} q = {agent, theme, }
ϕ ≈ language {smash, agent animate, theme concrete} ∪ q
Main ideas Centralized abstraction - from DFA’s initial state Identity of indiscernibles (Leibniz) - relativize to finite set Σ of attributes Open-endedness - let Σ vary over Fin(A) within an institution • Run many finite automata ; timeline causal temporal
≈ ≈Σ
mechanism language ≈Σ timeline string type generic ≈Σ particular episodic
See: Tense & aspect chapter of Lappin & Fox’s Semantics Handbk ESSLLI course: Finite-state methods for subatomic semantics
Strings & mechanisms Jan Feb · · · Dec Jan,d1 Jan,d2 · · · Jan,d31 Feb,d1 · · · Dec,d31
months in a year + d1,d2,. . . d31
tense R S R,S
aspect E,R E R
it rained it has rained
E,R S E R,S
soup cool in an hour
x, d ≤ sDg d ≤ sDg hour(x), sDg < d
soup cool for an hour
x [w]sDg↓ hour(x), [w]sDg↓
Barsalou 1999
(2008: situated simulation)
two levels of structure are proposed: a deep set of generating mechanisms produces an infinite set of surface images. . . . Mental models tend not to address the underlying generative mechanisms that produce a family of related simulations.
Types, tokens & examples
Varying alphabets & signatures
Conclusion
Types from frames as finite automata Tim Fernando Formal Grammar 2015, Barcelona
animate agent
smash
smash agent theme
animate concrete
theme
concrete
{smash, agent animate, theme concrete}
1 / 24
Types? Petersen (sortal) λe (smash(e) ∧ animate(agent(e)) ∧ concrete(theme(e)))
{smash, agent animate, theme concrete}
Or Muskens λyx λf ∃e [smash e ◦ agent ex ◦ theme ey ]f
[[L]] :=
T
s∈L
domain([[s]])
domain([[smash]]) ∩ domain([[agent]]; [[animate]]) ∩ domain([[theme]]; [[concrete]])
[[]] := λx.x [[sa]] := [[s]]; [[a]] = λx.[[a]]([[s]](x))
Record types?
agent theme
= =
b c
:
agent theme
: :
animate concrete
iff b : animate and c : concrete P λe (smash(e) ∧ animate(agent(e)) ∧ concrete(theme(e))) Cooper’s meaning function (λr : bg ) ϕ with agent : Ind - background bg = (presuppositions) theme : Ind p1 : smash(r ) dependent on - type ϕ = p2 : animate(r .agent) r : bg p3 : concrete(r .theme) bg ≈ signature/state q = {agent, theme, } ϕ ≈ language {smash, agent animate, theme concrete} ∪ q
Finite-state calculus Minimal DFA (Myhill-Nerode) via Brzozowski derivatives La La := {s | as ∈ L} Identity as indiscernibility (Leibniz) wrt Hennessy-Milner 1985 (Blackburn 1993)
L =
X
aLa + o(L) (Taylor series – Conway)
a∈Σ
Open-endedness of signatures (institution, Goguen & Burstall) Z Sign = Q (Grothendieck construction)
Link frames with timelines as strings (runs of automata)
Open-endedness (1) Jones did it slowly, deliberately, in the bathroom, with a knife, at midnight. (Davidson 1967) a q0 ;i qi (3) a1 q1 (2) agent jones . . slow . how deliberate ak qk h i where bathroom h i when midnight h i with-what knife - set Σ of labels a a
- relations ; ⊆ Q × Q for a ∈ Σ
Deterministic systems & languages
Types, tokens & examples
1
Deterministic systems & languages
2
Types, tokens & examples
3
Varying alphabets & signatures
4
Conclusion
Varying alphabets & signatures
Conclusion
6 / 24
Hennessy-Milner & traces Σ-deterministic system δ : Q × Σ + Q
a
q → δ(q, a)
trace δ (q) := domain(δq ) ⊆ Σ∗ where δq : Σ∗ + Q, 7→ q, sa 7→ δ(δq (s), a) (ΦΣ )
↓ a0 δq (aa0 )
ϕ ::= > | haiϕ | ϕ ∧ ϕ0 | ¬ϕ
(a ∈ Σ)
q |= haiϕ iff (q, a) ∈ domain(δ) and δ(q, a) |= ϕ iff a ∈ trace δ (q) and δq (a) |= ϕ
hiϕ := ϕ hasiϕ := haihsiϕ q |= hsiϕ iff s ∈ trace δ (q) and δq (s) |= ϕ
Identity of indiscernibles (Leibniz) trace δ (q) = {s ∈ Σ∗ | q |= hsi>} Does |= depend on more than trace δ (q)? ΦΣ (q) := {ϕ ∈ ΦΣ | q |= ϕ} trace δ (q) = {s ∈ Σ∗ | hsi> ∈ ΦΣ (q)}
Fact.
ΦΣ (q) = ΦΣ (q 0 )
iff trace δ (q) = trace δ (q 0 )
Holds also with ΦΣ closed under 3 where
q |= 3ϕ iff (∃s ∈ trace δ (q)) δq (s) |= ϕ _ 3ϕ ≈ hsiϕ s∈traceδ (q)
Components as derivatives (Brzozowski) Ls := {s 0 | ss 0 ∈ L}
L = L Lsa = (Ls )a
L = {s | ∈ Ls } aa0 a00 ∈ L iff a0 a00 ∈ La iff a00 ∈ Laa0 iff ∈ Laa0 a00 a0
a
a00
L → La → Laa0 → Laa0 a00
Minimal DFA & finality For all s, s 0 ∈ Σ∗ and L ⊆ Σ∗ , Ls = Ls 0 iff (∀w ∈ Σ∗ ) (sw ∈ L iff s 0 w ∈ L) so that the Myhill-Nerode Theorem says: L is regular
iff {Ls | s ∈ Σ∗ } is finite.
Finality: given a relation ; ⊆ Q × Σ × Q and q ∈ Q, let a
a
a
1 2 n L := {a1 · · · an ∈ Σ∗ | q ∈ domain(;; ;; · · · ; ;)}
for a unique morphism to {Ls | s ∈ L} [ a1 a 2 a {(q 0 , La1 ···an ) | q ;; ;; · · · ; ;n q 0 } a1 ···an ∈Σ∗
from the subset of Q accessible from q via ;.
Prefix-closed languages & coderivatives Fact. For L ⊆ Σ∗ , L =
X
aLa + o(L)
where o(L) :=
a∈Σ
and (i) (ii) (iii)
∅
if ∈ L otherwise
the following are equivalent L = trace δ (q) for some δ : Q × Σ + Q and q ∈ Q L is prefix-closed (s ∈ L whenever sa ∈ L) and non-empty ∈ L = pref(L) where pref(L) := {s | Ls 6= ∅}.
a-coderivative of L Fact.
aL
:= {s | sa ∈ L}
For any L ⊆ Σ∗ and a 6∈ Σ, L =
Deterministic systems & languages
a pref(La).
Types, tokens & examples
1
Deterministic systems & languages
2
Types, tokens & examples
3
Varying alphabets & signatures
4
Conclusion
Varying alphabets & signatures
Conclusion
12 / 24
Types as formulas Encode a type t as wff (t) — e.g., hat i> a particular a as singleton type {a} subtypeΣ (t, t 0 ) := {¬hsi(wff (t) ∧ ¬wff (t 0 )) | s ∈ Σ∗ } ≡ ¬3(wff (t) ∧ ¬wff (t 0 )) ≡ 2(wff (t) ⊃ wff (t 0 )) subtype as entailment inΣ (a, t) := subtypeΣ ({a}, t) ∪ nominalΣ (wff ({a})) {z } | sortal presupposition in Hybrid Logic nominalΣ (ϕ) := {¬(hs 0 i(ϕ ∧ hsi>) ∧ hs 00 i(ϕ ∧ ¬hsi>)) | s, s 0 , s 00 ∈ Σ∗ } ≡ {3(ϕ ∧ ψ) ⊃ 2(ϕ ⊃ ψ) | ψ ∈ ΦΣ }
Singletons, terminals & record labels For L ⊆ Σ∗ with aL 6∈ Σ singletonΣ (L) := {2(haL i> ⊃ hsi>) | s ∈ L} ∪ {2(haL i> ⊃ ¬hsi>) | s ∈ Σ∗ − L} L 7→ L + aL
terminalΣ (a) := {¬hsabi> | s ∈ Σ∗ and b ∈ Σ} ^ ≡ ¬3habi> b∈Σ
hsmashi> ∧ hagentihanimatei> ∧ hthemeihconcretei> L 7→
V
s∈L hsi>
smash agent theme
animate concrete
Record types from relations
x loc e
x L( loc e
Real Loc temp(loc,x)
: : : : : :
Real ) Loc temp(loc,x)
[[temp(loc,x)]]r = √ {(c, ) | [[temp]]([[loc]]r , [[x]]r , c)}
=
x L(Real) + loc L(Loc) + e L(temp(loc,x)) +
with T ∈ L(T ) and
x loc L( e l
: : : :
for T ∈ {Real, Loc, temp(loc,x)}
Real Loc ) temp(loc,x) R
Deterministic systems & languages
=
Types, tokens & examples
1
Deterministic systems & languages
2
Types, tokens & examples
3
Varying alphabets & signatures
4
Conclusion
x L(Real) + loc L(Loc) + e L(temp(loc,x)) + l L(R) +
Varying alphabets & signatures
Conclusion
16 / 24
A monster A-deterministic system δˆ Fin(A) := {Σ ⊆ A | Σ is finite} For X ∈ Fin(A) ∪ {A}, an X -state is a non-empty prefix-closed subset q of X ∗ δˆ = {(q, a, qa ) | q is an A-state and a ∈ q ∩ A} making δˆq = {(s, qs ) | s ∈ q} (sen(Σ))
ϕ ::= > | haiϕ | ϕ ∧ ϕ0 | ¬ϕ | 3Y ϕ
(a ∈ Σ, Y ⊆ Σ)
q |= 3Y ϕ iff (∃s ∈ q ∩ Y ∗ ) qs |= ϕ
Shorten 3Σ to 3
Σ-reducts for satisfaction For Σ ⊆ Σ0 ∈ Fin(A) and Σ0 -state q, sen(Σ) ⊆ sen(Σ0 ) q ∩ Σ∗ is a Σ-state qs is a Σ0 -state,
for s ∈ q
Fact. For every Σ ∈ Fin(A), ϕ ∈ sen(Σ) and A-state q, q |= ϕ iff
q ∩ Σ∗ |= ϕ
and if, moreover, s ∈ q ∩ Σ∗ , then q |= hsiϕ iff (q ∩ Σ∗ )s |= ϕ.
The functor Q : Fin(A)op → Cat For Σ ∈ Fin(A), Q(Σ) is the category with object non-empty prefix-closed q ⊆ Σ∗ morphisms (q, s) from q to qs , for q ∈ |Q(Σ)| and s ∈ q (q, s); (qs , s 0 ) = (q, ss 0 ) with identities (q, ) Q(Σ0 , Σ) : Q(Σ0 ) → Q(Σ)
for Σ ⊆ Σ0 ∈ Fin(A)
q 7→ q ∩ Σ∗ (q, s) 7→ (q ∩ Σ∗ , πΣ (s)) where πΣ (s) is the longest prefix of s in Σ∗ πΣ () := a πΣ (s) πΣ (as) :=
R
if a ∈ Σ otherwise.
Q (Grothendieck) & institutions (Goguen & Burstall) Signop =
R
Q
- objects (Σ, q) where Σ ∈ Fin(A) and q ∈ |Q(Σ)| - morphisms from (Σ0 , q 0 ) to (Σ, q) are pairs ((Σ0 , Σ), (q 00 , s)) of Fin(A)op -morphisms (Σ0 , Σ) and Q(Σ)-morphisms (q 00 , s) s.t. q 00 = q 0 ∩ Σ∗ and q = qs00 sen : Sign → Set - sen(Σ, q) := sen(Σ) - sen((Σ0 , Σ), (q 00 , s)) : ϕ 7→ hsiϕ Mod : Signop → Cat - |Mod(Σ, q)| := {q 0 ∈ |Q(Σ)| : q ⊆ q 0 } - Mod((Σ0 , Σ), (q 00 , s)) : qˆ 7→ (ˆ q ∩ Σ∗ )s
Deterministic systems & languages
Types, tokens & examples
1
Deterministic systems & languages
2
Types, tokens & examples
3
Varying alphabets & signatures
4
Conclusion
Varying alphabets & signatures
Conclusion
21 / 24
Back to smash animate agent
P
theme
concrete
λe (smash(e) ∧ animate(agent(e)) ∧ concrete(theme(e)))
C
smash
λr :
agent : theme : | {z bg
bg ≈ signature
p Ind 1 p2 Ind p3 } |
: : :
smash(r ) animate(r .agent) concrete(r .theme) {z } ϕ
Σ = {agent, theme, smash, animate, concrete} q = {agent, theme, }
ϕ ≈ language {smash, agent animate, theme concrete} ∪ q
Main ideas Centralized abstraction - from DFA’s initial state Identity of indiscernibles (Leibniz) - relativize to finite set Σ of attributes Open-endedness - let Σ vary over Fin(A) within an institution • Run many finite automata ; timeline causal temporal
≈ ≈Σ
mechanism language ≈Σ timeline string type generic ≈Σ particular episodic
See: Tense & aspect chapter of Lappin & Fox’s Semantics Handbk ESSLLI course: Finite-state methods for subatomic semantics
Strings & mechanisms Jan Feb · · · Dec Jan,d1 Jan,d2 · · · Jan,d31 Feb,d1 · · · Dec,d31
months in a year + d1,d2,. . . d31
tense R S R,S
aspect E,R E R
it rained it has rained
E,R S E R,S
soup cool in an hour
x, d ≤ sDg d ≤ sDg hour(x), sDg < d
soup cool for an hour
x [w]sDg↓ hour(x), [w]sDg↓
Barsalou 1999
(2008: situated simulation)
two levels of structure are proposed: a deep set of generating mechanisms produces an infinite set of surface images. . . . Mental models tend not to address the underlying generative mechanisms that produce a family of related simulations.
