Functional Reachability Luke Ong and Nikos Tzevelekos University of Oxford Shonan, September 2011
What this talk is about Reachability in functional computation. • Consider a term M of a higher-order functional programming language. • Now consider a point p inside M . • Is there a program context C such that computation of C[M ] reaches p?
Functional Reachability – 2
What this talk is about Reachability in functional computation. • Consider a term M of a higher-order functional programming language. • Now consider a point p inside M . •
Is there a program context C such that computation of C[M ] reaches p?
Functional Reachability – 2
What this talk is about Reachability in functional computation. • Consider a term M of a higher-order functional programming language. • Now consider a point p inside M . •
Is there a program context C such that computation of C[M ] reaches p?
Surprisingly, (Contextual) Reachability per se had not been studied in HO functional languages. Idea: Use Games, Traversals, Automata.
Functional Reachability – 2
Relevant work • Control Flow Analysis. Compute at compile time the flow of control that is going to happen at run time. • Reynolds (’70), Jones (’80), Shivers (’90), many more people in this room, . . . Using game semantics: Malacaria & Hankin (late 90’s). • CFA > Reach: more general. Reach > CFA: open vs closed world approach.
Functional Reachability – 3
Relevant work • Control Flow Analysis. Compute at compile time the flow of control that is going to happen at run time. • Reynolds (’70), Jones (’80), Shivers (’90), many more people in this room, . . . Using game semantics: Malacaria & Hankin (late 90’s). • CFA > Reach: more general. Reach > CFA: open vs closed world approach. • Useless code detection. • Strictness analysis, etc. Functional Reachability – 3
PCF The examined language: (binary) PCF. • lambda-calculus, • Boolean base type, • recursion at all types.
A, B ::= o | A → B v ::= t | f M, N ::= v | x | λx.M | M N | if M N1 N2 | YA
Functional Reachability – 4
PCF
A, B ::= o | A → B v ::= t | f M, N ::= v | x | λx.M | M N | if M N1 N2 | YA
Functional Reachability – 5
PCF
A, B ::= o | A → B v ::= t | f M, N ::= v | x | λx.M | M N | if M N1 N2 | YA (λx.M )N → M {N/x} YM → M (YM )
if t → λxy.x if f → λxy.y
M → N =⇒ E[M ] → E[N ] E ::= [−] | E M | if E Functional Reachability – 5
PCF
A, B ::= o | A → B
Call-by-name λ-calculus +if + Y
v ::= t | f M, N ::= v | x | λx.M | M N | if M N1 N2 | YA • Write (A1 , . . . , An , o) for A1 → · · · → An → o. • Divergence definable, e.g. ⊥ := Yo (λx.x).
Functional Reachability – 5
PCF
A, B ::= o | A → B
Call-by-name λ-calculus +if + Y
v ::= t | f M, N ::= v | x | λx.M | M N | if M N1 N2 | YA • Write (A1 , . . . , An , o) for A1 → · · · → An → o. • Divergence definable, e.g. ⊥ := Yo (λx.x). • Finitary restrictions (i.e. no Y): fPCF
M, N ::= v | x | λx.M | M N | if M N1 N2
fPCF⊥
M, N ::= v | x | λx.M | M N | if M N1 N2 | ⊥ Functional Reachability – 5
Reachability Using a Context Lemma we can restate Reachability as:
• Given a closed PCF-term M : (A1 , ..., An , o) and a coloured subterm L of M , • Are there closed PCF-terms N1 : A1 , . . . , Nn : An and a coloured term L′ such that
~ ։ E[L′ ] ? MN
Functional Reachability – 6
Reachability Using a Context Lemma we can restate Reachability as:
• Given a closed PCF-term M : (A1 , ..., An , o) and a coloured subterm L of M , • Are there closed PCF-terms N1 : A1 , . . . , Nn : An and a coloured term L′ such that
~ ։ E[L′ ] ? MN We can make things simpler
Functional Reachability – 6
PCF-with-error: PCF⋆ • Include an error constant: o = {t, f, ⋆} • New rule: if ⋆ → λxy.⋆
(i.e. E[⋆] ։ ⋆)
Functional Reachability – 7
PCF-with-error: PCF⋆ • Include an error constant: o = {t, f, ⋆} • New rule: if ⋆ → λxy.⋆
(i.e. E[⋆] ։ ⋆)
⋆-Reachability: • Given a closed PCF⋆ -term M with exactly one ⋆ ,
~ ։ ⋆? • Are there closed PCF-terms N1 , ..., Nn such that M N
Functional Reachability – 7
PCF-with-error: PCF⋆ • Include an error constant: o = {t, f, ⋆} • New rule: if ⋆ → λxy.⋆
(i.e. E[⋆] ։ ⋆)
⋆-Reachability: • Given a closed PCF⋆ -term M with exactly one ⋆ ,
~ ։ ⋆? • Are there closed PCF-terms N1 , ..., Nn such that M N
Reachability ∼ = ⋆-Reachability
Functional Reachability – 7
Reach template v-Reach [L1 , L2 ]:
v ∈ {t, f, ⋆} and L1 , L2 ⊆ PCF⋆
• Given a closed L1 -term M ,
~ ։ v? • Are there closed L2 -terms N1 , ..., Nn such that M N
E.g.
⋆-Reachability = ⋆-Reach [PCF1⋆ , PCF]
Functional Reachability – 8
Reach template v-Reach [L1 , L2 ]:
v ∈ {t, f, ⋆} and L1 , L2 ⊆ PCF⋆
• Given a closed L1 -term M ,
~ ։ v? • Are there closed L2 -terms N1 , ..., Nn such that M N
E.g.
⋆-Reachability = ⋆-Reach [PCF1⋆ , PCF]
v-Reach [L, PCF] = v-Reach [L, fPCF]
Functional Reachability – 8
Reach template v-Reach [L1 , L2 ]:
v ∈ {t, f, ⋆} and L1 , L2 ⊆ PCF⋆
• Given a closed L1 -term M ,
~ ։ v? • Are there closed L2 -terms N1 , ..., Nn such that M N
E.g.
⋆-Reachability = ⋆-Reach [PCF1⋆ , PCF]
v-Reach [L, PCF] = v-Reach [L, fPCF] From [Loader]: Observational equivalence in fPCF⊥ is undecidable. therefore: t-Reach [fPCF⊥ , fPCF] is undecidable. Functional Reachability – 8
Undecidability The following problems are undecidable. • t-Reach [fPCF⊥ , fPCF] • ⋆-Reach [fPCF1⊥⋆ , fPCF] • ⋆-Reachability, i.e. ⋆-Reach [PCF1⋆ , PCF] • Reachability
Functional Reachability – 9
Undecidability The following problems are undecidable. • t-Reach [fPCF⊥ , fPCF] • ⋆-Reach [fPCF1⊥⋆ , fPCF] • ⋆-Reachability, i.e. ⋆-Reach [PCF1⋆ , PCF] • Reachability Not all is lost • Reachability for finitary M ? • ⋆-Reach [fPCF1⋆ , fPCF] ? • ⋆-Reach [fPCF⋆ , fPCF] ? Functional Reachability – 9
Our approach We focus on v-Reach [fPCF⋆ , fPCF]
Computations of fPCF⋆ -term P : o O
Traversals over its computation tree, λ(P ) O
Runs of an Alternating Tree Automaton (ATA) on λ(P )
Functional Reachability – 10
Our approach We focus on v-Reach [fPCF⋆ , fPCF]
Computations of fPCF⋆ -term P : o O
Traversals over its computation tree, λ(P ) O
Runs of an Alternating Tree Automaton (ATA) on λ(P )
P ։ v iff an ATA accepts λ(P ) on initial state with value v Functional Reachability – 10
Computation trees Starting from a fPCF⋆ -term M , • take its η-long form, • add application symbols (@), • view the result as a tree, λ(M ).
Functional Reachability – 11
Computation trees Starting from a fPCF⋆ -term M ,
λ
• take its η-long form,
• add application symbols (@),
?? NNN NN
λf x λ
λΦz
• view the result as a tree, λ(M ). ( λΦz. Φ(λy. ify ⋆ z)t ) (λf x. f x) t
@ ?NNN
7−→
λy
~~ ~ ~
>> >
if C C
λ y
λ ⋆
CC
f
Φ>
λ
λ
t
x
t
λ z
Functional Reachability – 11
Traversals A traversal [Blum, Ong] over a full computation tree: • follows the flow of control within it, • seen from the perspective of Game Semantics.
Functional Reachability – 12
Traversals A traversal [Blum, Ong] over a full computation tree: λ • follows the flow of control within it,
@ JTJTTTT t JJ TTTT tt t JJ TTTT t t t TTT
• seen from the perspective of Game Semantics. λΦz
λy
λ y
Φ II t t II t t II t t II t t
v if LLLL v LL vv v LL vv
λ
λ
⋆
z
λf x
λ
f
t
λ
λ
t
x
Functional Reachability – 12
Traversals A traversal [Blum, Ong] over a full computation tree: λ • follows the flow of control within it,
@ JTJTTTT t JJ TTTT tt t JJ TTTT t t t TTT
• seen from the perspective of Game Semantics. λΦz
λy
λ y
Φ II t t II t t II t t II t t
v if LLLL v LL vv v LL vv
λ
λ
⋆
z
λf x
λ
f
t
λ
λ
t
x
Functional Reachability – 12
Traversals A traversal [Blum, Ong] over a full computation tree: λ • follows the flow of control within it,
@ JTJTTTT t JJ TTTT tt t JJ TTTT t t t TTT
• seen from the perspective of Game Semantics. λΦz
λy
λ y
Φ II t t II t t II t t II t t
v if LLLL v LL vv v LL vv
λ
λ
⋆
z
λf x
λ
f
t
λ
λ
t
x
Functional Reachability – 12
Traversals A traversal [Blum, Ong] over a full computation tree: λ • follows the flow of control within it,
@ JTJTTTT t JJ TTTT tt t JJ TTTT t t t TTT
• seen from the perspective of Game Semantics. λΦz
λy
λ y
Φ II t t II t t II t t II t t
v if LLLL v LL vv v LL vv
λ
λ
⋆
z
λf x
λ
f
t
λ
λ
t
x
Functional Reachability – 12
Traversals A traversal [Blum, Ong] over a full computation tree: λ • follows the flow of control within it,
@ JTJTTTT t JJ TTTT tt t JJ TTTT t t t TTT
• seen from the perspective of Game Semantics. λΦz
λy
λ y
Φ II t t II t t II t t II t t
v if LLLL v LL vv v LL vv
λ
λ
⋆
z
λf x
λ
f
t
λ
λ
t
x
Functional Reachability – 12
Traversals A traversal [Blum, Ong] over a full computation tree: λ • follows the flow of control within it,
@ JTJTTTT t JJ TTTT tt t JJ TTTT t t t TTT
• seen from the perspective of Game Semantics. λΦz
λy
λ y
Φ II t t II t t II t t II t t
v if LLLL v LL vv v LL vv
λ
λ
⋆
z
λf x
λ
f
t
λ
λ
t
x
Functional Reachability – 12
Traversals A traversal [Blum, Ong] over a full computation tree: λ • follows the flow of control within it,
@ JTJTTTT t JJ TTTT tt t JJ TTTT t t t TTT
• seen from the perspective of Game Semantics. λΦz
λy
λ y
Φ II t t II t t II t t II t t
v if LLLL v LL vv v LL vv
λ
λ
⋆
z
λf x
λ
f
t
λ
λ
t
x
Functional Reachability – 12
Traversals A traversal [Blum, Ong] over a full computation tree: λ • follows the flow of control within it,
@ JTJTTTT t JJ TTTT tt t JJ TTTT t t t TTT
• seen from the perspective of Game Semantics. λΦz
λy
λ y
Φ II t t II t t II t t II t t
v if LLLL v LL vv v LL vv
λ
λ
⋆
z
λf x
λ
f
t
λ
λ
t
x
Functional Reachability – 12
Traversals A traversal [Blum, Ong] over a full computation tree: λ • follows the flow of control within it,
@ JTJTTTT t JJ TTTT tt t JJ TTTT t t t TTT
• seen from the perspective of Game Semantics. λΦz
λy
λ y
Φ II t t II t t II t t II t t
v if LLLL v LL vv v LL vv
λ
λ
⋆
z
λf x
λ
f
t
λ
λ
t
x
Functional Reachability – 12
Traversals A traversal [Blum, Ong] over a full computation tree: λ • follows the flow of control within it,
@ JTJTTTT t JJ TTTT tt t JJ TTTT t t t TTT
• seen from the perspective of Game Semantics. λΦz
λy
λ y
Φ II t t II t t II t t II t t
v if LLLL v LL vv v LL vv
λ
λ
⋆
z
λf x
λ
f
t
λ
λ
t
x
Functional Reachability – 12
Traversals A traversal [Blum, Ong] over a full computation tree: λ • follows the flow of control within it,
@ JTJTTTT t JJ TTTT tt t JJ TTTT t t t TTT
• seen from the perspective of Game Semantics. λΦz
λy
λ y
Φ II t t II t t II t t II t t
v if LLLL v LL vv v LL vv
λ
λ
⋆
z
λf x
λ
f
t
λ
λ
t
x
Functional Reachability – 12
Traversals A traversal [Blum, Ong] over a full computation tree: λ • follows the flow of control within it,
@ JTJTTTT t JJ TTTT tt t JJ TTTT t t t TTT
• seen from the perspective of Game Semantics. λΦz
λy
λ y
Φ II t t II t t II t t II t t
v if LLLL v LL vv v LL vv
λ
λ
⋆
z
λf x
λ
f
t
λ
λ
t
x
Functional Reachability – 12
Traversals A traversal [Blum, Ong] over a full computation tree: λ • follows the flow of control within it,
@ JTJTTTT t JJ TTTT tt t JJ TTTT t t t TTT
• seen from the perspective of Game Semantics. λΦz
λy
λ y
Φ II t t II t t II t t II t t
v if LLLL v LL vv v LL vv
λ
λ
⋆
z
λf x
λ
f
t
λ
λ
t
x
Functional Reachability – 12
Traversals A traversal [Blum, Ong] over a full computation tree: λ • follows the flow of control within it,
@ JTJTTTT t JJ TTTT tt t JJ TTTT t t t TTT
• seen from the perspective of Game Semantics. λΦz
λy
λ y
Φ II t t II t t II t t II t t
v if LLLL v LL vv v LL vv
λ
λ
⋆
z
λf x
λ
f
t
λ
λ
t
x
Functional Reachability – 12
Traversals A traversal [Blum, Ong] over a full computation tree: λ • follows the flow of control within it,
@ JTJTTTT t JJ TTTT tt t JJ TTTT t t t TTT
• seen from the perspective of Game Semantics. λΦz
λy
λ y
Φ II t t II t t II t t II t t
v if LLLL v LL vv v LL vv
λ
λ
⋆
z
λf x
λ
f
t
λ
λ
t
x
Functional Reachability – 12
Traversals A traversal [Blum, Ong] over a full computation tree: λ • follows the flow of control within it,
@ JTJTTTT t JJ TTTT tt t JJ TTTT t t t TTT
• seen from the perspective of Game Semantics. λΦz
λy
λ y
Φ II t t II t t II t t II t t
v if LLLL v LL vv v LL vv
λ
λ
⋆
z
λf x
λ
f
t
λ
λ
t
x
Functional Reachability – 12
Traversals A traversal [Blum, Ong] over a full computation tree: λ • follows the flow of control within it,
@ JTJTTTT t JJ TTTT tt t JJ TTTT t t t TTT
• seen from the perspective of Game Semantics. λΦz
λy
λ y
Φ II t t II t t II t t II t t
v if LLLL v LL vv v LL vv
λ
λ
⋆
z
λf x
λ
f
t
λ
λ
t
x
Functional Reachability – 12
Traversals A traversal [Blum, Ong] over a full computation tree: λ • follows the flow of control within it,
@ JTJTTTT t JJ TTTT tt t JJ TTTT t t t TTT
• seen from the perspective of Game Semantics. λΦz
λy
λ y
Φ II t t II t t II t t II t t
v if LLLL v LL vv v LL vv
λ ⋆
λf x
λ
f
t
λ
λ
t
x
λ z
⋆-complete traversal
Functional Reachability – 12
Traversals A traversal [Blum, Ong] over a full computation tree: • follows the flow of control within it, • seen from the perspective of Game Semantics. A traversal is v -complete if: • every question (red visit) has been answered (green visit), • and the root question has been answered with v . For any P : o and v , P ։ v iff there is a v -complete traversal over λ(P ).
Functional Reachability – 12
Alternating Tree Automata An ATA is a quadruple A = hQ, Σ, q0 , ∆i where: • Q is a finite set of states, • Σ is a finite ranked alphabet, • q0 ∈ Q is the initial state, • ∆ is a finite transition relation: q
s∈Σ q∈Q s → (Q1 , . . . , Qk ). Q1 , ... , Qk ⊆ Q
Functional Reachability – 13
Alternating Tree Automata An ATA is a quadruple A = hQ, Σ, q0 , ∆i where: • Q is a finite set of states, • Σ is a finite ranked alphabet, • q0 ∈ Q is the initial state, • ∆ is a finite transition relation: q
s1 ...
s∈Σ q∈Q s → (Q1 , . . . , Qk ). Q1 , ... , Qk ⊆ Q
ly s OOOO l l OOO ll yy l l y OOO ll yy l l OOO y ll l y O l l ...
s2
...
...
sk ... Functional Reachability – 13
Alternating Tree Automata An ATA is a quadruple A = hQ, Σ, q0 , ∆i where: • Q is a finite set of states, • Σ is a finite ranked alphabet, • q0 ∈ Q is the initial state, • ∆ is a finite transition relation: q
s∈Σ q∈Q s → (Q1 , . . . , Qk ). Q1 , ... , Qk ⊆ Q
A(q)
s1 ...
ly s OOOO l l OOO ll yy l l y OOO ll yy l l OOO y ll l y O l l ...
s2
...
...
sk ... Functional Reachability – 13
Alternating Tree Automata An ATA is a quadruple A = hQ, Σ, q0 , ∆i where: • Q is a finite set of states, • Σ is a finite ranked alphabet, • q0 ∈ Q is the initial state, • ∆ is a finite transition relation: q
s∈Σ q∈Q s → (Q1 , . . . , Qk ). Q1 , ... , Qk ⊆ Q
ly s OOOO l l OOO ll yy l l y OOO ll yy l l OOO y 2) ll1 ) A(Q l A(Q y O l l ...
s1
s2
...
...
...
A(Qk )
sk
... Functional Reachability – 13
Traversal-simulating ATA’s λ
How can we simulate a complete traversal by an ATA?
@-;
; --- ;;; - ;
λf x λ
λΦz
f
Φ-
--
λy
λ
λ
if 0
t
x
0 000
λ λ
λ
y ⋆
z
t
Functional Reachability – 14
Traversal-simulating ATA’s How can we simulate a complete traversal by an ATA?
λ
• By guessing the number of visits of each node.
@-;
; --- ;;; - ;
• By guessing the profile of each variable per visit.
λf x λ
λΦz
• By verifying these guesses. f
Φ-
--
λy
λ
λ
if 0
t
x
0 000
λ λ
λ
y ⋆
z
t
Functional Reachability – 14
Variable profiles Introduced in [Ong’06]. • VPΣ (A1 , . . . , An , o) :=
Var A Σ
× Val × P(
Sn
i=1 VPΣ (Ai ))
• Notation: (x, v), (x, v | π1 , . . . , πn )
Functional Reachability – 15
VP(A1 , . . . , An , o) := Var × Val × P( Notation: (x, v), (x, v | π1 , . . . , πn )
Variable profiles
Sn
i=1 VP(Ai ))
λ
λΦz
λy
λ y
@ TIITTTT u III TTT uu u II TTTTT u u TTT u
Φ HH u u HH u u HH u u HH uu
w if KKKK w KK ww w KK ww
λ
λ
⋆
z
λf x
λ
f
t
λ
λ
t
x
Functional Reachability – 15
VP(A1 , . . . , An , o) := Var × Val × P( Notation: (x, v), (x, v | π1 , . . . , πn )
Variable profiles
Sn
i=1 VP(Ai ))
λ
λΦz
f
λ y
λy
@ TIITTTT u III TTT uu u II TTTTT u u TTT u Φ
Φ HH u u HH u u HH u u HH uu
w if KKKK w KK ww w KK ww
xλ t
λ
λ
⋆
z
λf x
λ
f
t
y
λ x
Functional Reachability – 15
VP(A1 , . . . , An , o) := Var × Val × P( Notation: (x, v), (x, v | π1 , . . . , πn )
Variable profiles
Sn
i=1 VP(Ai ))
λ
λΦz
f
λ y
λy
@ TIITTTT u III TTT uu u II TTTTT u u TTT u Φ
Φ HH u u HH u u HH u u HH uu
w if KKKK w KK ww w KK ww
xλ t
λ
λ
⋆
z
λf x
λ
f
t
y
λ x
Functional Reachability – 15
VP(A1 , . . . , An , o) := Var × Val × P( Notation: (x, v), (x, v | π1 , . . . , πn )
Variable profiles
Sn
i=1 VP(Ai ))
⋆λ
⋆ λΦz
f ⋆ λy
tλ ty
@ TIITTTT ⋆ u III TTT uu u II TTTTT u u TTT u Φ
Φ HH ⋆ u u HH u u HH u u HH uu
wt if KKKK w KK ww w KK ww
⋆λ
λ
⋆
z
⋆ λf x
λ
⋆f
t
xλ t
y tλ
t
tx
Functional Reachability – 15
VP(A1 , . . . , An , o) := Var × Val × P( Notation: (x, v), (x, v | π1 , . . . , πn )
Variable profiles
Sn
i=1 VP(Ai ))
⋆λ
⋆ λΦz
f ⋆ λy
tλ ty
@ TIITTTT ⋆ u III TTT uu u II TTTTT u u TTT u Φ
Φ HH ⋆ u u HH u u HH u u HH uu
wt if KKKK w KK ww w KK ww
⋆λ
λ
⋆
z
⋆ λf x
λ
⋆f
t
xλ t
y tλ
t
tx
Functional Reachability – 15
VP(A1 , . . . , An , o) := Var × Val × P( Notation: (x, v), (x, v | π1 , . . . , πn )
Variable profiles
Sn
i=1 VP(Ai ))
λ
λΦz
f
λ ty
λy
@ TIITTTT u III TTT uu u II TTTTT u u TTT u Φ
Φ HH ⋆ u u HH u u HH u u HH uu
w if KKKK w KK ww w KK ww
xλ t
λ
λ
⋆
z
λf x
λ
⋆f
t
y
λ
tx
Functional Reachability – 15
VP(A1 , . . . , An , o) := Var × Val × P( Notation: (x, v), (x, v | π1 , . . . , πn )
Variable profiles
Sn
i=1 VP(Ai ))
λ
λΦz
f
λ (y, t)
y
λy
@ TIITTTT u III TTT uu u II TTTTT u u TTT u Φ
Φ HH ⋆ u u HH u u HH u u HH uu
w if KKKK w KK ww w KK ww
xλ t
λ
λ
⋆
z
λf x
λ
⋆f
t
y
(x, t)
λ x
Functional Reachability – 15
VP(A1 , . . . , An , o) := Var × Val × P( Notation: (x, v), (x, v | π1 , . . . , πn )
Variable profiles
Sn
i=1 VP(Ai ))
λ
λΦz
f
λ (y, t)
y
λy
@ TIITTTT u III TTT uu u II TTTTT u u TTT u Φ λf x
f Φ HH ⋆ u u (f, ⋆ | (y, t)) H u HH uu HH u H uu
w if KKKK w KK ww w KK ww
xλ t
λ
λ
⋆
z
y
(x, t)
λ t
λ x
Functional Reachability – 15
VP(A1 , . . . , An , o) := Var × Val × P( Notation: (x, v), (x, v | π1 , . . . , πn )
Variable profiles
Sn
i=1 VP(Ai ))
λ
λΦz
@ TIITTTT u III TTT uu u II TTTTT u u TTT u Φ λf x
f Φ HH u u (f, ⋆ | (y, t)) H (Φ, ⋆ | (f, ⋆ | (y, t)), (x,ut)) u HH u HH u H uu f
λ (y, t)
y
xλ
λy
w if KKKK w KK ww w KK ww
t
λ
λ
⋆
z
y
(x, t)
λ t
λ x
Functional Reachability – 15
ATA correspondence Given a finite fPCF⋆ -alphabet Σ, the states of the traveral-simulating ATA AΣ are:
Q := Val × P(VPΣ ) × P(VPΣ ) P ։ v iff AΣ accepts λ(P ) on initial state with value v . Any tree accepted by AΣ on closed initial state represents a closed fPCF⋆ term over Σ.
Functional Reachability – 16
Results Theorem: M : (A1 , . . . , An , o) ∈ v-Reach [fPCF⋆Σ , fPCFΣ ] iff there is a closed initial state q0 with value v such that: • AΣ (q0 ) accepts λ(M ), • ∀i, the language accepted by AΣ ˜ (q0 ↾ Ai ) is non-empty.
Functional Reachability – 17
Results Theorem: M : (A1 , . . . , An , o) ∈ v-Reach [fPCF⋆Σ , fPCFΣ ] iff there is a closed initial state q0 with value v such that: • AΣ (q0 ) accepts λ(M ), • ∀i, the language accepted by AΣ ˜ (q0 ↾ Ai ) is non-empty. Corollary: ⋆-Reach [fPCF⋆ , fPCF(n)] is decidable. Corollary: ⋆-Reach [fPCF⋆ , fPCF] is decidable up to order 3.
Functional Reachability – 17
Results Theorem: M : (A1 , . . . , An , o) ∈ v-Reach [fPCF⋆Σ , fPCFΣ ] iff there is a closed initial state q0 with value v such that: • AΣ (q0 ) accepts λ(M ), • ∀i, the language accepted by AΣ ˜ (q0 ↾ Ai ) is non-empty. Corollary: ⋆-Reach [fPCF⋆ , fPCF(n)] is decidable. Corollary: ⋆-Reach [fPCF⋆ , fPCF] is decidable up to order 3.
• For the general case (also with ⊥) we use Alternating Dependency Tree Automata [Stirling’09]. • Corollary: Emptiness problem is undecidable for ADTA’s. Functional Reachability – 17
Last slide • A new kind of Reachability problems. • Some undecidability results. • Some technology from game semantics. • Characterisation by ATA’s and ADTA’s. • Some (relativised) decidability results.
Functional Reachability – 18
Last slide • A new kind of Reachability problems. • Some undecidability results. • Some technology from game semantics. • Characterisation by ATA’s and ADTA’s. • Some (relativised) decidability results.
• Conjecture: ⋆-Reach [fPCF⋆ , fPCF] ? • Can this yield a (semantic) CFA? • Extensions: expressivity, algorithms, abstractions.
Functional Reachability – 18
Last slide • A new kind of Reachability problems.
Thank you!
• Some undecidability results. • Some technology from game semantics. • Characterisation by ATA’s and ADTA’s. • Some (relativised) decidability results.
• Conjecture: ⋆-Reach [fPCF⋆ , fPCF] ? • Can this yield a (semantic) CFA? • Extensions: expressivity, algorithms, abstractions.
Functional Reachability – 18
