Luke Ong Nikos Tzevelekos Oxford University Computing

Ong & Tzevelekos Functional Reachability – 1

Functional Reachability

Luke Ong Nikos Tzevelekos

Oxford University Computing Laboratory

24th Symposium on Logic in Computer Science

Los Angeles, August 2009.

The Problem

The Problem

Relevant workThe examined

language: PCF

Reachability

PCF-with-error:

PCF⋆

Reach template

An undecidability

result

Our approach

Computation trees

TraversalsAlternating Tree

AutomataTraversal-simulating

ATA’s

Variable profiles

ATA correspondence

ResultsAlternating

Dependency Tree

Automata

Conclusion and on


Reachability in functional computation.

■ Consider a term M of a higher-order functionalprogramming language.

■ Now consider a point p inside M .

� Is there a program context C such that thecomputation of C[M ] reaches p?

The Problem

The Problem


language: PCF

Reachability

PCF-with-error:

PCF⋆

Reach template

An undecidability

result

Our approach

Computation trees



ATA’s

Variable profiles

ATA correspondence

ResultsAlternating

Dependency Tree

Automata

Conclusion and on






The Problem

The Problem


language: PCF

Reachability

PCF-with-error:

PCF⋆

Reach template

An undecidability

result

Our approach

Computation trees



ATA’s

Variable profiles

ATA correspondence

ResultsAlternating

Dependency Tree

Automata

Conclusion and on






Surprisingly, (Contextual) Reachability per se hadnot been studied in HO functional languages.

Relevant work

The Problem


language: PCF

Reachability

PCF-with-error:

PCF⋆

Reach template

An undecidability

result

Our approach

Computation trees



ATA’s

Variable profiles

ATA correspondence

ResultsAlternating

Dependency Tree

Automata

Conclusion and on


■ Control Flow Analysis: Approximate at compiletime the flow of control to happen at run time.

◆ In a HO-setting, the crucial element is that ofclosures.

◆ Reynolds (’70), Jones (’80), Shivers (’90), . . .Malacaria & Hankin (late 90’s).

◆ CFA > Reach: more general.Reach > CFA: open vs closed world approach.

■ Useless code detection, etc.

The examined language: PCF


Types: A,B ::= o | A → B

Terms: M,N ::= x | λx.M | MN | t | f | if M N1 N2 | YA

Contexts: C ::= . . .






Reductions: (λx.M)N → M{N/x} if t → λxy.x

YM → M(YM) if f → λxy.y

M → N =⇒ E[M ] → E[N ]

Ev. Contexts: E ::= [−] | E M | if E






Reductions: Call-by-name λ-calculus + if + Y

■ Write (A1, . . . , An, o) for A1 → · · · → An → o.

■ Divergence definable, e.g. ⊥ := Yo(λx.x).

■ Finitary restrictions (i.e. no Y): fPCF, fPCF⊥.

Reachability

The Problem


language: PCF

Reachability

PCF-with-error:

PCF⋆

Reach template

An undecidability

result

Our approach

Computation trees



ATA’s

Variable profiles

ATA correspondence

ResultsAlternating

Dependency Tree

Automata

Conclusion and on


■ Given a PCF-term M and a coloured subterm Lof M ,

� Is there a program context C such thatC[M ] ։ E[L′] with L′ coloured?

Reachability

The Problem


language: PCF

Reachability

PCF-with-error:

PCF⋆

Reach template

An undecidability

result

Our approach

Computation trees



ATA’s

Variable profiles

ATA correspondence

ResultsAlternating

Dependency Tree

Automata

Conclusion and on


■ Given a PCF-term M and a coloured subterm Lof M ,

� Is there a program context C such thatC[M ] ։ E[L′] with L′ coloured?

Equivalently:

■ Given a closed PCF-term M : (A1, ..., An, o) anda coloured subterm L of M ,

� Are there closed PCF-terms N1, . . . , Nn such that

M ~N ։ E[L′]

with L′ coloured?

PCF-with-error: PCF⋆

The Problem


language: PCF

Reachability

PCF-with-error:

PCF⋆

Reach template

An undecidability

result

Our approach

Computation trees



ATA’s

Variable profiles

ATA correspondence

ResultsAlternating

Dependency Tree

Automata

Conclusion and on


Take base type o = {t, f, ⋆} with ⋆ an error constant:

E[⋆] ։ ⋆

⋆-Reachability:

■ Given a closed PCF⋆-term M : (A1, ..., An, o) thathas exactly one occurrence of ⋆,

� are there closed PCF-terms N1, ..., Nn such thatM ~N ։ ⋆?

PCF-with-error: PCF⋆

The Problem


language: PCF

Reachability

PCF-with-error:

PCF⋆

Reach template

An undecidability

result

Our approach

Computation trees



ATA’s

Variable profiles

ATA correspondence

ResultsAlternating

Dependency Tree

Automata

Conclusion and on


Take base type o = {t, f, ⋆} with ⋆ an error constant:

E[⋆] ։ ⋆

⋆-Reachability:

■ Given a closed PCF⋆-term M : (A1, ..., An, o) thathas exactly one occurrence of ⋆,

� are there closed PCF-terms N1, ..., Nn such thatM ~N ։ ⋆?

Lemma: Reachability ∼= ⋆-Reachability.

Reach template


For v ∈ {t, f, ⋆} and L1,L2 ⊆ PCF⋆:

v-Reach [L1,L2]:Given a closed L1-term M : (A1, ..., An, o), are there closed

L2-terms N1, ..., Nn such that M ~N ։ v?

e.g. ⋆-Reachability = ⋆-Reach [PCF1⋆,PCF].

Reach template


For v ∈ {t, f, ⋆} and L1,L2 ⊆ PCF⋆:

v-Reach [L1,L2]:Given a closed L1-term M : (A1, ..., An, o), are there closed

L2-terms N1, ..., Nn such that M ~N ։ v?

Three classes of problems:

Reachability

⋆-Reachability

⋆-Reach [PCF1⋆,PCF]

⋆-Reach [PCF1⋆, fPCF]

⋆-Reach [fPCF1⋆, fPCF]⋆-Reach [fPCF⋆, fPCF]. . .

⋆-Reach [fPCF1⋆⊥, fPCF]

⋆-Reach [fPCF⋆⊥, fPCF]

. . .

An undecidability result


Lemma: ⋆-Reach [fPCF1⋆⊥, fPCF] is undecidable.

Proof: By reduction of solvability of systems of fPCF⊥-equations(proved undecidable by [Loader’01]).

Reachability

⋆-Reachability

⋆-Reach [PCF1⋆,PCF]

⋆-Reach [PCF1⋆, fPCF]

⋆-Reach [fPCF1⋆, fPCF]⋆-Reach [fPCF⋆, fPCF]. . .

⋆-Reach [fPCF1⋆⊥, fPCF]

⋆-Reach [fPCF⋆⊥, fPCF]

. . .

Our approach


■ We focus on v-Reach [fPCF⋆, fPCF].

■ For fPCF⋆-term P : o,

Computations of P oo //Traversals over its compu-tation tree, λ(P )

88

xxqqqqqqqqqq

Runs of an Alternating TreeAutomaton (ATA) on λ(P )

Our approach


■ We focus on v-Reach [fPCF⋆, fPCF].

■ For fPCF⋆-term P : o,

Computations of P oo //Traversals over its compu-tation tree, λ(P )

88

xxqqqqqqqqqq

Runs of an Alternating TreeAutomaton (ATA) on λ(P )

� P ։ v iff an ATA accepts λ(P ) on initial state with value v.

Computation trees


Starting from a fPCF⋆-term M ,

■ take its η-long form,

■ add application symbols (@),

■ view the result as a tree, λ(M).

Computation trees


Starting from a fPCF⋆-term M ,

■ take its η-long form,

■ add application symbols (@),

■ view the result as a tree, λ(M).

( λΦz.Φ(λy. ify ⋆ z)t ) (λϕx. ϕx) t 7−→

λ

@�� ??

?

OOOOOOOO

λΦz λϕx λ

Φ}}}

}??

??ϕ t

λy λ λ

if�� CC

CCt x

λ λ λ

y ⋆ z

Traversals


A traversal [Blum, Ong] over a computation tree,

■ follows the flow of control within it,

■ seen from the perspective of Game Semantics.

Traversals





λ

@

tttt

ttt

JJJJJJ

TTTTTTTTTTTTTTTT

λΦz λϕx λ

Φ

ssssss

ss

JJJJJJJJ ϕ t

λy λ λ

if

vvvvvvv

LLLLLLLL t x

λ λ λ

y ⋆ z

Traversals





λ

@

tttt

ttt

JJJJJJ

TTTTTTTTTTTTTTTT

λΦz λϕx λ

Φ

ssssss

ss

JJJJJJJJ ϕ t

λy λ λ

if

vvvvvvv

LLLLLLLL t x

λ λ λ

y ⋆ z

Traversals





λ

@

tttt

ttt

JJJJJJ

TTTTTTTTTTTTTTTT

λΦz λϕx λ

Φ

ssssss

ss

JJJJJJJJ ϕ t

λy λ λ

if

vvvvvvv

LLLLLLLL t x

λ λ λ

y ⋆ z

Traversals





λ

@

tttt

ttt

JJJJJJ

TTTTTTTTTTTTTTTT

λΦz λϕx λ

Φ

ssssss

ss

JJJJJJJJ ϕ t

λy λ λ

if

vvvvvvv

LLLLLLLL t x

λ λ λ

y ⋆ z

Traversals





λ

@

tttt

ttt

JJJJJJ

TTTTTTTTTTTTTTTT

λΦz λϕx λ

Φ

ssssss

ss

JJJJJJJJ ϕ t

λy λ λ

if

vvvvvvv

LLLLLLLL t x

λ λ λ

y ⋆ z

Traversals





λ

@

tttt

ttt

JJJJJJ

TTTTTTTTTTTTTTTT

λΦz λϕx λ

Φ

ssssss

ss

JJJJJJJJ ϕ t

λy λ λ

if

vvvvvvv

LLLLLLLL t x

λ λ λ

y ⋆ z

Traversals





λ

@

tttt

ttt

JJJJJJ

TTTTTTTTTTTTTTTT

λΦz λϕx λ

Φ

ssssss

ss

JJJJJJJJ ϕ t

λy λ λ

if

vvvvvvv

LLLLLLLL t x

λ λ λ

y ⋆ z

Traversals





λ

@

tttt

ttt

JJJJJJ

TTTTTTTTTTTTTTTT

λΦz λϕx λ

Φ

ssssss

ss

JJJJJJJJ ϕ t

λy λ λ

if

vvvvvvv

LLLLLLLL t x

λ λ λ

y ⋆ z

Traversals





λ

@

tttt

ttt

JJJJJJ

TTTTTTTTTTTTTTTT

λΦz λϕx λ

Φ

ssssss

ss

JJJJJJJJ ϕ t

λy λ λ

if

vvvvvvv

LLLLLLLL t x

λ λ λ

y ⋆ z

Traversals





λ

@

tttt

ttt

JJJJJJ

TTTTTTTTTTTTTTTT

λΦz λϕx λ

Φ

ssssss

ss

JJJJJJJJ ϕ t

λy λ λ

if

vvvvvvv

LLLLLLLL t x

λ λ λ

y ⋆ z

Traversals





λ

@

tttt

ttt

JJJJJJ

TTTTTTTTTTTTTTTT

λΦz λϕx λ

Φ

ssssss

ss

JJJJJJJJ ϕ t

λy λ λ

if

vvvvvvv

LLLLLLLL t x

λ λ λ

y ⋆ z

Traversals





λ

@

tttt

ttt

JJJJJJ

TTTTTTTTTTTTTTTT

λΦz λϕx λ

Φ

ssssss

ss

JJJJJJJJ ϕ t

λy λ λ

if

vvvvvvv

LLLLLLLL t x

λ λ λ

y ⋆ z

Traversals





λ

@

tttt

ttt

JJJJJJ

TTTTTTTTTTTTTTTT

λΦz λϕx λ

Φ

ssssss

ss

JJJJJJJJ ϕ t

λy λ λ

if

vvvvvvv

LLLLLLLL t x

λ λ λ

y ⋆ z

Traversals





λ

@

tttt

ttt

JJJJJJ

TTTTTTTTTTTTTTTT

λΦz λϕx λ

Φ

ssssss

ss

JJJJJJJJ ϕ t

λy λ λ

if

vvvvvvv

LLLLLLLL t x

λ λ λ

y ⋆ z

Traversals





λ

@

tttt

ttt

JJJJJJ

TTTTTTTTTTTTTTTT

λΦz λϕx λ

Φ

ssssss

ss

JJJJJJJJ ϕ t

λy λ λ

if

vvvvvvv

LLLLLLLL t x

λ λ λ

y ⋆ z

Traversals





λ

@

tttt

ttt

JJJJJJ

TTTTTTTTTTTTTTTT

λΦz λϕx λ

Φ

ssssss

ss

JJJJJJJJ ϕ t

λy λ λ

if

vvvvvvv

LLLLLLLL t x

λ λ λ

y ⋆ z

Traversals





λ

@

tttt

ttt

JJJJJJ

TTTTTTTTTTTTTTTT

λΦz λϕx λ

Φ

ssssss

ss

JJJJJJJJ ϕ t

λy λ λ

if

vvvvvvv

LLLLLLLL t x

λ λ λ

y ⋆ z

Traversals





λ

@

tttt

ttt

JJJJJJ

TTTTTTTTTTTTTTTT

λΦz λϕx λ

Φ

ssssss

ss

JJJJJJJJ ϕ t

λy λ λ

if

vvvvvvv

LLLLLLLL t x

λ λ λ

y ⋆ z⋆-complete traversal

Traversals





A traversal is v-complete if every question (red visit) has beenanswered (green visit), and the root question has been an-swered with v.

Theorem: For any P : o and value v, P ։ v iff there is acomplete v-traversal over λ(P ).

Alternating Tree Automata


An ATA is a quadruple A = 〈Q,Σ, q0,∆〉 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: qs→ (Q1, . . . , Qk).

s ∈ Σq ∈ Q

Q1, ... , Qk ⊆ Q








s ∈ Σq ∈ Q

Q1, ... , Qk ⊆ Q

s

llllllllllllllll

yyyy

yyyy

OOOOOOOOOOOOOO

s1 s2 . . . sk

. . . . . . . . . . . .








s ∈ Σq ∈ Q

Q1, ... , Qk ⊆ Q

s

llllllllllllllll

yyyy

yyyy

OOOOOOOOOOOOOO

s1 s2 . . . sk

. . . . . . . . . . . .

A(q)








s ∈ Σq ∈ Q

Q1, ... , Qk ⊆ Q

s

llllllllllllllll

yyyy

yyyy

OOOOOOOOOOOOOO

s1 s2 . . . sk

. . . . . . . . . . . .

A(Q1) A(Q2) A(Qk)

Traversal-simulating ATA’s


λ

@

��

....

.

;;;;

;;

λΦz λϕx λ

Φ

��

....

. ϕ t

λy λ λ

if

�� 00

000 t x

λ λ λ

y ⋆ z

How can we simulate a complete traversal by an ATA?

Traversal-simulating ATA’s


λ

@

��

....

.

;;;;

;;

λΦz λϕx λ

Φ

��

....

. ϕ t

λy λ λ

if

�� 00

000 t x

λ λ λ

y ⋆ z

How can we simulate a complete traversal by an ATA?

■ By guessing the number of visits of each node.

■ By guessing the profile of each variable per visit.

■ By verifying these guesses.

Variable profiles


■ Introduced by [Ong’06].

■ VP(A1, . . . , An, o) := Var × Val ×P(⋃n

i=1VP(Ai))

■ Notation: (x, v), (x, v | π1, . . . , πn)

Variable profiles


λ

@

uuuuuuuII

IIII

I

TTTTTTTTTTTTTTTT

λΦz λϕx λ

Φ

tttttt

tt

IIIIIIII ϕ t

λy λ λ

if

wwwwwww

KKKKKKKK t x

λ λ λ

y ⋆ z

Variable profiles


λ

@

uuuuuuuII

IIII

I

TTTTTTTTTTTTTTTT

λΦz λϕx λ

Φ

tttttt

tt

IIIIIIII ϕ t

λy λ λ

if

wwwwwww

KKKKKKKK t x

λ λ λ

y ⋆ z

Φ

ϕ x y

Variable profiles


λ

@

uuuuuuuII

IIII

I

TTTTTTTTTTTTTTTT

λΦz λϕx λ

Φ

tttttt

tt

IIIIIIII ϕ t

λy λ λ

if

wwwwwww

KKKKKKKK t x

λ λ λ

y ⋆ z

Φ

ϕ x yt

t

t

t

t

t ⋆

⋆

⋆

⋆

⋆

⋆

⋆

⋆

Variable profiles


λ

@

uuuuuuuII

IIII

I

TTTTTTTTTTTTTTTT

λΦz λϕx λ

Φ

tttttt

tt

IIIIIIII ϕ t

λy λ λ

if

wwwwwww

KKKKKKKK t x

λ λ λ

y ⋆ z

Φ

ϕ x y

t

t

⋆⋆

Variable profiles


λ

@

uuuuuuuII

IIII

I

TTTTTTTTTTTTTTTT

λΦz λϕx λ

Φ

tttttt

tt

IIIIIIII ϕ t

λy λ λ

if

wwwwwww

KKKKKKKK t x

λ λ λ

y ⋆ z

Φ

ϕ x y

⋆⋆

(x, t)

(y, t)

Variable profiles


λ

@

uuuuuuuII

IIII

I

TTTTTTTTTTTTTTTT

λΦz λϕx λ

Φ

tttttt

tt

IIIIIIII ϕ t

λy λ λ

if

wwwwwww

KKKKKKKK t x

λ λ λ

y ⋆ z

Φ

ϕ x y

⋆

(x, t)

(y, t)

(ϕ, ⋆ | (y, t))

Variable profiles


λ

@

uuuuuuuII

IIII

I

TTTTTTTTTTTTTTTT

λΦz λϕx λ

Φ

tttttt

tt

IIIIIIII ϕ t

λy λ λ

if

wwwwwww

KKKKKKKK t x

λ λ λ

y ⋆ z

Φ

ϕ x y

(x, t)

(y, t)

(ϕ, ⋆ | (y, t))(Φ, ⋆ | (ϕ, ⋆ | (y, t)), (x, t))

ATA correspondence


Given a finite fPCF⋆-alphabet Σ, the states of thetraveral-simulating ATA AΣ are:

Q := Val ×P(VPΣ) ×P(VPΣ)

ATA correspondence


Given a finite fPCF⋆-alphabet Σ, the states of thetraveral-simulating ATA AΣ are:

Q := Val ×P(VPΣ) ×P(VPΣ)

■ M ~N ։ v iff AΣ accepts λ(M ~N) on initial state with value v.

■ Any tree accepted by AΣ is a closed fPCF-term.

Results

The Problem


language: PCF

Reachability

PCF-with-error:

PCF⋆

Reach template

An undecidability

result

Our approach

Computation trees



ATA’s

Variable profiles

ATA correspondence

ResultsAlternating

Dependency Tree

Automata

Conclusion and on


Theorem: M ∈ v-Reach [fPCF⋆Σ, fPCFΣ] iff there is

an initial state q0 with value v such that:

■ AΣ(qo) accepts λ(M),

■ ∀i, the language accepted by AΣ(q0 ↾ Ai) isnon-empty.

Results

The Problem


language: PCF

Reachability

PCF-with-error:

PCF⋆

Reach template

An undecidability

result

Our approach

Computation trees



ATA’s

Variable profiles

ATA correspondence

ResultsAlternating

Dependency Tree

Automata

Conclusion and on


Theorem: M ∈ v-Reach [fPCF⋆Σ, fPCFΣ] iff there is

an initial state q0 with value v such that:

■ AΣ(qo) accepts λ(M),

■ ∀i, the language accepted by AΣ(q0 ↾ Ai) isnon-empty.

Corollary: ⋆-Reach [fPCF⋆, fPCF(n)] is decidable.

Corollary: ⋆-Reach [fPCF⋆, fPCF] is decidable up toorder 3.

Alternating Dependency Tree Automata

The Problem


language: PCF

Reachability

PCF-with-error:

PCF⋆

Reach template

An undecidability

result

Our approach

Computation trees



ATA’s

Variable profiles

ATA correspondence

ResultsAlternating

Dependency Tree

Automata

Conclusion and on


■ For the general case we can use AlternatingDependency Tree Automata [Stirling’09].

■ Corollary: Emptiness problem is undecidable forADTA’s.

Conclusion and on

The Problem


language: PCF

Reachability

PCF-with-error:

PCF⋆

Reach template

An undecidability

result

Our approach

Computation trees



ATA’s

Variable profiles

ATA correspondence

ResultsAlternating

Dependency Tree

Automata

Conclusion and on


■ 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.

Conclusion and on

The Problem


language: PCF

Reachability

PCF-with-error:

PCF⋆

Reach template

An undecidability

result

Our approach

Computation trees



ATA’s

Variable profiles

ATA correspondence

ResultsAlternating

Dependency Tree

Automata

Conclusion and on







� Revisit (semantic) CFA?

� Reachability through intersection types?

� Conjecture: ⋆-Reach [fPCF⋆, fPCF] ?

Conclusion and on

The Problem


language: PCF

Reachability

PCF-with-error:

PCF⋆

Reach template

An undecidability

result

Our approach

Computation trees



ATA’s

Variable profiles

ATA correspondence

ResultsAlternating

Dependency Tree

Automata

Conclusion and on







� Revisit (semantic) CFA?

� Reachability through intersection types?

� Conjecture: ⋆-Reach [fPCF⋆, fPCF] ?

THANKS!

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Luke Ong Nikos Tzevelekos Oxford University Computing