Verification of PCP-Related Computational Reductions in Coqforster/downloads/slides-itp18.pdf · Yannick Forster, Edith Heiter, Gert Smolka PCP-Related Computational Reductions ITP

Verification of PCP-RelatedComputational Reductions in Coq

Yannick Forster, Edith Heiter, Gert Smolka

ITP 2018July 12

computer science

saarlanduniversity

Yannick Forster, Edith Heiter, Gert Smolka PCP-Related Computational Reductions ITP 2018 – July 12 1

http://www.cs.uni-saarland.de/

computer science

saarlanduniversityDecidability

A problem P : X → P is decidable if . . .

Classically Fix a model of computation M:there is a decider in M

For the cbv λ-calculus ∃u : T.∀x : X . (ux . T ∧ Px)∨ (ux . F ∧ ¬Px)

Type Theory ∃f : X → B. ∀x : X . Px ↔ fx = true

dependent version(Coq, Agda, Lean, . . . ) ∀x : X . {Px}+ {¬Px}


computer science








computer science








computer science








computer science

saarlanduniversityUndecidability

A problem P : X → P is undecidable if . . .

Classically If there is no decider u in M

For the cbv λ-calculus ¬∃u : T.∀x : X . (ux . T ∧ Px)∨ (ux . F ∧ ¬Px)

Type Theory

In practice: most proofs are by reduction

Definition

P undecidable := Halting problem reduces to P


computer science





Type Theory


Definition



computer science





Type Theory ¬(∀x : X . {Px}+ {¬Px})


Definition



computer science





Type Theory((((((((((((¬(∀x : X . {Px}+ {¬Px})


Definition



computer science





Type Theory((((((((((((¬(∀x : X . {Px}+ {¬Px})


Definition



computer science

saarlanduniversityReduction

A problem is a type X and a unary predicate P : X → P

A reduction of (X ,P) to (Y ,Q) is a function f : X → Y s.t.∀x . Px ↔ Q(fx)

Write

P � Q


computer science

saarlanduniversity


computer science

saarlanduniversityPCP

C2

LoC2018

xfor nf

d

FLo

F

d

ord

018inO

inOxf

FLo

F

C2

LoC2018

018inO

inOxf

xfor d

ord

FLoC2018inOxford

FLoC2018inOxford

Symbols a, b, c: NStrings x , y , z : lists of symbols

Card c: pairs of strings

Stacks A,P: lists of cards

A ⊆ P: list inclusion

[ ]1 := ε [ ]2 := ε

(x/y :: A)1 := x(A1) (x/y :: A)2 := y(A2)

PCP (P) := ∃A ⊆ P. A 6= [ ]∧ A1 = A2


computer science


C2

LoC2018

xfor nf

d

FLo

F

d

ord

018inO

inOxf

FLo

F

C2

LoC2018

018inO

inOxf

xfor d

ord

FLoC2018inOxford

FLoC2018inOxford





[ ]1 := ε [ ]2 := ε

(x/y :: A)1 := x(A1) (x/y :: A)2 := y(A2)

PCP (P) := ∃A ⊆ P. A 6= [ ]∧ A1 = A2


computer science


C2

LoC2018

xfor nf

d

FLo

F

d

ord

018inO

inOxf

FLo

F

C2

LoC2018

018inO

inOxf

xfor d

ord

FLoC2018inOxford

FLoC2018inOxford





[ ]1 := ε [ ]2 := ε

(x/y :: A)1 := x(A1) (x/y :: A)2 := y(A2)

PCP (P) := ∃A ⊆ P. A 6= [ ]∧ A1 = A2


computer science


C2

LoC2018

xfor nf

d

FLo

F

d

ord

018inO

inOxf

FLo

F

C2

LoC2018

018inO

inOxf

xfor d

ord

FLoC2018inOxford

FLoC2018inOxford





[ ]1 := ε [ ]2 := ε

(x/y :: A)1 := x(A1) (x/y :: A)2 := y(A2)

PCP (P) := ∃A ⊆ P. A 6= [ ]∧ A1 = A2


computer science


C2

LoC2018

xfor nf

d

FLo

F

d

ord

018inO

inOxf

FLo

F

C2

LoC2018

018inO

inOxf

xfor

d

ord

FLoC2018inOxford

FLoC2018inOxford





[ ]1 := ε [ ]2 := ε

(x/y :: A)1 := x(A1) (x/y :: A)2 := y(A2)

PCP (P) := ∃A ⊆ P. A 6= [ ]∧ A1 = A2


computer science


C2

LoC2018

xfor nf

d

FLo

F

d

ord

018inO

inOxf

FLo

F

C2

LoC2018

018inO

inOxf

xfor d

ord

FLoC2018inOxford

FLoC2018inOxford





[ ]1 := ε [ ]2 := ε

(x/y :: A)1 := x(A1) (x/y :: A)2 := y(A2)

PCP (P) := ∃A ⊆ P. A 6= [ ]∧ A1 = A2


computer science


C2

LoC2018

xfor nf

d

FLo

F

d

ord

018inO

inOxf

FLo

F

C2

LoC2018

018inO

inOxf

xfor d

ord

FLoC2018inOxford

FLoC2018inOxford





[ ]1 := ε [ ]2 := ε

(x/y :: A)1 := x(A1) (x/y :: A)2 := y(A2)

PCP (P) := ∃A ⊆ P. A 6= [ ]∧ A1 = A2


computer science


C2

LoC2018

xfor nf

d

FLo

F

d

ord

018inO

inOxf

FLo

F

C2

LoC2018

018inO

inOxf

xfor d

ord

FLoC2018inOxford

FLoC2018inOxford





[ ]1 := ε [ ]2 := ε

(x/y :: A)1 := x(A1) (x/y :: A)2 := y(A2)

PCP (P) := ∃A ⊆ P. A 6= [ ]∧ A1 = A2


computer science


C2

LoC2018

xfor nf

d

FLo

F

d

ord

018inO

inOxf

FLo

F

C2

LoC2018

018inO

inOxf

xfor d

ord

FLoC2018inOxford

FLoC2018inOxford





[ ]1 := ε [ ]2 := ε

(x/y :: A)1 := x(A1) (x/y :: A)2 := y(A2)

PCP (P) := ∃A ⊆ P. A 6= [ ]∧ A1 = A2


computer science

saarlanduniversityHistory

TM halting � MPCP

SR � PCP PCP � CFI

At best proof sketches, no inductions are given


computer science


TM halting � MPCP SR � PCP

PCP � CFI



computer science


TM halting � MPCP SR � PCP PCP � CFI



computer science


TM halting � MPCP SR � PCP PCP � CFI



computer science

saarlanduniversityContribution

CFI

TM SRH SR MPCP PCP

CFP


computer science

saarlanduniversityContribution

CFI

TM SRH SR MPCP PCP

CFP


computer science

saarlanduniversity

MPCP � PCP


computer science

saarlanduniversityMPCP

C2

LoC2018

xfor FLo

F

d

ord

018inO

inOxf

C2

LoC2018

xfor FLo

F

d

ord

018inO

inOxf

FLo

F

C2

LoC2018

018inO

inOxf

xfor d

ord

FLoC2018inOxford

FLoC2018inOxford

MPCP (x/y ,P) := ∃A ⊆ x/y :: P. xA1 = yA2


computer science


C2

LoC2018

xfor FLo

F

d

ord

018inO

inOxf

FLo

F

C2

LoC2018

018inO

inOxf

xfor d

ord

FLoC2018inOxford

FLoC2018inOxford



computer science


C2

LoC2018

xfor FLo

F

d

ord

018inO

inOxf

FLo

F

C2

LoC2018

018inO

inOxf

xfor d

ord

FLoC2018inOxford

FLoC2018inOxford



computer science


C2

LoC2018

xfor FLo

F

d

ord

018inO

inOxf

FLo

F

C2

LoC2018

018inO

inOxf

xfor d

ord

FLoC2018inOxford

FLoC2018inOxford



computer science

saarlanduniversityMPCP � PCP

FLo

F

C2

LoC2018

018inO

inOxf

xfor d

ord

FLoC2018inOxford

FLoC2018inOxford

$#F#L#o#C#2#0#1#8#i#n#O#x#f#o#r#d#$

$#F#L#o#C#2#0#1#8#i#n#O#x#f#o#r#d#$

$#F#L#o

$#F#

#C#2

L#o#C#2#0#1#8#

#0#1#8#i#n#O

i#n#O#x#f#

#x#f#o#r #d

o#r#d#

#$

$


computer science


FLo

F

C2

LoC2018

018inO

inOxf

xfor d

ord

FLoC2018inOxford

FLoC2018inOxford

$#F#L#o#C#2#0#1#8#i#n#O#x#f#o#r#d#$

$#F#L#o#C#2#0#1#8#i#n#O#x#f#o#r#d#$

$#F#L#o

$#F#

#C#2

L#o#C#2#0#1#8#

#0#1#8#i#n#O

i#n#O#x#f#

#x#f#o#r #d

o#r#d#

#$

$


computer science


FLo

F

C2

LoC2018

018inO

inOxf

xfor d

ord

FLoC2018inOxford

FLoC2018inOxford

$#F#L#o#C#2#0#1#8#i#n#O#x#f#o#r#d#$

$#F#L#o#C#2#0#1#8#i#n#O#x#f#o#r#d#$

$#F#L#o

$#F#

#C#2

L#o#C#2#0#1#8#

#0#1#8#i#n#O

i#n#O#x#f#

#x#f#o#r #d

o#r#d#

#$

$


computer science


$#F#L#o

$#F#

#C#2

L#o#C#2#0#1#8#

#0#1#8#i#n#O

i#n#O#x#f#

#x#f#o#r #d

o#r#d#

#$

#

#ε := ε ε# := ε#(ax) := #a(#x) (ax)# := a#(x#)

We define:

d := $(#x0) / $#(y#0 )

e := #$ / $

P := {d , e}++ {#x / y# | x/y ∈ x0/y0 :: R ∧ (x/y) 6= (ε/ε)}


computer science



computer science

saarlanduniversity

SR � MPCP


computer science

saarlanduniversitySR � MPCP

Let R = [ab/ba︸︷︷︸1

, aa/ab︸︷︷︸2

] and aab �∗R bab with

aab �1 aba �1 baa �2 bab

$

$aab#

a

a

ab

ba

#

#

ab

ba

a

a

#

#

b

b

aa

ab

#

#

bab#$

$

copy cards transfer unchanged symbols to the next string

rewrite cards simulate a single rewrite

consecutive strings are separated by #

d := $ / $x0#

e := y0#$ / $

P := [d , e] ++R ++ [#/#] ++ [a/a | a ∈ Σ]


computer science



, aa/ab︸︷︷︸2



$

$aab#

a

a

ab

ba

#

#

ab

ba

a

a

#

#

b

b

aa

ab

#

#

bab#$

$

$

$aab#




d := $ / $x0#

e := y0#$ / $

P := [d , e] ++R ++ [#/#] ++ [a/a | a ∈ Σ]


computer science



, aa/ab︸︷︷︸2



$

$aab#

a

a

ab

ba

#

#

ab

ba

a

a

#

#

b

b

aa

ab

#

#

bab#$

$

$

$aab#




d := $ / $x0#

e := y0#$ / $

P := [d , e] ++R ++ [#/#] ++ [a/a | a ∈ Σ]


computer science



, aa/ab︸︷︷︸2



$

$aab#

a

a

ab

ba

#

#

ab

ba

a

a

#

#

b

b

aa

ab

#

#

bab#$

$

$

$aab#




d := $ / $x0#

e := y0#$ / $

P := [d , e] ++R ++ [#/#] ++ [a/a | a ∈ Σ]


computer science



, aa/ab︸︷︷︸2



$

$aab#

a

a

ab

ba

#

#

ab

ba

a

a

#

#

b

b

aa

ab

#

#

bab#$

$

$aab#

$aab#aba#




d := $ / $x0#

e := y0#$ / $

P := [d , e] ++R ++ [#/#] ++ [a/a | a ∈ Σ]


computer science



, aa/ab︸︷︷︸2



$

$aab#

a

a

ab

ba

#

#

ab

ba

a

a

#

#

b

b

aa

ab

#

#

bab#$

$

$aab#aba#

$aab#aba#baa#




d := $ / $x0#

e := y0#$ / $

P := [d , e] ++R ++ [#/#] ++ [a/a | a ∈ Σ]


computer science



, aa/ab︸︷︷︸2



$

$aab#

a

a

ab

ba

#

#

ab

ba

a

a

#

#

b

b

aa

ab

#

#

bab#$

$

$aab#aba#baa#

$aab#aba#baa#bab#




d := $ / $x0#

e := y0#$ / $

P := [d , e] ++R ++ [#/#] ++ [a/a | a ∈ Σ]


computer science



, aa/ab︸︷︷︸2



$

$aab#

a

a

ab

ba

#

#

ab

ba

a

a

#

#

b

b

aa

ab

#

#

bab#$

$

$aab#aba#baa#bab#$

$aab#aba#baa#bab#$




d := $ / $x0#

e := y0#$ / $

P := [d , e] ++R ++ [#/#] ++ [a/a | a ∈ Σ]


computer science



, aa/ab︸︷︷︸2



$

$aab#

a

a

ab

ba

#

#

ab

ba

a

a

#

#

b

b

aa

ab

#

#

bab#$

$

$aab#aba#baa#bab#$

$aab#aba#baa#bab#$




d := $ / $x0#

e := y0#$ / $

P := [d , e] ++R ++ [#/#] ++ [a/a | a ∈ Σ]Yannick Forster, Edith Heiter, Gert Smolka PCP-Related Computational Reductions ITP 2018 – July 12 13

computer science


Given R and x0, y0 construct P such that x0 �∗ y0 ↔ PCP(P):

d := $ / $x0#

e := y0#$ / $

P := [d , e] ++R ++ [#/#] ++ [a/a | a ∈ Σ]

Theorem

x0 �∗R y0 if and only if there exists a stack A ⊆ P such that d :: A matches.

Lemma

Let x ⊆ Σ and x �∗R y0. Then there exists A ⊆ P such that A1 = x#A2.

Lemma

Let A ⊆ P, A1 = x#yA2, and x , y ⊆ Σ. Then yx �∗R y0.


computer science



d := $ / $x0#

e := y0#$ / $

P := [d , e] ++R ++ [#/#] ++ [a/a | a ∈ Σ]

Theorem


Lemma


Lemma



computer science



d := $ / $x0#

e := y0#$ / $

P := [d , e] ++R ++ [#/#] ++ [a/a | a ∈ Σ]

Theorem


Lemma


Lemma



computer science



d := $ / $x0#

e := y0#$ / $

P := [d , e] ++R ++ [#/#] ++ [a/a | a ∈ Σ]

Theorem


Lemma


Lemma



computer science

saarlanduniversityPost Correspondence Problems are special CFGs

C2

LoC2018

xfor nf

d

FLo

F

d

ord

018inO

inOxf

Produce grammars:

G1(P) : N → C2 M C2#LoC2018# G2(P) : N → LoC2018 M C2#LoC2018#

N → xfor M xfor## N → M xfor##

N → nf M nf#d# N → d M nf#d#

N → FLo M FLo#F# N → F M FLo#F#

N → d M d#ord# N → ord M d#ord#

N → 018inO M 018inO#inOxf# N → inOxf M 018inO#inOxf#

M → N | # M → N | #

FLo

F

C2

LoC2018

018inO

inOxf

xfor d

ord

PCP(P)↔ L(G1(P))∩L(G2(P)) 6= ∅


computer science


C2

LoC2018

xfor nf

d

FLo

F

d

ord

018inO

inOxf

Produce grammars:







M → N | # M → N | #

FLo

F

C2

LoC2018

018inO

inOxf

xfor d

ord

PCP(P)↔ L(G1(P))∩L(G2(P)) 6= ∅


computer science


C2

LoC2018

xfor nf

d

FLo

F

d

ord

018inO

inOxf

Produce grammars:







M → N | # M → N | #

FLo

F

C2

LoC2018

018inO

inOxf

xfor d

ord

FLo N FLo#F#

F N FLo#F#

PCP(P)↔ L(G1(P))∩L(G2(P)) 6= ∅Yannick Forster, Edith Heiter, Gert Smolka PCP-Related Computational Reductions ITP 2018 – July 12 15

computer science


C2

LoC2018

xfor nf

d

FLo

F

d

ord

018inO

inOxf

Produce grammars:







M → N | # M → N | #

FLo

F

C2

LoC2018

018inO

inOxf

xfor d

ord

FLoC2 N C2#LoC2018#FLo#F#

FLoC2018 N C2#LoC2018#FLo#F#


computer science


C2

LoC2018

xfor nf

d

FLo

F

d

ord

018inO

inOxf

Produce grammars:







M → N | # M → N | #

FLo

F

C2

LoC2018

018inO

inOxf

xfor d

ord

FLoC2018InO N 018inO#inOxf#C2#LoC2018#FLo#F#

FLoC2018InOxf N 018inO#inOxf#C2#LoC2018#FLo#F#


computer science


C2

LoC2018

xfor nf

d

FLo

F

d

ord

018inO

inOxf

Produce grammars:







M → N | # M → N | #

FLo

F

C2

LoC2018

018inO

inOxf

xfor

d

ord

FLoC2018InOxfor N xfor#018inO#inOxf#C2#LoC2018#FLo#F#

FLoC2018InOxf N xfor#018inO#inOxf#C2#LoC2018#FLo#F#


computer science


C2

LoC2018

xfor nf

d

FLo

F

d

ord

018inO

inOxf

Produce grammars:







M → N | # M → N | #

FLo

F

C2

LoC2018

018inO

inOxf

xfor d

ord

FLoC2018InOxford # d#ord#xfor#018inO#inOxf#C2#LoC2018#FLo#F#

FLoC2018InOxford # d#ord#xfor#018inO#inOxf#C2#LoC2018#FLo#F#


computer science

saarlanduniversityTM halting � SR

We use the TM definition from Asperti, Ricciotti (2015, in Matita)

Turing machines are special forms of rewriting systems:

Huge proof, essentially a big case distinction, no insight


computer science






computer science






computer science






computer science

saarlanduniversityFuture Work

CFP CFI

TM PCP

Forster, Kunze: Automated extraction from Coq to cbv λ-calculus yields

computability proofs for all reductions


computer science


CFP CFI ILL

TM PCP BPCP BSM MM eILLForster, Larchey-Wendling (LOLA18)




computer science


FXP FAM

CFP CFI ILL

cbvλ 2SM mTM TM PCP BPCP BSM MM eILL

Larchey-Wendling

Forster, Kunze, SmolkaForster, Wuttke

Forster, Kunze

Forster, Larchey-Wendling (LOLA18)




computer science


FXP FAM

CFP CFI ILL


(i)FOL

Larchey-Wendling


Forster, Kunze

Forster, KirstForster, Larchey-Wendling (LOLA18)




computer science


FXP FAM

CFP CFI ILL


(i)FOL

HOL IPC2

Larchey-Wendling


Forster, Kunze





computer science


FXP FAM

CFP CFI 3rd-ord. unif ILL


(i)FOL

HOL IPC2

Larchey-Wendling


Forster, Kunze





computer science


FXP FAM



(i)FOL

HOL IPC2

tiling problems

Larchey-Wendling


Forster, Kunze





computer science


FXP FAM



(i)FOL

HOL IPC2

tiling problems

System F inhab.

diophantine eqs

Larchey-Wendling


Forster, Kunze





computer science


FXP FAM



(i)FOL

IMP HOL IPC2

µ rec. functions tiling problems

System F inhab.

diophantine eqs

Larchey-Wendling


Forster, Kunze





computer science


FXP FAM



(i)FOL

IMP HOL IPC2

µ rec. functions tiling problems

System F inhab.

diophantine eqs

Larchey-Wendling


Forster, Kunze





computer science

saarlanduniversityWrap-up

A novel way to prove undecidability in Coq

Transparent, explainable reduction from TM to PCP

Enabling loads of future work. Add your own undecidable problems!

https://www.ps.uni-saarland.de/extras/PCP/


https://www.ps.uni-saarland.de/extras/PCP/

Documents

Verification of PCP-Related Computational Reductions in Coqforster/downloads/slides-itp18.pdf · Yannick Forster, Edith Heiter, Gert Smolka PCP-Related Computational Reductions ITP