[PPT] - Verification of PCP-Related Computational Reductions in Coq Yannick PowerPoint Presentation

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Verification of PCP-Related Computational Reductions in Coq

Yannick Forster, Edith Heiter, Gert Smolka ITP 2018 July 12

computer science

saarland

university

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

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computer science

saarland

university

Decidability

A problem P : X → P is decidable if . . . Classically Fix a model of computation M: there is a decider in M

Yannick Forster, Edith Heiter, Gert Smolka PCP-Related Computational Reductions ITP 2018 – July 12 2

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computer science

saarland

university

Decidability

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)

Yannick Forster, Edith Heiter, Gert Smolka PCP-Related Computational Reductions ITP 2018 – July 12 2

SLIDE 4

computer science

saarland

university

Decidability

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

Yannick Forster, Edith Heiter, Gert Smolka PCP-Related Computational Reductions ITP 2018 – July 12 2

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computer science

saarland

university

Decidability

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}

Yannick Forster, Edith Heiter, Gert Smolka PCP-Related Computational Reductions ITP 2018 – July 12 2

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computer science

saarland

university

Undecidability

A problem P : X → P is undecidable if . . . Classically If there is no decider u in M

Yannick Forster, Edith Heiter, Gert Smolka PCP-Related Computational Reductions ITP 2018 – July 12 3

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computer science

saarland

university

Undecidability

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)

Yannick Forster, Edith Heiter, Gert Smolka PCP-Related Computational Reductions ITP 2018 – July 12 3

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computer science

saarland

university

Undecidability

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 ¬(∀x : X. {Px} + {¬Px})

Yannick Forster, Edith Heiter, Gert Smolka PCP-Related Computational Reductions ITP 2018 – July 12 3

SLIDE 9

computer science

saarland

university

Undecidability

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

✭✭✭✭✭✭✭✭✭✭✭ ✭

¬(∀x : X. {Px} + {¬Px})

Yannick Forster, Edith Heiter, Gert Smolka PCP-Related Computational Reductions ITP 2018 – July 12 3

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computer science

saarland

university

Undecidability

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

✭✭✭✭✭✭✭✭✭✭✭ ✭

¬(∀x : X. {Px} + {¬Px}) In practice: most proofs are by reduction

Definition

P undecidable := Halting problem reduces to P

Yannick Forster, Edith Heiter, Gert Smolka PCP-Related Computational Reductions ITP 2018 – July 12 3

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computer science

saarland

university

Reduction

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

Yannick Forster, Edith Heiter, Gert Smolka PCP-Related Computational Reductions ITP 2018 – July 12 4

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computer science

saarland

university Yannick Forster, Edith Heiter, Gert Smolka PCP-Related Computational Reductions ITP 2018 – July 12 5

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computer science

saarland

university

PCP

C2 LoC2018 xfor nf d FLo F d

rd

018inO inOxf Yannick Forster, Edith Heiter, Gert Smolka PCP-Related Computational Reductions ITP 2018 – July 12 6

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computer science

saarland

university

PCP

C2 LoC2018 xfor nf d FLo F d

rd

018inO inOxf FLo F Yannick Forster, Edith Heiter, Gert Smolka PCP-Related Computational Reductions ITP 2018 – July 12 6

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computer science

saarland

university

PCP

C2 LoC2018 xfor nf d FLo F d

rd

018inO inOxf FLo F C2 LoC2018 Yannick Forster, Edith Heiter, Gert Smolka PCP-Related Computational Reductions ITP 2018 – July 12 6

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computer science

saarland

university

PCP

C2 LoC2018 xfor nf d FLo F d

rd

018inO inOxf FLo F C2 LoC2018 018inO inOxf Yannick Forster, Edith Heiter, Gert Smolka PCP-Related Computational Reductions ITP 2018 – July 12 6

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computer science

saarland

university

PCP

C2 LoC2018 xfor nf d FLo F d

rd

018inO inOxf FLo F C2 LoC2018 018inO inOxf xfor Yannick Forster, Edith Heiter, Gert Smolka PCP-Related Computational Reductions ITP 2018 – July 12 6

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computer science

saarland

university

PCP

C2 LoC2018 xfor nf d FLo F d

rd

018inO inOxf FLo F C2 LoC2018 018inO inOxf xfor d

rd

Yannick Forster, Edith Heiter, Gert Smolka PCP-Related Computational Reductions ITP 2018 – July 12 6

SLIDE 19

computer science

saarland

university

PCP

C2 LoC2018 xfor nf d FLo F d

rd

018inO inOxf FLo F C2 LoC2018 018inO inOxf xfor d

rd

FLoC2018inOxford FLoC2018inOxford Yannick Forster, Edith Heiter, Gert Smolka PCP-Related Computational Reductions ITP 2018 – July 12 6

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computer science

saarland

university

PCP

C2 LoC2018 xfor nf d FLo F d

rd

018inO inOxf FLo F C2 LoC2018 018inO inOxf xfor d

rd

FLoC2018inOxford FLoC2018inOxford

Symbols a, b, c: N Strings x, y, z: lists of symbols Card c: pairs of strings Stacks A, P: lists of cards A ⊆ P: list inclusion

Yannick Forster, Edith Heiter, Gert Smolka PCP-Related Computational Reductions ITP 2018 – July 12 6

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computer science

saarland

university

PCP

C2 LoC2018 xfor nf d FLo F d

rd

018inO inOxf FLo F C2 LoC2018 018inO inOxf xfor d

rd

FLoC2018inOxford FLoC2018inOxford

Symbols a, b, c: N Strings x, y, z: lists of symbols Card c: pairs of strings Stacks A, P: lists of cards A ⊆ P: list inclusion [ ]1 := ǫ [ ]2 := ǫ (x/y :: A)1 := x(A1) (x/y :: A)2 := y(A2) PCP (P) := ∃A ⊆ P. A = [ ] ∧ A1 = A2

Yannick Forster, Edith Heiter, Gert Smolka PCP-Related Computational Reductions ITP 2018 – July 12 6

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computer science

saarland

university

History

TM halting MPCP

Yannick Forster, Edith Heiter, Gert Smolka PCP-Related Computational Reductions ITP 2018 – July 12 7

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computer science

saarland

university

History

TM halting MPCP SR PCP

Yannick Forster, Edith Heiter, Gert Smolka PCP-Related Computational Reductions ITP 2018 – July 12 7

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computer science

saarland

university

History

TM halting MPCP SR PCP PCP CFI

Yannick Forster, Edith Heiter, Gert Smolka PCP-Related Computational Reductions ITP 2018 – July 12 7

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computer science

saarland

university

History

TM halting MPCP SR PCP PCP CFI At best proof sketches, no inductions are given

Yannick Forster, Edith Heiter, Gert Smolka PCP-Related Computational Reductions ITP 2018 – July 12 7

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computer science

saarland

university

Contribution

CFI TM SRH SR MPCP PCP CFP

Yannick Forster, Edith Heiter, Gert Smolka PCP-Related Computational Reductions ITP 2018 – July 12 8

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computer science

saarland

university

Contribution

CFI TM SRH SR MPCP PCP CFP

Yannick Forster, Edith Heiter, Gert Smolka PCP-Related Computational Reductions ITP 2018 – July 12 8

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computer science

saarland

university

MPCP PCP

Yannick Forster, Edith Heiter, Gert Smolka PCP-Related Computational Reductions ITP 2018 – July 12 8

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computer science

saarland

university

MPCP

C2 LoC2018 xfor FLo F d

rd

018inO inOxf Yannick Forster, Edith Heiter, Gert Smolka PCP-Related Computational Reductions ITP 2018 – July 12 9

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computer science

saarland

university

MPCP

C2 LoC2018 xfor FLo F d

rd

018inO inOxf Yannick Forster, Edith Heiter, Gert Smolka PCP-Related Computational Reductions ITP 2018 – July 12 9

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computer science

saarland

university

MPCP

C2 LoC2018 xfor FLo F d

rd

018inO inOxf FLo F C2 LoC2018 018inO inOxf xfor d

rd

FLoC2018inOxford FLoC2018inOxford Yannick Forster, Edith Heiter, Gert Smolka PCP-Related Computational Reductions ITP 2018 – July 12 9

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computer science

saarland

university

MPCP

C2 LoC2018 xfor FLo F d

rd

018inO inOxf FLo F C2 LoC2018 018inO inOxf xfor d

rd

FLoC2018inOxford FLoC2018inOxford

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

Yannick Forster, Edith Heiter, Gert Smolka PCP-Related Computational Reductions ITP 2018 – July 12 9

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computer science

saarland

university

MPCP PCP

FLo F C2 LoC2018 018inO inOxf xfor d

rd

FLoC2018inOxford FLoC2018inOxford Yannick Forster, Edith Heiter, Gert Smolka PCP-Related Computational Reductions ITP 2018 – July 12 10

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computer science

saarland

university

MPCP PCP

FLo F C2 LoC2018 018inO inOxf xfor d

rd

FLoC2018inOxford FLoC2018inOxford $#F#L#o#C#2#0#1#8#i#n#O#x#f #o#r#d#$ $#F#L#o#C#2#0#1#8#i#n#O#x#f #o#r#d#$ Yannick Forster, Edith Heiter, Gert Smolka PCP-Related Computational Reductions ITP 2018 – July 12 10

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computer science

saarland

university

MPCP PCP

FLo F C2 LoC2018 018inO inOxf xfor d

rd

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

#r#d#

#$ $ Yannick Forster, Edith Heiter, Gert Smolka PCP-Related Computational Reductions ITP 2018 – July 12 10

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computer science

saarland

university

MPCP PCP

$#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

#r#d#

#$ #

#ǫ := ǫ

ǫ# := ǫ

#(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) = (ǫ/ǫ)}

Yannick Forster, Edith Heiter, Gert Smolka PCP-Related Computational Reductions ITP 2018 – July 12 11

SLIDE 37

computer science

saarland

university

MPCP PCP

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

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computer science

saarland

university

SR MPCP

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

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computer science

saarland

university

SR MPCP

Let R = [ab/ba

1

, aa/ab

2

] and aab ≻∗

R bab with

aab ≻1 aba ≻1 baa ≻2 bab

Yannick Forster, Edith Heiter, Gert Smolka PCP-Related Computational Reductions ITP 2018 – July 12 13

SLIDE 40

computer science

saarland

university

SR MPCP

Let R = [ab/ba

1

, aa/ab

2

] and aab ≻∗

R bab with

aab ≻1 aba ≻1 baa ≻2 bab $ $aab# $ $aab#

Yannick Forster, Edith Heiter, Gert Smolka PCP-Related Computational Reductions ITP 2018 – July 12 13

SLIDE 41

computer science

saarland

university

SR MPCP

Let R = [ab/ba

1

, aa/ab

2

] and aab ≻∗

R bab with

aab ≻1 aba ≻1 baa ≻2 bab $ $aab# a a $ $aab#

copy cards transfer unchanged symbols to the next string

Yannick Forster, Edith Heiter, Gert Smolka PCP-Related Computational Reductions ITP 2018 – July 12 13

SLIDE 42

computer science

saarland

university

SR 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 $ $aab#

copy cards transfer unchanged symbols to the next string rewrite cards simulate a single rewrite

Yannick Forster, Edith Heiter, Gert Smolka PCP-Related Computational Reductions ITP 2018 – July 12 13

SLIDE 43

computer science

saarland

university

SR 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 # # $aab# $aab#aba#

copy cards transfer unchanged symbols to the next string rewrite cards simulate a single rewrite consecutive strings are separated by #

Yannick Forster, Edith Heiter, Gert Smolka PCP-Related Computational Reductions ITP 2018 – July 12 13

SLIDE 44

computer science

saarland

university

SR 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 # # $aab#aba# $aab#aba#baa#

copy cards transfer unchanged symbols to the next string rewrite cards simulate a single rewrite consecutive strings are separated by #

Yannick Forster, Edith Heiter, Gert Smolka PCP-Related Computational Reductions ITP 2018 – July 12 13

SLIDE 45

computer science

saarland

university

SR 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 # # $aab#aba#baa# $aab#aba#baa#bab#

copy cards transfer unchanged symbols to the next string rewrite cards simulate a single rewrite consecutive strings are separated by #

Yannick Forster, Edith Heiter, Gert Smolka PCP-Related Computational Reductions ITP 2018 – July 12 13

SLIDE 46

computer science

saarland

university

SR MPCP

Let R = [ab/ba

1

, aa/ab

2

] and aab ≻∗

R bab with

aab ≻1 aba ≻1 baa ≻2 bab $ $aab# a a ab ba # # ab ba a a # # b b aa ab # # bab#$ $ $aab#aba#baa#bab#$ $aab#aba#baa#bab#$

copy cards transfer unchanged symbols to the next string rewrite cards simulate a single rewrite consecutive strings are separated by #

Yannick Forster, Edith Heiter, Gert Smolka PCP-Related Computational Reductions ITP 2018 – July 12 13

SLIDE 47

computer science

saarland

university

SR MPCP

Let R = [ab/ba

1

, aa/ab

2

] and aab ≻∗

R bab with

aab ≻1 aba ≻1 baa ≻2 bab $ $aab# a a ab ba # # ab ba a a # # b b aa ab # # bab#$ $ $aab#aba#baa#bab#$ $aab#aba#baa#bab#$

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]

Yannick Forster, Edith Heiter, Gert Smolka PCP-Related Computational Reductions ITP 2018 – July 12 13

SLIDE 48

computer science

saarland

university

SR MPCP

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]

Yannick Forster, Edith Heiter, Gert Smolka PCP-Related Computational Reductions ITP 2018 – July 12 14

SLIDE 49

computer science

saarland

university

SR MPCP

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.

Yannick Forster, Edith Heiter, Gert Smolka PCP-Related Computational Reductions ITP 2018 – July 12 14

SLIDE 50

computer science

saarland

university

SR MPCP

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.

Yannick Forster, Edith Heiter, Gert Smolka PCP-Related Computational Reductions ITP 2018 – July 12 14

SLIDE 51

computer science

saarland

university

SR MPCP

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.

Yannick Forster, Edith Heiter, Gert Smolka PCP-Related Computational Reductions ITP 2018 – July 12 14

SLIDE 52

computer science

saarland

university

Post Correspondence Problems are special CFGs

C2 LoC2018 xfor nf d FLo F d

rd

018inO inOxf Yannick Forster, Edith Heiter, Gert Smolka PCP-Related Computational Reductions ITP 2018 – July 12 15

SLIDE 53

computer science

saarland

university

Post Correspondence Problems are special CFGs

C2 LoC2018 xfor nf d FLo F d

rd

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 | #

Yannick Forster, Edith Heiter, Gert Smolka PCP-Related Computational Reductions ITP 2018 – July 12 15

SLIDE 54

computer science

saarland

university

Post Correspondence Problems are special CFGs

C2 LoC2018 xfor nf d FLo F d

rd

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

FLo N FLo#F# F N FLo#F# PCP(P) ↔ L(G1(P)) ∩ L(G2(P)) = ∅

Yannick Forster, Edith Heiter, Gert Smolka PCP-Related Computational Reductions ITP 2018 – July 12 15

SLIDE 55

computer science

saarland

university

Post Correspondence Problems are special CFGs

C2 LoC2018 xfor nf d FLo F d

rd

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

FLoC2 N C2#LoC2018#FLo#F# FLoC2018 N C2#LoC2018#FLo#F# PCP(P) ↔ L(G1(P)) ∩ L(G2(P)) = ∅

Yannick Forster, Edith Heiter, Gert Smolka PCP-Related Computational Reductions ITP 2018 – July 12 15

SLIDE 56

computer science

saarland

university

Post Correspondence Problems are special CFGs

C2 LoC2018 xfor nf d FLo F d

rd

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

FLoC2018InO N 018inO#inOxf#C2#LoC2018#FLo#F# FLoC2018InOxf N 018inO#inOxf#C2#LoC2018#FLo#F# PCP(P) ↔ L(G1(P)) ∩ L(G2(P)) = ∅

Yannick Forster, Edith Heiter, Gert Smolka PCP-Related Computational Reductions ITP 2018 – July 12 15

SLIDE 57

computer science

saarland

university

Post Correspondence Problems are special CFGs

C2 LoC2018 xfor nf d FLo F d

rd

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

FLoC2018InOxfor N xfor#018inO#inOxf#C2#LoC2018#FLo#F# FLoC2018InOxf N xfor#018inO#inOxf#C2#LoC2018#FLo#F# PCP(P) ↔ L(G1(P)) ∩ L(G2(P)) = ∅

Yannick Forster, Edith Heiter, Gert Smolka PCP-Related Computational Reductions ITP 2018 – July 12 15

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computer science

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university

Post Correspondence Problems are special CFGs

C2 LoC2018 xfor nf d FLo F d

rd

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

rd

FLoC2018InOxford # d#ord#xfor#018inO#inOxf#C2#LoC2018#FLo#F# FLoC2018InOxford # d#ord#xfor#018inO#inOxf#C2#LoC2018#FLo#F# PCP(P) ↔ L(G1(P)) ∩ L(G2(P)) = ∅

Yannick Forster, Edith Heiter, Gert Smolka PCP-Related Computational Reductions ITP 2018 – July 12 15

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TM halting SR

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

Yannick Forster, Edith Heiter, Gert Smolka PCP-Related Computational Reductions ITP 2018 – July 12 16

SLIDE 60

computer science

saarland

university

TM halting SR

We use the TM definition from Asperti, Ricciotti (2015, in Matita) Turing machines are special forms of rewriting systems:

Yannick Forster, Edith Heiter, Gert Smolka PCP-Related Computational Reductions ITP 2018 – July 12 16

SLIDE 61

computer science

saarland

university

TM halting SR

We use the TM definition from Asperti, Ricciotti (2015, in Matita) Turing machines are special forms of rewriting systems:

Yannick Forster, Edith Heiter, Gert Smolka PCP-Related Computational Reductions ITP 2018 – July 12 16

SLIDE 62

computer science

saarland

university

TM 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

Yannick Forster, Edith Heiter, Gert Smolka PCP-Related Computational Reductions ITP 2018 – July 12 16

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Future Work

CFP CFI TM PCP Yannick Forster, Edith Heiter, Gert Smolka PCP-Related Computational Reductions ITP 2018 – July 12 17

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Future Work

CFP CFI ILL TM PCP BPCP BSM MM eILL

Forster, Larchey-Wendling (LOLA18)

Yannick Forster, Edith Heiter, Gert Smolka PCP-Related Computational Reductions ITP 2018 – July 12 17

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computer science

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Future Work

FXP FAM CFP CFI ILL cbvλ 2SM mTM TM PCP BPCP BSM MM eILL

Larchey-Wendling Forster, Kunze, Smolka Forster, Wuttke Forster, Kunze Forster, Larchey-Wendling (LOLA18)

Yannick Forster, Edith Heiter, Gert Smolka PCP-Related Computational Reductions ITP 2018 – July 12 17

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computer science

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Future Work

FXP FAM CFP CFI ILL cbvλ 2SM mTM TM PCP BPCP BSM MM eILL (i)FOL

Larchey-Wendling Forster, Kunze, Smolka Forster, Wuttke Forster, Kunze Forster, Kirst Forster, Larchey-Wendling (LOLA18)

Yannick Forster, Edith Heiter, Gert Smolka PCP-Related Computational Reductions ITP 2018 – July 12 17

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computer science

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Future Work

FXP FAM CFP CFI ILL cbvλ 2SM mTM TM PCP BPCP BSM MM eILL (i)FOL HOL IPC2

Larchey-Wendling Forster, Kunze, Smolka Forster, Wuttke Forster, Kunze Forster, Kirst Forster, Larchey-Wendling (LOLA18)

Yannick Forster, Edith Heiter, Gert Smolka PCP-Related Computational Reductions ITP 2018 – July 12 17

SLIDE 68

computer science

saarland

university

Future Work

FXP FAM CFP CFI 3rd-ord. unif ILL cbvλ 2SM mTM TM PCP BPCP BSM MM eILL (i)FOL HOL IPC2

Larchey-Wendling Forster, Kunze, Smolka Forster, Wuttke Forster, Kunze Forster, Kirst Forster, Larchey-Wendling (LOLA18)

Yannick Forster, Edith Heiter, Gert Smolka PCP-Related Computational Reductions ITP 2018 – July 12 17

SLIDE 69

computer science

saarland

university

Future Work

FXP FAM CFP CFI 3rd-ord. unif ILL cbvλ 2SM mTM TM PCP BPCP BSM MM eILL (i)FOL HOL IPC2 tiling problems

Larchey-Wendling Forster, Kunze, Smolka Forster, Wuttke Forster, Kunze Forster, Kirst Forster, Larchey-Wendling (LOLA18)

Yannick Forster, Edith Heiter, Gert Smolka PCP-Related Computational Reductions ITP 2018 – July 12 17

SLIDE 70

computer science

saarland

university

Future Work

FXP FAM CFP CFI 3rd-ord. unif ILL cbvλ 2SM mTM TM PCP BPCP BSM MM eILL (i)FOL HOL IPC2 tiling problems System F inhab. diophantine eqs

Larchey-Wendling Forster, Kunze, Smolka Forster, Wuttke Forster, Kunze Forster, Kirst Forster, Larchey-Wendling (LOLA18)

Yannick Forster, Edith Heiter, Gert Smolka PCP-Related Computational Reductions ITP 2018 – July 12 17

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computer science

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Future Work

FXP FAM CFP CFI 3rd-ord. unif ILL cbvλ 2SM mTM TM PCP BPCP BSM MM eILL (i)FOL IMP HOL IPC2 µ rec. functions tiling problems System F inhab. diophantine eqs

Larchey-Wendling Forster, Kunze, Smolka Forster, Wuttke Forster, Kunze Forster, Kirst Forster, Larchey-Wendling (LOLA18)

Yannick Forster, Edith Heiter, Gert Smolka PCP-Related Computational Reductions ITP 2018 – July 12 17

SLIDE 72

computer science

saarland

university

Future Work

FXP FAM CFP CFI 3rd-ord. unif ILL cbvλ 2SM mTM TM PCP BPCP BSM MM eILL (i)FOL IMP HOL IPC2 µ rec. functions tiling problems System F inhab. diophantine eqs

Larchey-Wendling Forster, Kunze, Smolka Forster, Wuttke Forster, Kunze Forster, Kirst Forster, Larchey-Wendling (LOLA18)

Forster, Kunze: Automated extraction from Coq to cbv λ-calculus yields computability proofs for all reductions

Yannick Forster, Edith Heiter, Gert Smolka PCP-Related Computational Reductions ITP 2018 – July 12 17

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Wrap-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/

Yannick Forster, Edith Heiter, Gert Smolka PCP-Related Computational Reductions ITP 2018 – July 12 18