Big Picture TRS yes implement no Literature CSI maybe - - PowerPoint PPT Presentation

Layer Systems for Confluence — Formalized1

Bertram Felgenhauer, Franziska Rapp

University of Innsbruck, Allgemeines Rechenzentrum Innsbruck

ICTAC, Stellenbosch 2018-10-16

1supported by FWF project P27528

Motivation

Big Picture

TRS Literature CSI yes no maybe implement

• automated confluence of first-order term rewrite systems

Bertram Felgenhauer (UIBK) Layer Systems — Formalized ICTAC 2018 2/22

Motivation

Big Picture

TRS Literature CSI yes no maybe proof Ce T A yes no maybe IsaFoR implement formalize generate

• certified automated confluence of first-order term rewrite systems

Bertram Felgenhauer (UIBK) Layer Systems — Formalized ICTAC 2018 2/22

Motivation

Big Picture

TRS Literature CSI yes no maybe proof Ce T A yes no maybe IsaFoR implement formalize generate

• certified automated confluence of first-order term rewrite systems
• here: formalizing layer systems

Bertram Felgenhauer (UIBK) Layer Systems — Formalized ICTAC 2018 2/22

Motivation

Term Rewriting

+(x, 0) → x +(x, S(y)) → S(+(x, y)) +(0, +(0, S(S(0))))

Bertram Felgenhauer (UIBK) Layer Systems — Formalized ICTAC 2018 3/22

Motivation

Term Rewriting

+(x, 0) → x +(x, S(y)) → S(+(x, y)) +(0, +(0, S(S(0)))) +(0, S(+(0, S(0))))

Bertram Felgenhauer (UIBK) Layer Systems — Formalized ICTAC 2018 3/22

Motivation

Term Rewriting

+(x, 0) → x +(x, S(y)) → S(+(x, y)) +(0, +(0, S(S(0)))) +(0, S(+(0, S(0)))) S(+(0, +(0, S(0)))) +(0, S(S(+(0, 0))))

Bertram Felgenhauer (UIBK) Layer Systems — Formalized ICTAC 2018 3/22

Motivation

Term Rewriting

+(x, 0) → x +(x, S(y)) → S(+(x, y)) +(0, +(0, S(S(0)))) +(0, S(+(0, S(0)))) S(+(0, +(0, S(0)))) +(0, S(S(+(0, 0)))) S(+(0, S(+(0, 0)))) +(0, S(S(0))) S(S(+(0, +(0, 0)))) S(+(0, S(0))) S(S(+(0, 0))) S(S(0))

Bertram Felgenhauer (UIBK) Layer Systems — Formalized ICTAC 2018 3/22

Motivation

Confluence

s t u v

∗ ∗ ∗ ∗

Definition

• ∗← · →∗ ⊆ →∗ · ∗←

Bertram Felgenhauer (UIBK) Layer Systems — Formalized ICTAC 2018 4/22

Motivation

Confluence

s t u v

∗ ∗ ∗ ∗

Definition

• ∗← · →∗ ⊆ →∗ · ∗←

Criteria for TRSs

• orthogonality: left-linear, no critical pairs
• Knuth-Bendix: terminating, joinable critical pairs
• . . .

Bertram Felgenhauer (UIBK) Layer Systems — Formalized ICTAC 2018 4/22

Motivation

Example

@(@(K, x), y) → x @(@(@(S, x), y), z) → @(@(x, z), @(y, z)) e(x, x) → ⊤ Orthogonal?

• not left-linear
• no critical pairs

Bertram Felgenhauer (UIBK) Layer Systems — Formalized ICTAC 2018 5/22

Motivation

Example

@(@(K, x), y) → x @(@(@(S, x), y), z) → @(@(x, z), @(y, z)) e(x, x) → ⊤ Orthogonal?

• not left-linear
• no critical pairs

Knuth-Bendix?

• non-terminating

@(@(@(S, I), I), @(@(S, I), I)) →+ @(@(@(S, I), I), @(@(S, I), I)) where I = @(@(S, K), K)

• joinable critical pairs

Bertram Felgenhauer (UIBK) Layer Systems — Formalized ICTAC 2018 5/22

Motivation

Modularity

Theorem

Let R1, R2 be TRSs over disjoint signatures. Then CR(R1 ∪ R2) ⇐ ⇒ CR(R1) ∧ CR(R2)

Bertram Felgenhauer (UIBK) Layer Systems — Formalized ICTAC 2018 6/22

Motivation

Modularity

Theorem

Let R1, R2 be TRSs over disjoint signatures. Then CR(R1 ∪ R2) ⇐ ⇒ CR(R1) ∧ CR(R2) Example @(@(K, x), y) → x @(@(@(S, x), y), z) → @(@(x, z), @(y, z)) e(x, x) → ⊤

Bertram Felgenhauer (UIBK) Layer Systems — Formalized ICTAC 2018 6/22

Motivation

Modularity

Theorem

Let R1, R2 be TRSs over disjoint signatures. Then CR(R1 ∪ R2) ⇐ ⇒ CR(R1) ∧ CR(R2) Example @(@(K, x), y) → x @(@(@(S, x), y), z) → @(@(x, z), @(y, z)) e(x, x) → ⊤

• first two rules are orthogonal

Bertram Felgenhauer (UIBK) Layer Systems — Formalized ICTAC 2018 6/22

Motivation

Modularity

Theorem

Let R1, R2 be TRSs over disjoint signatures. Then CR(R1 ∪ R2) ⇐ ⇒ CR(R1) ∧ CR(R2) Example @(@(K, x), y) → x @(@(@(S, x), y), z) → @(@(x, z), @(y, z)) e(x, x) → ⊤

• first two rules are orthogonal
• last rule is terminating, and has no critical pairs

Bertram Felgenhauer (UIBK) Layer Systems — Formalized ICTAC 2018 6/22

Motivation

Modularity

Theorem

Let R1, R2 be TRSs over disjoint signatures. Then CR(R1 ∪ R2) ⇐ ⇒ CR(R1) ∧ CR(R2) Example @(@(K, x), y) → x @(@(@(S, x), y), z) → @(@(x, z), @(y, z)) e(x, x) → ⊤

• first two rules are orthogonal
• last rule is terminating, and has no critical pairs
• disjoint signatures =

⇒ confluent by modularity

Bertram Felgenhauer (UIBK) Layer Systems — Formalized ICTAC 2018 6/22

Layer Systems

Table of Contents

Motivation Layer Systems Formalization Implementation

Bertram Felgenhauer (UIBK) Layer Systems — Formalized ICTAC 2018 7/22

Layer Systems

Proving Modularity

History

• Toyama 1987
• Klop et al. 1994
• van Oostrom 2008
• . . .

Bertram Felgenhauer (UIBK) Layer Systems — Formalized ICTAC 2018 8/22

Layer Systems

Proving Modularity

History

• Toyama 1987
• Klop et al. 1994
• van Oostrom 2008
• . . .

Proof idea

• =

⇒ is easy (homogeneous terms are closed under rewriting)

Bertram Felgenhauer (UIBK) Layer Systems — Formalized ICTAC 2018 8/22

Layer Systems

Proving Modularity

History

• Toyama 1987
• Klop et al. 1994
• van Oostrom 2008
• . . .

Proof idea

• =

⇒ is easy (homogeneous terms are closed under rewriting)

• decompose terms into maximal top and aliens recursively

Bertram Felgenhauer (UIBK) Layer Systems — Formalized ICTAC 2018 8/22

Layer Systems

Example

@(@(K, x), y) → x @(@(@(S, x), y), z) → @(@(x, z), @(y, z)) e(x, x) → ⊤

• e(@(x, e(S, x)), K)

e @ x e S x K

Bertram Felgenhauer (UIBK) Layer Systems — Formalized ICTAC 2018 9/22

Layer Systems

Example

@(@(K, x), y) → x @(@(@(S, x), y), z) → @(@(x, z), @(y, z)) e(x, x) → ⊤

• e(@(x, e(S, x)), K)

e @ x e S x K e @ x e S x K

Bertram Felgenhauer (UIBK) Layer Systems — Formalized ICTAC 2018 9/22

Layer Systems

Example

@(@(K, x), y) → x @(@(@(S, x), y), z) → @(@(x, z), @(y, z)) e(x, x) → ⊤

• e(@(x, e(S, x)), K)

e @ x e S x K e @ x e S x K

• max-top e(, ), aliens @(x, e(S, x)) and K, rank 4

Bertram Felgenhauer (UIBK) Layer Systems — Formalized ICTAC 2018 9/22

Layer Systems

Proving Modularity

History

• Toyama 1987
• Klop et al. 1994
• van Oostrom 2008
• . . .

Proof idea

• =

⇒ is easy (homogeneous terms are closed under rewriting)

• decompose terms into maximal top and aliens recursively
• use induction on rank

Bertram Felgenhauer (UIBK) Layer Systems — Formalized ICTAC 2018 10/22

Layer Systems

Proving Modularity

History

• Toyama 1987
• Klop et al. 1994
• van Oostrom 2008
• . . .

Proof idea

• =

⇒ is easy (homogeneous terms are closed under rewriting)

• decompose terms into maximal top and aliens recursively
• use induction on rank
• . . . details are complicated

Bertram Felgenhauer (UIBK) Layer Systems — Formalized ICTAC 2018 10/22

Layer Systems

Related Results

Results

• persistence (Aoto and Toyama 1997)
• layer preservation (Ohlebusch 1994)
• currying (Kahrs 1995)
• . . .

Bertram Felgenhauer (UIBK) Layer Systems — Formalized ICTAC 2018 11/22

Layer Systems

Related Results

Results

• persistence (Aoto and Toyama 1997)
• layer preservation (Ohlebusch 1994)
• currying (Kahrs 1995)
• . . .

Proof idea

• similar to modularity
• different decomposition into max-top and aliens

Bertram Felgenhauer (UIBK) Layer Systems — Formalized ICTAC 2018 11/22

Layer Systems

Layer Systems in a Nutshell

Idea

• layer system L: set of admissible tops

Bertram Felgenhauer (UIBK) Layer Systems — Formalized ICTAC 2018 12/22

Layer Systems

Layer Systems in a Nutshell

Idea

• layer system L: set of admissible tops
• theorem: if R is confluent on L then R is confluent

Bertram Felgenhauer (UIBK) Layer Systems — Formalized ICTAC 2018 12/22

Layer Systems

Layer Systems in a Nutshell

Idea

• layer system L: set of admissible tops
• theorem: if R is confluent on L then R is confluent
• adapt modularity proof

Bertram Felgenhauer (UIBK) Layer Systems — Formalized ICTAC 2018 12/22

Layer Systems

Layer Systems in a Nutshell

Idea

• layer system L: set of admissible tops
• theorem: if R is confluent on L then R is confluent
• adapt modularity proof

Complications

• max-tops must be unique

Bertram Felgenhauer (UIBK) Layer Systems — Formalized ICTAC 2018 12/22

Layer Systems

Layer Systems in a Nutshell

Idea

• layer system L: set of admissible tops
• theorem: if R is confluent on L then R is confluent
• adapt modularity proof

Complications

• max-tops must be unique
• rewriting must not increase rank

Bertram Felgenhauer (UIBK) Layer Systems — Formalized ICTAC 2018 12/22

Layer Systems

Layer Systems in a Nutshell

Idea

• layer system L: set of admissible tops
• theorem: if R is confluent on L then R is confluent
• adapt modularity proof

Complications

• max-tops must be unique
• rewriting must not increase rank
• several restrictions on fusion

f H c →H(x)→x f c

• f

c

Bertram Felgenhauer (UIBK) Layer Systems — Formalized ICTAC 2018 12/22

Layer Systems

Layer Systems Definition

Definition

Under the following conditions, the TRS R is layered wrt L ⊆ C(F, V): L1 Every term in T (F, V) has a non-empty top L2 If x ∈ V and C ∈ C, then C[x]p ∈ L if and only if C[]p ∈ L L3 If L, N ∈ L, p ∈ PosF(L), and L|p ⊔ N is defined then L[L|p ⊔ N]p ∈ L W If M is the max-top of s, p ∈ PosF(M), and s →p,ℓ→r t with ℓ → r ∈ R, then M →p,ℓ→r L for some L ∈ L C1 In (W), either L is the max-top of t or L = C2 If L, N ∈ L and L ⊑ N, then L[N|p]p ∈ L for any p ∈ Pos(L)

Bertram Felgenhauer (UIBK) Layer Systems — Formalized ICTAC 2018 13/22

Layer Systems

Layer Systems Definition

Definition

Under the following conditions, the TRS R is layered wrt L ⊆ C(F, V): L1 Every term in T (F, V) has a non-empty top L2 If x ∈ V and C ∈ C, then C[x]p ∈ L if and only if C[]p ∈ L L3 If L, N ∈ L, p ∈ PosF(L), and L|p ⊔ N is defined then L[L|p ⊔ N]p ∈ L W If M is the max-top of s, p ∈ PosF(M), and s →p,ℓ→r t with ℓ → r ∈ R, then M →p,ℓ→r L for some L ∈ L C1 In (W), either L is the max-top of t or L = C2 If L, N ∈ L and L ⊑ N, then L[N|p]p ∈ L for any p ∈ Pos(L)

DON’T PANIC

it’s formalized

Bertram Felgenhauer (UIBK) Layer Systems — Formalized ICTAC 2018 13/22

Layer Systems

Using Layer Systems

Main result(s)

• R layered by L and rank 1 terms confluent =

⇒ R confluent

• . . .

Bertram Felgenhauer (UIBK) Layer Systems — Formalized ICTAC 2018 14/22

Layer Systems

Using Layer Systems

Main result(s)

• R layered by L and rank 1 terms confluent =

⇒ R confluent

• . . .

Applications

• modularity: R1 ∪ R2 is layered by C(F1, V) ∪ C(F2, V).
• persistence: R is layered by well-sorted contexts
• currying: PP(R) is layered by. . .

Bertram Felgenhauer (UIBK) Layer Systems — Formalized ICTAC 2018 14/22

Formalization

Table of Contents

Motivation Layer Systems Formalization Implementation

Bertram Felgenhauer (UIBK) Layer Systems — Formalized ICTAC 2018 15/22

Formalization

Challenges

Software engineering

• interface between confluence result and applications

(separation of concerns)

• structuring the big induction proof

Bertram Felgenhauer (UIBK) Layer Systems — Formalized ICTAC 2018 16/22

Formalization

Challenges

Software engineering

• interface between confluence result and applications

(separation of concerns)

• structuring the big induction proof

Miscellanea

• obvious
• express properties algebraically
• open: nice abstraction for multi-hole contexts

Bertram Felgenhauer (UIBK) Layer Systems — Formalized ICTAC 2018 16/22

Formalization

Using Locales

layer system sig layer system (L1,L2,L3) weakly layered (W) layered (C1,C2) Locales

• bundle assumptions and conclusions (the interface)
• can be instantiated (for applications)

Bertram Felgenhauer (UIBK) Layer Systems — Formalized ICTAC 2018 17/22

Formalization

Using Locales

layer system sig layer system (L1,L2,L3) weakly layered (W) layered (C1,C2) Locales

• bundle assumptions and conclusions (the interface)
• can be instantiated (for applications)
• main result is in layered locale

Bertram Felgenhauer (UIBK) Layer Systems — Formalized ICTAC 2018 17/22

Formalization

Using Locales

layer system sig layer system (L1,L2,L3) weakly layered (W) layered (C1,C2) Locales

• bundle assumptions and conclusions (the interface)
• can be instantiated (for applications)
• main result is in layered locale
• we also use locales for the induction on rank

Bertram Felgenhauer (UIBK) Layer Systems — Formalized ICTAC 2018 17/22

Formalization

Results and Effort

• definitions, basic results about layers

3.2k

• if R is layered by L, and R is confluent on L, then R is confluent

2.0k

• for disjoint R1 and R2, R1 ∪ R2 is layered by homogeneous terms

= ⇒ modularity 0.8k

• for many-sorted R, R is layered by well-typed terms

= ⇒ persistence 1.5k

• for any R, Cu(R) is layered by a layer system

= ⇒ preservation of confluence by currying 3.8k

• executable persistent decomposition check for Ce

T A 0.6k Σ 12k lines of Isar

Bertram Felgenhauer (UIBK) Layer Systems — Formalized ICTAC 2018 18/22

Formalization

Results and Effort

• definitions, basic results about layers

3.2k (20×)

• if R is layered by L, and R is confluent on L, then R is confluent

2.0k (13×)

• for disjoint R1 and R2, R1 ∪ R2 is layered by homogeneous terms

= ⇒ modularity 0.8k (30×)

• for many-sorted R, R is layered by well-typed terms

= ⇒ persistence 1.5k (50×)

• for any R, Cu(R) is layered by a layer system

= ⇒ preservation of confluence by currying 3.8k (40×)

• executable persistent decomposition check for Ce

T A 0.6k Σ 12k lines of Isar

Bertram Felgenhauer (UIBK) Layer Systems — Formalized ICTAC 2018 18/22

Implementation

Table of Contents

Motivation Layer Systems Formalization Implementation

Bertram Felgenhauer (UIBK) Layer Systems — Formalized ICTAC 2018 19/22

Implementation

Implementation

Ce T A

• extend CPF format
• formalize persistent decomposition
• define executable code

Bertram Felgenhauer (UIBK) Layer Systems — Formalized ICTAC 2018 20/22

Implementation

Implementation

Ce T A

• extend CPF format
• formalize persistent decomposition
• define executable code

CSI

• order-sorted persistence was there
• add many-sorted persistence
• add proof output

Bertram Felgenhauer (UIBK) Layer Systems — Formalized ICTAC 2018 20/22

Implementation

Experiments

CSI +pd CSI yes 148 154 244 no 162 162 162 maybe 127 121 31 total 437 437 437

http://cl-informatik.uibk.ac.at/software/lisa/ictac2018/

Bertram Felgenhauer (UIBK) Layer Systems — Formalized ICTAC 2018 21/22

Implementation

Conclusion

Contributions

• formalization of layer systems in Isabelle/HOL
• first formalization of Toyama’s theorem
• persistence, currying
• certification for persistence-based decomposition

Bertram Felgenhauer (UIBK) Layer Systems — Formalized ICTAC 2018 22/22

Implementation

Conclusion

Contributions

• formalization of layer systems in Isabelle/HOL
• first formalization of Toyama’s theorem
• persistence, currying
• certification for persistence-based decomposition

Future work

• order-sorted persistence
• further applications
• currying is foundation for efficient ground TRS confluence check

Bertram Felgenhauer (UIBK) Layer Systems — Formalized ICTAC 2018 22/22

Implementation

Conclusion

Contributions

• formalization of layer systems in Isabelle/HOL
• first formalization of Toyama’s theorem
• persistence, currying
• certification for persistence-based decomposition

Future work

• order-sorted persistence
• further applications
• currying is foundation for efficient ground TRS confluence check

Thanks!

Bertram Felgenhauer (UIBK) Layer Systems — Formalized ICTAC 2018 22/22

Implementation

Non-Modularity of Termination

F(0, 1, x) → F(x, x, x) h(x, y) → x h(x, y) → y F(0, 1, h(0, 1)) − → F(h(0, 1), h(0, 1), h(0, 1)) ∗ − → F(0, 1, h(0, 1)) − → . . .

Bertram Felgenhauer (UIBK) Layer Systems — Formalized ICTAC 2018 23/22

Implementation

Related Work — First-Order TRSs

Formalization and certification

• CiME3 – generates Coq scripts (Knuth-Bendix criterion)
• trs – PVS (Knuth-Bendix criterion, orthogonality)
• Ruiz-Reina et al., 2002 – ACL2 (Knuth-Bendix criterion)

Tools

• ACP
• CoLL-Saigawa

Bertram Felgenhauer (UIBK) Layer Systems — Formalized ICTAC 2018 24/22