Unix filesystem and graph constraints Nicolas Jeannerod Journ ees - - PowerPoint PPT Presentation

▶
unix filesystem and graph constraints
SMART_READER_LITE
LIVE PREVIEW

Unix filesystem and graph constraints Nicolas Jeannerod Journ ees - - PowerPoint PPT Presentation

Dec 13, 2022 423 likes •1.08k views

Unix filesystem and graph constraints Nicolas Jeannerod Journ ees PPS, October 12, 2017 Nicolas Jeannerod Journ ees PPS October 12, 2017 1 / 21 The CoLiS project T1 Shell Script P Paris L Lille S Saclay P Compiler S,P CoLiS

slide-1
SLIDE 1

Unix filesystem and graph constraints

Nicolas Jeannerod Journ´ ees PPS, October 12, 2017

Nicolas Jeannerod Journ´ ees PPS October 12, 2017 1 / 21

slide-2
SLIDE 2

The CoLiS project

T1

Shell Script Compiler P CoLiS Program S,P

T3

Interpreter Deep Embedding S,P Symbolic Execution

T2

Translator Tree Transducers L,P Equivalence Test Tree Constraints Incremental Simplification P Paris L Lille S Saclay

Nicolas Jeannerod Journ´ ees PPS October 12, 2017 2 / 21

slide-3
SLIDE 3

The CoLiS project

T1

Shell Script Compiler P CoLiS Program S,P

T3

Interpreter Deep Embedding S,P Symbolic Execution

T2

Translator Tree Transducers L,P Equivalence Test Tree Constraints Incremental Simplification P Paris L Lille S Saclay

Nicolas Jeannerod Journ´ ees PPS October 12, 2017 2 / 21

slide-4
SLIDE 4

Description of file systems Unix file system

Table of Contents

  • 1. Description of file systems

Unix file system Static description Directory update

  • 2. Tree constraints

Definitions Basic constraints Existential and first order constraints

Nicolas Jeannerod Journ´ ees PPS October 12, 2017 3 / 21

slide-5
SLIDE 5

Description of file systems Unix file system

Unix file system

/ usr etc lib libc.so libc.so.6 lib root Basically a tree with labelled nodes and edges; There can be sharing at the leafs (hard link between files); There can be pointers to other parts of the tree (symbolic links) which may form cycles.

Nicolas Jeannerod Journ´ ees PPS October 12, 2017 4 / 21

slide-6
SLIDE 6

Description of file systems Unix file system

Unix file system

/ usr etc lib libc.so libc.so.6 lib root Basically a tree with labelled nodes and edges; There can be sharing at the leafs (hard link between files); There can be pointers to other parts of the tree (symbolic links) which may form cycles.

Nicolas Jeannerod Journ´ ees PPS October 12, 2017 4 / 21

slide-7
SLIDE 7

Description of file systems Unix file system

Unix file system

/ usr etc lib libc.so libc.so.6 lib root Basically a tree with labelled nodes and edges; There can be sharing at the leafs (hard link between files); There can be pointers to other parts of the tree (symbolic links) which may form cycles.

Nicolas Jeannerod Journ´ ees PPS October 12, 2017 4 / 21

slide-8
SLIDE 8

Description of file systems Unix file system

Unix file system

/ usr etc lib libc.so libc.so.6 lib root Basically a tree with labelled nodes and edges; There can be sharing at the leafs (hard link between files); There can be pointers to other parts of the tree (symbolic links) which may form cycles.

Nicolas Jeannerod Journ´ ees PPS October 12, 2017 4 / 21

slide-9
SLIDE 9

Description of file systems Static description

Table of Contents

  • 1. Description of file systems

Unix file system Static description Directory update

  • 2. Tree constraints

Definitions Basic constraints Existential and first order constraints

Nicolas Jeannerod Journ´ ees PPS October 12, 2017 5 / 21

slide-10
SLIDE 10

Description of file systems Static description

Static description

/ v w x u∅ usr etc lib skel

  • caml

r

  • ∃u, v, x, w. r[usr]v ∧ v[lib]x ∧ x[ocaml] ↑

∧ r[etc]w ∧ w[skel]u ∧ u∅

Nicolas Jeannerod Journ´ ees PPS October 12, 2017 6 / 21

slide-11
SLIDE 11

Description of file systems Static description

Static description

/ v w x u∅ usr etc lib skel

  • caml

r

  • ∃u, v, x, w. r[usr]v ∧ v[lib]x ∧ x[ocaml] ↑

∧ r[etc]w ∧ w[skel]u ∧ u∅

Nicolas Jeannerod Journ´ ees PPS October 12, 2017 6 / 21

slide-12
SLIDE 12

Description of file systems Static description

Static description

r v w x u∅ usr etc lib skel

  • caml

r

  • ∃u, v, x, w. r[usr]v ∧ v[lib]x ∧ x[ocaml] ↑

∧ r[etc]w ∧ w[skel]u ∧ u∅

Nicolas Jeannerod Journ´ ees PPS October 12, 2017 6 / 21

slide-13
SLIDE 13

Description of file systems Static description

Static description

r v w x u∅ usr etc lib skel

  • caml

r

  • ∃u, v, x, w. r[usr]v ∧ v[lib]x ∧ x[ocaml] ↑

∧ r[etc]w ∧ w[skel]u ∧ u∅

Nicolas Jeannerod Journ´ ees PPS October 12, 2017 6 / 21

slide-14
SLIDE 14

Description of file systems Static description

Static description

r v w x u∅ usr etc lib skel

  • caml

r

  • ∃u, v, x, w. r[usr]v ∧ v[lib]x ∧ x[ocaml] ↑

∧ r[etc]w ∧ w[skel]u ∧ u∅

Nicolas Jeannerod Journ´ ees PPS October 12, 2017 6 / 21

slide-15
SLIDE 15

Description of file systems Static description

Static description

r v w x u∅ usr etc lib skel

  • caml

r

  • ∃u, v, x, w. r[usr]v ∧ v[lib]x ∧ x[ocaml] ↑

∧ r[etc]w ∧ w[skel]u ∧ u∅

Nicolas Jeannerod Journ´ ees PPS October 12, 2017 6 / 21

slide-16
SLIDE 16

Description of file systems Static description

Static description

r v w x u∅ usr etc lib skel

  • caml

r

  • ∃u, v, x, w. r[usr]v ∧ v[lib]x ∧ x[ocaml] ↑

∧ r[etc]w ∧ w[skel]u ∧ u∅

Nicolas Jeannerod Journ´ ees PPS October 12, 2017 6 / 21

slide-17
SLIDE 17

Description of file systems Directory update

Table of Contents

  • 1. Description of file systems

Unix file system Static description Directory update

  • 2. Tree constraints

Definitions Basic constraints Existential and first order constraints

Nicolas Jeannerod Journ´ ees PPS October 12, 2017 7 / 21

slide-18
SLIDE 18

Description of file systems Directory update

Directory update

r v w x usr etc lib

  • caml

mkdir /usr/lib/ocaml

r′ v′ w x′ y′∅ usr etc lib

  • caml

We want something like: r′ = r[usr → v′] ∧ v′ = v[lib → x′] ∧ x′ = x[ocaml → y′] ∧ y′∅

Nicolas Jeannerod Journ´ ees PPS October 12, 2017 8 / 21

slide-19
SLIDE 19

Description of file systems Directory update

Directory update

r v w x usr etc lib

  • caml

mkdir /usr/lib/ocaml

r′ v′ w x′ y′∅ usr etc lib

  • caml

We want something like: r′ = r[usr → v′] ∧ v′ = v[lib → x′] ∧ x′ = x[ocaml → y′] ∧ y′∅

Nicolas Jeannerod Journ´ ees PPS October 12, 2017 8 / 21

slide-20
SLIDE 20

Description of file systems Directory update

Directory update

r v w x usr etc lib

  • caml

mkdir /usr/lib/ocaml

r′ v′ w x′ y′∅ usr etc lib

  • caml

We want something like: r′ = r[usr → v′] ∧ v′ = v[lib → x′] ∧ x′ = x[ocaml → y′] ∧ y′∅

Nicolas Jeannerod Journ´ ees PPS October 12, 2017 8 / 21

slide-21
SLIDE 21

Description of file systems Directory update

Directory update

r v w x usr etc lib

  • caml

mkdir /usr/lib/ocaml

r′ v′ w x′ y′∅ usr etc lib

  • caml

We want something like: r′ = r[usr → v′] ∧ v′ = v[lib → x′] ∧ x′ = x[ocaml → y′] ∧ y′∅

Nicolas Jeannerod Journ´ ees PPS October 12, 2017 8 / 21

slide-22
SLIDE 22

Description of file systems Directory update

Er.. is that really what we want?

Asymmetric: y = x[f → v] Makes it hard to eliminate variables: y = x[f → v] ∧ z = x[g → w] Contains in fact two pieces of information:

“y and x are different in f, identical everywhere else”: y ∼f x “y points to v through f”: y[f]v

Nicolas Jeannerod Journ´ ees PPS October 12, 2017 9 / 21

slide-23
SLIDE 23

Description of file systems Directory update

Er.. is that really what we want?

Asymmetric: y = x[f → v] Makes it hard to eliminate variables: y = x[f → v] ∧ z = x[g → w] Contains in fact two pieces of information:

“y and x are different in f, identical everywhere else”: y ∼f x “y points to v through f”: y[f]v

Nicolas Jeannerod Journ´ ees PPS October 12, 2017 9 / 21

slide-24
SLIDE 24

Description of file systems Directory update

Er.. is that really what we want?

Asymmetric: y = x[f → v] Makes it hard to eliminate variables: y = x[f → v] ∧ z = x[g → w] Contains in fact two pieces of information:

“y and x are different in f, identical everywhere else”: y ∼f x “y points to v through f”: y[f]v

Nicolas Jeannerod Journ´ ees PPS October 12, 2017 9 / 21

slide-25
SLIDE 25

Description of file systems Directory update

Er.. is that really what we want?

Asymmetric: y = x[f → v] Makes it hard to eliminate variables: y = x[f → v] ∧ z = x[g → w] Contains in fact two pieces of information:

“y and x are different in f, identical everywhere else”: y ∼f x “y points to v through f”: y[f]v

Nicolas Jeannerod Journ´ ees PPS October 12, 2017 9 / 21

slide-26
SLIDE 26

Description of file systems Directory update

Er.. is that really what we want?

Asymmetric: y = x[f → v] Makes it hard to eliminate variables: y = x[f → v] ∧ z = x[g → w] Contains in fact two pieces of information:

“y and x are different in f, identical everywhere else”: y ∼f x “y points to v through f”: y[f]v

Nicolas Jeannerod Journ´ ees PPS October 12, 2017 9 / 21

slide-27
SLIDE 27

Description of file systems Directory update

Er.. is that really what we want?

Asymmetric: y = x[f → v] Makes it hard to eliminate variables: y = x[f → v] ∧ z = x[g → w] Contains in fact two pieces of information:

“y and x are different in f, identical everywhere else”: y ∼f x “y points to v through f”: y[f]v

Nicolas Jeannerod Journ´ ees PPS October 12, 2017 9 / 21

slide-28
SLIDE 28

Description of file systems Directory update

Er.. is that really what we want?

Asymmetric: y = x[f → v] Makes it hard to eliminate variables: y = x[f → v] ∧ z = x[g → w] Contains in fact two pieces of information:

“y and x are different in f, identical everywhere else”: y ∼f x “y points to v through f”: y[f]v

Nicolas Jeannerod Journ´ ees PPS October 12, 2017 9 / 21

slide-29
SLIDE 29

Description of file systems Directory update

Much better

Allows to express the update: y = x[f → v] := y ∼f x ∧ y[f]v Symmetric: y ∼f x ⇐ ⇒ x ∼f y Transitive: y ∼f x ∧ z ∼f x = ⇒ y ∼f z

Nicolas Jeannerod Journ´ ees PPS October 12, 2017 10 / 21

slide-30
SLIDE 30

Description of file systems Directory update

Much better

Allows to express the update: y = x[f → v] := y ∼f x ∧ y[f]v Symmetric: y ∼f x ⇐ ⇒ x ∼f y Transitive: y ∼f x ∧ z ∼f x = ⇒ y ∼f z

Nicolas Jeannerod Journ´ ees PPS October 12, 2017 10 / 21

slide-31
SLIDE 31

Description of file systems Directory update

Much better

Allows to express the update: y = x[f → v] := y ∼f x ∧ y[f]v Symmetric: y ∼f x ⇐ ⇒ x ∼f y Transitive: y ∼f x ∧ z ∼f x = ⇒ y ∼f z

Nicolas Jeannerod Journ´ ees PPS October 12, 2017 10 / 21

slide-32
SLIDE 32

Tree constraints Definitions

Table of Contents

  • 1. Description of file systems

Unix file system Static description Directory update

  • 2. Tree constraints

Definitions Basic constraints Existential and first order constraints

Nicolas Jeannerod Journ´ ees PPS October 12, 2017 11 / 21

slide-33
SLIDE 33

Tree constraints Definitions

Constraints

K finite set; Dir ∈ K; F infinite set Finite trees labelled with K on nodes and F on edges x, y variables; K ∈ K, f ∈ F, F ⊆ F Equality x . = y K(x) Kind Feature xfy xf ↑ Absence Fence xF x ∼F y Similarity Composed with ¬, ∧, ∨, ∃x, ∀x No quantification on kinds and features Wanted: (un)satisfiability of these constraints Bonus point for incremental procedures

Nicolas Jeannerod Journ´ ees PPS October 12, 2017 12 / 21

slide-34
SLIDE 34

Tree constraints Definitions

Constraints

K finite set; Dir ∈ K; F infinite set Finite trees labelled with K on nodes and F on edges x, y variables; K ∈ K, f ∈ F, F ⊆ F Equality x . = y K(x) Kind Feature xfy xf ↑ Absence Fence xF x ∼F y Similarity Composed with ¬, ∧, ∨, ∃x, ∀x No quantification on kinds and features Wanted: (un)satisfiability of these constraints Bonus point for incremental procedures

Nicolas Jeannerod Journ´ ees PPS October 12, 2017 12 / 21

slide-35
SLIDE 35

Tree constraints Definitions

Constraints

K finite set; Dir ∈ K; F infinite set Finite trees labelled with K on nodes and F on edges x, y variables; K ∈ K, f ∈ F, F ⊆ F Equality x . = y K(x) Kind Feature xfy xf ↑ Absence Fence xF x ∼F y Similarity Composed with ¬, ∧, ∨, ∃x, ∀x No quantification on kinds and features Wanted: (un)satisfiability of these constraints Bonus point for incremental procedures

Nicolas Jeannerod Journ´ ees PPS October 12, 2017 12 / 21

slide-36
SLIDE 36

Tree constraints Definitions

Constraints

K finite set; Dir ∈ K; F infinite set Finite trees labelled with K on nodes and F on edges x, y variables; K ∈ K, f ∈ F, F ⊆ F Equality x . = y K(x) Kind Feature xfy xf ↑ Absence Fence xF x ∼F y Similarity Composed with ¬, ∧, ∨, ∃x, ∀x No quantification on kinds and features Wanted: (un)satisfiability of these constraints Bonus point for incremental procedures

Nicolas Jeannerod Journ´ ees PPS October 12, 2017 12 / 21

slide-37
SLIDE 37

Tree constraints Definitions

Constraints

K finite set; Dir ∈ K; F infinite set Finite trees labelled with K on nodes and F on edges x, y variables; K ∈ K, f ∈ F, F ⊆ F Equality x . = y K(x) Kind Feature xfy xf ↑ Absence Fence xF x ∼F y Similarity Composed with ¬, ∧, ∨, ∃x, ∀x No quantification on kinds and features Wanted: (un)satisfiability of these constraints Bonus point for incremental procedures

Nicolas Jeannerod Journ´ ees PPS October 12, 2017 12 / 21

slide-38
SLIDE 38

Tree constraints Definitions

Game plan

  • 1. Write a system of rewriting rules;
  • 2. Prove that the system terminates (help it if needed);
  • 3. Prove that the rules respect equivalences:

Lemma If φ reduces to ψ, then | = φ ↔ ψ.

  • 4. Prove nice properties on the normal forms:

Lemma If φ is in normal form, then it is either satisfiable or ⊥.

Nicolas Jeannerod Journ´ ees PPS October 12, 2017 13 / 21

slide-39
SLIDE 39

Tree constraints Definitions

Game plan

  • 1. Write a system of rewriting rules;
  • 2. Prove that the system terminates (help it if needed);
  • 3. Prove that the rules respect equivalences:

Lemma If φ reduces to ψ, then | = φ ↔ ψ.

  • 4. Prove nice properties on the normal forms:

Lemma If φ is in normal form, then it is either satisfiable or ⊥.

Nicolas Jeannerod Journ´ ees PPS October 12, 2017 13 / 21

slide-40
SLIDE 40

Tree constraints Definitions

Game plan

  • 1. Write a system of rewriting rules;
  • 2. Prove that the system terminates (help it if needed);
  • 3. Prove that the rules respect equivalences:

Lemma If φ reduces to ψ, then | = φ ↔ ψ.

  • 4. Prove nice properties on the normal forms:

Lemma If φ is in normal form, then it is either satisfiable or ⊥.

Nicolas Jeannerod Journ´ ees PPS October 12, 2017 13 / 21

slide-41
SLIDE 41

Tree constraints Definitions

Game plan

  • 1. Write a system of rewriting rules;
  • 2. Prove that the system terminates (help it if needed);
  • 3. Prove that the rules respect equivalences:

Lemma If φ reduces to ψ, then | = φ ↔ ψ.

  • 4. Prove nice properties on the normal forms:

Lemma If φ is in normal form, then it is either satisfiable or ⊥.

Nicolas Jeannerod Journ´ ees PPS October 12, 2017 13 / 21

slide-42
SLIDE 42

Tree constraints Basic constraints

Table of Contents

  • 1. Description of file systems

Unix file system Static description Directory update

  • 2. Tree constraints

Definitions Basic constraints Existential and first order constraints

Nicolas Jeannerod Journ´ ees PPS October 12, 2017 14 / 21

slide-43
SLIDE 43

Tree constraints Basic constraints

Basic rules

Basic constraints: conjunction of positive atoms.

Simpl-Feats

xfy ∧ xfz xfy ∧ y . = z

C-Feat-Abs

xfy ∧ xf ↑ ⊥

Intro-Feat-Sim

x ∼F y ∧ xfz x ∼F y ∧ xfz ∧ yfz

f / ∈ F

Intro-Sim-Sims

x ∼F y ∧ y ∼G z x ∼F y ∧ y ∼G z ∧ x ∼(F∪G) z

Nicolas Jeannerod Journ´ ees PPS October 12, 2017 15 / 21

slide-44
SLIDE 44

Tree constraints Basic constraints

Basic rules

Basic constraints: conjunction of positive atoms.

Simpl-Feats

xfy ∧ xfz xfy ∧ y . = z

C-Feat-Abs

xfy ∧ xf ↑ ⊥

Intro-Feat-Sim

x ∼F y ∧ xfz x ∼F y ∧ xfz ∧ yfz

f / ∈ F

Intro-Sim-Sims

x ∼F y ∧ y ∼G z x ∼F y ∧ y ∼G z ∧ x ∼(F∪G) z

Nicolas Jeannerod Journ´ ees PPS October 12, 2017 15 / 21

slide-45
SLIDE 45

Tree constraints Basic constraints

Basic rules

Basic constraints: conjunction of positive atoms.

Simpl-Feats

xfy ∧ xfz xfy ∧ y . = z

C-Feat-Abs

xfy ∧ xf ↑ ⊥

Intro-Feat-Sim

x ∼F y ∧ xfz x ∼F y ∧ xfz ∧ yfz

f / ∈ F

Intro-Sim-Sims

x ∼F y ∧ y ∼G z x ∼F y ∧ y ∼G z ∧ x ∼(F∪G) z

Nicolas Jeannerod Journ´ ees PPS October 12, 2017 15 / 21

slide-46
SLIDE 46

Tree constraints Basic constraints

Basic rules

Basic constraints: conjunction of positive atoms.

Simpl-Feats

xfy ∧ xfz xfy ∧ y . = z

C-Feat-Abs

xfy ∧ xf ↑ ⊥

Intro-Feat-Sim

x ∼F y ∧ xfz x ∼F y ∧ xfz ∧ yfz

f / ∈ F

Intro-Sim-Sims

x ∼F y ∧ y ∼G z x ∼F y ∧ y ∼G z ∧ x ∼(F∪G) z

Nicolas Jeannerod Journ´ ees PPS October 12, 2017 15 / 21

slide-47
SLIDE 47

Tree constraints Basic constraints

Basic constraints

Basic constraints: conjunction of positive atoms Equality: rewritten Kind: Static “positive” information Feature: Static “positive” information Absence: Static “negative” information Fence: Static “negative” information Similarity: Dynamic information r[usr]v ∧ v[lib]x ∧ x[ocaml] ↑ ∧r[etc]w ∧ w[skel]u ∧ u∅ ∧ . . . r v w x u usr etc lib skel

Nicolas Jeannerod Journ´ ees PPS October 12, 2017 16 / 21

slide-48
SLIDE 48

Tree constraints Basic constraints

Basic constraints

Basic constraints: conjunction of positive atoms Equality: rewritten Kind: Static “positive” information Feature: Static “positive” information Absence: Static “negative” information Fence: Static “negative” information Similarity: Dynamic information r[usr]v ∧ v[lib]x ∧ x[ocaml] ↑ ∧r[etc]w ∧ w[skel]u ∧ u∅ ∧ . . . r v w x u usr etc lib skel

Nicolas Jeannerod Journ´ ees PPS October 12, 2017 16 / 21

slide-49
SLIDE 49

Tree constraints Basic constraints

Basic constraints

Basic constraints: conjunction of positive atoms Equality: rewritten Kind: Static “positive” information Feature: Static “positive” information Absence: Static “negative” information Fence: Static “negative” information Similarity: Dynamic information r[usr]v ∧ v[lib]x ∧ x[ocaml] ↑ ∧r[etc]w ∧ w[skel]u ∧ u∅ ∧ . . . r v w x u usr etc lib skel

Nicolas Jeannerod Journ´ ees PPS October 12, 2017 16 / 21

slide-50
SLIDE 50

Tree constraints Basic constraints

Basic constraints

Basic constraints: conjunction of positive atoms Equality: rewritten Kind: Static “positive” information Feature: Static “positive” information Absence: Static “negative” information Fence: Static “negative” information Similarity: Dynamic information r[usr]v ∧ v[lib]x ∧ x[ocaml] ↑ ∧r[etc]w ∧ w[skel]u ∧ u∅ ∧ . . . r v w x u usr etc lib skel

Nicolas Jeannerod Journ´ ees PPS October 12, 2017 16 / 21

slide-51
SLIDE 51

Tree constraints Basic constraints

Basic constraints

Basic constraints: conjunction of positive atoms Equality: rewritten Kind: Static “positive” information Feature: Static “positive” information Absence: Static “negative” information Fence: Static “negative” information Similarity: Dynamic information r[usr]v ∧ v[lib]x ∧ x[ocaml] ↑ ∧r[etc]w ∧ w[skel]u ∧ u∅ ∧ . . . r v w x u usr etc lib skel

Nicolas Jeannerod Journ´ ees PPS October 12, 2017 16 / 21

slide-52
SLIDE 52

Tree constraints Basic constraints

Basic constraints

Basic constraints: conjunction of positive atoms Equality: rewritten Kind: Static “positive” information Feature: Static “positive” information Absence: Static “negative” information Fence: Static “negative” information Similarity: Dynamic information r[usr]v ∧ v[lib]x ∧ x[ocaml] ↑ ∧r[etc]w ∧ w[skel]u ∧ u∅ ∧ . . . r v w x u usr etc lib skel

Nicolas Jeannerod Journ´ ees PPS October 12, 2017 16 / 21

slide-53
SLIDE 53

Tree constraints Basic constraints

Basic constraints

Basic constraints: conjunction of positive atoms Equality: rewritten Kind: Static “positive” information Feature: Static “positive” information Absence: Static “negative” information Fence: Static “negative” information Similarity: Dynamic information r[usr]v ∧ v[lib]x ∧ x[ocaml] ↑ ∧r[etc]w ∧ w[skel]u ∧ u∅ ∧ . . . r v w x u usr etc lib skel

Nicolas Jeannerod Journ´ ees PPS October 12, 2017 16 / 21

slide-54
SLIDE 54

Tree constraints Basic constraints

Basic constraints

Basic constraints: conjunction of positive atoms Equality: rewritten Kind: Static “positive” information Feature: Static “positive” information Absence: Static “negative” information Fence: Static “negative” information Similarity: Dynamic information r[usr]v ∧ v[lib]x ∧ x[ocaml] ↑ ∧r[etc]w ∧ w[skel]u ∧ u∅ ∧ . . . r v w x u usr etc lib skel

Nicolas Jeannerod Journ´ ees PPS October 12, 2017 16 / 21

slide-55
SLIDE 55

Tree constraints Existential and first order constraints

Table of Contents

  • 1. Description of file systems

Unix file system Static description Directory update

  • 2. Tree constraints

Definitions Basic constraints Existential and first order constraints

Nicolas Jeannerod Journ´ ees PPS October 12, 2017 17 / 21

slide-56
SLIDE 56

Tree constraints Existential and first order constraints

Negation: new players, new rules

¬xF: there exist a g / ∈ F such that xg↓; x ∼F y: there exist a g / ∈ F such that x . =g y;

Repl-NKind

¬K(x)

  • L∈K

L(x)

Repl-NAbs

¬xf ↑ ∃z.xfz

Repl-NSim-NFence

xF ∧ x ∼G y xF ∧ ¬xG

F ⊆ G

Nicolas Jeannerod Journ´ ees PPS October 12, 2017 18 / 21

slide-57
SLIDE 57

Tree constraints Existential and first order constraints

Negation: new players, new rules

¬xF: there exist a g / ∈ F such that xg↓; x ∼F y: there exist a g / ∈ F such that x . =g y;

Repl-NKind

¬K(x)

  • L∈K

L(x)

Repl-NAbs

¬xf ↑ ∃z.xfz

Repl-NSim-NFence

xF ∧ x ∼G y xF ∧ ¬xG

F ⊆ G

Nicolas Jeannerod Journ´ ees PPS October 12, 2017 18 / 21

slide-58
SLIDE 58

Tree constraints Existential and first order constraints

Negation: new players, new rules

¬xF: there exist a g / ∈ F such that xg↓; x ∼F y: there exist a g / ∈ F such that x . =g y;

Repl-NKind

¬K(x)

  • L∈K

L(x)

Repl-NAbs

¬xf ↑ ∃z.xfz

Repl-NSim-NFence

xF ∧ x ∼G y xF ∧ ¬xG

F ⊆ G

Nicolas Jeannerod Journ´ ees PPS October 12, 2017 18 / 21

slide-59
SLIDE 59

Tree constraints Existential and first order constraints

Negation: new players, new rules

¬xF: there exist a g / ∈ F such that xg↓; x ∼F y: there exist a g / ∈ F such that x . =g y;

Repl-NKind

¬K(x)

  • L∈K

L(x)

Repl-NAbs

¬xf ↑ ∃z.xfz

Repl-NSim-NFence

xF ∧ x ∼G y xF ∧ ¬xG

F ⊆ G

Nicolas Jeannerod Journ´ ees PPS October 12, 2017 18 / 21

slide-60
SLIDE 60

Tree constraints Existential and first order constraints

Quantifier elimination

Goal: be able to change an existentially quantified block into a universally quantified one.

  • ∃

X.

  • . . .
  • ↔
  • ∀

Y . . . .

  • Special rules:

Enlarg-Feat-Local

∃x.∃ X.(yfx ∧ φ(x, X)) yf ↓ ∧∀x.∃ X.(yfx → φ(x, X))

Nicolas Jeannerod Journ´ ees PPS October 12, 2017 19 / 21

slide-61
SLIDE 61

Tree constraints Existential and first order constraints

Quantifier elimination

Goal: be able to change an existentially quantified block into a universally quantified one.

  • ∃

X.

  • . . .
  • ↔
  • ∀

Y . . . .

  • Special rules:

Enlarg-Feat-Local

∃x.∃ X.(yfx ∧ φ(x, X)) yf ↓ ∧∀x.∃ X.(yfx → φ(x, X))

Nicolas Jeannerod Journ´ ees PPS October 12, 2017 19 / 21

slide-62
SLIDE 62

Tree constraints Existential and first order constraints

Lemma of 31 August

Lemma (31 August) Let φ be a conjunction of the form: φ( X, Y ) =

  • stuff about

X

  • ∧ ψ(

Y ) in normal form for our system. Then we have: | = ∀ Y .

  • ∃
  • X. φ(

X, Y )

  • ↔ ψ(

Y ) The system propagates all the useful information We can just remove what we don’t need!

Nicolas Jeannerod Journ´ ees PPS October 12, 2017 20 / 21

slide-63
SLIDE 63

Tree constraints Existential and first order constraints

Lemma of 31 August

Lemma (31 August) Let φ be a conjunction of the form: φ( X, Y ) =

  • stuff about

X

  • ∧ ψ(

Y ) in normal form for our system. Then we have: | = ∀ Y .

  • ∃
  • X. φ(

X, Y )

  • ↔ ψ(

Y ) The system propagates all the useful information We can just remove what we don’t need!

Nicolas Jeannerod Journ´ ees PPS October 12, 2017 20 / 21

slide-64
SLIDE 64

Thank you for your attention!

Recap’: Need constraints on graphs to represent relations on file systems; Extend “feature trees” with x ∼F y (“x and y are the same, except maybe for the features in F”); Use a system of rewrite rules whose normal forms have nice properties. Future work: Cleanup, formalise in a technical report; Add inodes, permissions, timestamps, etc. Implement an efficient version for the existential subset.

Nicolas Jeannerod Journ´ ees PPS October 12, 2017 21 / 21