Logics for Petri nets with propagating failures Leandro Gomes, - - PowerPoint PPT Presentation

Logics for Petri nets with propagating failures

Leandro Gomes, Alexandre Madeira and Mario Benevides 8th IPM International Conference on Fundamentals of Software Engineering, 2nd May 2019

Outline

Background and motivations Petri nets with A-failures Dynamic logics for Petri nets with A-failures: GP(A) Conclusions

Outline

Background and motivations Petri nets with A-failures Dynamic logics for Petri nets with A-failures: GP(A) Conclusions

Propositional dynamic logic (in a rush)

Signatures Are pairs (Prop, Π) where Prop and Π are disjoint sets of propositions, and atomic programs

Propositional dynamic logic (in a rush)

Signatures Are pairs (Prop, Π) where Prop and Π are disjoint sets of propositions, and atomic programs Sentences ϕ ::= p | πϕ | [π]ϕ | ¬ϕ | ϕ ∨ ϕ | ϕ ∧ ϕ π ::= π0 | π + π | π; π | π∗ | ?ϕ and p ∈ Prop

Propositional dynamic logic (in a rush)

Signatures Are pairs (Prop, Π) where Prop and Π are disjoint sets of propositions, and atomic programs Sentences ϕ ::= p | πϕ | [π]ϕ | ¬ϕ | ϕ ∨ ϕ | ϕ ∧ ϕ π ::= π0 | π + π | π; π | π∗ | ?ϕ and p ∈ Prop Models Models are Kripke structures, i.e. tuples

• W , V , R
• W is a set

V : Prop → P(W ) is a function R = (Rπ ⊆ W × W )π∈Π is an Π-family of binary relations

Propositional dynamic logic (in a rush)

Satisfaction M, w | = p iff w ∈ V (p) M, w | = ϕ ∧ ϕ′ iff M, w | = ϕ and M, w | = ϕ′ M, w | = ϕ ∨ ϕ′ iff M, w | = ϕ or M, w | = ϕ′ M, w | = ¬ϕ iff it is false that M, w | = ϕ

Propositional dynamic logic (in a rush)

Satisfaction M, w | = p iff w ∈ V (p) M, w | = ϕ ∧ ϕ′ iff M, w | = ϕ and M, w | = ϕ′ M, w | = ϕ ∨ ϕ′ iff M, w | = ϕ or M, w | = ϕ′ M, w | = ¬ϕ iff it is false that M, w | = ϕ M, w | = πϕ iff there is a w′ ∈ W such that (w, w′) ∈ Ra and M, w′ | = ϕ; M, w | = [π]ϕ iff for any w′ ∈ W such that (w, w′) ∈ Ra we have M, w′ | = ϕ;

Petri Nets

• It is a tuple of the form (P, T , w, m0) where:
• P represents a finite set of places;
• T represents a finite set of transitions;
• w: (P × T ) ∪ (T × P) → N0 attributes to each arc a

non-negative integer, representing its multiplicity.

• m0: P → N0 defines an Initial Marking

Example

ℓ m x y c t2 t3 t1

Figure: Example of a Petri net

Proposition dynamic logic for Petri nets: Petri-PDL⋆

(Bruno Lopes, 2014) and (Mario Benevides, 2016)

Proposition dynamic logic for Petri nets: Petri-PDL⋆

(Bruno Lopes, 2014) and (Mario Benevides, 2016) Basic programs Π0(P): π ::= at1b | abt2c | at3bc where ti is of type Ti, i = 1, 2, 3 and a, b, c ∈ P

Proposition dynamic logic for Petri nets: Petri-PDL⋆

(Bruno Lopes, 2014) and (Mario Benevides, 2016) Basic programs Π0(P): π ::= at1b | abt2c | at3bc where ti is of type Ti, i = 1, 2, 3 and a, b, c ∈ P Petri net Programs Π(P): η ::= π | π ⊙ η | η⋆ for π ∈ Π0(P)

Proposition dynamic logic for Petri nets: Petri-PDL⋆

(Bruno Lopes, 2014) and (Mario Benevides, 2016) Basic programs Π0(P): π ::= at1b | abt2c | at3bc where ti is of type Ti, i = 1, 2, 3 and a, b, c ∈ P Petri net Programs Π(P): η ::= π | π ⊙ η | η⋆ for π ∈ Π0(P) Formulas FmPetri−PDL⋆(Prop): ρ ::= p | ⊤ | ¬ρ | ρ ∧ ρ | s, ηρ where p ∈ Prop.

Petri-PDL⋆ dynamics – firing function

f : S × Π0 → S f (s, abt2c) = s2 | c ∈ s2, if a, b ∈ s ǫ, if a / ∈ s or b / ∈ s

• ℓ

m x y c t2 t3 t1

Petri-PDL⋆ dynamics – firing function

f : S × Π0 → S f (s, abt2c) = s2 | c ∈ s2, if a, b ∈ s ǫ, if a / ∈ s or b / ∈ s

• ℓ

m x y c t2 t3 t1

− →

ℓ m x y c t2 t3 t1

Figure: Dynamics for transitions of type abt2c

Models: (W, Rπ, M), V

• W is a non-empty set of states, M : W → S, Rπ is a binary

relation over W .

• V is a valuation function V: Prop → 2W .

Models: (W, Rπ, M), V

• W is a non-empty set of states, M : W → S, Rπ is a binary

relation over W .

• V is a valuation function V: Prop → 2W .

Satisfaction: V: Prop → 2W

• M, w |

= p iff w ∈ V(p);

• M, w |

= ⊤ always;

• M, w |

= ¬ρ iff M, w | = ρ;

• M, w |

= ρ ∧ ρ′ iff M, w | = ρ and M, w | = ρ′;

• M, w |

= s, ηρ if there exists w′ ∈ W , wRηw′, s M(w) and M, w′ | = ρ.

Example

ℓ m x y c t2 t3 t1

Example

ℓ m x y c t2 t3 t1

− →

ℓ m x y c t2 t3 t1

Example

ℓ m x y c t2 t3 t1

− →

ℓ m x y c t2 t3 t1

− →

ℓ m x y c t2 t3 t1

Figure: Execution of the program ℓmt2x; xt3yc in a chocolate vending machine

M, ℓm | = ℓmt2x; xt3yc⊤ = ⊤

Failures in machines

After putting a coin in a vending machine, the desired chocolate gets stuck behind of the glass

Failures in machines

After putting a coin in a vending machine, the desired chocolate gets stuck behind of the glass How to model these phenomena?

Outline

Background and motivations Petri nets with A-failures Dynamic logics for Petri nets with A-failures: GP(A) Conclusions

Construction parameter

Action Lattice Klenne Algebra Residuated Lattice Generic model for computations Generic truth space

Construction parameter

Action Lattice Klenne Algebra Residuated Lattice Generic model for computations Generic truth space

Action lattice (Pratt 90, Kozen 91)

A = (A, +, ; , 0, 1, ∗, →, ·) (A, +, ; , 0, 1, ∗) is a Kleene algebra; → is a residue wrt ; (A, +, .) is a lattice wrt relation a ≤ b ≡ a + b = b

Examples: 2 - linear two-values lattice.

2 = ({⊤, ⊥}, ∨, ∧, ⊥, ⊤, ∗, →, ∧) ∨ ⊥ ⊤ ⊥ ⊥ ⊤ ⊤ ⊤ ⊤ ∧ ⊥ ⊤ ⊥ ⊥ ⊥ ⊤ ⊥ ⊤ → ⊥ ⊤ ⊥ ⊤ ⊤ ⊤ ⊥ ⊤ ∗ ⊥ ⊤ ⊤ ⊤

Examples: Wk finite Wajsberg hoops

For a fixed natural k > 0 and a generator a, Wk = (Wk, + , ; , 0, 1, ∗, →, ·) where

• Wk = {a0, a1, · · · , ak}, 1 = a0 and 0 = ak,
• For any m, n ≤ k:

am + an = amin{m,n}, am; an = amin{m+n,k}, (am)∗ = a0, am → an = amax{n−m,0}, am · an = amax{m,n}

Examples: L - the Lukasiewicz arithmetic lattice

• L = ([0, 1], max, ⊙, 0, 1, ∗, → , min)

where x ⊙ y = max{0, y + x − 1}, x → y = min{1, 1 − x + y} and ∗ maps each point of [0, 1] to 1.

Petri net with A-failures

P =

• P, S, Π0, I, M0
• P is a set of places,

S ⊆ P is the set of (admissible) markings, Π0 ⊆ Π(P) is the set of atomic programs, I : Π0 → A is the atomic programs reliability degree and M0 ∈ S is the initial marking.

Dynamics of Petri nets with A-failures - firing function

f I(π)

π

: S × S → A f α

abt2c(s, s′) =

   α, if a, b ∈ s and c ∈ s′ α → 0, if a, b ∈ s and a, b ∈ s′ 0 if a, b / ∈ s   

ℓ m x y c t2 t3 t1

Dynamics of Petri nets with A-failures - firing function

f I(π)

π

: S × S → A f α

abt2c(s, s′) =

   α, if a, b ∈ s and c ∈ s′ α → 0, if a, b ∈ s and a, b ∈ s′ 0 if a, b / ∈ s   

ℓ m x y c t2 t3 t1 α

− →

ℓ m x y c t2 t3 t1

Figure: Dynamics of transition of type abt2c

Interpretation of Petri net programs

The algebra of A-firing functions F = (F, ∪, ◦, ∅, χ, ∗) where: F is the universe of all the α-firing functions, for all α ∈ A

Interpretation of Petri net programs

The algebra of A-firing functions F = (F, ∪, ◦, ∅, χ, ∗) where: F is the universe of all the α-firing functions, for all α ∈ A

Theorem

F is a Kleene Algebra

Proof.

See (Gomes, 2017).

Interpretation of Petri net programs

The algebra of A-firing functions F = (F, ∪, ◦, ∅, χ, ∗) where: F is the universe of all the α-firing functions, for all α ∈ A

Theorem

F is a Kleene Algebra

Proof.

See (Gomes, 2017). A-interpretations of programs

• for any atomic program π ∈ Π0, π(s, s′) = f F(π)

π

(s, s′)

• π1; π2(s, s′) = (π1 ◦ π2)(s, s′)
• η∗(s, s′) = η∗(s, s′), for η∗(s, s′) =

i≥0ηi(s, s′)

where η0(s, s′) = χ(s, s′) and for any i ≥ 0, ηi+1(s, s′) =

• ηi ◦ η
• (s, s′)

Outline

Background and motivations Petri nets with A-failures Dynamic logics for Petri nets with A-failures: GP(A) Conclusions

PDL for Petri Nets with A-failures – GP(A)

The method is based on the one introduced in [Madeira, 2016].

PDL for Petri Nets with A-failures – GP(A)

The method is based on the one introduced in [Madeira, 2016]. Formulas: The set of formulas FmGP(A)(Prop, P): ρ ::= p | ⊤ | ⊥ | ρ ∧ ρ | ρ ∨ ρ | ρ → ρ | ηρ | [η]ρ where p ∈ Prop, η ::= π | π; η | η⋆ for π ::= at1b | abt2c | at3bc, and a, b, c ∈ P.

PDL for Petri Nets with A-failures – GP(A)

The method is based on the one introduced in [Madeira, 2016]. Formulas: The set of formulas FmGP(A)(Prop, P): ρ ::= p | ⊤ | ⊥ | ρ ∧ ρ | ρ ∨ ρ | ρ → ρ | ηρ | [η]ρ where p ∈ Prop, η ::= π | π; η | η⋆ for π ::= at1b | abt2c | at3bc, and a, b, c ∈ P. Models: M = P, V P is a Petri net with A-failures and V is a valuation function V : Prop × S → A.

PDL for Petri Nets with A-failures – GP(A)

(Graded) Satisfaction: | = : (M × S) × FmGP(A)(Prop) → A

• (M, s |

= p) = V(p, s);

• (M, s |

= ⊤) = ⊤;

• (M, s |

= ⊥) = ⊥;

• (M, s |

= ρ ∧ ρ′) = (M, s | = ρ) · (M, s | = ρ′);

• (M, s |

= ρ ∨ ρ′) = (M, s | = ρ) + (M, s | = ρ′);

• (M, s |

= ρ → ρ′) = (M, s | = ρ) → (M, s | = ρ′);

• (M, s |

= ηρ) =

s′∈S

• η(s, s′); (M, s′ |

= ρ)

• ;
• (M, s |

= [η]ρ) =

s′∈S

• η(s, s′) → (M, s′ |

= ρ)

Example

ℓ m x y c t2 t3 t1 a

− →

ℓ m x y c t2 t3 t1 b

− →

ℓ m x y c t2 t3 t1

Figure: A Petri net for a (possibly) defective chocolate vending machine

Example

ℓ m x y c t2 t3 t1 a

− →

ℓ m x y c t2 t3 t1 b

− →

ℓ m x y c t2 t3 t1

Figure: A Petri net for a (possibly) defective chocolate vending machine

GP(2)

Example

ℓ m x y c t2 t3 t1 a

− →

ℓ m x y c t2 t3 t1 b

− →

ℓ m x y c t2 t3 t1

Figure: A Petri net for a (possibly) defective chocolate vending machine

GP(2) a = ⊤, b = ⊤

M, ℓm | = ℓmt2x; xt3yc⊤ = =

• s∈S
• ℓmt2x(ℓm, x); xt3yc(x, s) + ℓmt2x(ℓm, ℓm); xt3yc(ℓm, s)
• ; (M, s |

= ⊤) = ⊤

GP(W10)

GP(W10) a = 8, b = 9

M, ℓm | = ℓmt2x; xt3yc⊤ = =

• 8; 9 + (8 → 0); 0
• ; 10 +
• (8 → 0); 0
• ; 10 +
• 8; (9 → 0) + (8 → 0); 0
• ; 10

= 7.

GP(W10) a = 8, b = 9

M, ℓm | = ℓmt2x; xt3yc⊤ = =

• 8; 9 + (8 → 0); 0
• ; 10 +
• (8 → 0); 0
• ; 10 +
• 8; (9 → 0) + (8 → 0); 0
• ; 10

= 7.

GP( L)

GP(W10) a = 8, b = 9

M, ℓm | = ℓmt2x; xt3yc⊤ = =

• 8; 9 + (8 → 0); 0
• ; 10 +
• (8 → 0); 0
• ; 10 +
• 8; (9 → 0) + (8 → 0); 0
• ; 10

= 7.

GP( L) a = 0.78, b = 0.93

M, ℓm | = ℓmt2x; xt3yc⊤ = =max{max{0.78 ⊙ 0.93, 0} ⊙ 1, 0, max{0.78 ⊙ (0.93 → 0), 0} ⊙ 1} =0.71

Outline

Background and motivations Petri nets with A-failures Dynamic logics for Petri nets with A-failures: GP(A) Conclusions

Conclusions

We achieved Generalisation of Petri PDL⋆: Petri net with A-failures; A class of Kleene algebras; GP(A)

Conclusions

We achieved Generalisation of Petri PDL⋆: Petri net with A-failures; A class of Kleene algebras; GP(A) . . . and we may enrich with Developing a proof calculi and model checking; Vary the parameter A to handle costs and time.

Thank you for your attention!