Lecture Slides for MAT-60556 PART V: Temporal logic, with - - PowerPoint PPT Presentation

Lecture Slides for MAT-60556 PART V: Temporal logic, with applications to program verification (and other sciences!)

Henri Hansen October 10, 2013

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Modal logics

• We have seen first order logic as an extension of

propositional logic, to make it more expressive

• However, Herbrand theorem essentiall says that FOL

is reducible to propositional logic, albeit in an infi- nite manner

• Another way to increase expressiveness is through

modalities, i.e., operators that need richer models than merely valuations of propositions

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Temporal Logic

• Temporal logic is a modal logic intended as a formal

system for reasoning about “time”.

• More accurately, the reasoning is about sequences
• f events or propositions, not real-time
• We focus here on propositional temporal logic
• We can think of temporal logics talking about things

that can be expressed in propositional logic, but where propositions change their truth values over time

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Temporal Logic (contd.)

• Consider the following statements:

– After the administration of sodium cyanide, cel- lular respiration halts – The output line maintains its value until the set-line is asserted. Afterwards they are com- plemented – The operating system never deadlocks

• we use temporal modalities to express temporal ar-

rangements such as those mentioned above

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Syntax and Semantics (part I)

• Propositional temporal logic (PTL) is defined with

the syntax of propositional logic, with the addition

• f the following two operators

– , which means “always” – ⋄, which means “eventually”

• p is true if p holds now and will never become

false, and ⋄p is true if p is true at any point in time in the future

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Interpretation of PTL formulas

• Interpretations of PTL formulas could be given over

many different structures, but we restrict here to state-transition diagrams

• An Interpretation for a formula A is a pair, (I, ρ)

where I = {s1, . . . , sn} is a set of states, each of which is an assignment to truth values of the propo- sitions in A, i.e., si : P → {T, F} and ρ is a binary relation ρ ⊆ S × S

• we define ρ(s) = {s′ | (s, s′) ∈ ρ}

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Interpretation of PTL formulas (contd.)

• Given an interpretation (I, ρ) and a state s, the

truth value of a PTL formula A in a state s is de- fined as follows – If A is p ∈ P, then ν(A, s) = s(p) – If A is ¬A′ then ν(A, s) = T iff ν(A′, s) = F (etc, just like in propositional logic) – If A is A′, then ν(A, s) = T iff ν(A′, s′) = T for all s′ ∈ ρ(s) – If A is ⋄A′, then ν(A, s) = T iff ν(A′, s′) = T for some s′ ∈ ρ(s)

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Satisfiability and validity

• A PTL formula A is satisfiable iff there is an inter-

pretation (I, ρ) such that for some state s, s | = A.

• The formula is valid iff it is true in all states of all

interpretations

• Theorem (duality) A ⇔ ¬ ⋄ ¬A
• Theorem: (p → q) ⇒ p → q

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Models of time

• An interpretation (I, ρ) is said to be

– Reflexive, iff for all s ∈ I, s ∈ ρ(s) – Transitive, iff s2 ∈ ρ(s1) ∧ s3 ∈ ρ(s2) imples s3 ∈ ρ(s1) – Linear, iff for every s ∈ I, there is at most one s′ such that s′ = ρ(s)

• Linearity may seem a strange property, but it will

become clear when we talk of LTL

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Linear Temporal Logic

• Linear temporal logic (LTL) is PTL (possibly etended

with some operators) whose interpretations are lim- ited to transition functions that are reflexive, tran- sitive and linear

• The interpretations of LTL can be represented as

paths ρ = s0s1 · · ·, where si : P → {T, F}, we write σi = sisi+1 · · ·

• Truth value is defined similarly:

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– For propositional formula A, ν(A, σ) = T iff s0 | = A – If A is A′ then ν(A, σ) = T iff ν(A, σi) = T for i ≥ 0 – If A is ◦A′, then ν(A, σ) = T iff ν(A, σ1) = T, etc.

• if ν(A, σ) = T we also write σ |

= A.

Side note: Models of modal logics

• A canonical model for temporal logics and also for

epistemic logic ∗ is a Kripke structure

• A general Kripke structure is a tuple (S, L, R1, . . . , Rn)

where S is a set of states, L : S → 2P and each Ri ⊆ S × S are relations over states.

• For temporal logics there is just one relation in the

model, but for instance in multi agent epistemic logics, each agent has its own relation.

∗see, e.g., Fagin, Halpern, Moses, Vardi: Reasoning about knowl-

edge MIT Press, 1995

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LTL (contd)

• Even though semantics of LTL are given over paths,

we can extend them to Kripke structures, so that M = (S, L, R) is a model for A at state s, written (M, s) | = A if all paths σ that start at s have σ | = A.

• Validity and satisfiability are not sensitive to whether

we think of models of formulas to be paths or Kripke structures, but it is possible that neither (M, s) | = A nor (M, s) | = ¬A holds for a given Kripke structure, even if it is true that σ | = A or σ | = ¬A does always hold.

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Tableaux for LTL

• We can construct tableaux just like with proposi-

tional formulas, but we need new rules for temporal

• perators
• A is an α-formula, that results in A and ◦A: A

must hold here, and A must hold in the next state

• ⋄A is a β-formula, whose descendants are A and
• ⋄ A:

A must hold here, or ⋄A must hold in the next state

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• ◦A has a new rule, the X-rule, which says that A

must hold in the next state, and ¬ ◦ A says that A must not hold in the next state.

Tableaux for LTL (contd)

• When a node in a tableaux contains only literals
• r and X-formulas, then we say that the node is a

state node

• A state node has a given set of literals, but when we

apply the X-rule, these literals are not copied; the application of the X-rule means that we are moving from one state to another

• This breaks down the proof of completness and

soundness that were proven using Hintikka sets ear- lier.

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slide LTL Tableau algorithm for formula A

• The root is labelled A.
• At each iteration, choose a leaf that is not yet han-

dled and do one of the following:

• 1. If it has a complementary pair of literals, mark it
• closed. If the node consists of only literals, but

no complementary pairs, mark it open.

• 2. Apply α or β rules accordingly, if possible
• 3. If it is a state node, generate the successor: use

the X-rule on all X formulas simultaneously and

remove all the other labels. If the resulting node is an existing state node, connect; otherwise cre- ate The tableau is closed if all leaves are closed and there are no cycles. Otherwise it is open.

Hintikka Structure

• A tableau structure for a given LTL Formula A is

like a Kripke structure, but its labels are sets of formulas built from the propositions of A

• A state path is a path l0, . . . , lk through a tableau,

such that l0 is a state node (or the ruut) and lk is a state node, and none of l1, . . . , lk−1 are state nodes.

• A tableau structure built from a tableau is defined

so that S is the set of state nodes of the tableau

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• Each state s is labelled by the union of formulas that

appear on non-state nodes of a state path leading to s

• (s, s′) ∈ R iff there is a state path leading from s to

s′ in the tableau

• we say that s′ is reachable from s iff s′ = s or there

is some s′′ such that (s, s′′) ∈ R and s′ is reachable from s′′.

Hintikka Structure (contd.)

• A tableau structure is a Hintikka structure for A iff

A ∈ L(s0) and for all states si the following hold:

• 1. For all propositions of A either p /

∈ L(si)or¬p / ∈ L(si)

• 2. If α ∈ L(si) then α1 ∈ L(si) and α2 ∈ L(si)
• 3. If β ∈ L(si) then β1 ∈ L(si) and β2 ∈ L(si)
• 4. If X ∈ L(si) then for every sj such that (si, sj) ∈

R, X1 ∈ L(sj).

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• Theorem:

The Structure created from an open tableau is a Hintikka structure

Hintikka Structure (contd.)

• A Hintikka structure (S, L, R) is linear iff for every

s there exists exactly one s′ such that (s, s′) ∈ R

• Lemma: An infinite path through a Hintikka struc-

ture of A is itself a linear Hintikka structure

• A linear Hintikka structure is fulfilling iff for all fu-

ture formulas ⋄A and for all states s of the structure, if ⋄A ∈ L(s) then A ∈ L(s′) for some state s′ such that s′ is reachable from s.

• Theorem: If there exists a linear fulfilling Hintikka

structure for A then A is satisfiable

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Deduction in Temporal Logic

• Valid propositional logic formulas and deducion rules

remain valid for temporal logic

• Additional valid formulas can be taken as axioms:
• 1. (A → B) → (A → B)
• 2. ◦(A → B) → (◦A → ◦B)
• 3. A → (A ∧ ◦A ∧ ◦A)
• 4. (A → ◦A) → (A → A)

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• 5. ◦A ↔ ¬ ◦ ¬A
• The rules of inference are the standard Modus Po-

nens, and generalization: A ⇒ A, again with the same caveat as in FOL: A must be valid and not part of the assumptions

Binary temporal operators

• The unary temporal operators can be expressed with

the help of one binary operator U.

• AUB means that A must be true until B becomes

true

• A characterization of U is the following:

AUB ↔ (B ∨ (A ∧ ◦(AUB) ∧ ⋄B))

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