[PPT] - Metric Temporal Logic With Counting S.N.Krishna, Khushraj Madnani, PowerPoint Presentation

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Metric Temporal Logic With Counting

S.N.Krishna, Khushraj Madnani, Paritosh.K.Pandya February 1, 2016

S.N.Krishna, Khushraj Madnani, Paritosh.K.Pandya Metric Temporal Logic With Counting

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Introduction

S.N.Krishna, Khushraj Madnani, Paritosh.K.Pandya Metric Temporal Logic With Counting

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Introduction

Metric Temporal Logic is extensively studied Real time Logic in the literature.

S.N.Krishna, Khushraj Madnani, Paritosh.K.Pandya Metric Temporal Logic With Counting

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Introduction

Metric Temporal Logic is extensively studied Real time Logic in the literature. Allows timing constraints to be specified along with the temporal ordering.

S.N.Krishna, Khushraj Madnani, Paritosh.K.Pandya Metric Temporal Logic With Counting

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Introduction

Metric Temporal Logic is extensively studied Real time Logic in the literature. Allows timing constraints to be specified along with the temporal ordering. Exhibits considerable diversity in expressiveness and decidability properties based on restriction on modalities and type of timing constraints.

S.N.Krishna, Khushraj Madnani, Paritosh.K.Pandya Metric Temporal Logic With Counting

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Introduction

Metric Temporal Logic is extensively studied Real time Logic in the literature. Allows timing constraints to be specified along with the temporal ordering. Exhibits considerable diversity in expressiveness and decidability properties based on restriction on modalities and type of timing constraints. In general satisfiability checking for MTL is undecidable.

S.N.Krishna, Khushraj Madnani, Paritosh.K.Pandya Metric Temporal Logic With Counting

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Introduction

Metric Temporal Logic is extensively studied Real time Logic in the literature. Allows timing constraints to be specified along with the temporal ordering. Exhibits considerable diversity in expressiveness and decidability properties based on restriction on modalities and type of timing constraints. In general satisfiability checking for MTL is undecidable. Counting within a given time slot is a very natural and useful property in real time systems.

S.N.Krishna, Khushraj Madnani, Paritosh.K.Pandya Metric Temporal Logic With Counting

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Introduction

Metric Temporal Logic is extensively studied Real time Logic in the literature. Allows timing constraints to be specified along with the temporal ordering. Exhibits considerable diversity in expressiveness and decidability properties based on restriction on modalities and type of timing constraints. In general satisfiability checking for MTL is undecidable. Counting within a given time slot is a very natural and useful property in real time systems. Thus it becomes interesting to study satisfiability checking for its fragments and their extensions with ability to count.

S.N.Krishna, Khushraj Madnani, Paritosh.K.Pandya Metric Temporal Logic With Counting

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Presentation Flow

Model : Timed Words Timed Logic with Counting : Syntax and Semantics Temporal Projections : Simple and Oversampled Expressiveness Relations with Counting Extensions Satisfiability Checking: Decidability Conclusion Future Work

S.N.Krishna, Khushraj Madnani, Paritosh.K.Pandya Metric Temporal Logic With Counting

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Model : Timed Word

S.N.Krishna, Khushraj Madnani, Paritosh.K.Pandya Metric Temporal Logic With Counting

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Model : Timed Word

Models over which pointwise MTL Formula is being evaluated .

S.N.Krishna, Khushraj Madnani, Paritosh.K.Pandya Metric Temporal Logic With Counting

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Model : Timed Word

Models over which pointwise MTL Formula is being evaluated . Finite sequence of symbols along with their corresponding timestamps.

S.N.Krishna, Khushraj Madnani, Paritosh.K.Pandya Metric Temporal Logic With Counting

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Model : Timed Word

Models over which pointwise MTL Formula is being evaluated . Finite sequence of symbols along with their corresponding timestamps.In general, timestamps monotonically increases

S.N.Krishna, Khushraj Madnani, Paritosh.K.Pandya Metric Temporal Logic With Counting

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Model : Timed Word

Models over which pointwise MTL Formula is being evaluated . Finite sequence of symbols along with their corresponding timestamps.In general, timestamps monotonically increases For the purpose of this presentation we will restrict our timed words to be strictly monotonic.

S.N.Krishna, Khushraj Madnani, Paritosh.K.Pandya Metric Temporal Logic With Counting

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Model : Timed Word

Models over which pointwise MTL Formula is being evaluated . Finite sequence of symbols along with their corresponding timestamps.In general, timestamps monotonically increases For the purpose of this presentation we will restrict our timed words to be strictly monotonic.

Figure: A finite timed word over Σ = {a, b, c}. A strictly monotonic timed word can be seen as a real line annotated with symbols from Σ

S.N.Krishna, Khushraj Madnani, Paritosh.K.Pandya Metric Temporal Logic With Counting

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Presentation Flow

Model : Timed Words Timed Logic with Counting : Syntax and Semantics Temporal Projections : Simple and Oversampled Expressiveness Relations with Counting Extensions Satisfiability Checking: Decidability Conclusion Future Work

S.N.Krishna, Khushraj Madnani, Paritosh.K.Pandya Metric Temporal Logic With Counting

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Time Logic : Metric Temporal Logic

MTL Syntax

S.N.Krishna, Khushraj Madnani, Paritosh.K.Pandya Metric Temporal Logic With Counting

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Time Logic : Metric Temporal Logic

MTL Syntax φ ::= AP | φ ∧ φ | φ ∨ φ | ¬ φ | φ UI φ | φ SI φ where I is interval of the form x, y, x ∈ N ∪ {0}, y, x ∈ N ∪ {0, ∞} and ... ∈ {[...], (...), [...), (...]}

S.N.Krishna, Khushraj Madnani, Paritosh.K.Pandya Metric Temporal Logic With Counting

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Metric Temporal Logic : Semantics

S.N.Krishna, Khushraj Madnani, Paritosh.K.Pandya Metric Temporal Logic With Counting

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Metric Temporal Logic : Semantics

MTL Semantics

S.N.Krishna, Khushraj Madnani, Paritosh.K.Pandya Metric Temporal Logic With Counting

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Metric Temporal Logic : Semantics

MTL Semantics ρ, i | = φ1 UIφ2 ⇐ ⇒ ∃j > i ρ, j | = φ2 and τj − τi ∈ I and ∀ i < k < j ρ, k | = φ1

S.N.Krishna, Khushraj Madnani, Paritosh.K.Pandya Metric Temporal Logic With Counting

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Metric Temporal Logic : Semantics

MTL Semantics ρ, i | = φ1 UIφ2 ⇐ ⇒ ∃j > i ρ, j | = φ2 and τj − τi ∈ I and ∀ i < k < j ρ, k | = φ1 ρ, i | = φ1 SIφ2 ⇐ ⇒ ∃j < i ρ, j | = φ2 and τi − τj ∈ I and ∀ i > k > j ρ, k | = φ1

S.N.Krishna, Khushraj Madnani, Paritosh.K.Pandya Metric Temporal Logic With Counting

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Metric Temporal Logic : Semantics

MTL Semantics ρ, i | = φ1 UIφ2 ⇐ ⇒ ∃j > i ρ, j | = φ2 and τj − τi ∈ I and ∀ i < k < j ρ, k | = φ1 ρ, i | = φ1 SIφ2 ⇐ ⇒ ∃j < i ρ, j | = φ2 and τi − τj ∈ I and ∀ i > k > j ρ, k | = φ1 Ex: work-hard U[5,10] giving-up

S.N.Krishna, Khushraj Madnani, Paritosh.K.Pandya Metric Temporal Logic With Counting

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Time Logic : MTL Fragments

Subclasses:

S.N.Krishna, Khushraj Madnani, Paritosh.K.Pandya Metric Temporal Logic With Counting

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Time Logic : MTL Fragments

Subclasses: By restricting set of allowed intervals. e.g. MTL[Unp, Snp], where np refers to non-punctual intervals. It is well known as MITL in the literature.

S.N.Krishna, Khushraj Madnani, Paritosh.K.Pandya Metric Temporal Logic With Counting

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Time Logic : MTL Fragments

Subclasses: By restricting set of allowed intervals. e.g. MTL[Unp, Snp], where np refers to non-punctual intervals. It is well known as MITL in the literature. By restricting set of operators. We denote MTL[W] for subclass of MTL restricted to operators in W . e.g. MTL[ UI] where only until operator is allowed.

S.N.Krishna, Khushraj Madnani, Paritosh.K.Pandya Metric Temporal Logic With Counting

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Time Logic : MTL Fragments

Subclasses: By restricting set of allowed intervals. e.g. MTL[Unp, Snp], where np refers to non-punctual intervals. It is well known as MITL in the literature. By restricting set of operators. We denote MTL[W] for subclass of MTL restricted to operators in W . e.g. MTL[ UI] where only until operator is allowed. We will restrict to future only fragment of MTL.

S.N.Krishna, Khushraj Madnani, Paritosh.K.Pandya Metric Temporal Logic With Counting

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Time Logic with counting: CTMTL

We introduce two new modal operators for counting C and UT.

S.N.Krishna, Khushraj Madnani, Paritosh.K.Pandya Metric Temporal Logic With Counting

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Time Logic with counting: CTMTL

We introduce two new modal operators for counting C and UT. CTMTL Syntax φ ::= AP | φ ∧ φ | φ ∨ φ | ¬ φ | φ UI,#φ∼n φ | Cn

I φ

where I is interval of the form x, y, x ∈ N ∪ {0}, y, x ∈ N ∪ {0, ∞}, ... ∈ {[...], (...), [...), (...]},

S.N.Krishna, Khushraj Madnani, Paritosh.K.Pandya Metric Temporal Logic With Counting

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Time Logic with counting: CTMTL

We introduce two new modal operators for counting C and UT. CTMTL Syntax φ ::= AP | φ ∧ φ | φ ∨ φ | ¬ φ | φ UI,#φ∼n φ | Cn

I φ

where I is interval of the form x, y, x ∈ N ∪ {0}, y, x ∈ N ∪ {0, ∞}, ... ∈ {[...], (...), [...), (...]}, ∼= {≥, ≤} and

S.N.Krishna, Khushraj Madnani, Paritosh.K.Pandya Metric Temporal Logic With Counting

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Time Logic with counting: CTMTL

We introduce two new modal operators for counting C and UT. CTMTL Syntax φ ::= AP | φ ∧ φ | φ ∨ φ | ¬ φ | φ UI,#φ∼n φ | Cn

I φ

where I is interval of the form x, y, x ∈ N ∪ {0}, y, x ∈ N ∪ {0, ∞}, ... ∈ {[...], (...), [...), (...]}, ∼= {≥, ≤} and n ∈ N ∪ {0}

S.N.Krishna, Khushraj Madnani, Paritosh.K.Pandya Metric Temporal Logic With Counting

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CTMTL: Semantics

S.N.Krishna, Khushraj Madnani, Paritosh.K.Pandya Metric Temporal Logic With Counting

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CTMTL: Semantics

ρ, i | = C ∼n

l,uφ

⇐ ⇒ Nφ(τi + l, τi + u) ∼ n

S.N.Krishna, Khushraj Madnani, Paritosh.K.Pandya Metric Temporal Logic With Counting

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CTMTL: Semantics

ρ, i | = C ∼n

l,uφ

⇐ ⇒ Nφ(τi + l, τi + u) ∼ n ρ, i | = φ1 UI,#η∼nφ2 ⇐ ⇒ ∃j > i ρ, j | = φ2 ∧ τj − τi ∈ I ∧ ∀ ∧ i < k < j ρ, k | = φ1 ∧ N ′φ(i, j) ∼ n

S.N.Krishna, Khushraj Madnani, Paritosh.K.Pandya Metric Temporal Logic With Counting

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Subclasses of CTMTL

C(0,1)MTL: Counting of the form C ∼n

(0,1).

S.N.Krishna, Khushraj Madnani, Paritosh.K.Pandya Metric Temporal Logic With Counting

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Subclasses of CTMTL

C(0,1)MTL: Counting of the form C ∼n

(0,1).

C(0,u)MTL: Counting of the form C ∼n

(0,u).

S.N.Krishna, Khushraj Madnani, Paritosh.K.Pandya Metric Temporal Logic With Counting

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Subclasses of CTMTL

C(0,1)MTL: Counting of the form C ∼n

(0,1).

C(0,u)MTL: Counting of the form C ∼n

(0,u).

CMTL: Counting with C modality only.

S.N.Krishna, Khushraj Madnani, Paritosh.K.Pandya Metric Temporal Logic With Counting

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Subclasses of CTMTL

C(0,1)MTL: Counting of the form C ∼n

(0,1).

C(0,u)MTL: Counting of the form C ∼n

(0,u).

CMTL: Counting with C modality only. TMTL: Counting with UT Modality only.

S.N.Krishna, Khushraj Madnani, Paritosh.K.Pandya Metric Temporal Logic With Counting

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Scheduling HVAC in Demand Response: An Example

In Demand Response system an important requirement is to reduce the Peak Power Demand.

S.N.Krishna, Khushraj Madnani, Paritosh.K.Pandya Metric Temporal Logic With Counting

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Scheduling HVAC in Demand Response: An Example

In Demand Response system an important requirement is to reduce the Peak Power Demand. Scheduling of HVAC to limit peak power demand below threshold. HVAC are more flexible as compared to devices like microwave

ven.

Constant mode switching (OFF-¿ON) causes wear and tear and more power consumption due to transient currents.

No. of Switch yet another important parameter to grade such

scheduling algorithms.

S.N.Krishna, Khushraj Madnani, Paritosh.K.Pandya Metric Temporal Logic With Counting

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Scheduling HVAC in Demand Response: An Example

Temperature 24 25 26 27 28 29 30 Time 2 4 6 8 10 26.5 Upper Bound 24.5 Lower Bound AC1 AC2 AC3

S.N.Krishna, Khushraj Madnani, Paritosh.K.Pandya Metric Temporal Logic With Counting

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Scheduling HVAC in Demand Response: An Example

♦(0,3),#Switch−ON−AC≤3(Comfort − AC1 ∧ Comfort − AC2 ∧ Comfort − AC3)

Temperature 24 25 26 27 28 29 30 Time 2 4 6 8 10 26.5 Upper Bound 24.5 Lower Bound AC1 AC2 AC3

S.N.Krishna, Khushraj Madnani, Paritosh.K.Pandya Metric Temporal Logic With Counting

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Presentation Flow

Model : Timed Words Timed Logic with Counting : Syntax and Semantics Temporal Projections : Simple and Oversampled Expressiveness Relations with Counting Extensions Decidability : Satisfiability Checking Conclusion Future Work

S.N.Krishna, Khushraj Madnani, Paritosh.K.Pandya Metric Temporal Logic With Counting

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Definitions: Simple Projection

S.N.Krishna, Khushraj Madnani, Paritosh.K.Pandya Metric Temporal Logic With Counting

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Definitions: Simple Projection

Let Σ, X be finite disjoint set.

S.N.Krishna, Khushraj Madnani, Paritosh.K.Pandya Metric Temporal Logic With Counting

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Definitions: Simple Projection

Let Σ, X be finite disjoint set. Simple Extension A (Σ, X)-simple extension is a timed word ρ over 2X ∪ Σ such that at any point i ∈ dom(ρ), σi ∩ Σ = ∅

S.N.Krishna, Khushraj Madnani, Paritosh.K.Pandya Metric Temporal Logic With Counting

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Definitions: Simple Projection

Let Σ, X be finite disjoint set. Simple Extension A (Σ, X)-simple extension is a timed word ρ over 2X ∪ Σ such that at any point i ∈ dom(ρ), σi ∩ Σ = ∅ Simple Projection A timed word ρ over Σ obtained by deleting symbols in X from (Σ, X) extension ρ′ is called its Simple Projection.

S.N.Krishna, Khushraj Madnani, Paritosh.K.Pandya Metric Temporal Logic With Counting

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Definitions: Simple Projection

Let Σ, X be finite disjoint set. Simple Extension A (Σ, X)-simple extension is a timed word ρ over 2X ∪ Σ such that at any point i ∈ dom(ρ), σi ∩ Σ = ∅ Simple Projection A timed word ρ over Σ obtained by deleting symbols in X from (Σ, X) extension ρ′ is called its Simple Projection.

S.N.Krishna, Khushraj Madnani, Paritosh.K.Pandya Metric Temporal Logic With Counting

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Definitions: Simple Projection

Let Σ, X be finite disjoint set. Simple Extension A (Σ, X)-simple extension is a timed word ρ over 2X ∪ Σ such that at any point i ∈ dom(ρ), σi ∩ Σ = ∅ Simple Projection A timed word ρ over Σ obtained by deleting symbols in X from (Σ, X) extension ρ′ is called its Simple Projection.

S.N.Krishna, Khushraj Madnani, Paritosh.K.Pandya Metric Temporal Logic With Counting

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Definitions: Oversampled Projection

S.N.Krishna, Khushraj Madnani, Paritosh.K.Pandya Metric Temporal Logic With Counting

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Definitions: Oversampled Projection

Let Σ, X be finite disjoint set.

S.N.Krishna, Khushraj Madnani, Paritosh.K.Pandya Metric Temporal Logic With Counting

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Definitions: Oversampled Projection

Let Σ, X be finite disjoint set. Oversampled Behaviour A (Σ, X)-oversampled behaviour is a timed word over 2X ∪ Σ, such that σ′

1 ∩ Σ = ∅ and

σ′

|dom(ρ′)| ∩ Σ = ∅.

S.N.Krishna, Khushraj Madnani, Paritosh.K.Pandya Metric Temporal Logic With Counting

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Definitions: Oversampled Projection

Let Σ, X be finite disjoint set. Oversampled Behaviour A (Σ, X)-oversampled behaviour is a timed word over 2X ∪ Σ, such that σ′

1 ∩ Σ = ∅ and

σ′

|dom(ρ′)| ∩ Σ = ∅.

Oversampled Projection A timed word ρ over Σ obtained by deleting symbols in X (and thus deleting the points containing only X)from (Σ, X) oversampled behaviour ρ′ is called its Oversampled Projection.

S.N.Krishna, Khushraj Madnani, Paritosh.K.Pandya Metric Temporal Logic With Counting

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Definitions: Oversampled Projection

Let Σ, X be finite disjoint set. Oversampled Behaviour A (Σ, X)-oversampled behaviour is a timed word over 2X ∪ Σ, such that σ′

1 ∩ Σ = ∅ and

σ′

|dom(ρ′)| ∩ Σ = ∅.

Oversampled Projection A timed word ρ over Σ obtained by deleting symbols in X (and thus deleting the points containing only X)from (Σ, X) oversampled behaviour ρ′ is called its Oversampled Projection.

S.N.Krishna, Khushraj Madnani, Paritosh.K.Pandya Metric Temporal Logic With Counting

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Definitions: Oversampled Projection

Let Σ, X be finite disjoint set. Oversampled Behaviour A (Σ, X)-oversampled behaviour is a timed word over 2X ∪ Σ, such that σ′

1 ∩ Σ = ∅ and

σ′

|dom(ρ′)| ∩ Σ = ∅.

Oversampled Projection A timed word ρ over Σ obtained by deleting symbols in X (and thus deleting the points containing only X)from (Σ, X) oversampled behaviour ρ′ is called its Oversampled Projection.

S.N.Krishna, Khushraj Madnani, Paritosh.K.Pandya Metric Temporal Logic With Counting

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Definitions: Equisaitisfiability modulo Temporal Projection

We say that ϕ over Σ is equisatisfiable modulo temporal projection ψ over Σ ∪ 2X iff:

Figure: Figure Illustrating ϕ is equisatisfiable to ψ. Arrow represents the temporal(simple or oversampled) projection function

S.N.Krishna, Khushraj Madnani, Paritosh.K.Pandya Metric Temporal Logic With Counting

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Flattening: An example

S.N.Krishna, Khushraj Madnani, Paritosh.K.Pandya Metric Temporal Logic With Counting

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Flattening: An example

Flattening is a technique to reduce the modal depth of the formula preserving satisfiability.

S.N.Krishna, Khushraj Madnani, Paritosh.K.Pandya Metric Temporal Logic With Counting

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Flattening: An example

Flattening is a technique to reduce the modal depth of the formula preserving satisfiability. Example Let φ = a U[2,5](b U[2,3]c)

S.N.Krishna, Khushraj Madnani, Paritosh.K.Pandya Metric Temporal Logic With Counting

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Flattening: An example

Flattening is a technique to reduce the modal depth of the formula preserving satisfiability. Example Let φ = a U[2,5](b U[2,3]c) φflat = a U[2,5]d ∧ (d ↔ (b U[2,3]c)) ∧ (d → a ∨ b ∨ c)

S.N.Krishna, Khushraj Madnani, Paritosh.K.Pandya Metric Temporal Logic With Counting

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Flattening: An example

Flattening is a technique to reduce the modal depth of the formula preserving satisfiability. Example Let φ = a U[2,5](b U[2,3]c) φflat = a U[2,5]d ∧ (d ↔ (b U[2,3]c)) ∧ (d → a ∨ b ∨ c)

S.N.Krishna, Khushraj Madnani, Paritosh.K.Pandya Metric Temporal Logic With Counting

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Flattening: An example

Flattening is a technique to reduce the modal depth of the formula preserving satisfiability. Example Let φ = a U[2,5](b U[2,3]c) φflat = a U[2,5]d ∧ (d ↔ (b U[2,3]c)) ∧ (d → a ∨ b ∨ c)

S.N.Krishna, Khushraj Madnani, Paritosh.K.Pandya Metric Temporal Logic With Counting

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Flattening: An example

Flattening is a technique to reduce the modal depth of the formula preserving satisfiability. Example Let φ = a U[2,5](b U[2,3]c) φflat = a U[2,5]d ∧ (d ↔ (b U[2,3]c)) ∧ (d → a ∨ b ∨ c) Thus flattening is an example of a reduction preserving satisfiability modulo simple projections.

S.N.Krishna, Khushraj Madnani, Paritosh.K.Pandya Metric Temporal Logic With Counting

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Related Work

S.N.Krishna, Khushraj Madnani, Paritosh.K.Pandya Metric Temporal Logic With Counting

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Related Work

Satisfiability Checking of MITL is decidable with EXPSPACE

complexity. [Alur et al. J.ACM 1996]

S.N.Krishna, Khushraj Madnani, Paritosh.K.Pandya Metric Temporal Logic With Counting

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Related Work

Satisfiability Checking of MITL is decidable with EXPSPACE

complexity. [Alur et al. J.ACM 1996]

Satisfiability problem for MTL[ UI] is decidable by reducing it to reachability problem for 1-clock Timed Alternating Automata.[Ouaknine et al. LICS 2005]

S.N.Krishna, Khushraj Madnani, Paritosh.K.Pandya Metric Temporal Logic With Counting

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Related Work

Satisfiability Checking of MITL is decidable with EXPSPACE

complexity. [Alur et al. J.ACM 1996]

Satisfiability problem for MTL[ UI] is decidable by reducing it to reachability problem for 1-clock Timed Alternating Automata.[Ouaknine et al. LICS 2005] Satisfiability Checking of QTL with counting is decidable with EXPSPACE complexity. [Rabinovich et.al. FORMATS 2008]

S.N.Krishna, Khushraj Madnani, Paritosh.K.Pandya Metric Temporal Logic With Counting

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Related Work

Satisfiability Checking of MITL is decidable with EXPSPACE

complexity. [Alur et al. J.ACM 1996]

Satisfiability problem for MTL[ UI] is decidable by reducing it to reachability problem for 1-clock Timed Alternating Automata.[Ouaknine et al. LICS 2005] Satisfiability Checking of QTL with counting is decidable with EXPSPACE complexity. [Rabinovich et.al. FORMATS 2008] Counting adds expressiveness to MITL over signals [Rabinovich FORMATS 2008].

S.N.Krishna, Khushraj Madnani, Paritosh.K.Pandya Metric Temporal Logic With Counting

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Related Work

Satisfiability Checking of MITL is decidable with EXPSPACE

complexity. [Alur et al. J.ACM 1996]

Satisfiability problem for MTL[ UI] is decidable by reducing it to reachability problem for 1-clock Timed Alternating Automata.[Ouaknine et al. LICS 2005] Satisfiability Checking of QTL with counting is decidable with EXPSPACE complexity. [Rabinovich et.al. FORMATS 2008] Counting adds expressiveness to MITL over signals [Rabinovich FORMATS 2008]. MTL with counting over signals is expressively complete with FO[<, +1] over reals [Hunter CSL 2013].

S.N.Krishna, Khushraj Madnani, Paritosh.K.Pandya Metric Temporal Logic With Counting

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Related Work

Satisfiability Checking of MITL is decidable with EXPSPACE

complexity. [Alur et al. J.ACM 1996]

Satisfiability problem for MTL[ UI] is decidable by reducing it to reachability problem for 1-clock Timed Alternating Automata.[Ouaknine et al. LICS 2005] Satisfiability Checking of QTL with counting is decidable with EXPSPACE complexity. [Rabinovich et.al. FORMATS 2008] Counting adds expressiveness to MITL over signals [Rabinovich FORMATS 2008]. MTL with counting over signals is expressively complete with FO[<, +1] over reals [Hunter CSL 2013]. Counting LTL is equivalent to LTL and has EXP − SPACE complete satisfiability checking.[Laroussinie et. al. TIME 2010].

S.N.Krishna, Khushraj Madnani, Paritosh.K.Pandya Metric Temporal Logic With Counting

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Our Results

S.N.Krishna, Khushraj Madnani, Paritosh.K.Pandya Metric Temporal Logic With Counting

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Our Results

Satisfiability Checking for CTMTL is decidable.

S.N.Krishna, Khushraj Madnani, Paritosh.K.Pandya Metric Temporal Logic With Counting

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Our Results

Satisfiability Checking for CTMTL is decidable. Exploring Expressiveness relations amongst fragments of MTL with counting over timed words(Pointwise Semantics).

S.N.Krishna, Khushraj Madnani, Paritosh.K.Pandya Metric Temporal Logic With Counting

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Presentation Flow

Model : Timed Words Timed Logic with Counting : Syntax and Semantics Temporal Projections : Simple and Oversampled Expressiveness Relations with Counting Extensions Satisfiability Checking: Decidability Discussion Future Work

S.N.Krishna, Khushraj Madnani, Paritosh.K.Pandya Metric Temporal Logic With Counting

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Expressiveness Heirarchy : Logic with counting

S.N.Krishna, Khushraj Madnani, Paritosh.K.Pandya Metric Temporal Logic With Counting

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Expressiveness Heirarchy : Non-Punctual Fragments

S.N.Krishna, Khushraj Madnani, Paritosh.K.Pandya Metric Temporal Logic With Counting

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TMTL − CMTL = ∅

ϕ = ♦(0,1),#a≥3b

S.N.Krishna, Khushraj Madnani, Paritosh.K.Pandya Metric Temporal Logic With Counting

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Presentation Flow

Model : Timed Words Timed Logic with Counting : Syntax and Semantics Temporal Projections : Simple and Oversampled Expressiveness Relations with Counting Extensions Satisfiability Checking: Decidability Discussion Future Work

S.N.Krishna, Khushraj Madnani, Paritosh.K.Pandya Metric Temporal Logic With Counting

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Satisfiability Checking : Decidability

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Satisfiability Checking : Decidability

Flatten the formula

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Satisfiability Checking : Decidability

Flatten the formula All the counting modalities are of the form (w ↔ C∼n

I

a) and (w ↔ aUI,#x∼nb)

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Satisfiability Checking : Decidability

Flatten the formula All the counting modalities are of the form (w ↔ C∼n

I

a) and (w ↔ aUI,#x∼nb) Next we eliminate counting modalities from the above flattened formula preserving satisfiability to show decidability.

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Eliminating C≥n

I b modality

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Eliminating C≥n

I b modality

Given a word ρ over Σ we construct a simple extension ρ′ over Σ ∪ {b0, b1, . . . , bn−1}

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Eliminating C≥n

I b modality

Given a word ρ over Σ we construct a simple extension ρ′ over Σ ∪ {b0, b1, . . . , bn−1} {b0, b1, . . . , bn−1} works as a counter. Using their behaviour we precisely mark a as the witness for C≥n

I

b.

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Construction of ρ′

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Construction of ρ′

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Construction of ρ′

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Construction of ρ′

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Construction of ρ′

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Construction of ρ′

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Construction of ρ′

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Construction of ρ′

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Construction of ρ′

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Construction of ρ′

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Construction of ρ′

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Construction of ρ′

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Construction of ρ′

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Construction of ρ′

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Construction of ρ′

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Eliminating UT modality

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Eliminating UT modality

Given a word ρ over Σ we construct a oversampling ρ′ over Σ ∪ C ∪ B

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Eliminating UT modality

Given a word ρ over Σ we construct a oversampling ρ′ over Σ ∪ C ∪ B

C = {c0, . . . , cu}:

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Eliminating UT modality

Given a word ρ over Σ we construct a oversampling ρ′ over Σ ∪ C ∪ B

C = {c0, . . . , cu}:These propositions oversample the model at integer time stamps.

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Eliminating UT modality

Given a word ρ over Σ we construct a oversampling ρ′ over Σ ∪ C ∪ B

C = {c0, . . . , cu}:These propositions oversample the model at integer time stamps. B = u

i=0 Bi where Bi = {bi 0, . . . bi n} : These propositions are

used as counters for b. Counter Bi resets at integer point marked ci and saturates once the value reaches n till the next reset.

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Construction of ρ′

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Construction of ρ′

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Construction of ρ′

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Construction of ρ′

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Construction of ρ′

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Construction of ρ′

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Construction of ρ′

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Construction of ρ′

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Marking Witness for UT sub-formula

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Marking Witness for UT sub-formula

We count the number of occurrence of b in two stages:

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Marking Witness for UT sub-formula

We count the number of occurrence of b in two stages:

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Marking Witness for UT sub-formula

We count the number of occurrence of b in two stages:

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Marking Witness for UT sub-formula

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Marking Witness for UT sub-formula

n1 ≥ n ∨ n2 ≥ n

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Marking Witness for UT sub-formula

n1 ≥ n ∨ n2 ≥ n Or, n1 < n ∧ n2 < n and thus bounded number of cases (disjunctions).

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Presentation Flow

Model : Timed Words Timed Logic with Counting : Syntax and Semantics Temporal Projections : Simple and Oversampled Expressiveness Relations with Counting Extensions Satisfiability Checking: Decidability Conclusion Future Work

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Conclusion

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Conclusion

Two ways of extending MTL with counting threshold constraints is studied.

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Conclusion

Two ways of extending MTL with counting threshold constraints is studied. Both ways add expressiveness to MTL orthogonally.

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Conclusion

Two ways of extending MTL with counting threshold constraints is studied. Both ways add expressiveness to MTL orthogonally. Satisfiability checking for the logic CTMTL is decidable.

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Conclusion

Two ways of extending MTL with counting threshold constraints is studied. Both ways add expressiveness to MTL orthogonally. Satisfiability checking for the logic CTMTL is decidable. Both the extensions enjoy benefits of relaxing punctuality.

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Conclusion

Two ways of extending MTL with counting threshold constraints is studied. Both ways add expressiveness to MTL orthogonally. Satisfiability checking for the logic CTMTL is decidable. Both the extensions enjoy benefits of relaxing punctuality. Unlike continuous semantics, pointwise semantics creates a zoo of sub-logics in the expressiveness hierarchy.

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Presentation Flow

Model : Timed Words Timed Logic with Counting : Syntax and Semantics Temporal Projections : Simple and Oversampled Expressiveness Relations with Counting Extensions Satisfiability Checking: Decidability Conclusion Future Work

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Future Work

Exploring complexity results for satisfiability checking of CTMTL. Extending logics with modulo counting and study the expressiveness and satisfiability checking for those extensions. Complete picture of expressiveness of these counting extensions with different versions of past operators. Study model checking and synthesis problems for these extensions.

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Thank You

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References

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