labeled deduction systems for temporal...
TRANSCRIPT
Labeled Natural Deduction for Modal and Temporal LogicsA System for a Branching Temporal Logic
Labeled Deduction Systems for Temporal Logics
Marco Volpe
Dipartimento di InformaticaUniversita degli Studi di Verona
Italy
Marco Volpe Labeled Deduction Systems for Temporal Logics
Labeled Natural Deduction for Modal and Temporal LogicsA System for a Branching Temporal Logic
Outline
1 Labeled Natural Deduction for Modal and Temporal LogicsNatural DeductionModal LogicLabeled Natural DeductionTemporal Logic
2 A System for a Branching Temporal LogicThe Logic CTL∗
The Logic BCTL∗−The System
Marco Volpe Labeled Deduction Systems for Temporal Logics
Labeled Natural Deduction for Modal and Temporal LogicsA System for a Branching Temporal Logic
Natural DeductionModal LogicLabeled Natural DeductionTemporal Logic
A Map
Marco Volpe Labeled Deduction Systems for Temporal Logics
Labeled Natural Deduction for Modal and Temporal LogicsA System for a Branching Temporal Logic
Natural DeductionModal LogicLabeled Natural DeductionTemporal Logic
A Map
Marco Volpe Labeled Deduction Systems for Temporal Logics
Labeled Natural Deduction for Modal and Temporal LogicsA System for a Branching Temporal Logic
Natural DeductionModal LogicLabeled Natural DeductionTemporal Logic
A Map
Marco Volpe Labeled Deduction Systems for Temporal Logics
Labeled Natural Deduction for Modal and Temporal LogicsA System for a Branching Temporal Logic
Natural DeductionModal LogicLabeled Natural DeductionTemporal Logic
Natural Deduction: Rules.
Proposed by Gentzen as a “natural” formalization of thehuman process of making proofs.
Rules for introduction/elimination of connectives.
Possibility of reasoning under assumptions.
Example (Propositional Classical Logic)
[α ⊃⊥]...
⊥(⊥E)
α
[α]...
β(⊃I )
α ⊃ β
α ⊃ β α(⊃E )
β
Marco Volpe Labeled Deduction Systems for Temporal Logics
Labeled Natural Deduction for Modal and Temporal LogicsA System for a Branching Temporal Logic
Natural DeductionModal LogicLabeled Natural DeductionTemporal Logic
Natural Deduction: Derivations.
A derivation has a tree-like structure:
the open assumptions are the leaves;the conclusion is the root.
Example (Composition of derivations)
Γ1
π1
α1 ⊃ α2
Γ2
π2
α1(⊃E )
α2
by usingα ⊃ β α
(⊃E )β
Marco Volpe Labeled Deduction Systems for Temporal Logics
Labeled Natural Deduction for Modal and Temporal LogicsA System for a Branching Temporal Logic
Natural DeductionModal LogicLabeled Natural DeductionTemporal Logic
Natural Deduction: Derivations.
Example (An example of derivation)
[α ⊃ (β ⊃ γ)]1
[α ∧ β]2
(∧E )α
(⊃E )
β ⊃ γ
[α ∧ β]2
(∧E )
β(⊃E )
γ(⊃I )2
α ∧ β ⊃ γ(⊃I )1
(α ⊃ (β ⊃ γ)) ⊃ (α ∧ β ⊃ γ)
Marco Volpe Labeled Deduction Systems for Temporal Logics
Labeled Natural Deduction for Modal and Temporal LogicsA System for a Branching Temporal Logic
Natural DeductionModal LogicLabeled Natural DeductionTemporal Logic
A Map
Marco Volpe Labeled Deduction Systems for Temporal Logics
Labeled Natural Deduction for Modal and Temporal LogicsA System for a Branching Temporal Logic
Natural DeductionModal LogicLabeled Natural DeductionTemporal Logic
A Map
Marco Volpe Labeled Deduction Systems for Temporal Logics
Labeled Natural Deduction for Modal and Temporal LogicsA System for a Branching Temporal Logic
Natural DeductionModal LogicLabeled Natural DeductionTemporal Logic
Modal Logic
The true/false approach used in classical logic is substitutedby the notions of necessarily true and possibly true.
The syntax is enriched with two modal operators: � and ♦.
Marco Volpe Labeled Deduction Systems for Temporal Logics
Labeled Natural Deduction for Modal and Temporal LogicsA System for a Branching Temporal Logic
Natural DeductionModal LogicLabeled Natural DeductionTemporal Logic
Modal Logic
The true/false approach used in classical logic is substitutedby the notions of necessarily true and possibly true.
The syntax is enriched with two modal operators: � and ♦.
Syntax
α ::= p | ⊥ | α ⊃ α
where p is an atomic proposition.
Marco Volpe Labeled Deduction Systems for Temporal Logics
Labeled Natural Deduction for Modal and Temporal LogicsA System for a Branching Temporal Logic
Natural DeductionModal LogicLabeled Natural DeductionTemporal Logic
Modal Logic
The true/false approach used in classical logic is substitutedby the notions of necessarily true and possibly true.
The syntax is enriched with two modal operators: � and ♦.
Syntax
α ::= p | ⊥ | α ⊃ α | �α | ♦α ,
where p is an atomic proposition.
Marco Volpe Labeled Deduction Systems for Temporal Logics
Labeled Natural Deduction for Modal and Temporal LogicsA System for a Branching Temporal Logic
Natural DeductionModal LogicLabeled Natural DeductionTemporal Logic
Modal Logic
A modal logic is interpreted within Kripke Models.
Marco Volpe Labeled Deduction Systems for Temporal Logics
Labeled Natural Deduction for Modal and Temporal LogicsA System for a Branching Temporal Logic
Natural DeductionModal LogicLabeled Natural DeductionTemporal Logic
Modal Logic
A modal logic is interpreted within Kripke Models.A frame is a pair (W,R) where:
W is a non-empty set of worlds;R is a binary relation on W .
Marco Volpe Labeled Deduction Systems for Temporal Logics
Labeled Natural Deduction for Modal and Temporal LogicsA System for a Branching Temporal Logic
Natural DeductionModal LogicLabeled Natural DeductionTemporal Logic
Modal Logic
A modal logic is interpreted within Kripke Models.A frame is a pair (W,R) where:
W is a non-empty set of worlds;R is a binary relation on W .
A model is a triple M = (W,R,V) where:(W ,R) is a frame;V is a function mapping worlds and atoms to truth values.
Marco Volpe Labeled Deduction Systems for Temporal Logics
Labeled Natural Deduction for Modal and Temporal LogicsA System for a Branching Temporal Logic
Natural DeductionModal LogicLabeled Natural DeductionTemporal Logic
Modal Logic
A modal logic is interpreted within Kripke Models.A frame is a pair (W,R) where:
W is a non-empty set of worlds;R is a binary relation on W .
A model is a triple M = (W,R,V) where:(W ,R) is a frame;V is a function mapping worlds and atoms to truth values.
Marco Volpe Labeled Deduction Systems for Temporal Logics
Labeled Natural Deduction for Modal and Temporal LogicsA System for a Branching Temporal Logic
Natural DeductionModal LogicLabeled Natural DeductionTemporal Logic
Modal Logic
Semantics
M,w |= p iff V(w , p) = 1;M,w |= α ⊃ β iff M,w |= α implies M,w |= β ;M,w |= �α iff ∀w ′ ∈ W, wRw ′ implies M,w ′ |= α;M,w |= ♦α iff ∃w ′ ∈ W such that wRw ′ and M,w ′ |= α.
Marco Volpe Labeled Deduction Systems for Temporal Logics
Labeled Natural Deduction for Modal and Temporal LogicsA System for a Branching Temporal Logic
Natural DeductionModal LogicLabeled Natural DeductionTemporal Logic
Modal Logic
Semantics
M,w |= p iff V(w , p) = 1;M,w |= α ⊃ β iff M,w |= α implies M,w |= β ;M,w |= �α iff ∀w ′ ∈ W, wRw ′ implies M,w ′ |= α;M,w |= ♦α iff ∃w ′ ∈ W such that wRw ′ and M,w ′ |= α.
An example
M,w � �p
M,w 2 �p′
Marco Volpe Labeled Deduction Systems for Temporal Logics
Labeled Natural Deduction for Modal and Temporal LogicsA System for a Branching Temporal Logic
Natural DeductionModal LogicLabeled Natural DeductionTemporal Logic
Modal Logic
Most common modal logics are frame logics, i.e. valid formulas are
those formulas which are true at every world of every model based on a frame
with a given property .Different modal logics can be obtained by placing conditionson the accessibility relation of the frames.
Marco Volpe Labeled Deduction Systems for Temporal Logics
Labeled Natural Deduction for Modal and Temporal LogicsA System for a Branching Temporal Logic
Natural DeductionModal LogicLabeled Natural DeductionTemporal Logic
Modal Logic
Most common modal logics are frame logics, i.e. valid formulas are
those formulas which are true at every world of every model based on a frame
with a given property .Different modal logics can be obtained by placing conditionson the accessibility relation of the frames.
Example (The logic K4)
The logic K4 is defined by the class of all transitive frames.
Marco Volpe Labeled Deduction Systems for Temporal Logics
Labeled Natural Deduction for Modal and Temporal LogicsA System for a Branching Temporal Logic
Natural DeductionModal LogicLabeled Natural DeductionTemporal Logic
Modal Logic
Most common modal logics are frame logics, i.e. valid formulas are
those formulas which are true at every world of every model based on a frame
with a given property .Different modal logics can be obtained by placing conditionson the accessibility relation of the frames.
Example (The logic K4)
The logic K4 is defined by the class of all transitive frames.
Marco Volpe Labeled Deduction Systems for Temporal Logics
Labeled Natural Deduction for Modal and Temporal LogicsA System for a Branching Temporal Logic
Natural DeductionModal LogicLabeled Natural DeductionTemporal Logic
A Map
Marco Volpe Labeled Deduction Systems for Temporal Logics
Labeled Natural Deduction for Modal and Temporal LogicsA System for a Branching Temporal Logic
Natural DeductionModal LogicLabeled Natural DeductionTemporal Logic
A Map
Marco Volpe Labeled Deduction Systems for Temporal Logics
Labeled Natural Deduction for Modal and Temporal LogicsA System for a Branching Temporal Logic
Natural DeductionModal LogicLabeled Natural DeductionTemporal Logic
Labeled deduction systems
The application of standard deduction systems, e.g. naturaldeduction, to non-classical logics is not straightforward.
Labeled deduction approach: we encode in the syntaxadditional information (e.g. of a semantic nature).
Two classes of formulas:1 Labeled formulas, e.g. b : α2 Relational formulas, e.g. bRc
The basic idea is:
each label c refers to a world c in the semanticseach relational symbol R refers to a semantic relation R
From standard to labeled logics
M |=lab b : α iff M, b |= α
Marco Volpe Labeled Deduction Systems for Temporal Logics
Labeled Natural Deduction for Modal and Temporal LogicsA System for a Branching Temporal Logic
Natural DeductionModal LogicLabeled Natural DeductionTemporal Logic
Labeled deduction systems
Rules for classical connectives
[ α ⊃⊥]....⊥α
⊥E
[ α]....β
α ⊃ β⊃I
α ⊃ β α
β⊃E
Marco Volpe Labeled Deduction Systems for Temporal Logics
Labeled Natural Deduction for Modal and Temporal LogicsA System for a Branching Temporal Logic
Natural DeductionModal LogicLabeled Natural DeductionTemporal Logic
Labeled deduction systems
Rules for classical connectives
[b : α ⊃⊥]....
c : ⊥b : α
⊥E
[b : α]....
b : β
b : α ⊃ β⊃I
b : α ⊃ β b : α
b : β⊃E
Marco Volpe Labeled Deduction Systems for Temporal Logics
Labeled Natural Deduction for Modal and Temporal LogicsA System for a Branching Temporal Logic
Natural DeductionModal LogicLabeled Natural DeductionTemporal Logic
Labeled deduction systems
Rules for classical connectives
[b : α ⊃⊥]....
c : ⊥b : α
⊥E
[b : α]....
b : β
b : α ⊃ β⊃I
b : α ⊃ β b : α
b : β⊃E
An example of a relational rule
b1Rb2 b2Rb3
[b1Rb3]....b : α
b : αtransR
Marco Volpe Labeled Deduction Systems for Temporal Logics
Labeled Natural Deduction for Modal and Temporal LogicsA System for a Branching Temporal Logic
Natural DeductionModal LogicLabeled Natural DeductionTemporal Logic
Labeled deduction systems for modal logics
Labeling provides a clean way of dealing with modalities and givesrise to systems with good meta and proof-theoretical properties.
Semantics
M, b |= �α iff for all c . bRc implies M, c |= α.
Rules for modal operators
[bRc]....
c : αb : �α
�I ∗ b : �α bRcc : α �E
*In �I , the label c is fresh.
Marco Volpe Labeled Deduction Systems for Temporal Logics
Labeled Natural Deduction for Modal and Temporal LogicsA System for a Branching Temporal Logic
Natural DeductionModal LogicLabeled Natural DeductionTemporal Logic
A Map
Marco Volpe Labeled Deduction Systems for Temporal Logics
Labeled Natural Deduction for Modal and Temporal LogicsA System for a Branching Temporal Logic
Natural DeductionModal LogicLabeled Natural DeductionTemporal Logic
A Map
Marco Volpe Labeled Deduction Systems for Temporal Logics
Labeled Natural Deduction for Modal and Temporal LogicsA System for a Branching Temporal Logic
Natural DeductionModal LogicLabeled Natural DeductionTemporal Logic
Temporal Logic
Temporal logics can be seen as a branch of modal logic:
worlds in a frame are seen as time instants;the accessibility relation is used to model the flow of time.
We can define several temporal logics by:1 modifying the properties of the accessibility relation;2 introducing new modal (temporal) operators.
Marco Volpe Labeled Deduction Systems for Temporal Logics
Labeled Natural Deduction for Modal and Temporal LogicsA System for a Branching Temporal Logic
Natural DeductionModal LogicLabeled Natural DeductionTemporal Logic
Linear vs. Branching time
Semantics
The flow of time can be linear, branching, discrete, dense, with first/final point...
•
•
OO
•
OO
•
OO
•
OO
•
•
OO
•
• •
[[888CC���
•
•
OO
•
OO
• •
OO
•
•
eeJJJJJOO 99ttttt
•
[[888CC���
•
eeJJJJJ77nnnnnnn
Marco Volpe Labeled Deduction Systems for Temporal Logics
Labeled Natural Deduction for Modal and Temporal LogicsA System for a Branching Temporal Logic
Natural DeductionModal LogicLabeled Natural DeductionTemporal Logic
Linear vs. Branching time
Semantics
The flow of time can be linear, branching, discrete, dense, with first/final point...
•
•
OO
•
OO
•
OO
•
OO
•
•
OO
•
• •
[[888CC���
•
•
OO
•
OO
• •
OO
•
•
eeJJJJJOO 99ttttt
•
[[888CC���
•
eeJJJJJ77nnnnnnn
Syntax
α ::= p | ⊥ | α ⊃ α
Marco Volpe Labeled Deduction Systems for Temporal Logics
Labeled Natural Deduction for Modal and Temporal LogicsA System for a Branching Temporal Logic
Natural DeductionModal LogicLabeled Natural DeductionTemporal Logic
Linear vs. Branching time
Semantics
The flow of time can be linear, branching, discrete, dense, with first/final point...
•
•
OO
•
OO
•
OO
•
OO
•
•
OO
•
• •
[[888CC���
•
•
OO
•
OO
• •
OO
•
•
eeJJJJJOO 99ttttt
•
[[888CC���
•
eeJJJJJ77nnnnnnn
Syntax
α ::= p | ⊥ | α ⊃ α | Xα
Marco Volpe Labeled Deduction Systems for Temporal Logics
Labeled Natural Deduction for Modal and Temporal LogicsA System for a Branching Temporal Logic
Natural DeductionModal LogicLabeled Natural DeductionTemporal Logic
Linear vs. Branching time
Semantics
The flow of time can be linear, branching, discrete, dense, with first/final point...
•
•
OO
•
OO
•
OO
•
OO
•
•
OO
•
• •
[[888CC���
•
•
OO
•
OO
• •
OO
•
•
eeJJJJJOO 99ttttt
•
[[888CC���
•
eeJJJJJ77nnnnnnn
Syntax
α ::= p | ⊥ | α ⊃ α | Xα | Gα
Marco Volpe Labeled Deduction Systems for Temporal Logics
Labeled Natural Deduction for Modal and Temporal LogicsA System for a Branching Temporal Logic
Natural DeductionModal LogicLabeled Natural DeductionTemporal Logic
Linear vs. Branching time
Semantics
The flow of time can be linear, branching, discrete, dense, with first/final point...
•
•
OO
•
OO
•
OO
•
OO
•
•
OO
•
• •
[[888CC���
•
•
OO
•
OO
• •
OO
•
•
eeJJJJJOO 99ttttt
•
[[888CC���
•
eeJJJJJ77nnnnnnn
Syntax
α ::= p | ⊥ | α ⊃ α | Xα | Gα | αUα
Marco Volpe Labeled Deduction Systems for Temporal Logics
Labeled Natural Deduction for Modal and Temporal LogicsA System for a Branching Temporal Logic
Natural DeductionModal LogicLabeled Natural DeductionTemporal Logic
Linear vs. Branching time
Semantics
The flow of time can be linear, branching, discrete, dense, with first/final point...
•
•
OO
•
OO
•
OO
•
OO
•
•
OO
•
• •
[[888CC���
•
•
OO
•
OO
• •
OO
•
•
eeJJJJJOO 99ttttt
•
[[888CC���
•
eeJJJJJ77nnnnnnn
Syntax
α ::= p | ⊥ | α ⊃ α | Xα | Gα | αUα | ∀α
Marco Volpe Labeled Deduction Systems for Temporal Logics
Labeled Natural Deduction for Modal and Temporal LogicsA System for a Branching Temporal Logic
The Logic CTL∗
The Logic BCTL∗−
The System
Outline
1 Labeled Natural Deduction for Modal and Temporal LogicsNatural DeductionModal LogicLabeled Natural DeductionTemporal Logic
2 A System for a Branching Temporal LogicThe Logic CTL∗
The Logic BCTL∗−The System
Marco Volpe Labeled Deduction Systems for Temporal Logics
Labeled Natural Deduction for Modal and Temporal LogicsA System for a Branching Temporal Logic
The Logic CTL∗
The Logic BCTL∗−
The System
Overview
The Motivations
Branching-time logics are of great relevance in computer science(specification and verification), but there are still many open issues.
The Aim
Define a deduction system for (a fragment of) CTL∗ with goodmeta-theoretical and proof-theoretical properties.
The Instrument
Labeled deduction: an approach to deduction succesfully applied tomany modal (and in general non-classical) logics.
Marco Volpe Labeled Deduction Systems for Temporal Logics
Labeled Natural Deduction for Modal and Temporal LogicsA System for a Branching Temporal Logic
The Logic CTL∗
The Logic BCTL∗−
The System
Outline
1 Labeled Natural Deduction for Modal and Temporal LogicsNatural DeductionModal LogicLabeled Natural DeductionTemporal Logic
2 A System for a Branching Temporal LogicThe Logic CTL∗
The Logic BCTL∗−The System
Marco Volpe Labeled Deduction Systems for Temporal Logics
Labeled Natural Deduction for Modal and Temporal LogicsA System for a Branching Temporal Logic
The Logic CTL∗
The Logic BCTL∗−
The System
The logic CTL∗
Syntax
α ::= p | ⊥ | α ⊃ α | Xα | Gα | αUα | ∀α.
Marco Volpe Labeled Deduction Systems for Temporal Logics
Labeled Natural Deduction for Modal and Temporal LogicsA System for a Branching Temporal Logic
The Logic CTL∗
The Logic BCTL∗−
The System
The logic CTL∗
Syntax
α ::= p | ⊥ | α ⊃ α | Xα | Gα | αUα | ∀α.
Semantics
Defined on frames (S,Π) where:
1 S is a set of states;
2 Π is a set of paths, i.e. of ω-sequences, over S.
Marco Volpe Labeled Deduction Systems for Temporal Logics
Labeled Natural Deduction for Modal and Temporal LogicsA System for a Branching Temporal Logic
The Logic CTL∗
The Logic BCTL∗−
The System
The logic CTL∗
Syntax
α ::= p | ⊥ | α ⊃ α | Xα | Gα | αUα | ∀α.
Semantics
Defined on frames (S,Π) where the set Π of paths over S satisfythe following constraints:
Marco Volpe Labeled Deduction Systems for Temporal Logics
Labeled Natural Deduction for Modal and Temporal LogicsA System for a Branching Temporal Logic
The Logic CTL∗
The Logic BCTL∗−
The System
The logic CTL∗
Syntax
α ::= p | ⊥ | α ⊃ α | Xα | Gα | αUα | ∀α.
Semantics
Defined on frames (S,Π) where the set Π of paths over S satisfythe following constraints:
1 suffix-closure: every suffix of a path is itself a path
Marco Volpe Labeled Deduction Systems for Temporal Logics
Labeled Natural Deduction for Modal and Temporal LogicsA System for a Branching Temporal Logic
The Logic CTL∗
The Logic BCTL∗−
The System
The suffix-closure property
Every suffix of a path is itself a path.
•
•
OO
•
•
XX111
• •
XX111
Marco Volpe Labeled Deduction Systems for Temporal Logics
Labeled Natural Deduction for Modal and Temporal LogicsA System for a Branching Temporal Logic
The Logic CTL∗
The Logic BCTL∗−
The System
The suffix-closure property
Every suffix of a path is itself a path.
•
•
OO
•
•
XX111
• •
Marco Volpe Labeled Deduction Systems for Temporal Logics
Labeled Natural Deduction for Modal and Temporal LogicsA System for a Branching Temporal Logic
The Logic CTL∗
The Logic BCTL∗−
The System
The suffix-closure property
Every suffix of a path is itself a path.
•
•
OO
•
•
• •
Marco Volpe Labeled Deduction Systems for Temporal Logics
Labeled Natural Deduction for Modal and Temporal LogicsA System for a Branching Temporal Logic
The Logic CTL∗
The Logic BCTL∗−
The System
The logic CTL∗
Syntax
α ::= p | ⊥ | α ⊃ α | αUα | Gα | Xα | ∀α.
Semantics
Defined on frames (S,Π) where the set Π of paths over S satisfythe following constraints:
1 suffix-closure: every suffix of a path is itself a path
2 fusion-closure: we can always put together a finite prefix ofone path with the suffix of any other path such that the prefixends at the same state as the suffix begins
Marco Volpe Labeled Deduction Systems for Temporal Logics
Labeled Natural Deduction for Modal and Temporal LogicsA System for a Branching Temporal Logic
The Logic CTL∗
The Logic BCTL∗−
The System
The fusion-closure property
We can always put together a finite prefix of one path with thesuffix of any other path such that the prefix ends at the same stateas the suffix begins.
•
•
OO
•
•
XX111
• •
XX111
Marco Volpe Labeled Deduction Systems for Temporal Logics
Labeled Natural Deduction for Modal and Temporal LogicsA System for a Branching Temporal Logic
The Logic CTL∗
The Logic BCTL∗−
The System
The fusion-closure property
We can always put together a finite prefix of one path with thesuffix of any other path such that the prefix ends at the same stateas the suffix begins.
•
• •
•
FF
•
FF •
Marco Volpe Labeled Deduction Systems for Temporal Logics
Labeled Natural Deduction for Modal and Temporal LogicsA System for a Branching Temporal Logic
The Logic CTL∗
The Logic BCTL∗−
The System
The fusion-closure property
We can always put together a finite prefix of one path with thesuffix of any other path such that the prefix ends at the same stateas the suffix begins.
•
•
OO
•
•
XX111
•
FF •
Marco Volpe Labeled Deduction Systems for Temporal Logics
Labeled Natural Deduction for Modal and Temporal LogicsA System for a Branching Temporal Logic
The Logic CTL∗
The Logic BCTL∗−
The System
The fusion-closure property
We can always put together a finite prefix of one path with thesuffix of any other path such that the prefix ends at the same stateas the suffix begins.
•
• •
•
FF
• •
XX111
Marco Volpe Labeled Deduction Systems for Temporal Logics
Labeled Natural Deduction for Modal and Temporal LogicsA System for a Branching Temporal Logic
The Logic CTL∗
The Logic BCTL∗−
The System
The logic CTL∗
Syntax
α ::= p | ⊥ | α ⊃ α | Xα | Gα | αUα | ∀α.
Semantics
Defined on frames (S,Π) where the set Π of paths over S satisfythe following constraints:
1 suffix-closure: every suffix of a path is itself a path
2 fusion-closure: we can always put together a finite prefix ofone path with the suffix of any other path such that the prefixends at the same state as the suffix begins
3 limit-closure: if every finite prefix of a path σ is a prefix ofsome path, then σ itself is a path
Marco Volpe Labeled Deduction Systems for Temporal Logics
Labeled Natural Deduction for Modal and Temporal LogicsA System for a Branching Temporal Logic
The Logic CTL∗
The Logic BCTL∗−
The System
The limit-closure property
If every finite prefix of a path σ is a prefix of some path, then σitself is a path.
•
OO
• •
•
OO
• •
Marco Volpe Labeled Deduction Systems for Temporal Logics
Labeled Natural Deduction for Modal and Temporal LogicsA System for a Branching Temporal Logic
The Logic CTL∗
The Logic BCTL∗−
The System
The limit-closure property
If every finite prefix of a path σ is a prefix of some path, then σitself is a path.
• •
OO
•
• // •
OO
•
Marco Volpe Labeled Deduction Systems for Temporal Logics
Labeled Natural Deduction for Modal and Temporal LogicsA System for a Branching Temporal Logic
The Logic CTL∗
The Logic BCTL∗−
The System
The limit-closure property
If every finite prefix of a path σ is a prefix of some path, then σitself is a path.
• • •
OO
• // • // •
OO
Marco Volpe Labeled Deduction Systems for Temporal Logics
Labeled Natural Deduction for Modal and Temporal LogicsA System for a Branching Temporal Logic
The Logic CTL∗
The Logic BCTL∗−
The System
The limit-closure property
If every finite prefix of a path σ is a prefix of some path, then σitself is a path.
• • •
• // • // • //
Marco Volpe Labeled Deduction Systems for Temporal Logics
Labeled Natural Deduction for Modal and Temporal LogicsA System for a Branching Temporal Logic
The Logic CTL∗
The Logic BCTL∗−
The System
Outline
1 Labeled Natural Deduction for Modal and Temporal LogicsNatural DeductionModal LogicLabeled Natural DeductionTemporal Logic
2 A System for a Branching Temporal LogicThe Logic CTL∗
The Logic BCTL∗−The System
Marco Volpe Labeled Deduction Systems for Temporal Logics
Labeled Natural Deduction for Modal and Temporal LogicsA System for a Branching Temporal Logic
The Logic CTL∗
The Logic BCTL∗−
The System
The logic CTL∗
Syntax
α ::= p | ⊥ | α ⊃ α | Xα | Gα | αUα | ∀α.
Semantics
Defined on frames (S,Π) where the set Π of paths over S satisfythe following constraints:
1 suffix-closure: every suffix of a path is itself a path
2 fusion-closure: we can always put together a finite prefix ofone path with the suffix of any other path such that the prefixends at the same state as the suffix begins
3 limit-closure: if every finite prefix of a path σ is a prefix ofsome path, then σ itself is a path
Marco Volpe Labeled Deduction Systems for Temporal Logics
Labeled Natural Deduction for Modal and Temporal LogicsA System for a Branching Temporal Logic
The Logic CTL∗
The Logic BCTL∗−
The System
The logic CTL∗−
Syntax
α ::= p | ⊥ | α ⊃ α | Xα | Gα | ∀α.
Semantics
Defined on frames (S,Π) where the set Π of paths over S satisfythe following constraints:
1 suffix-closure: every suffix of a path is itself a path
2 fusion-closure: we can always put together a finite prefix ofone path with the suffix of any other path such that the prefixends at the same state as the suffix begins
3 limit-closure: if every finite prefix of a path σ is a prefix ofsome path, then σ itself is a path
Marco Volpe Labeled Deduction Systems for Temporal Logics
Labeled Natural Deduction for Modal and Temporal LogicsA System for a Branching Temporal Logic
The Logic CTL∗
The Logic BCTL∗−
The System
The logic BCTL∗−
Syntax
α ::= p | ⊥ | α ⊃ α | Xα | Gα | ∀α.
Semantics
Defined on frames (S,Π) where the set Π of paths over S satisfythe following constraints:
1 suffix-closure: every suffix of a path is itself a path
2 fusion-closure: we can always put together a finite prefix ofone path with the suffix of any other path such that the prefixends at the same state as the suffix begins
Marco Volpe Labeled Deduction Systems for Temporal Logics
Labeled Natural Deduction for Modal and Temporal LogicsA System for a Branching Temporal Logic
The Logic CTL∗
The Logic BCTL∗−
The System
The logic BCTL∗−
Marco Volpe Labeled Deduction Systems for Temporal Logics
Labeled Natural Deduction for Modal and Temporal LogicsA System for a Branching Temporal Logic
The Logic CTL∗
The Logic BCTL∗−
The System
The logic BCTL∗−
Marco Volpe Labeled Deduction Systems for Temporal Logics
Labeled Natural Deduction for Modal and Temporal LogicsA System for a Branching Temporal Logic
The Logic CTL∗
The Logic BCTL∗−
The System
The logic BCTL∗−
Marco Volpe Labeled Deduction Systems for Temporal Logics
Labeled Natural Deduction for Modal and Temporal LogicsA System for a Branching Temporal Logic
The Logic CTL∗
The Logic BCTL∗−
The System
The logic BCTL∗−
Applications: useful when only some of the paths are countedas legitimate computations (e.g. fairness constraints).
Used as a simpler variant on which to work towards CTL∗.
Marco Volpe Labeled Deduction Systems for Temporal Logics
Labeled Natural Deduction for Modal and Temporal LogicsA System for a Branching Temporal Logic
The Logic CTL∗
The Logic BCTL∗−
The System
An equivalent semantical characterization
Highlight the modal nature of this logic: it is possible to give asemantics only in terms of paths.
•
•
OO
•
• •
XX111FF
•
•
OO
•
OO
• •
OO
•
•
``BBBBOO >>||||
•
XX111FF
•
``BBBB;;vvvvv
⇒
•
•
`
•
• •
`
≃•
`
•
•
`
•
`
≃•
`
• •
`
•
•
`
≃•
`
≃•
`
≃•
`
•
`
≃•
`
•
`
≃•
`
≃•
`
≃•
`
≃•
`
≃•
`
Marco Volpe Labeled Deduction Systems for Temporal Logics
Labeled Natural Deduction for Modal and Temporal LogicsA System for a Branching Temporal Logic
The Logic CTL∗
The Logic BCTL∗−
The System
An equivalent semantical characterization
Highlight the modal nature of this logic: it is possible to give asemantics only in terms of paths.
•
•
OO
•
• •
XX111FF
•
•
OO
•
OO
• •
OO
•
•
``BBBBOO >>||||
•
XX111FF
•
``BBBB;;vvvvv
⇒
•
•
`
•
• •
`
≃•
`
•
•
`
•
`
≃•
`
• •
`
•
•
`
≃•
`
≃•
`
≃•
`
•
`
≃•
`
•
`
≃•
`
≃•
`
≃•
`
≃•
`
≃•
`
Marco Volpe Labeled Deduction Systems for Temporal Logics
Labeled Natural Deduction for Modal and Temporal LogicsA System for a Branching Temporal Logic
The Logic CTL∗
The Logic BCTL∗−
The System
An equivalent semantical characterization
Highlight the modal nature of this logic: it is possible to give asemantics only in terms of paths.
•
•
OO
•
• •
XX111FF
•
•
OO
•
OO
• •
OO
•
•
``BBBBOO >>||||
•
XX111FF
•
``BBBB;;vvvvv
⇒
•
•
`
•
• •
`
≃•
`
•
•
`
•
`
≃•
`
• •
`
•
•
`
≃•
`
≃•
`
≃•
`
•
`
≃•
`
•
`
≃•
`
≃•
`
≃•
`
≃•
`
≃•
`
Also the path quantifier can be considered as a pure modaloperator with respect to the equivalence relation ≃.
Marco Volpe Labeled Deduction Systems for Temporal Logics
Labeled Natural Deduction for Modal and Temporal LogicsA System for a Branching Temporal Logic
The Logic CTL∗
The Logic BCTL∗−
The System
Outline
1 Labeled Natural Deduction for Modal and Temporal LogicsNatural DeductionModal LogicLabeled Natural DeductionTemporal Logic
2 A System for a Branching Temporal LogicThe Logic CTL∗
The Logic BCTL∗−The System
Marco Volpe Labeled Deduction Systems for Temporal Logics
Labeled Natural Deduction for Modal and Temporal LogicsA System for a Branching Temporal Logic
The Logic CTL∗
The Logic BCTL∗−
The System
Modularity of the system
Each operator is seen as a modal operator with:
a proper accessibility relation
proper relational rules
Operators and accessibility relations
operator relation
X ⊳
G 4
∀ ≃
•
•
`
•
• •
`
≃•
`
•
•
`
•
`
≃•
`
• •
`
•
•
`
≃•
`
≃•
`
≃•
`
•
`
≃•
`
•
`
≃•
`
≃•
`
≃•
`
≃•
`
≃•
`
Marco Volpe Labeled Deduction Systems for Temporal Logics
Labeled Natural Deduction for Modal and Temporal LogicsA System for a Branching Temporal Logic
The Logic CTL∗
The Logic BCTL∗−
The System
Modularity of the system
Each operator is seen as a modal operator with:
a proper accessibility relation
proper relational rules
Operators and accessibility relations
operator relation
X ⊳
G 4
∀ ≃
•
•
`
•
• •
`
≃•
`
•
•
`
•
`
≃•
`
• •
`
•
•
`
≃•
`
≃•
`
≃•
`
•
`
≃•
`
•
`
≃•
`
≃•
`
≃•
`
≃•
`
≃•
`
Marco Volpe Labeled Deduction Systems for Temporal Logics
Labeled Natural Deduction for Modal and Temporal LogicsA System for a Branching Temporal Logic
The Logic CTL∗
The Logic BCTL∗−
The System
Modularity of the system
Each operator is seen as a modal operator with:
a proper accessibility relation
proper relational rules
Operators and accessibility relations
operator relation
X ⊳
G 4
∀ ≃
•
•
`
•
• •
`
≃•
`
•
•
`
•
`
≃•
`
• •
`
•
•
`
≃•
`
≃•
`
≃•
`
•
`
≃•
`
•
`
≃•
`
≃•
`
≃•
`
≃•
`
≃•
`
Marco Volpe Labeled Deduction Systems for Temporal Logics
Labeled Natural Deduction for Modal and Temporal LogicsA System for a Branching Temporal Logic
The Logic CTL∗
The Logic BCTL∗−
The System
Modularity of the system
Each operator is seen as a modal operator with:
a proper accessibility relation
proper relational rules
Operators and accessibility relations
operator relation
X ⊳
G 4
∀ ≃
•
•
`
•
• •
`
≃•
`
•
•
`
•
`
≃•
`
• •
`
•
•
`
≃•
`
≃•
`
≃•
`
•
`
≃•
`
•
`
≃•
`
≃•
`
≃•
`
≃•
`
≃•
`
Marco Volpe Labeled Deduction Systems for Temporal Logics
Labeled Natural Deduction for Modal and Temporal LogicsA System for a Branching Temporal Logic
The Logic CTL∗
The Logic BCTL∗−
The System
The system for BCTL∗−
Marco Volpe Labeled Deduction Systems for Temporal Logics
Labeled Natural Deduction for Modal and Temporal LogicsA System for a Branching Temporal Logic
The Logic CTL∗
The Logic BCTL∗−
The System
The system for BCTL∗−
The system consists of:
1 rules for introduction/elimination of classical connectives
Marco Volpe Labeled Deduction Systems for Temporal Logics
Labeled Natural Deduction for Modal and Temporal LogicsA System for a Branching Temporal Logic
The Logic CTL∗
The Logic BCTL∗−
The System
Rules for classical connectives
[b :α ⊃⊥]....
c : ⊥b :α
⊥E
[b :α]....
b :β
b :α ⊃ β⊃I
b :α ⊃ β b :α
b :β⊃E
Marco Volpe Labeled Deduction Systems for Temporal Logics
Labeled Natural Deduction for Modal and Temporal LogicsA System for a Branching Temporal Logic
The Logic CTL∗
The Logic BCTL∗−
The System
The system for BCTL∗−
The system consists of:
1 rules for introduction/elimination of classical connectives
2 rules for introduction/elimination of temporal operators
Marco Volpe Labeled Deduction Systems for Temporal Logics
Labeled Natural Deduction for Modal and Temporal LogicsA System for a Branching Temporal Logic
The Logic CTL∗
The Logic BCTL∗−
The System
Rules for the temporal operators
Same pattern of introduction/elimination rules for all the operators.
Semantics
M, b |= Xα iff for all c . b ⊳ c implies M, c |= α.
Rules for the temporal operator X
[b ⊳ c]....
c : αb : Xα
XI ∗ b : Xα b ⊳ cc : α XE
*In XI , the label c is fresh.
Marco Volpe Labeled Deduction Systems for Temporal Logics
Labeled Natural Deduction for Modal and Temporal LogicsA System for a Branching Temporal Logic
The Logic CTL∗
The Logic BCTL∗−
The System
Rules for the temporal operators
Same pattern of introduction/elimination rules for all the operators.
Semantics
M, b |= Gα iff for all c . b 4 c implies M, c |= α.
Rules for the temporal operator G
[b 4 c]....
c : αb : Gα
GI ∗b : Gα b 4 c
c : α GE
*In GI , the label c is fresh.
Marco Volpe Labeled Deduction Systems for Temporal Logics
Labeled Natural Deduction for Modal and Temporal LogicsA System for a Branching Temporal Logic
The Logic CTL∗
The Logic BCTL∗−
The System
Rules for the temporal operators
Same pattern of introduction/elimination rules for all the operators.
Semantics
M, b |= ∀α iff for all c . b ≃ c implies M, c |= α.
Rules for the path quantifier ∀
[b ≃ c]....
c : αb : ∀α
∀I ∗ b : ∀α b ≃ cc : α ∀E
*In ∀I , the label c is fresh.
Marco Volpe Labeled Deduction Systems for Temporal Logics
Labeled Natural Deduction for Modal and Temporal LogicsA System for a Branching Temporal Logic
The Logic CTL∗
The Logic BCTL∗−
The System
The system for BCTL∗−
The system consists of:
1 rules for introduction/elimination of classical connectives
2 rules for introduction/elimination of temporal operators
3 rules modeling properties of accessibility relations
Marco Volpe Labeled Deduction Systems for Temporal Logics
Labeled Natural Deduction for Modal and Temporal LogicsA System for a Branching Temporal Logic
The Logic CTL∗
The Logic BCTL∗−
The System
Relational rules
Properties of accessibility relations are modeled by means ofrelational rules.
Example: rules concerning the equivalence relation ≃
[b1 ≃ b1]....b : αb : α
refl ≃b1 ≃ b2
[b2 ≃ b1]....b : α
b : αsymm ≃
b1 ≃ b2 b2 ≃ b3
[b1 ≃ b3]....b : α
b : αtrans ≃
Marco Volpe Labeled Deduction Systems for Temporal Logics
Labeled Natural Deduction for Modal and Temporal LogicsA System for a Branching Temporal Logic
The Logic CTL∗
The Logic BCTL∗−
The System
The system for BCTL∗−
The system consists of:
1 rules for introduction/elimination of classical connectives
2 rules for introduction/elimination of temporal operators
3 rules modeling properties of accessibility relations
4 rules modeling interactions between the operators
Marco Volpe Labeled Deduction Systems for Temporal Logics
Labeled Natural Deduction for Modal and Temporal LogicsA System for a Branching Temporal Logic
The Logic CTL∗
The Logic BCTL∗−
The System
Interaction rules
Interactions between the operators are modeled by rules that donot involve the operators themselves directly.
Example: a rule modeling the induction principle
b0 : α b0 4 b
[b0 4 bi ] [bi ⊳ bj ] [bi : α]....
bj : α
b : αind
Marco Volpe Labeled Deduction Systems for Temporal Logics
Labeled Natural Deduction for Modal and Temporal LogicsA System for a Branching Temporal Logic
The Logic CTL∗
The Logic BCTL∗−
The System
Interaction rules
Interactions between the operators are modeled by rules that donot involve the operators themselves directly.
Example: a rule modeling the fusion-closure property
Fusion-Closure: we can always put together a finite prefix of one path
with the suffix of any other path such that the prefix ends at the same
state as the suffix begins.
b1 ⊳ b2 b2 ≃ b3
[b′ ≃ b1] [b′ ⊳ b3]....b : α
b : αfusion
Marco Volpe Labeled Deduction Systems for Temporal Logics
Labeled Natural Deduction for Modal and Temporal LogicsA System for a Branching Temporal Logic
The Logic CTL∗
The Logic BCTL∗−
The System
Soundness and completeness
Theorem
Our system for Kl is sound: Γ ⊢ b : α ⇒ Γ |= b : αand (weakly) complete: |= b : α ⇒ ⊢ b : α.
Proof of soundness is standard (for labeled systems).
Completeness can be proved
either by deriving all the axioms and rules of a Hilbert-styleaxiomatization for the same logic,or by a Lindenbaum-Henkin style construction.
Marco Volpe Labeled Deduction Systems for Temporal Logics
Labeled Natural Deduction for Modal and Temporal LogicsA System for a Branching Temporal Logic
The Logic CTL∗
The Logic BCTL∗−
The System
Normalization
We have also shown a form of normalization for our system.
The main difficulty is given by the induction rule modeling thetemporal induction principle (relating the operators X and G).
The procedure is inspired by those for
other labeled systemsnatural deduction systems for Heyting Arithmetic(Prawitz, Troelstra, Girard).
Standard subformula property cannot hold.
Marco Volpe Labeled Deduction Systems for Temporal Logics
Labeled Natural Deduction for Modal and Temporal LogicsA System for a Branching Temporal Logic
The Logic CTL∗
The Logic BCTL∗−
The System
grazie!
Marco Volpe Labeled Deduction Systems for Temporal Logics