Labeled Natural Deduction for Modal and Temporal Logics A System for a Branching Temporal Logic Labeled Deduction Systems for Temporal Logics Marco Volpe Dipartimento di Informatica Universit` a degli Studi di Verona Italy Marco Volpe Labeled Deduction Systems for Temporal Logics
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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
The flow of time can be linear, branching, discrete, dense, with first/final point...
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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.
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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
⇒
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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
⇒
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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
∀ ≃
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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
∀ ≃
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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
∀ ≃
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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
∀ ≃
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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