Logic as language
TDDC05: Embedded Systems
• Propositional logic (sv. Satslogik)
• Defined in terms of syntax and
semantics
• Syntax defined over a given vocabulary
• A vocabulary is a set
Simulation and Verification
Lecture 4: Propositional logic
Σ = {A, B, C,...}
Simin Nadjm-Tehrani
of proposition symbols
Real-time Systems Laboratory
Department of Computer and Information Science
Embedded systems simulation and verification
Linköping university
18 pages
Spring term 2006
Embedded systems simulation and verification
Linköping university
Definition: formula
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The semantics
Let Σ be a vocabulary.
• Defined given an evaluation
• if A ∈ Σ then A is a formula
• if F is a formula then ¬F is a formula
• if F and G are formulas then (F∧G),
(F∨G), (F→G) and (F↔G) are formulas
• ⊥ is a formula
• (nothing else is a formula)
V: Σ → {0,1} , i.e. associates every
proposition symbol to false or true
• Formulas are interpreted via an
evaluation
Embedded systems simulation and verification
Linköping university
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• An evaluation (of Σ) is a function
Embedded systems simulation and verification
Linköping university
Definition: Interpretation
Valuation V can be extended over
formulas, and is then called an
interpretation:
•
•
•
•
•
V(⊥)=0, V(¬F)=1 iff V(F)=0
V(F∧G)=1 iff V(F)=1 and V(G)=1
V(F∨G)=1 iff V(F)=1 or V(G)=1
V(F→G)=1 iff V(F)=0 or V(G)=1
V(F↔G)=1 iff V(F)=V(G)
Embedded systems simulation and verification
Linköping university
Truth tables
• To show the interpretation of formulas
given each valuation:
F
0
0
1
1
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G
0
1
0
1
F→ G
1
1
0
1
Embedded systems simulation and verification
Linköping university
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Models
Let F be a formula and Γ a set of
formulas.
• An interpretation V such that V(F)=1 is
a model for F
• A model for Γ is an interpretation V such
that V is a model for each formula in Γ
• Mod(F) resp. Mod(Γ) denote the set of
all models for F resp. Γ
Embedded systems simulation and verification
Linköping university
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• How does one model a (static) world
using logic?
Embedded systems simulation and verification
Linköping university
Example
Let P1 ,…, P4 denote:
• P1: All students registered on the course are
present here
• P2: Logic is boring
• P3: Logic is fun
• P4: Logic is simple
• How can one describe this world?
• Which conclusions can be derived?
Embedded systems simulation and verification
Linköping university
Axiomatisation
(1)
(2)
(3)
(4)
¬P1 → ¬P3 ∨ P4
P3 → ¬ P 2
P4 → P 2
¬P1
Is P4 a valid conclusion?
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Embedded systems simulation and verification
Linköping university
• Tautology: if F is true in all
interpretations then F is a tautology
• Contradiction: if F is false in all
interpretations then F is a contradiction
• Satisfiable: if F is true in some
interpretation (has at least one model)
then F is satisfiable
• Falsifiable: if F is false in at least one
interpretation then F is falsifiable
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Extension to Γ
Properties of formulas
Embedded systems simulation and verification
Linköping university
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• Γ is satisfiable if Mod(Γ) ≠ ∅
• Γ is unsatisfiable if Γ has no models
• NOTE! Γ can be unsatisfiable even if
none of the formulas in Γ are
contradictions
Embedded systems simulation and verification
Linköping university
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Logical consequence
• F is a logical consequence of Γ if every
model for Γ is also a model for F, i.e.
Mod(Γ) ⊆ Mod(F)
• Denoted Γ
F
• F is a tautology denoted
F
• F and G are (logically) equivalent iff
Mod(F)=Mod(G)
• Denoted F ⇔ G
Embedded systems simulation and verification
Linköping university
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Theorems
•
•
•
•
•
Γ
F iff Γ ∪ {¬F} is unsatisfiable
Γ is unsatisfiable iff Γ ⊥
F is satisfiable iff ¬F is falsifiable
F G iff
F→G
F is a tautology iff ¬F is a
contradiction
Embedded systems simulation and verification
Linköping university
Example: Train crossing
Approaching
In-crossing
Leaving
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Modelling (1)
Let F denote ”Far”, that is, when train is
outside the crossing area.
• Train
Up
T1 ↔ ((F ∨ A) ∨ (I ∨ L))
T2 ↔ (¬(A ∧ I) ∧ ¬(A ∧ L) ∧ ¬(I ∧ L) ∧
¬(F ∧ I) ∧ ¬(F ∧ L) ∧ ¬(A ∧ F) )
Down
• Gate
B1 ↔ (U ∨ D)
B2 ↔ ¬ (U ∧ D)
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Linköping university
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Linköping university
Modelling (2)
Analysis
• Show that:
• Control system:
S1 ↔ ((A ∨ I) → D)
S2 ↔ ((F ∨ L) → U)
((T1 ∧ T2) ∧ (B1 ∧ B2)) ∧ (S1 ∧ S2) → K
• That is
((T1 ∧ T2) ∧ (B1 ∧ B2)) ∧ (S1 ∧ S2)
K
• That is
Mod(((T1 ∧ T2) ∧ (B1 ∧ B2)) ∧ (S1 ∧ S2))
⊆ Mod(K)
• Safety property:
K ↔ ¬(I ∧ U)
Embedded systems simulation and verification
Linköping university
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