Propositional Logic Sequent Calculus
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1 1 / 16 Propositional Logic Sequent Calculus Mario Alviano University of Calabria, Italy A.Y. 2017/2018
2 Outline 2 / 16 1 Intuition 2 The LK system 3 Derivation 4 Summary 5 Exercises
3 Outline 3 / 16 1 Intuition 2 The LK system 3 Derivation 4 Summary 5 Exercises
4 Intuition 4 / 16 Idea Define inference rules for sequents Γ where Γ and are sequences of formulas
5 Intuition 4 / 16 Idea Define inference rules for sequents Γ where Γ and are sequences of formulas Intuition Γ is the syntactic counterpart of Γ = Goal 1 Γ holds if Γ = (completeness) Goal 2 Γ implies Γ = (soundness)
6 Intuition 4 / 16 Idea Define inference rules for sequents Γ where Γ and are sequences of formulas Intuition Γ is the syntactic counterpart of Γ = Goal 1 Γ holds if Γ = (completeness) Goal 2 Γ implies Γ = (soundness) Notation S 1 S or S 1 S 2 S From sequents S 1 (and S 2 ) conclude sequent S.
7 Intuition Idea Define inference rules for sequents Γ where Γ and are sequences of formulas Intuition Γ is the syntactic counterpart of Γ = Goal 1 Γ holds if Γ = (completeness) Goal 2 Γ implies Γ = (soundness) Notation S 1 S or S 1 S 2 S From sequents S 1 (and S 2 ) conclude sequent S. System considered here: LK, defined by Gerhard Gentzen 4 / 16
8 Outline 5 / 16 1 Intuition 2 The LK system 3 Derivation 4 Summary 5 Exercises
9 LK Logistischer Klassischer Kalkül (1) 6 / 16 Axioms Structural rules
10 LK Logistischer Klassischer Kalkül (1) 6 / 16 Axioms A A (ax) Structural rules
11 LK Logistischer Klassischer Kalkül (1) 6 / 16 Axioms A A (ax) Structural rules permutation Γ, A, B, Γ Γ, B, A, Γ (p.l) Γ, A, B, Γ, B, A, (p.r)
12 LK Logistischer Klassischer Kalkül (1) 6 / 16 Axioms A A (ax) Structural rules permutation contraction Γ, A, B, Γ Γ, B, A, Γ (p.l) Γ, A, B, Γ, B, A, (p.r) Γ, A, A Γ, A (c.l) Γ, A, A Γ, A (c.r)
13 LK Logistischer Klassischer Kalkül (1) 6 / 16 Axioms A A (ax) Structural rules permutation contraction weakening Γ, A, B, Γ Γ, B, A, Γ (p.l) Γ, A, B, Γ, B, A, (p.r) Γ, A, A Γ, A (c.l) Γ, A, A Γ, A (c.r) Γ Γ, A (w.l) Γ Γ, A (w.r)
14 LK Logistischer Klassischer Kalkül (1) Axioms A A (ax) Structural rules permutation contraction weakening cut Γ, A, B, Γ Γ, B, A, Γ (p.l) Γ, A, B, Γ, B, A, (p.r) Γ, A, A Γ, A (c.l) Γ, A, A Γ, A (c.r) Γ Γ, A (w.l) Γ Γ, A (w.r) Γ, A Γ, A Γ, Γ, (cut) 6 / 16
15 LK Logistischer Klassischer Kalkül (1) Axioms A A (ax) Structural rules What are A and B? What Γ, Γ,,? permutation contraction weakening cut Γ, A, B, Γ Γ, B, A, Γ (p.l) Γ, A, B, Γ, B, A, (p.r) Γ, A, A Γ, A (c.l) Γ, A, A Γ, A (c.r) Γ Γ, A (w.l) Γ Γ, A (w.r) Γ, A Γ, A Γ, Γ, (cut) 6 / 16
16 LK Logistischer Klassischer Kalkül (2) 7 / 16 Logical rules
17 LK Logistischer Klassischer Kalkül (2) 7 / 16 Logical rules Γ, A Γ, A B ( l.1) Γ, A Γ, B ( r) Γ, A B Γ, B Γ, A B ( l.2)
18 LK Logistischer Klassischer Kalkül (2) Logical rules Γ, A Γ, A B ( l.1) Γ, A Γ, B ( r) Γ, A B Γ, B Γ, A B ( l.2) Γ, A Γ, B ( l) Γ, A B Γ, A Γ, A B ( r.1) Γ, B Γ, A B ( r.2) 7 / 16
19 LK Logistischer Klassischer Kalkül (2) Logical rules Γ, A Γ, A B ( l.1) Γ, A Γ, B ( r) Γ, A B Γ, B Γ, A B ( l.2) Γ, A Γ, B ( l) Γ, A B Γ, A Γ, A B ( r.1) Γ, B Γ, A B ( r.2) Γ, A Γ, A ( l) Γ, A Γ, A ( r) 7 / 16
20 LK Logistischer Klassischer Kalkül (2) Logical rules Γ, A Γ, A B ( l.1) Γ, A Γ, B ( r) Γ, A B Γ, B Γ, A B ( l.2) Γ, A Γ, B ( l) Γ, A B Γ, A Γ, A B ( r.1) Γ, B Γ, A B ( r.2) Γ, A Γ, A ( l) Γ, A Γ, A ( r) Γ, A Γ, B ( l) Γ, A B Γ, A, B Γ, A B ( r) 7 / 16
21 Outline 8 / 16 1 Intuition 2 The LK system 3 Derivation 4 Summary 5 Exercises
22 Derivation 9 / 16 Use these inference rules consecutively
23 Derivation 9 / 16 Example Use these inference rules consecutively A A (ax) A, A ( r)
24 Derivation 9 / 16 Example Use these inference rules consecutively A A (ax) A, A ( r) If on top there are only axioms then it is a derivation of the bottom sequent
25 Derivation 9 / 16 Example Use these inference rules consecutively A A (ax) A, A ( r) If on top there are only axioms then it is a derivation of the bottom sequent Theorem Sequent Calculus is sound and complete, i.e., if we can derive Γ then Γ =, and if Γ = then there is a derivation for Γ.
26 Outline 10 / 16 1 Intuition 2 The LK system 3 Derivation 4 Summary 5 Exercises
27 LK Summary A A (ax) Γ, A, B, Γ Γ, B, A, Γ (p.l) Γ, A, B, Γ, B, A, (p.r) Γ Γ, A (w.l) Γ Γ, A (w.r) Γ, A, A Γ, A (c.l) Γ, A, A Γ, A (c.r) Γ, A Γ, A Γ, Γ, (cut) Γ, A Γ, A B ( l.1) Γ, A Γ, B ( r) Γ, A B Γ, B Γ, A B ( l.2) Γ, A Γ, B ( l) Γ, A B Γ, A Γ, A B ( r.1) Γ, B Γ, A B ( r.2) Γ, A Γ, A ( l) Γ, A Γ, A ( r) Γ, A Γ, B ( l) Γ, A B Γ, A, B ( r) Γ, A B 11 / 16
28 12 / 16 LK Example Example Do the following entailments hold? 1 (A B) B, A = 2 A C, A B = (On the blackboard.)
29 Outline 13 / 16 1 Intuition 2 The LK system 3 Derivation 4 Summary 5 Exercises
30 14 / 16 Exercises (1) (From Logic for Computer Science: Foundations of Automatic Theorem Proving) Give proof trees for the following tautologies: 1 A (B A) 2 (A B) ((A (B C)) (A C)) 3 A (B A B) 4 A A B 5 B A B 6 (A B) ((A B) A) 7 A B A 8 A B B 9 (A C) ((B C) (A B C)) 10 A A
31 Exercises (2) (From Logic for Computer Science: Foundations of Automatic Theorem Proving) Give proof trees for the following equivalences: 1 (A B) C A (B C) (associativity) 2 (A B) C A (B C) (associativity) 3 A B B A (commutativity) 4 A B B A (commutativity) 5 A (B C) (A B) (A C) (distributivity) 6 A (B C) (A B) (A C) (distributivity) 7 (A B) A B (De Morgan) 8 (A B) A B (De Morgan) 9 A A A (idempotency) 10 A A A (idempotency) 11 A A (double negation) 12 (A B) ( A C) (A B) ( A C) (B C) (resolution) 15 / 16
32 END OF THE LECTURE 16 / 16
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