OpenHPI 4.9 - Tableaux Algorithm
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Dr. Harald Sack
Hasso Plattner Institute for IT Systems Engineering
University of Potsdam
Spring 2013
Semantic Web Technologies
Lecture 4: Knowledge Representations I09: Tableaux Algorithm
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Semantic Web Technologies , Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam
2
Lecture 4: Knowledge Representations I
Open HPI - Course: Semantic Web Technologies
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Vorlesung Semantic Web, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam
3
09 Tableaux AlgorithmOpen HPI - Course: Semantic Web Technologies - Lecture 4: Knowledge Representations I
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Lecture: Semantic Web Technologies, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam, WS 2012/13
4
Tableaux Algorithm
•For automated reasoning, we need syntactic algorithms to check the consistency of logical assertions
•To apply resolution, formulas have to be in clausal form
•The method of analytical tableaux is based on disjunctive normal form
• invented by Dutch logician Evert Willem Beth in 1955 and simplified by Raymond Smullyan.Evert Willem Beth
(1908-1964)
Raymond Merrill Smullyan
Beth, Evert W., 1955. "Semantic entailment and formal derivability", Mededlingen van de Koninklijke Nederlandse Akademie van Wetenschappen, Afdeling Letterkunde, N.R. Vol 18, no 13, 1955, pp 309–42.
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Lecture: Semantic Web Technologies, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam, WS 2012/13
4
Tableaux Algorithm
•For automated reasoning, we need syntactic algorithms to check the consistency of logical assertions
•To apply resolution, formulas have to be in clausal form
•The method of analytical tableaux is based on disjunctive normal form
• invented by Dutch logician Evert Willem Beth in 1955 and simplified by Raymond Smullyan.
•Basic Idea of Tableaux Algorithm (similar to Resolution):
Evert Willem Beth(1908-1964)
Raymond Merrill Smullyan
Beth, Evert W., 1955. "Semantic entailment and formal derivability", Mededlingen van de Koninklijke Nederlandse Akademie van Wetenschappen, Afdeling Letterkunde, N.R. Vol 18, no 13, 1955, pp 309–42.
![Page 6: OpenHPI 4.9 - Tableaux Algorithm](https://reader034.fdocuments.in/reader034/viewer/2022051014/54b8ac5b4a7959c6438b45a1/html5/thumbnails/6.jpg)
Lecture: Semantic Web Technologies, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam, WS 2012/13
4
Tableaux Algorithm
•For automated reasoning, we need syntactic algorithms to check the consistency of logical assertions
•To apply resolution, formulas have to be in clausal form
•The method of analytical tableaux is based on disjunctive normal form
• invented by Dutch logician Evert Willem Beth in 1955 and simplified by Raymond Smullyan.
•Basic Idea of Tableaux Algorithm (similar to Resolution):
•Proof algorithm (decision procedure) to check the consistency of a logical formula by inferring that its negation is a contradiction (proof by refutation).
Evert Willem Beth(1908-1964)
Raymond Merrill Smullyan
Beth, Evert W., 1955. "Semantic entailment and formal derivability", Mededlingen van de Koninklijke Nederlandse Akademie van Wetenschappen, Afdeling Letterkunde, N.R. Vol 18, no 13, 1955, pp 309–42.
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Lecture: Semantic Web Technologies, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam, WS 2012/13
5
Tableaux Algorithm for Propositional Logic
(1) Construct Decision Tree, where each node is marked with a logical formula.
•A path from the root to a leaf is the conjunction of all formulas represented within the nodes of the path;
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Lecture: Semantic Web Technologies, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam, WS 2012/13
5
Tableaux Algorithm for Propositional Logic
(1) Construct Decision Tree, where each node is marked with a logical formula.
•A path from the root to a leaf is the conjunction of all formulas represented within the nodes of the path;
•a branch of the path represents a disjunction.
![Page 9: OpenHPI 4.9 - Tableaux Algorithm](https://reader034.fdocuments.in/reader034/viewer/2022051014/54b8ac5b4a7959c6438b45a1/html5/thumbnails/9.jpg)
Lecture: Semantic Web Technologies, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam, WS 2012/13
5
Tableaux Algorithm for Propositional Logic
(1) Construct Decision Tree, where each node is marked with a logical formula.
•A path from the root to a leaf is the conjunction of all formulas represented within the nodes of the path;
•a branch of the path represents a disjunction.
(q ∧ r) ∨ (p ∧ ¬ r) ∨ r
(q ∧ r) (p ∧ ¬ r) ∨ r
(p ∧ ¬ r) rq
r p
¬ r
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Lecture: Semantic Web Technologies, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam, WS 2012/13
6
Tableaux Algorithm for Propositional Logic
(1) Construct Decision Tree, where each node is marked with a logical formula.
•A path from the root to a leaf is the conjunction of all formulas represented within the nodes of the path;
•a branch of the path represents a disjunction.
(2) The tree is created by successive application of the Tableaux Extension Rules.
(3) A path in the Tableaux is closed,
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Lecture: Semantic Web Technologies, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam, WS 2012/13
6
Tableaux Algorithm for Propositional Logic
(1) Construct Decision Tree, where each node is marked with a logical formula.
•A path from the root to a leaf is the conjunction of all formulas represented within the nodes of the path;
•a branch of the path represents a disjunction.
(2) The tree is created by successive application of the Tableaux Extension Rules.
(3) A path in the Tableaux is closed,
• if along the path as well X as ¬X for a formula X occurs (X doesn‘t have to be atomic) or
![Page 12: OpenHPI 4.9 - Tableaux Algorithm](https://reader034.fdocuments.in/reader034/viewer/2022051014/54b8ac5b4a7959c6438b45a1/html5/thumbnails/12.jpg)
Lecture: Semantic Web Technologies, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam, WS 2012/13
6
Tableaux Algorithm for Propositional Logic
(1) Construct Decision Tree, where each node is marked with a logical formula.
•A path from the root to a leaf is the conjunction of all formulas represented within the nodes of the path;
•a branch of the path represents a disjunction.
(2) The tree is created by successive application of the Tableaux Extension Rules.
(3) A path in the Tableaux is closed,
• if along the path as well X as ¬X for a formula X occurs (X doesn‘t have to be atomic) or
• if false occurs .
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Lecture: Semantic Web Technologies, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam, WS 2012/13
7
Tableaux Algorithm for Propositional Logic
(4) A tableaux is fully expanded, if no more extension rules can be applied.
(5) A tableaux is called closed, if all its paths are closed.
(6) A Tableaux Proof for a formula X is a closed tableaux for ¬X.
•The selection of the tableaux extension rules to be applied in the tableaux is not deterministic.
•There are heuristics for the propositional logic tableaux to select which extension rules to be applied best
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•for PL:
•for conjunctive Formula (α-Rules):
•for disjunctive formula (β-Rules):
Lecture: Semantic Web Technologies, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam, WS 2012/13
8
Tableaux Extension Rules - PL
¬¬XX
¬TF
¬FT
α α1 α2
X∧YXY
¬(X∨Y)¬X¬Y
¬(X⇒Y)X¬Y
β β1 | β2
X∨YX | Y
¬(X∧Y)¬X | ¬Y
(X⇒Y)¬X | Y
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Lecture: Semantic Web Technologies, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam, WS 2012/13
9
Tableaux Algorithm (PL) - Example (1):
¬(X⇒Y)X¬Y
α-Rule
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Lecture: Semantic Web Technologies, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam, WS 2012/13
9
Tableaux Algorithm (PL) - Example (1):
proof: ((q ∧ r) ⇒ (¬q ∨ r))
¬(X⇒Y)X¬Y
α-Rule
![Page 17: OpenHPI 4.9 - Tableaux Algorithm](https://reader034.fdocuments.in/reader034/viewer/2022051014/54b8ac5b4a7959c6438b45a1/html5/thumbnails/17.jpg)
Lecture: Semantic Web Technologies, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam, WS 2012/13
9
Tableaux Algorithm (PL) - Example (1):
proof: ((q ∧ r) ⇒ (¬q ∨ r))
(1) ¬((q ∧ r) ⇒ (¬q ∨ r)) ¬(X⇒Y)
X¬Y
α-Rule
![Page 18: OpenHPI 4.9 - Tableaux Algorithm](https://reader034.fdocuments.in/reader034/viewer/2022051014/54b8ac5b4a7959c6438b45a1/html5/thumbnails/18.jpg)
Lecture: Semantic Web Technologies, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam, WS 2012/13
9
Tableaux Algorithm (PL) - Example (1):
proof: ((q ∧ r) ⇒ (¬q ∨ r))
(1) ¬((q ∧ r) ⇒ (¬q ∨ r))
(2) α,1: (q ∧ r)¬(X⇒Y)
X¬Y
α-Rule
![Page 19: OpenHPI 4.9 - Tableaux Algorithm](https://reader034.fdocuments.in/reader034/viewer/2022051014/54b8ac5b4a7959c6438b45a1/html5/thumbnails/19.jpg)
Lecture: Semantic Web Technologies, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam, WS 2012/13
9
Tableaux Algorithm (PL) - Example (1):
proof: ((q ∧ r) ⇒ (¬q ∨ r))
(1) ¬((q ∧ r) ⇒ (¬q ∨ r))
(2) α,1: (q ∧ r)
(3) α,1: ¬(¬q ∨ r) = q ∧ ¬r
¬(X⇒Y)X¬Y
α-Rule
![Page 20: OpenHPI 4.9 - Tableaux Algorithm](https://reader034.fdocuments.in/reader034/viewer/2022051014/54b8ac5b4a7959c6438b45a1/html5/thumbnails/20.jpg)
Lecture: Semantic Web Technologies, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam, WS 2012/13
10
Tableaux Algorithm (PL) - Example (1):
proof: ((q ∧ r) ⇒ (¬q ∨ r))
(1) ¬((q ∧ r) ⇒ (¬q ∨ r))
(2) α,1: (q ∧ r)
(3) α,1: ¬(¬q ∨ r) = q ∧ ¬r
X ∧ YXY
α-Rule
![Page 21: OpenHPI 4.9 - Tableaux Algorithm](https://reader034.fdocuments.in/reader034/viewer/2022051014/54b8ac5b4a7959c6438b45a1/html5/thumbnails/21.jpg)
Lecture: Semantic Web Technologies, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam, WS 2012/13
10
Tableaux Algorithm (PL) - Example (1):
proof: ((q ∧ r) ⇒ (¬q ∨ r))
(1) ¬((q ∧ r) ⇒ (¬q ∨ r))
(2) α,1: (q ∧ r)
(3) α,1: ¬(¬q ∨ r) = q ∧ ¬r
(4) α,2: qX ∧ Y
XY
α-Rule
![Page 22: OpenHPI 4.9 - Tableaux Algorithm](https://reader034.fdocuments.in/reader034/viewer/2022051014/54b8ac5b4a7959c6438b45a1/html5/thumbnails/22.jpg)
Lecture: Semantic Web Technologies, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam, WS 2012/13
10
Tableaux Algorithm (PL) - Example (1):
proof: ((q ∧ r) ⇒ (¬q ∨ r))
(1) ¬((q ∧ r) ⇒ (¬q ∨ r))
(2) α,1: (q ∧ r)
(3) α,1: ¬(¬q ∨ r) = q ∧ ¬r
(4) α,2: q
(5) α,2: rX ∧ YXY
α-Rule
![Page 23: OpenHPI 4.9 - Tableaux Algorithm](https://reader034.fdocuments.in/reader034/viewer/2022051014/54b8ac5b4a7959c6438b45a1/html5/thumbnails/23.jpg)
Lecture: Semantic Web Technologies, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam, WS 2012/13
10
Tableaux Algorithm (PL) - Example (1):
proof: ((q ∧ r) ⇒ (¬q ∨ r))
(1) ¬((q ∧ r) ⇒ (¬q ∨ r))
(2) α,1: (q ∧ r)
(3) α,1: ¬(¬q ∨ r) = q ∧ ¬r
(4) α,2: q
(5) α,2: r
(6) α,3: q
X ∧ YXY
α-Rule
![Page 24: OpenHPI 4.9 - Tableaux Algorithm](https://reader034.fdocuments.in/reader034/viewer/2022051014/54b8ac5b4a7959c6438b45a1/html5/thumbnails/24.jpg)
Lecture: Semantic Web Technologies, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam, WS 2012/13
10
Tableaux Algorithm (PL) - Example (1):
proof: ((q ∧ r) ⇒ (¬q ∨ r))
(1) ¬((q ∧ r) ⇒ (¬q ∨ r))
(2) α,1: (q ∧ r)
(3) α,1: ¬(¬q ∨ r) = q ∧ ¬r
(4) α,2: q
(5) α,2: r
(6) α,3: q
(7) α,3: ¬r
X ∧ YXY
α-Rule
![Page 25: OpenHPI 4.9 - Tableaux Algorithm](https://reader034.fdocuments.in/reader034/viewer/2022051014/54b8ac5b4a7959c6438b45a1/html5/thumbnails/25.jpg)
Lecture: Semantic Web Technologies, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam, WS 2012/13
10
Tableaux Algorithm (PL) - Example (1):
proof: ((q ∧ r) ⇒ (¬q ∨ r))
(1) ¬((q ∧ r) ⇒ (¬q ∨ r))
(2) α,1: (q ∧ r)
(3) α,1: ¬(¬q ∨ r) = q ∧ ¬r
(4) α,2: q
(5) α,2: r
(6) α,3: q
(7) α,3: ¬r
X ∧ YXY
α-Rule
• path is closed
• tableaux is closed
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Lecture: Semantic Web Technologies, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam, WS 2012/13
11
Tableaux Algorithm (PL) - Example (2):
¬(X⇒Y)X
¬Y
α-Rule
proof: (p⇒(q⇒r)) ⇒ ((p⇒q)⇒(p⇒r))
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Lecture: Semantic Web Technologies, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam, WS 2012/13
11
Tableaux Algorithm (PL) - Example (2):
¬(X⇒Y)X
¬Y
α-Rule
proof: (p⇒(q⇒r)) ⇒ ((p⇒q)⇒(p⇒r))
(1) ¬((p⇒(q⇒r)) ⇒ ((p⇒q)⇒(p⇒r)))
![Page 28: OpenHPI 4.9 - Tableaux Algorithm](https://reader034.fdocuments.in/reader034/viewer/2022051014/54b8ac5b4a7959c6438b45a1/html5/thumbnails/28.jpg)
Lecture: Semantic Web Technologies, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam, WS 2012/13
11
Tableaux Algorithm (PL) - Example (2):
¬(X⇒Y)X
¬Y
α-Rule
proof: (p⇒(q⇒r)) ⇒ ((p⇒q)⇒(p⇒r))
(1) ¬((p⇒(q⇒r)) ⇒ ((p⇒q)⇒(p⇒r)))(2|α from 1) (p⇒(q⇒r))
![Page 29: OpenHPI 4.9 - Tableaux Algorithm](https://reader034.fdocuments.in/reader034/viewer/2022051014/54b8ac5b4a7959c6438b45a1/html5/thumbnails/29.jpg)
Lecture: Semantic Web Technologies, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam, WS 2012/13
11
Tableaux Algorithm (PL) - Example (2):
¬(X⇒Y)X
¬Y
α-Rule
proof: (p⇒(q⇒r)) ⇒ ((p⇒q)⇒(p⇒r))
(1) ¬((p⇒(q⇒r)) ⇒ ((p⇒q)⇒(p⇒r)))(2|α from 1) (p⇒(q⇒r))(3|α from 1) ¬((p⇒q)⇒(p⇒r))
![Page 30: OpenHPI 4.9 - Tableaux Algorithm](https://reader034.fdocuments.in/reader034/viewer/2022051014/54b8ac5b4a7959c6438b45a1/html5/thumbnails/30.jpg)
Lecture: Semantic Web Technologies, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam, WS 2012/13
11
Tableaux Algorithm (PL) - Example (2):
¬(X⇒Y)X
¬Y
α-Rule
proof: (p⇒(q⇒r)) ⇒ ((p⇒q)⇒(p⇒r))
(1) ¬((p⇒(q⇒r)) ⇒ ((p⇒q)⇒(p⇒r)))(2|α from 1) (p⇒(q⇒r))(3|α from 1) ¬((p⇒q)⇒(p⇒r))(4|α from 3) (p⇒q)
![Page 31: OpenHPI 4.9 - Tableaux Algorithm](https://reader034.fdocuments.in/reader034/viewer/2022051014/54b8ac5b4a7959c6438b45a1/html5/thumbnails/31.jpg)
Lecture: Semantic Web Technologies, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam, WS 2012/13
11
Tableaux Algorithm (PL) - Example (2):
¬(X⇒Y)X
¬Y
α-Rule
proof: (p⇒(q⇒r)) ⇒ ((p⇒q)⇒(p⇒r))
(1) ¬((p⇒(q⇒r)) ⇒ ((p⇒q)⇒(p⇒r)))(2|α from 1) (p⇒(q⇒r))(3|α from 1) ¬((p⇒q)⇒(p⇒r))(4|α from 3) (p⇒q)(5|α from 3) ¬(p⇒r)
![Page 32: OpenHPI 4.9 - Tableaux Algorithm](https://reader034.fdocuments.in/reader034/viewer/2022051014/54b8ac5b4a7959c6438b45a1/html5/thumbnails/32.jpg)
Lecture: Semantic Web Technologies, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam, WS 2012/13
11
Tableaux Algorithm (PL) - Example (2):
¬(X⇒Y)X
¬Y
α-Rule
proof: (p⇒(q⇒r)) ⇒ ((p⇒q)⇒(p⇒r))
(1) ¬((p⇒(q⇒r)) ⇒ ((p⇒q)⇒(p⇒r)))(2|α from 1) (p⇒(q⇒r))(3|α from 1) ¬((p⇒q)⇒(p⇒r))(4|α from 3) (p⇒q)(5|α from 3) ¬(p⇒r)(6|α from 5) p
![Page 33: OpenHPI 4.9 - Tableaux Algorithm](https://reader034.fdocuments.in/reader034/viewer/2022051014/54b8ac5b4a7959c6438b45a1/html5/thumbnails/33.jpg)
Lecture: Semantic Web Technologies, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam, WS 2012/13
11
Tableaux Algorithm (PL) - Example (2):
¬(X⇒Y)X
¬Y
α-Rule
proof: (p⇒(q⇒r)) ⇒ ((p⇒q)⇒(p⇒r))
(1) ¬((p⇒(q⇒r)) ⇒ ((p⇒q)⇒(p⇒r)))(2|α from 1) (p⇒(q⇒r))(3|α from 1) ¬((p⇒q)⇒(p⇒r))(4|α from 3) (p⇒q)(5|α from 3) ¬(p⇒r)(6|α from 5) p
(7|α from 5) ¬r
![Page 34: OpenHPI 4.9 - Tableaux Algorithm](https://reader034.fdocuments.in/reader034/viewer/2022051014/54b8ac5b4a7959c6438b45a1/html5/thumbnails/34.jpg)
proof: (p⇒(q⇒r)) ⇒ ((p⇒q)⇒(p⇒r))
Lecture: Semantic Web Technologies, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam, WS 2012/13
12
Tableaux Algorithm (PL) - Example (2):
(X⇒Y)¬X | Y
β-Rule
(1) ¬((p⇒(q⇒r)) ⇒ ((p⇒q)⇒(p⇒r)))(2|α from 1) (p⇒(q⇒r))(3|α from 1) ¬((p⇒q)⇒(p⇒r))(4|α from 3) (p⇒q)(5|α from 3) ¬(p⇒r)(6|α from 5) p
(7|α from 5) ¬r
![Page 35: OpenHPI 4.9 - Tableaux Algorithm](https://reader034.fdocuments.in/reader034/viewer/2022051014/54b8ac5b4a7959c6438b45a1/html5/thumbnails/35.jpg)
proof: (p⇒(q⇒r)) ⇒ ((p⇒q)⇒(p⇒r))
Lecture: Semantic Web Technologies, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam, WS 2012/13
12
Tableaux Algorithm (PL) - Example (2):
(X⇒Y)¬X | Y
β-Rule (8|β from 2) ¬p | (9|β from 2) (q⇒r)
(1) ¬((p⇒(q⇒r)) ⇒ ((p⇒q)⇒(p⇒r)))(2|α from 1) (p⇒(q⇒r))(3|α from 1) ¬((p⇒q)⇒(p⇒r))(4|α from 3) (p⇒q)(5|α from 3) ¬(p⇒r)(6|α from 5) p
(7|α from 5) ¬r
![Page 36: OpenHPI 4.9 - Tableaux Algorithm](https://reader034.fdocuments.in/reader034/viewer/2022051014/54b8ac5b4a7959c6438b45a1/html5/thumbnails/36.jpg)
proof: (p⇒(q⇒r)) ⇒ ((p⇒q)⇒(p⇒r))
Lecture: Semantic Web Technologies, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam, WS 2012/13
12
Tableaux Algorithm (PL) - Example (2):
(X⇒Y)¬X | Y
β-Rule
(10|β from 9) ¬q | (11|β from 9) r
(8|β from 2) ¬p | (9|β from 2) (q⇒r)
(1) ¬((p⇒(q⇒r)) ⇒ ((p⇒q)⇒(p⇒r)))(2|α from 1) (p⇒(q⇒r))(3|α from 1) ¬((p⇒q)⇒(p⇒r))(4|α from 3) (p⇒q)(5|α from 3) ¬(p⇒r)(6|α from 5) p
(7|α from 5) ¬r
![Page 37: OpenHPI 4.9 - Tableaux Algorithm](https://reader034.fdocuments.in/reader034/viewer/2022051014/54b8ac5b4a7959c6438b45a1/html5/thumbnails/37.jpg)
proof: (p⇒(q⇒r)) ⇒ ((p⇒q)⇒(p⇒r))
Lecture: Semantic Web Technologies, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam, WS 2012/13
12
Tableaux Algorithm (PL) - Example (2):
(X⇒Y)¬X | Y
β-Rule
(10|β from 9) ¬q | (11|β from 9) r
(12|β from 4) ¬p | (13|β from 4) q
(8|β from 2) ¬p | (9|β from 2) (q⇒r)
(1) ¬((p⇒(q⇒r)) ⇒ ((p⇒q)⇒(p⇒r)))(2|α from 1) (p⇒(q⇒r))(3|α from 1) ¬((p⇒q)⇒(p⇒r))(4|α from 3) (p⇒q)(5|α from 3) ¬(p⇒r)(6|α from 5) p
(7|α from 5) ¬r
![Page 38: OpenHPI 4.9 - Tableaux Algorithm](https://reader034.fdocuments.in/reader034/viewer/2022051014/54b8ac5b4a7959c6438b45a1/html5/thumbnails/38.jpg)
proof: (p⇒(q⇒r)) ⇒ ((p⇒q)⇒(p⇒r))
Lecture: Semantic Web Technologies, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam, WS 2012/13
12
Tableaux Algorithm (PL) - Example (2):
(X⇒Y)¬X | Y
β-Rule
(10|β from 9) ¬q | (11|β from 9) r
(12|β from 4) ¬p | (13|β from 4) q
(8|β from 2) ¬p | (9|β from 2) (q⇒r)
(1) ¬((p⇒(q⇒r)) ⇒ ((p⇒q)⇒(p⇒r)))(2|α from 1) (p⇒(q⇒r))(3|α from 1) ¬((p⇒q)⇒(p⇒r))(4|α from 3) (p⇒q)(5|α from 3) ¬(p⇒r)(6|α from 5) p
(7|α from 5) ¬r
![Page 39: OpenHPI 4.9 - Tableaux Algorithm](https://reader034.fdocuments.in/reader034/viewer/2022051014/54b8ac5b4a7959c6438b45a1/html5/thumbnails/39.jpg)
proof: (p⇒(q⇒r)) ⇒ ((p⇒q)⇒(p⇒r))
Lecture: Semantic Web Technologies, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam, WS 2012/13
12
Tableaux Algorithm (PL) - Example (2):
(X⇒Y)¬X | Y
β-Rule
(10|β from 9) ¬q | (11|β from 9) r
(12|β from 4) ¬p | (13|β from 4) q
(8|β from 2) ¬p | (9|β from 2) (q⇒r)
(1) ¬((p⇒(q⇒r)) ⇒ ((p⇒q)⇒(p⇒r)))(2|α from 1) (p⇒(q⇒r))(3|α from 1) ¬((p⇒q)⇒(p⇒r))(4|α from 3) (p⇒q)(5|α from 3) ¬(p⇒r)(6|α from 5) p
(7|α from 5) ¬r
![Page 40: OpenHPI 4.9 - Tableaux Algorithm](https://reader034.fdocuments.in/reader034/viewer/2022051014/54b8ac5b4a7959c6438b45a1/html5/thumbnails/40.jpg)
proof: (p⇒(q⇒r)) ⇒ ((p⇒q)⇒(p⇒r))
Lecture: Semantic Web Technologies, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam, WS 2012/13
12
Tableaux Algorithm (PL) - Example (2):
(X⇒Y)¬X | Y
β-Rule
(10|β from 9) ¬q | (11|β from 9) r
(12|β from 4) ¬p | (13|β from 4) q
(8|β from 2) ¬p | (9|β from 2) (q⇒r)
(1) ¬((p⇒(q⇒r)) ⇒ ((p⇒q)⇒(p⇒r)))(2|α from 1) (p⇒(q⇒r))(3|α from 1) ¬((p⇒q)⇒(p⇒r))(4|α from 3) (p⇒q)(5|α from 3) ¬(p⇒r)(6|α from 5) p
(7|α from 5) ¬r
![Page 41: OpenHPI 4.9 - Tableaux Algorithm](https://reader034.fdocuments.in/reader034/viewer/2022051014/54b8ac5b4a7959c6438b45a1/html5/thumbnails/41.jpg)
Lecture: Semantic Web Technologies, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam, WS 2012/13
13
Tableaux Algorithm Extensions for FOL
• as for propositional logic - X and Y stand for arbitrary (FOL) formulas
• Additional Rules for quantified formulas :
• γ for universally quantified formulas, δ existentially quantified formulas, with:
• t is an arbitrary ground term (i.e. doesn‘t contain variables that are bound in Φ),
• c is a „new“ constant
γ γ[t]
δ δ[c]
γ γ[t]
∀x.Φ Φ[x←t]
¬∃x.Φ ¬Φ[x←t]
δ δ[c]
∃x.Φ Φ[x←c]
¬∀x.Φ ¬Φ[x←c]
![Page 42: OpenHPI 4.9 - Tableaux Algorithm](https://reader034.fdocuments.in/reader034/viewer/2022051014/54b8ac5b4a7959c6438b45a1/html5/thumbnails/42.jpg)
Lecture: Semantic Web Technologies, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam, WS 2012/13
14Proof: (∀x.P(x)∨Q(x))⇒(∃x.P(x))∨(∀x.Q(x))
γ γ[t]
∀x.Φ Φ[x←t]
¬∃x.Φ ¬Φ[x←t]
δ δ[c]
∃x.Φ Φ[x←c]
¬∀x.Φ ¬Φ[x←c]
Tableaux Algorithm (FOL) - Example(3):
![Page 43: OpenHPI 4.9 - Tableaux Algorithm](https://reader034.fdocuments.in/reader034/viewer/2022051014/54b8ac5b4a7959c6438b45a1/html5/thumbnails/43.jpg)
Lecture: Semantic Web Technologies, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam, WS 2012/13
14Proof: (∀x.P(x)∨Q(x))⇒(∃x.P(x))∨(∀x.Q(x))
(1) ¬((∀x.P(x)∨Q(x))⇒(∃x.P(x))∨(∀x.Q(x)))
γ γ[t]
∀x.Φ Φ[x←t]
¬∃x.Φ ¬Φ[x←t]
δ δ[c]
∃x.Φ Φ[x←c]
¬∀x.Φ ¬Φ[x←c]
Tableaux Algorithm (FOL) - Example(3):
![Page 44: OpenHPI 4.9 - Tableaux Algorithm](https://reader034.fdocuments.in/reader034/viewer/2022051014/54b8ac5b4a7959c6438b45a1/html5/thumbnails/44.jpg)
Lecture: Semantic Web Technologies, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam, WS 2012/13
14Proof: (∀x.P(x)∨Q(x))⇒(∃x.P(x))∨(∀x.Q(x))
(1) ¬((∀x.P(x)∨Q(x))⇒(∃x.P(x))∨(∀x.Q(x)))(2|α from 1) (∀x.P(x)∨Q(x))
γ γ[t]
∀x.Φ Φ[x←t]
¬∃x.Φ ¬Φ[x←t]
δ δ[c]
∃x.Φ Φ[x←c]
¬∀x.Φ ¬Φ[x←c]
Tableaux Algorithm (FOL) - Example(3):
![Page 45: OpenHPI 4.9 - Tableaux Algorithm](https://reader034.fdocuments.in/reader034/viewer/2022051014/54b8ac5b4a7959c6438b45a1/html5/thumbnails/45.jpg)
Lecture: Semantic Web Technologies, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam, WS 2012/13
14Proof: (∀x.P(x)∨Q(x))⇒(∃x.P(x))∨(∀x.Q(x))
(1) ¬((∀x.P(x)∨Q(x))⇒(∃x.P(x))∨(∀x.Q(x)))(2|α from 1) (∀x.P(x)∨Q(x))
(3|α from 1) ¬((∃x.P(x))∨(∀x.Q(x)))
γ γ[t]
∀x.Φ Φ[x←t]
¬∃x.Φ ¬Φ[x←t]
δ δ[c]
∃x.Φ Φ[x←c]
¬∀x.Φ ¬Φ[x←c]
Tableaux Algorithm (FOL) - Example(3):
![Page 46: OpenHPI 4.9 - Tableaux Algorithm](https://reader034.fdocuments.in/reader034/viewer/2022051014/54b8ac5b4a7959c6438b45a1/html5/thumbnails/46.jpg)
Lecture: Semantic Web Technologies, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam, WS 2012/13
14Proof: (∀x.P(x)∨Q(x))⇒(∃x.P(x))∨(∀x.Q(x))
(1) ¬((∀x.P(x)∨Q(x))⇒(∃x.P(x))∨(∀x.Q(x)))(2|α from 1) (∀x.P(x)∨Q(x))
(3|α from 1) ¬((∃x.P(x))∨(∀x.Q(x)))
(4|α from 3) ¬(∃x.P(x))
γ γ[t]
∀x.Φ Φ[x←t]
¬∃x.Φ ¬Φ[x←t]
δ δ[c]
∃x.Φ Φ[x←c]
¬∀x.Φ ¬Φ[x←c]
Tableaux Algorithm (FOL) - Example(3):
![Page 47: OpenHPI 4.9 - Tableaux Algorithm](https://reader034.fdocuments.in/reader034/viewer/2022051014/54b8ac5b4a7959c6438b45a1/html5/thumbnails/47.jpg)
Lecture: Semantic Web Technologies, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam, WS 2012/13
14Proof: (∀x.P(x)∨Q(x))⇒(∃x.P(x))∨(∀x.Q(x))
(1) ¬((∀x.P(x)∨Q(x))⇒(∃x.P(x))∨(∀x.Q(x)))(2|α from 1) (∀x.P(x)∨Q(x))
(3|α from 1) ¬((∃x.P(x))∨(∀x.Q(x)))
(4|α from 3) ¬(∃x.P(x))
(5|α from 3) ¬(∀x.Q(x))
γ γ[t]
∀x.Φ Φ[x←t]
¬∃x.Φ ¬Φ[x←t]
δ δ[c]
∃x.Φ Φ[x←c]
¬∀x.Φ ¬Φ[x←c]
Tableaux Algorithm (FOL) - Example(3):
![Page 48: OpenHPI 4.9 - Tableaux Algorithm](https://reader034.fdocuments.in/reader034/viewer/2022051014/54b8ac5b4a7959c6438b45a1/html5/thumbnails/48.jpg)
Lecture: Semantic Web Technologies, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam, WS 2012/13
14Proof: (∀x.P(x)∨Q(x))⇒(∃x.P(x))∨(∀x.Q(x))
(1) ¬((∀x.P(x)∨Q(x))⇒(∃x.P(x))∨(∀x.Q(x)))(2|α from 1) (∀x.P(x)∨Q(x))
(3|α from 1) ¬((∃x.P(x))∨(∀x.Q(x)))
(4|α from 3) ¬(∃x.P(x))
(5|α from 3) ¬(∀x.Q(x))
(6|δ from 5) ¬Q(c)
γ γ[t]
∀x.Φ Φ[x←t]
¬∃x.Φ ¬Φ[x←t]
δ δ[c]
∃x.Φ Φ[x←c]
¬∀x.Φ ¬Φ[x←c]
Tableaux Algorithm (FOL) - Example(3):
![Page 49: OpenHPI 4.9 - Tableaux Algorithm](https://reader034.fdocuments.in/reader034/viewer/2022051014/54b8ac5b4a7959c6438b45a1/html5/thumbnails/49.jpg)
Lecture: Semantic Web Technologies, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam, WS 2012/13
14Proof: (∀x.P(x)∨Q(x))⇒(∃x.P(x))∨(∀x.Q(x))
(1) ¬((∀x.P(x)∨Q(x))⇒(∃x.P(x))∨(∀x.Q(x)))(2|α from 1) (∀x.P(x)∨Q(x))
(3|α from 1) ¬((∃x.P(x))∨(∀x.Q(x)))
(4|α from 3) ¬(∃x.P(x))
(5|α from 3) ¬(∀x.Q(x))
(6|δ from 5) ¬Q(c)
(7|γ from 4) ¬P(c)
γ γ[t]
∀x.Φ Φ[x←t]
¬∃x.Φ ¬Φ[x←t]
δ δ[c]
∃x.Φ Φ[x←c]
¬∀x.Φ ¬Φ[x←c]
Tableaux Algorithm (FOL) - Example(3):
![Page 50: OpenHPI 4.9 - Tableaux Algorithm](https://reader034.fdocuments.in/reader034/viewer/2022051014/54b8ac5b4a7959c6438b45a1/html5/thumbnails/50.jpg)
Lecture: Semantic Web Technologies, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam, WS 2012/13
14Proof: (∀x.P(x)∨Q(x))⇒(∃x.P(x))∨(∀x.Q(x))
(1) ¬((∀x.P(x)∨Q(x))⇒(∃x.P(x))∨(∀x.Q(x)))(2|α from 1) (∀x.P(x)∨Q(x))
(3|α from 1) ¬((∃x.P(x))∨(∀x.Q(x)))
(4|α from 3) ¬(∃x.P(x))
(5|α from 3) ¬(∀x.Q(x))
(6|δ from 5) ¬Q(c)
(7|γ from 4) ¬P(c)
(8|γ from 2) P(c)∨Q(c)
γ γ[t]
∀x.Φ Φ[x←t]
¬∃x.Φ ¬Φ[x←t]
δ δ[c]
∃x.Φ Φ[x←c]
¬∀x.Φ ¬Φ[x←c]
Tableaux Algorithm (FOL) - Example(3):
![Page 51: OpenHPI 4.9 - Tableaux Algorithm](https://reader034.fdocuments.in/reader034/viewer/2022051014/54b8ac5b4a7959c6438b45a1/html5/thumbnails/51.jpg)
Lecture: Semantic Web Technologies, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam, WS 2012/13
14Proof: (∀x.P(x)∨Q(x))⇒(∃x.P(x))∨(∀x.Q(x))
(9|β from 8) P(c) | (10|β from 8) Q(c)
(1) ¬((∀x.P(x)∨Q(x))⇒(∃x.P(x))∨(∀x.Q(x)))(2|α from 1) (∀x.P(x)∨Q(x))
(3|α from 1) ¬((∃x.P(x))∨(∀x.Q(x)))
(4|α from 3) ¬(∃x.P(x))
(5|α from 3) ¬(∀x.Q(x))
(6|δ from 5) ¬Q(c)
(7|γ from 4) ¬P(c)
(8|γ from 2) P(c)∨Q(c)
γ γ[t]
∀x.Φ Φ[x←t]
¬∃x.Φ ¬Φ[x←t]
δ δ[c]
∃x.Φ Φ[x←c]
¬∀x.Φ ¬Φ[x←c]
Tableaux Algorithm (FOL) - Example(3):
![Page 52: OpenHPI 4.9 - Tableaux Algorithm](https://reader034.fdocuments.in/reader034/viewer/2022051014/54b8ac5b4a7959c6438b45a1/html5/thumbnails/52.jpg)
Lecture: Semantic Web Technologies, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam, WS 2012/13
14Proof: (∀x.P(x)∨Q(x))⇒(∃x.P(x))∨(∀x.Q(x))
(9|β from 8) P(c) | (10|β from 8) Q(c)
(1) ¬((∀x.P(x)∨Q(x))⇒(∃x.P(x))∨(∀x.Q(x)))(2|α from 1) (∀x.P(x)∨Q(x))
(3|α from 1) ¬((∃x.P(x))∨(∀x.Q(x)))
(4|α from 3) ¬(∃x.P(x))
(5|α from 3) ¬(∀x.Q(x))
(6|δ from 5) ¬Q(c)
(7|γ from 4) ¬P(c)
(8|γ from 2) P(c)∨Q(c)
γ γ[t]
∀x.Φ Φ[x←t]
¬∃x.Φ ¬Φ[x←t]
δ δ[c]
∃x.Φ Φ[x←c]
¬∀x.Φ ¬Φ[x←c]
Tableaux Algorithm (FOL) - Example(3):
![Page 53: OpenHPI 4.9 - Tableaux Algorithm](https://reader034.fdocuments.in/reader034/viewer/2022051014/54b8ac5b4a7959c6438b45a1/html5/thumbnails/53.jpg)
Lecture: Semantic Web Technologies, Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam, WS 2012/13
14Proof: (∀x.P(x)∨Q(x))⇒(∃x.P(x))∨(∀x.Q(x))
(9|β from 8) P(c) | (10|β from 8) Q(c)
(1) ¬((∀x.P(x)∨Q(x))⇒(∃x.P(x))∨(∀x.Q(x)))(2|α from 1) (∀x.P(x)∨Q(x))
(3|α from 1) ¬((∃x.P(x))∨(∀x.Q(x)))
(4|α from 3) ¬(∃x.P(x))
(5|α from 3) ¬(∀x.Q(x))
(6|δ from 5) ¬Q(c)
(7|γ from 4) ¬P(c)
(8|γ from 2) P(c)∨Q(c)
γ γ[t]
∀x.Φ Φ[x←t]
¬∃x.Φ ¬Φ[x←t]
δ δ[c]
∃x.Φ Φ[x←c]
¬∀x.Φ ¬Φ[x←c]
Tableaux Algorithm (FOL) - Example(3):
![Page 54: OpenHPI 4.9 - Tableaux Algorithm](https://reader034.fdocuments.in/reader034/viewer/2022051014/54b8ac5b4a7959c6438b45a1/html5/thumbnails/54.jpg)
Semantic Web Technologies , Dr. Harald Sack, Hasso-Plattner-Institut, Universität Potsdam
15
Lecture 5: Knowledge Representations II
Open HPI - Course: Semantic Web Technologies