Introduction
Some methods for ATP in Propositional Classical Logic
Examples in Derive
A stepwise graphical approach for teaching
Automated Theorem Proving in Engineering
Gabriel Aguilera, José Luis Galán, Antonio Gálvez
Yolanda Padilla, Pedro Rodrı́guez, Ricardo Rodrı́guez
Department of Applied Mathematics.
University of Málaga - Spain
Technology and its Integration into Mathematics Education
TIME 2012
July 10-14. Tartu. Estonia
Aguilera, Galán, Gálvez, Padilla, Rodrı́guez, Rodrı́guez
A stepwise graphical approach for teaching ATPs
university-logo
1
Introduction
Some methods for ATP in Propositional Classical Logic
Examples in Derive
Contents
1
Introduction
Type of problems solved by ATP
Built-in boolean functions in Derive
university-logo
Aguilera, Galán, Gálvez, Padilla, Rodrı́guez, Rodrı́guez
A stepwise graphical approach for teaching ATPs
2
Introduction
Some methods for ATP in Propositional Classical Logic
Examples in Derive
Contents
1
Introduction
Type of problems solved by ATP
Built-in boolean functions in Derive
2
Some methods for ATP in Propositional Classical Logic
Quine’s Method
Short Normal Forms + Resolution
Semantic Tableaux
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Aguilera, Galán, Gálvez, Padilla, Rodrı́guez, Rodrı́guez
A stepwise graphical approach for teaching ATPs
3
Introduction
Some methods for ATP in Propositional Classical Logic
Examples in Derive
Contents
1
Introduction
Type of problems solved by ATP
Built-in boolean functions in Derive
2
Some methods for ATP in Propositional Classical Logic
Quine’s Method
Short Normal Forms + Resolution
Semantic Tableaux
3
Examples in Derive
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Aguilera, Galán, Gálvez, Padilla, Rodrı́guez, Rodrı́guez
A stepwise graphical approach for teaching ATPs
4
Introduction
Some methods for ATP in Propositional Classical Logic
Examples in Derive
Type of problems solved by ATP
Built-in boolean functions in Derive
Introduction
Automated Theorem Proving (ATP) is one of the most important concepts when teaching Propositional Classical Logic in
Computer Sciences degrees.
university-logo
Aguilera, Galán, Gálvez, Padilla, Rodrı́guez, Rodrı́guez
A stepwise graphical approach for teaching ATPs
5
Introduction
Some methods for ATP in Propositional Classical Logic
Examples in Derive
Type of problems solved by ATP
Built-in boolean functions in Derive
Introduction
Automated Theorem Proving (ATP) is one of the most important concepts when teaching Propositional Classical Logic in
Computer Sciences degrees.
An ATP is an algorithm that checks the validity of a formula.
university-logo
Aguilera, Galán, Gálvez, Padilla, Rodrı́guez, Rodrı́guez
A stepwise graphical approach for teaching ATPs
6
Introduction
Some methods for ATP in Propositional Classical Logic
Examples in Derive
Type of problems solved by ATP
Built-in boolean functions in Derive
Introduction
Automated Theorem Proving (ATP) is one of the most important concepts when teaching Propositional Classical Logic in
Computer Sciences degrees.
An ATP is an algorithm that checks the validity of a formula.
Some methods for ATP in Propositional Classical Logic will be
described.
university-logo
Aguilera, Galán, Gálvez, Padilla, Rodrı́guez, Rodrı́guez
A stepwise graphical approach for teaching ATPs
7
Introduction
Some methods for ATP in Propositional Classical Logic
Examples in Derive
Type of problems solved by ATP
Built-in boolean functions in Derive
Introduction
Automated Theorem Proving (ATP) is one of the most important concepts when teaching Propositional Classical Logic in
Computer Sciences degrees.
An ATP is an algorithm that checks the validity of a formula.
Some methods for ATP in Propositional Classical Logic will be
described.
Examples of executions of these methods will be shown using
Derive.
university-logo
Aguilera, Galán, Gálvez, Padilla, Rodrı́guez, Rodrı́guez
A stepwise graphical approach for teaching ATPs
8
Introduction
Some methods for ATP in Propositional Classical Logic
Examples in Derive
Type of problems solved by ATP
Built-in boolean functions in Derive
Introduction
Automated Theorem Proving (ATP) is one of the most important concepts when teaching Propositional Classical Logic in
Computer Sciences degrees.
An ATP is an algorithm that checks the validity of a formula.
Some methods for ATP in Propositional Classical Logic will be
described.
Examples of executions of these methods will be shown using
Derive.
A graphical approach of the execution of Semantic Tableaux
method will be also shown.
university-logo
Aguilera, Galán, Gálvez, Padilla, Rodrı́guez, Rodrı́guez
A stepwise graphical approach for teaching ATPs
9
Introduction
Some methods for ATP in Propositional Classical Logic
Examples in Derive
Type of problems solved by ATP
Built-in boolean functions in Derive
Type of problems solved by ATP
The kind of exercises that can be solved using an ATP can be
grouped in the following three Logic problems:
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Aguilera, Galán, Gálvez, Padilla, Rodrı́guez, Rodrı́guez
A stepwise graphical approach for teaching ATPs
10
Introduction
Some methods for ATP in Propositional Classical Logic
Examples in Derive
Type of problems solved by ATP
Built-in boolean functions in Derive
Type of problems solved by ATP
The kind of exercises that can be solved using an ATP can be
grouped in the following three Logic problems:
Checking the satisfiability of a formula (SAT). Satisfiability is
the problem of determining if there exists an interpretation that
assigns TRUE to the given formula.
university-logo
Aguilera, Galán, Gálvez, Padilla, Rodrı́guez, Rodrı́guez
A stepwise graphical approach for teaching ATPs
11
Introduction
Some methods for ATP in Propositional Classical Logic
Examples in Derive
Type of problems solved by ATP
Built-in boolean functions in Derive
Type of problems solved by ATP
The kind of exercises that can be solved using an ATP can be
grouped in the following three Logic problems:
Checking the satisfiability of a formula (SAT). Satisfiability is
the problem of determining if there exists an interpretation that
assigns TRUE to the given formula.
Checking the validity of a formula (TAUT). A formula of propositional logic is a tautology (or it is valid) if all possible interpretations assign TRUE to the formula.
university-logo
Aguilera, Galán, Gálvez, Padilla, Rodrı́guez, Rodrı́guez
A stepwise graphical approach for teaching ATPs
12
Introduction
Some methods for ATP in Propositional Classical Logic
Examples in Derive
Type of problems solved by ATP
Built-in boolean functions in Derive
Type of problems solved by ATP
The kind of exercises that can be solved using an ATP can be
grouped in the following three Logic problems:
Checking the satisfiability of a formula (SAT). Satisfiability is
the problem of determining if there exists an interpretation that
assigns TRUE to the given formula.
Checking the validity of a formula (TAUT). A formula of propositional logic is a tautology (or it is valid) if all possible interpretations assign TRUE to the formula.
Checking the correctness of an inference, that is, determining
if a formula (conclusion) can be logically deducted from a set
of formulae (hypothesis).
university-logo
Aguilera, Galán, Gálvez, Padilla, Rodrı́guez, Rodrı́guez
A stepwise graphical approach for teaching ATPs
13
Introduction
Some methods for ATP in Propositional Classical Logic
Examples in Derive
Type of problems solved by ATP
Built-in boolean functions in Derive
Built-in boolean functions in Derive
Derive deals with NOT, OR, AND, XOR, IMP and IFF connectives.
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Aguilera, Galán, Gálvez, Padilla, Rodrı́guez, Rodrı́guez
A stepwise graphical approach for teaching ATPs
14
Introduction
Some methods for ATP in Propositional Classical Logic
Examples in Derive
Type of problems solved by ATP
Built-in boolean functions in Derive
Built-in boolean functions in Derive
Derive deals with NOT, OR, AND, XOR, IMP and IFF connectives.
Formulae are simplified (or expanded) using different rules.
university-logo
Aguilera, Galán, Gálvez, Padilla, Rodrı́guez, Rodrı́guez
A stepwise graphical approach for teaching ATPs
15
Introduction
Some methods for ATP in Propositional Classical Logic
Examples in Derive
Type of problems solved by ATP
Built-in boolean functions in Derive
Built-in boolean functions in Derive
Derive deals with NOT, OR, AND, XOR, IMP and IFF connectives.
Formulae are simplified (or expanded) using different rules.
The result is not an ATP.
university-logo
Aguilera, Galán, Gálvez, Padilla, Rodrı́guez, Rodrı́guez
A stepwise graphical approach for teaching ATPs
16
Introduction
Some methods for ATP in Propositional Classical Logic
Examples in Derive
Type of problems solved by ATP
Built-in boolean functions in Derive
Built-in boolean functions in Derive
Derive deals with NOT, OR, AND, XOR, IMP and IFF connectives.
Formulae are simplified (or expanded) using different rules.
The result is not an ATP.
The only built-in method in DERIVE is the truth table method.
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Aguilera, Galán, Gálvez, Padilla, Rodrı́guez, Rodrı́guez
A stepwise graphical approach for teaching ATPs
17
Introduction
Some methods for ATP in Propositional Classical Logic
Examples in Derive
Quine’s Method
Short Normal Forms + Resolution
Semantic Tableaux
Quine’s Method
Quine’s method is a variant of the truth tables method but
improving it using partial interpretations.
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Aguilera, Galán, Gálvez, Padilla, Rodrı́guez, Rodrı́guez
A stepwise graphical approach for teaching ATPs
18
Introduction
Some methods for ATP in Propositional Classical Logic
Examples in Derive
Quine’s Method
Short Normal Forms + Resolution
Semantic Tableaux
Short Normal Forms + Resolution
Short Normal Forms: Renaming method due to Boy de la Tour.
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Aguilera, Galán, Gálvez, Padilla, Rodrı́guez, Rodrı́guez
A stepwise graphical approach for teaching ATPs
19
Introduction
Some methods for ATP in Propositional Classical Logic
Examples in Derive
Quine’s Method
Short Normal Forms + Resolution
Semantic Tableaux
Short Normal Forms + Resolution
Short Normal Forms: Renaming method due to Boy de la Tour.
Ground resolution: The basical resolution method.
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Aguilera, Galán, Gálvez, Padilla, Rodrı́guez, Rodrı́guez
A stepwise graphical approach for teaching ATPs
20
Introduction
Some methods for ATP in Propositional Classical Logic
Examples in Derive
Quine’s Method
Short Normal Forms + Resolution
Semantic Tableaux
Semantic Tableaux: Uniform notation (Smullyan)
Uniform Notation (Smullyan)
α
A∧B
¬(A ∨ B)
¬(A → B)
¬¬A
α1
A
¬A
A
A
α2
B
¬B
¬B
A
β
A∨B
¬(A ∧ B)
A→B
A↔B
¬(A ↔ B)
β1
A
¬A
¬A
A∧B
¬A ∧ B
β2
B
¬B
B
¬A ∧ ¬B
A ∧ ¬B
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Aguilera, Galán, Gálvez, Padilla, Rodrı́guez, Rodrı́guez
A stepwise graphical approach for teaching ATPs
21
Introduction
Some methods for ATP in Propositional Classical Logic
Examples in Derive
Quine’s Method
Short Normal Forms + Resolution
Semantic Tableaux
Semantic Tableaux (Rules)
α-rule
T =
α1
|
α2
β-rule
V
T =
β1 β2
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Aguilera, Galán, Gálvez, Padilla, Rodrı́guez, Rodrı́guez
A stepwise graphical approach for teaching ATPs
22
Introduction
Some methods for ATP in Propositional Classical Logic
Examples in Derive
Quine’s Method
Short Normal Forms + Resolution
Semantic Tableaux
Semantic Tableaux (Definitions)
A formula is marked when an α-rule or a β-rule is applied to it.
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Aguilera, Galán, Gálvez, Padilla, Rodrı́guez, Rodrı́guez
A stepwise graphical approach for teaching ATPs
23
Introduction
Some methods for ATP in Propositional Classical Logic
Examples in Derive
Quine’s Method
Short Normal Forms + Resolution
Semantic Tableaux
Semantic Tableaux (Definitions)
A formula is marked when an α-rule or a β-rule is applied to it.
A branch of a Jeffrey’s tree is close when a formula and its
negation occurs in the branch.
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Aguilera, Galán, Gálvez, Padilla, Rodrı́guez, Rodrı́guez
A stepwise graphical approach for teaching ATPs
24
Introduction
Some methods for ATP in Propositional Classical Logic
Examples in Derive
Quine’s Method
Short Normal Forms + Resolution
Semantic Tableaux
Semantic Tableaux (Definitions)
A formula is marked when an α-rule or a β-rule is applied to it.
A branch of a Jeffrey’s tree is close when a formula and its
negation occurs in the branch.
A branch of a Jeffrey’s tree is open when it is not close and all
the formulae in the branch are literals or marked.
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Aguilera, Galán, Gálvez, Padilla, Rodrı́guez, Rodrı́guez
A stepwise graphical approach for teaching ATPs
25
Introduction
Some methods for ATP in Propositional Classical Logic
Examples in Derive
Quine’s Method
Short Normal Forms + Resolution
Semantic Tableaux
Semantic Tableaux (Algorithm)
1
Take the one branch of the Jeffrey’s tree corresponding to S =
{A1 , A2 , . . . , An }.
2
If a branch of the Jeffrey’s tree is open then S is satisfiable,
literals in the branch correspond to a model for S and END.
3
If all the branches of the Jeffrey’s tree are closed then S is
unsatisfiable and END.
4
If a formula α without mark exists, apply an α-rule to the first
in deep formula α, else got to step 6.
5
Review open and closed branches and go to step 2.
6
If a formula β without mark exists, apply a β-rule to the first
in deep formula β.
7
Review open and closed branches and go to step 2.
university-logo
Aguilera, Galán, Gálvez, Padilla, Rodrı́guez, Rodrı́guez
A stepwise graphical approach for teaching ATPs
26
Introduction
Some methods for ATP in Propositional Classical Logic
Examples in Derive
Examples in Derive
university-logo
Aguilera, Galán, Gálvez, Padilla, Rodrı́guez, Rodrı́guez
A stepwise graphical approach for teaching ATPs
27
Introduction
Some methods for ATP in Propositional Classical Logic
Examples in Derive
A stepwise graphical approach for teaching
Automated Theorem Proving in Engineering
Gabriel Aguilera, José Luis Galán, Antonio Gálvez
Yolanda Padilla, Pedro Rodrı́guez, Ricardo Rodrı́guez
Department of Applied Mathematics.
University of Málaga - Spain
Technology and its Integration into Mathematics Education
TIME 2012
July 10-14. Tartu. Estonia
Aguilera, Galán, Gálvez, Padilla, Rodrı́guez, Rodrı́guez
A stepwise graphical approach for teaching ATPs
university-logo
28