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sav08:applications_of_quantifier_elimination [2009/04/22 23:26] vkuncak |
sav08:applications_of_quantifier_elimination [2010/02/22 11:30] vkuncak |
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| we obtain formula | we obtain formula | ||
| \[ | \[ | ||
| - | x \le z | + | y \le z |
| \] | \] | ||
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| \] | \] | ||
| - | ===== Deciding Formulas ===== | + | We review several uses of quantifier elimination. |
| + | |||
| + | ===== Deciding Formula Validity and Satisfiability using Quantifier Elimination ===== | ||
| **Quantified formulas:** | **Quantified formulas:** | ||
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| - | ===== Do we need quantifiers when there is QE? ===== | + | |
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| + | ===== The Purpose of Quantifiers ===== | ||
| Suppose we want to describe a set of states of an integer program manipulating varibles $x,y$. We describe it by a formula $F(x,y)$ in theory admitting QE. | Suppose we want to describe a set of states of an integer program manipulating varibles $x,y$. We describe it by a formula $F(x,y)$ in theory admitting QE. | ||
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| S = \{ (x,y) \mid F'(x,y) \} | S = \{ (x,y) \mid F'(x,y) \} | ||
| \] | \] | ||
| - | Are quantifiers useless? | + | Are quantifiers useful when there is qe? |
| - | * to an extent | + | * to an extent, qe does show quantifiers are not necessary |
| * $F'$ can be (more than) exponentially larger than $F$ | * $F'$ can be (more than) exponentially larger than $F$ | ||
| - | * sometimes quantifiers are natural | + | * sometimes quantifiers naturally come up when encoding the problem |
| + | |||
| ===== Quantifier Elimination in Relation Composition ===== | ===== Quantifier Elimination in Relation Composition ===== | ||
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| What class of relations is defined by integer programs without loops in the syntax of [[Simple Programming Language]] ? | What class of relations is defined by integer programs without loops in the syntax of [[Simple Programming Language]] ? | ||
| - | ++++| Formulas in Presburger arithmetic. | + | ++++| The class given by formulas in Presburger arithmetic. |
| ++++ | ++++ | ||
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| Example reference: | Example reference: | ||
| * [[http://dx.doi.org/10.1006/jsco.1997.0122|Simulation and Optimization by Quantifier Elimination]] | * [[http://dx.doi.org/10.1006/jsco.1997.0122|Simulation and Optimization by Quantifier Elimination]] | ||
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| 3) $FV(H)\subseteq FV(F) \cap FV(G)$ | 3) $FV(H)\subseteq FV(F) \cap FV(G)$ | ||
| + | {{sav08:mcmillaninterpolation.pdf|Paper on Interpolation in Verification}} | ||
| Here are two **simple ways to construct an interpolant:** | Here are two **simple ways to construct an interpolant:** | ||
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| * [[http://www.springerlink.com/content/n674875430157r80/|A New Approach for Automatic Theorem Proving in Real Geometry]] | * [[http://www.springerlink.com/content/n674875430157r80/|A New Approach for Automatic Theorem Proving in Real Geometry]] | ||
| + | * [[:Harrison textbook]] | ||