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sav08:introductory_remarks_on_smt_provers [2009/05/06 09:25] vkuncak |
sav08:introductory_remarks_on_smt_provers [2015/04/21 17:30] (current) |
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| ===== SMT Prover Architecture ===== | ===== SMT Prover Architecture ===== | ||
| - | Two layers: | + | Layers: |
| * fast solving of quantifier-free formulas (see [[http://citeseer.ist.psu.edu/671087.html|DPLL(T) paper]]) | * fast solving of quantifier-free formulas (see [[http://citeseer.ist.psu.edu/671087.html|DPLL(T) paper]]) | ||
| * SAT solver to 'enumerate' conjunctions of disjunctive normal form ([[sav09:Lecture 08a]]) | * SAT solver to 'enumerate' conjunctions of disjunctive normal form ([[sav09:Lecture 08a]]) | ||
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| Note: formulas that are valid in the combination of quantifiers, linear integers, and uninterpreted functions are not enumerable (and neither are formulas that are not valid). The situation is worse than in pure first-order logic. | Note: formulas that are valid in the combination of quantifiers, linear integers, and uninterpreted functions are not enumerable (and neither are formulas that are not valid). The situation is worse than in pure first-order logic. | ||
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| ===== Example ===== | ===== Example ===== | ||
| Consider the following formula, where $f$ denotes functions from integers to integers and $a,b$ are integers. | Consider the following formula, where $f$ denotes functions from integers to integers and $a,b$ are integers. | ||
| - | \[\begin{array}{l} | + | \begin{equation*}\begin{array}{l} |
| (\forall x.\forall y.\ x \le y \rightarrow f(x) \le f(y))\ \land \\ | (\forall x.\forall y.\ x \le y \rightarrow f(x) \le f(y))\ \land \\ | ||
| (\forall x.\forall y. f(x)=f(y) \rightarrow x=y)\ \land\\ | (\forall x.\forall y. f(x)=f(y) \rightarrow x=y)\ \land\\ | ||
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| b < a | b < a | ||
| \end{array} | \end{array} | ||
| - | \] | + | \end{equation*} |
| + | Is this formula satisfiable? | ||
| + | * over integers ++|No. Why?++ | ||
| + | * over reals ++|Yes. Why?++ | ||