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Some tweaks

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Lars Tveito 2019-11-17 20:22:12 +01:00
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@ -53,7 +53,7 @@
$\mathcal{M}$ such that $\mathcal{M} \models \phi$. In general, this is an $\mathcal{M}$ such that $\mathcal{M} \models \phi$. In general, this is an
undecidable problem. However, there are theories within first order logic undecidable problem. However, there are theories within first order logic
that are decidable. SMT solvers can produce models that satisfy a set of that are decidable. SMT solvers can produce models that satisfy a set of
formulas for many useful theories. formulas for many useful theories, some of which are satisfiable.
The solver we will be using is [[https://github.com/Z3Prover/z3][Z3]]. The solver we will be using is [[https://github.com/Z3Prover/z3][Z3]].
@ -121,14 +121,15 @@
#+END_EXAMPLE #+END_EXAMPLE
The first line ~sat~ indicates that the formula is satisfiable, and produce The first line ~sat~ indicates that the formula is satisfiable, and produce
a model where $a=3$, $b=4$ and $c=5$. a model where $a^\mathcal{M}=3$, $b^\mathcal{M}=4$ and $c^\mathcal{M}=5$.
** Many-sorted first order logic ** Many-sorted first order logic
Z3 implements [[http://smtlib.cs.uiowa.edu/papers/smt-lib-reference-v2.6-r2017-07-18.pdf][SMT-LIB]], a standardized syntax and semantics for SMT solvers. Z3 implements [[http://smtlib.cs.uiowa.edu/papers/smt-lib-reference-v2.6-r2017-07-18.pdf][SMT-LIB]], a standardized syntax and semantics for SMT solvers.
It's underlying logic is a /Many-sorted first order logic/, where values It's underlying logic is a /Many-sorted first order logic/, where values
must have an associated sort (which is a basic form of type). A signature in must have an associated sort (which is a basic form of type). Think of it as
a many-sorted first logic is defined as follows. partitioning the domain, where each sort corresponds to a part. A signature
in a many-sorted first logic is defined as follows.
#+BEGIN_definition #+BEGIN_definition
A signature $\Sigma = (S, F, P)$ consists of a countable set of A signature $\Sigma = (S, F, P)$ consists of a countable set of
@ -146,15 +147,11 @@
For example, the signature for the integers can be formalized as For example, the signature for the integers can be formalized as
$\Sigma_{int} = (S_{Int}, F_{Int}, P_{Int})$ where $\Sigma_{int} = (S_{Int}, F_{Int}, P_{Int})$ where
- $S_{Int} = \{Int\}$ - $S_{Int} = \{Int\}$
- $F_{Int} = \{0, 1, +, -, *\}$ where - $F_{Int} = \{0, 1, +, -, *\}$ where the constant symbols $0, 1$ has a type
- $0 : \to Int$ signature $\to Int$ and the function symbols $+,-,*$ has a type signature
- $1 : \to Int$ $Int \times Int \to Int$.
- $+ : Int \times Int \to Int$ - $P_{Int} = \{<, =\}$ where the predicate symbols $<, =$ has type signature
- $- : Int \times Int \to Int$ $Int \times Int$.
- $* : Int \times Int \to Int$
- $P_{Int} = \{<, =\}$ where
- $< : Int \times Int$
- $= : Int \times Int$
* Back to the problem * Back to the problem