

last edited 3 years ago by test1 
1  
Editor: test1
Time: 2019/07/26 22:38:48 GMT+0 

Note: 
changed:  FriCAS expression domains are based on notion of differential field. This have advantages, for example some otherwise unsolvable problems became solvable when dealing with differential fields. But this also means that FriCAS expression behave differently than function in real analysis. For example, in field quadratic equation have 0, 1 or 2 roots. In particular consider equation $y^2 = x^2$ where $y$ is variable of the equation and $x$ is element of differential field representing variable $x$. This equation have exactly two roots, namely $y = x$ and $y = x$. This means that absolute value of real function is excluded, since for real $x$ we have $x^2 = x^2$, so $x$ would be solution to $y^2 = x^2$, but $x$ is different than $x$ or $x$. In fact, forbiding absolute value is crucial to solvability  Richardson proved that equality for expressions build from variable $x$, rational constants, constant $\pi$, trigonometric functions and absolute values by composition and arithmetic operations is undecidable. When we put functions above inside a differential field equality becomes decidable (note: "put inside" means that we provide extra information). By Seidenberg theorem finitely generated differential field of characteristic $0$ is izomorphic to a subfield of field of meromorphic functions in some complex domain (open and connected set). And opposite: field of meromorphic functions in complex domain is a differential field. So differential field correspond very well to complex analysis. But there may be some mismatch when one wants to deal with real analysis. Dealing with differential field we implicitly remove inessential singularities, so we treat $1$ and $x/x$ as the same function (for more background see [Division by zero during evaluation]). Also, by principle of analytic continuation we may restrict functions to arbitrarily small open set and still get izomorphic field. Which means that to get interesting results we may be forced to perform analytic continuation. Since we work with connected sets, we have to chose some branch of function which otherwise is considered as multivalued function. Notably, we have to chose branches of logarithms and roots. Interestingly, in many cases choice of branches does not matter much, we get equivalent results for different choices. What matters are dependencies. For example at algebraic level it does not matter which square roots of $2$ and $3$ we chose, all four choices give isomorphic fields. However, when we add square root of $6$, we get dependence, either $\sqrt{6} = \sqrt{2}\sqrt{3}$ or $\sqrt{6} = \sqrt{2}\sqrt{3}$. In numerical computations it is usual to make choice of root, for example to use so called principal branch of square root. However, in purely algebraic context there is no canonical way to distinguish roots and all roots of the same irreducible equation play equivalent role. In context of differential fields logarithm is determined up to additive constant and there is no way to have canonical choice of constant, all choices lead to izomorphic differential fields. Compared to some other systems FriCAS approach have both advantages:  FriCAS avoids unsolvability due to Richardson theorem  there are powerful algorithms working in differential fields but also some disadvantages. Main limitation is that with FriCAS approach it is harder to handle solvable instances of problems involving absolute value.
FriCAS expression domains are based on notion of differential field. This have advantages, for example some otherwise unsolvable problems became solvable when dealing with differential fields. But this also means that FriCAS expression behave differently than function in real analysis. For example, in field quadratic equation have 0, 1 or 2 roots. In particular consider equation where is variable of the equation and is element of differential field representing variable . This equation have exactly two roots, namely and . This means that absolute value of real function is excluded, since for real we have , so would be solution to , but is different than or . In fact, forbiding absolute value is crucial to solvability  Richardson proved that equality for expressions build from variable , rational constants, constant , trigonometric functions and absolute values by composition and arithmetic operations is undecidable. When we put functions above inside a differential field equality becomes decidable (note: "put inside" means that we provide extra information).
By Seidenberg theorem finitely generated differential field of characteristic is izomorphic to a subfield of field of meromorphic functions in some complex domain (open and connected set). And opposite: field of meromorphic functions in complex domain is a differential field. So differential field correspond very well to complex analysis. But there may be some mismatch when one wants to deal with real analysis. Dealing with differential field we implicitly remove inessential singularities, so we treat and as the same function (for more background see Division by zero during evaluation). Also, by principle of analytic continuation we may restrict functions to arbitrarily small open set and still get izomorphic field. Which means that to get interesting results we may be forced to perform analytic continuation. Since we work with connected sets, we have to chose some branch of function which otherwise is considered as multivalued function. Notably, we have to chose branches of logarithms and roots. Interestingly, in many cases choice of branches does not matter much, we get equivalent results for different choices. What matters are dependencies. For example at algebraic level it does not matter which square roots of and we chose, all four choices give isomorphic fields. However, when we add square root of , we get dependence, either or . In numerical computations it is usual to make choice of root, for example to use so called principal branch of square root. However, in purely algebraic context there is no canonical way to distinguish roots and all roots of the same irreducible equation play equivalent role. In context of differential fields logarithm is determined up to additive constant and there is no way to have canonical choice of constant, all choices lead to izomorphic differential fields.
Compared to some other systems FriCAS approach have both advantages:
but also some disadvantages. Main limitation is that with FriCAS approach it is harder to handle solvable instances of problems involving absolute value.