5 ms·

Along the same lines, does there exist an "algebra checker" that could, say, take in two successive lines of latex, with perhaps a hint of how to get from one t

by mjfl 9y ago

Along the same lines, does there exist an "algebra checker" that could, say, take in two successive lines of latex, with perhaps a hint of how to get from one to the other, and confirm that there are no algebra errors?

chillydawg 9y ago
Mathematica can probably do that
- Smaug123 9y ago
  Correct: Implies[x^2 + 2 x + 1 == 0, x == -1] // FullSimplify
- glinia 9y ago
  [In] FullSimplify[expr1 == expr2] [Out] True
CodesInChaos 9y ago
The choice of LaTeX comes with two problems: 1) It's designed for describe graphical representation and not semantics. 2) parsing it properly is very difficult, especially considering how macros and other definitions can influence the syntax.
- yorwba 9y ago
  You could: 1) assume a standard interpretation of symbols (e.g. this site uses a-g for scalars, h-z for vectors and A-Z for matrices) and make those assumptions explicit and modifiable (e.g. WolframAlpha displays "assuming X is a Y, use as a Z instead"). 2) Support only the subset that MathJax can handle, which seems to be enough for most purposes. I definitely agree that LaTeX is not the optimal input format for that purpose, though.
  - qubex 9y ago
    See my comment above about my experiences with this: I tried, and a corner case killed me. There’s a reason why Mathematica and other CAS packages have invented their own idiosyncratic notations. LATEX and MathML are truly hopeless when it comes to communicating with algebra systems. It looks deceptively ’doable’, but really it’s a recipe for disaster.
eitland 9y ago
I think there used to be a open source tool called Maxima or something that would simplify equations. Havent used it for the last 10 years and never used it much anyway.
- lottin 9y ago
  I can recommended Maxima too. It's useful for checking for obvious errors and for simplifying expressions. It can be used for more advanced stuff, but I found it has a steep learning curve.
cosmic_ape 9y ago
SymPy does not take Latex as input, but it has symbolic equivalence checkers. Methods are highly heuristic, and some languages (all expressions with,say, \pi, and exp function, or something similar) are in general undecidable.
- rav 9y ago
  To elaborate: Symbolic equivalence checking for two expressions f1 and f2 can be done by seeing if sympy.simplify(sympy.Eq(f1, f2)) is sympy.true. import sympy from sympy.abc import x, y, alpha, s quad = s ** 2 - alpha * s - 2 # Let s1 and s2 be the two solutions to the quadratic equation 'quad == 0' s1, s2 = sympy.solve(quad, s) u = (x - s2) / (x - s1) * (y - s1) / (y - s2) f1 = (s2 - s1 * u) / (1 - u) f2 = (x * y - alpha * x - 2) / (y - x) # Claim: f1 is equal to f2 print(sympy.simplify(sympy.Eq(f1, f2))) # Prints "True"
  - yorwba 9y ago
    Is there something for implication or equivalence of equations? I was trying the example from Smaug123's sibling comment (https://news.ycombinator.com/item?id=15728598 https://news.ycombinator.com/item?id=15728598), but >>> import sympy as sp >>> x = sp.Symbol('x') >>> sp.simplify(sp.Implies(sp.Eq(x**2 + 2*x + 1, 0), sp.Eq(x, -1))) Eq(x, -1) | Ne(x**2 + 2*x + 1, 0) >>> sp.solve(sp.simplify(sp.Implies(sp.Eq(x**2 + 2*x + 1, 0), sp.Eq(x, -1)))) Traceback (most recent call last): File "<stdin>", line 1, in <module> File "/home/user/.local/lib/python3.5/site-packages/sympy/solvers/solvers.py", line 1065, in solve solution = _solve(f[0], *symbols, **flags) File "/home/user/.local/lib/python3.5/site-packages/sympy/solvers/solvers.py", line 1401, in _solve f_num, sol = solve_linear(f, symbols=symbols) File "/home/user/.local/lib/python3.5/site-packages/sympy/solvers/solvers.py", line 1971, in solve_linear eq = lhs - rhs TypeError: unsupported operand type(s) for -: 'Or' and 'int' >>> sp.solveset(sp.simplify(sp.Implies(sp.Eq(x**2 + 2*x + 1, 0), sp.Eq(x, -1)))) Traceback (most recent call last): File "<stdin>", line 1, in <module> File "/home/user/.local/lib/python3.5/site-packages/sympy/solvers/solveset.py", line 880, in solveset raise ValueError("%s is not a valid SymPy expression" % (f)) ValueError: Eq(x, -1) | Ne(x**2 + 2*x + 1, 0) is not a valid SymPy expression none of the obvious ways appear to work. Does Sympy not support this kind of equational reasoning?
ivan_ah 9y ago
I'm with cosmic_ape regarding the recommendation for SymPy. Write the two lines as symbolic expressions `expr1` and `expr2` then run `simplify(expr1-expr2)` to check if the answer is zero. If you want to start with LaTeX then the parser from this library might be useful: https://github.com/Khan/KAS https://github.com/Khan/KAS
qubex 9y ago
On a whim I programmed a symbolic integral-solving library in Python (I am not a professional programmer) and, as an ill-advised ’extension’, bolted on the ability to receive and emit LATEX expressions. Long story short, I was boasting about it on Nichols Nassim Taleb’s twitter feed, submitted an answer to a problem question he had posed that got mis-parsed by the front end, gave the wrong answer (which was entirely consistent, because the check stage compared the answer by deriving the flawed question), and got banned, flamed, and persecuted to Hell and back. True story. From this I learnt the hard way that tokenisation and representation (in LATEX or MathML) do not belong in the same place as a CAS (Computer-assisted Algebra System).
- yorwba 9y ago
  I believe that LaTeX/MathML absolutely do belong in a CAS, just not as the primary means of interaction. (I.e. you do your work in the CAS's own language and then export to LaTeX.) But maybe your example can convince me otherwise. Could you show the specific LaTeX code in question and describe how your program mishandled it?
  - qubex 9y ago
    I had set myself the target of implementing the Risch Algorithm (https://en.wikipedia.org/wiki/Risch_algorithm https://en.wikipedia.org/wiki/Risch_algorithm) in SymPy (3) but the event was so embarrassing I deleted my code and never published it publicly. I’m not a programmer. I erred in trying to do a programmer’s job. Also, I discovered why Nicholas Nassim Taleb has the reputation for being rather uncharitable (but deep down I feel like I deserved it, because hey, I stated a mathematical untruth).
SoerenL 9y ago
As one of the authors of this tool I understand that you would like to have some way of trusting the output. Internally we check it numerically, i.e., generate some random data for the given variables and check the derivative by comparing it to an approximation via finite differences. We will ship this code (hopefully soon) with one of the next versions. You can then check it yourself. Otherwise, as far as I know there does not exist any other tool that computes matrix derivatives so I understand it is hard to convince anyone of the correctness of the results. But I hope the numerical tests will be helpful.

by mjfl 9y ago

chillydawg 9y ago

Mathematica can probably do that

Smaug123 9y ago
Correct: Implies[x^2 + 2 x + 1 == 0, x == -1] // FullSimplify
glinia 9y ago
[In] FullSimplify[expr1 == expr2] [Out] True

CodesInChaos 9y ago

The choice of LaTeX comes with two problems: 1) It's designed for describe graphical representation and not semantics. 2) parsing it properly is very difficult, especially considering how macros and other definitions can influence the syntax.

yorwba 9y ago
You could: 1) assume a standard interpretation of symbols (e.g. this site uses a-g for scalars, h-z for vectors and A-Z for matrices) and make those assumptions explicit and modifiable (e.g. WolframAlpha displays "assuming X is a Y, use as a Z instead"). 2) Support only the subset that MathJax can handle, which seems to be enough for most purposes. I definitely agree that LaTeX is not the optimal input format for that purpose, though.
- qubex 9y ago
  See my comment above about my experiences with this: I tried, and a corner case killed me. There’s a reason why Mathematica and other CAS packages have invented their own idiosyncratic notations. LATEX and MathML are truly hopeless when it comes to communicating with algebra systems. It looks deceptively ’doable’, but really it’s a recipe for disaster.

eitland 9y ago

I think there used to be a open source tool called Maxima or something that would simplify equations. Havent used it for the last 10 years and never used it much anyway.

lottin 9y ago
I can recommended Maxima too. It's useful for checking for obvious errors and for simplifying expressions. It can be used for more advanced stuff, but I found it has a steep learning curve.

cosmic_ape 9y ago

SymPy does not take Latex as input, but it has symbolic equivalence checkers. Methods are highly heuristic, and some languages (all expressions with,say, \pi, and exp function, or something similar) are in general undecidable.

rav 9y ago
To elaborate: Symbolic equivalence checking for two expressions f1 and f2 can be done by seeing if sympy.simplify(sympy.Eq(f1, f2)) is sympy.true. import sympy from sympy.abc import x, y, alpha, s quad = s ** 2 - alpha * s - 2 # Let s1 and s2 be the two solutions to the quadratic equation 'quad == 0' s1, s2 = sympy.solve(quad, s) u = (x - s2) / (x - s1) * (y - s1) / (y - s2) f1 = (s2 - s1 * u) / (1 - u) f2 = (x * y - alpha * x - 2) / (y - x) # Claim: f1 is equal to f2 print(sympy.simplify(sympy.Eq(f1, f2))) # Prints "True"
- yorwba 9y ago
  Is there something for implication or equivalence of equations? I was trying the example from Smaug123's sibling comment (https://news.ycombinator.com/item?id=15728598 https://news.ycombinator.com/item?id=15728598), but >>> import sympy as sp >>> x = sp.Symbol('x') >>> sp.simplify(sp.Implies(sp.Eq(x**2 + 2*x + 1, 0), sp.Eq(x, -1))) Eq(x, -1) | Ne(x**2 + 2*x + 1, 0) >>> sp.solve(sp.simplify(sp.Implies(sp.Eq(x**2 + 2*x + 1, 0), sp.Eq(x, -1)))) Traceback (most recent call last): File "<stdin>", line 1, in <module> File "/home/user/.local/lib/python3.5/site-packages/sympy/solvers/solvers.py", line 1065, in solve solution = _solve(f[0], *symbols, **flags) File "/home/user/.local/lib/python3.5/site-packages/sympy/solvers/solvers.py", line 1401, in _solve f_num, sol = solve_linear(f, symbols=symbols) File "/home/user/.local/lib/python3.5/site-packages/sympy/solvers/solvers.py", line 1971, in solve_linear eq = lhs - rhs TypeError: unsupported operand type(s) for -: 'Or' and 'int' >>> sp.solveset(sp.simplify(sp.Implies(sp.Eq(x**2 + 2*x + 1, 0), sp.Eq(x, -1)))) Traceback (most recent call last): File "<stdin>", line 1, in <module> File "/home/user/.local/lib/python3.5/site-packages/sympy/solvers/solveset.py", line 880, in solveset raise ValueError("%s is not a valid SymPy expression" % (f)) ValueError: Eq(x, -1) | Ne(x**2 + 2*x + 1, 0) is not a valid SymPy expression none of the obvious ways appear to work. Does Sympy not support this kind of equational reasoning?

ivan_ah 9y ago

I'm with cosmic_ape regarding the recommendation for SymPy. Write the two lines as symbolic expressions `expr1` and `expr2` then run `simplify(expr1-expr2)` to check if the answer is zero. If you want to start with LaTeX then the parser from this library might be useful: https://github.com/Khan/KAS https://github.com/Khan/KAS

qubex 9y ago

On a whim I programmed a symbolic integral-solving library in Python (I am not a professional programmer) and, as an ill-advised ’extension’, bolted on the ability to receive and emit LATEX expressions. Long story short, I was boasting about it on Nichols Nassim Taleb’s twitter feed, submitted an answer to a problem question he had posed that got mis-parsed by the front end, gave the wrong answer (which was entirely consistent, because the check stage compared the answer by deriving the flawed question), and got banned, flamed, and persecuted to Hell and back. True story. From this I learnt the hard way that tokenisation and representation (in LATEX or MathML) do not belong in the same place as a CAS (Computer-assisted Algebra System).

yorwba 9y ago
I believe that LaTeX/MathML absolutely do belong in a CAS, just not as the primary means of interaction. (I.e. you do your work in the CAS's own language and then export to LaTeX.) But maybe your example can convince me otherwise. Could you show the specific LaTeX code in question and describe how your program mishandled it?
- qubex 9y ago
  I had set myself the target of implementing the Risch Algorithm (https://en.wikipedia.org/wiki/Risch_algorithm https://en.wikipedia.org/wiki/Risch_algorithm) in SymPy (3) but the event was so embarrassing I deleted my code and never published it publicly. I’m not a programmer. I erred in trying to do a programmer’s job. Also, I discovered why Nicholas Nassim Taleb has the reputation for being rather uncharitable (but deep down I feel like I deserved it, because hey, I stated a mathematical untruth).

SoerenL 9y ago

As one of the authors of this tool I understand that you would like to have some way of trusting the output. Internally we check it numerically, i.e., generate some random data for the given variables and check the derivative by comparing it to an approximation via finite differences. We will ship this code (hopefully soon) with one of the next versions. You can then check it yourself. Otherwise, as far as I know there does not exist any other tool that computes matrix derivatives so I understand it is hard to convince anyone of the correctness of the results. But I hope the numerical tests will be helpful.