## Check invariant property using SAGE - boolean-expression

I know that the XOR operation of three variables is invariant (symmetric) with respect to two of them. How Can I discover the invariance of any boolean function automatically for all possible variables combinations with sagemath? In the example of the xor operation, I am tried to use the method is_symmetric(), but this method check if the entire function is invariant, and it is not possible to specify as input the variables to check the invariance (symmetric). Here the function
from sage.crypto.boolean_function import BooleanFunction
R1 = BooleanPolynomialRing(3, ','.join(['x','y','z']))
f=BooleanFunction(R1(x*y+z))

## Related

### MATLAB set class `size` (overload operation upon class)

classdef Dog end d=Dog(); can size(d) be controlled? Is there some property to set or method to overload? Ultimately I have like d.data = [1, 2, 3] and want length(d) == 3. I'm aware I can make d.length(). As an aside, is there a list of MATLAB "magic methods", i.e. functions that control interaction with classes, like subsref?

In MATLAB, don't think of class method as similar to class methods in other languages. Really, what they are is overloaded versions of a function. size(d) is the same as d.size() (which is the same as d.size, parentheses are not needed to call a function), if d is an object of a custom class and size is overloaded for that class. So, you can define function size in the methods section of your classdef, to overload size for your class. You can also create a size.m file within a #Dog/ directory to accomplish the same thing. For example, if you create a file #char/size.m with a function definition inside, you will have overloaded size for char arrays. The above is true for any function. Some functions, when overloaded, can cause headaches. For example be careful when overloading numel, as it could cause indexed assignment expressions to fail. size is used by the whos command to display variable information, as well as by the similar functionality in the GUI, so it is important to have it behave in the expected way. The object behavior that you might want to change that is not obviously a function, relates to operators. Each operator is also defined by a function (including end in indexing!). See the docs for a full list.

### Is it possible to check for handle equality for a class that overloads eq?

My code has some handles for objects belonging to a third-party class that I can't modify. This class overloads eq so that rather than comparing whether the handles point to the same object, as the built-in version of == does, it does a different comparison based on the value of the input objects, irrespective of whether given two handles to the same object or not. I explicitly want to check if the two handles point to the same object. I thought perhaps builtin would rescue me. But builtin('eq',A,B) just results in the error: Error using builtin Undefined operator '==' for input arguments of type 'ThirdPartyClass'. It seems like builtin only likes "pure" functions, and handle.eq is a special method of the handle class that is distinct from the builtin function eq that operates on pure functions. Calling eq explicitly with the class name using handle.eq(A,B) doesn't work either - this yields the error No method 'eq' with matching signature found for class 'handle'. Curiously, invoking the overload using the same syntax ThirdPartyClass.eq(A,B) yields a different error: The class ThirdPartyClass has no Constant property or Static method named 'eq'. So it isn't exactly clear to me if handle.eq(A,B) is also necessarily interpreted as a call to a static method. But in this exact form at least it seems not to be a viable route to calling that (overloaded) regular method. So have I been done over by the very inconsiderate design choice of the authors of this class? Or is there some way to access the method of the superclass that has been so recklessly overloaded? Or indeed, a way to achieve what handle.eq does from scratch (despite this being one of MATLAB's built-in functions that doesn't have its implementation visible in m-code)?

Going over the list of public methods for handle led to one solution that is viable in this case. Since le and ge are not overloaded in this class, the expression A <= B & A >= B has the same result as A == B does for handle classes without an overload on eq.

### passing anonymous functions through inputParser (matlab)

I'm trying to use Matlab's inputParser for the first time, and I have to say I'm finding it a bit confusing. I am unable to successfully provide an anonymous function as an optional parameter. This is the function I am passing arguments to function myfun(str,bounds,varargin) p = inputParser; p.FunctionName = mfilename; p.addRequired('str',#isstr); p.addRequired('bounds',#isvector); p.addOptional('str_latex','',#isstr); p.addOptional('seed',[], #(x) isa(x,'function_handle')) p.parse(str,bounds,varargin{:}); p.Results % do something here end And I am calling it like this... myfun('str', 'epsilon',... 'str_latex', '\epsilon',... 'bounds', [0 1],... 'seed', #() betarnd(2,2)) But I get an error: Error using my fun The value of 'seed' is invalid. It must satisfy the function: #(x)isa(x,'function_handle'). I suspect a simple error, but I cannot figure it out.

Name-value pairs are declared using the addParameter method (R2013b+, addParamValue prior to that). addRequired and addOptional do not have name-value pairs associated with them, simply identifying/documenting argname inputs for internal use and association with the parsed struct. It appears you want to use all addParamter-s in this use case. The main idea behind the three input types is Required: the very first arguments with explicit, documented inputnames that absolutely need to be supplied by the user for the function to perform properly. Optional: arguments that typically follow Required arguments with explicit, documented inputnames that are often input by the user for customized behavior. Name-Value: arguments that typically follow Optional arguments with a name specifying the value to be set are often input by the user for customized behavior but not so often as to be given an upfront, explicit parameter like Optional arguments. In my experience, Required arguments are almost always obvious, for good, well-defined functions, while Optional and Name-Value is more experience-, complexity-, and aesthetics-based. A simple example would be linspace: the start and end of the interval are absolutely needed for the function to work, but not necessarily the number of points which can be left to 100 by default, but giving it an explicit name-value pair is a little overkill. A more complex example would be the plot function: at a minimum y data is needed, then x,y pairs of data, then x,y,linSpec sets of data, and then a whole list of specific name-value pairs for pinpoint customization that users can use if they so choose. With your input parser as written, the call sequence should be: myfun('epsilon',[0,1],'\epsilon',#() betarnd(2,2)); Since no name-value pairs were declared, none exist, but the Optional arguments still have a positional order associated with them. You can re-write your parser as: function myfun(varargin) p = inputParser; p.FunctionName = mfilename; p.addParameter('str',[],#isstr); p.addParameter('bounds',[],#isvector); p.addParameter('str_latex','',#isstr); p.addParameter('seed',[], #(x) isa(x,'function_handle')) p.parse(str, bounds, varargin{:}); p.Results % do something here end For something like the generic input sequence you may have been expecting. Notice that I used []-s to fail the simple validations without a good error message; you should add a good error message indicating that those name-value pairs are required for proper functionality, or do as you were doing and have explicit, upfront Required inputs with addRequired but without the name-value semantics.

### Should I use partial function for database calls

As per my understanding, partial functions are functions that are defined for a subset of input values. So should I use partial functions for DAO's. For example: getUserById(userId: Long): User There is always an input userId which does not exists in db. So can I say it is not defined. And lift it when I call this function. If yes where do I stop. Should I use partial functions for all methods which are not defined, say for null.

PartialFunction is used when function is undefined for some elements of input data (input data may be Seq etc.) For your case Option is better choice: it says that return data may be absent: getUserById(userId:Long):Option[User]

I would avoid using partial functions at all, because scala makes it very easy to call a partial function as though it were a total function. Instead it's better to use a function that returns Option as #Sergey suggests; that way the "partial-ness" is always explicit. Idiomatic scala does not use null so I wouldn't worry about methods which are not defined for null, but certainly it's worth returning Option for methods which are only defined for some of their possible input values. Better still, though, is to only accept suitable types as input. E.g. if you have a function that's only valid for non-empty lists, it should take (scalaz) NonEmptyList as input rather than List.

### Simplify boolean expression i.t.o variable occurrence

How to simplify a given boolean expression with many variables (>10) so that the number of occurrences of each variable is minimized? In my scenario, the value of a variable has to be considered ephemeral, that is, has to recomputed for each access (while still being static of course). I therefor need to minimize the number of times a variable has to be evaluated before trying to solve the function. Consider the function f(A,B,C,D,E,F) = (ABC)+(ABCD)+(ABEF) Recursively using the distributive and absorption law one comes up with f'(A,B,C,E,F) = AB(C+(EF)) I'm now wondering if there is an algorithm or method to solve this task in minimal runtime. Using only Quine-McCluskey in the example above gives f'(A,B,C,E,F) = (ABEF) + (ABC) which is not optimal for my case. Is it save to assume that simplifying with QM first and then use algebra like above to reduce further is optimal?

I usually use Wolfram Alpha for this sort of thing.

Try Logic Friday 1 It features multi-level design of boolean circuits. For your example, input and output look as follows:

You can use an online boolean expression calculator like https://www.dcode.fr/boolean-expressions-calculator You can refer to Any good boolean expression simplifiers out there? it will definitely help.