I am developing an expression evaluator. Which association is considered to be correct for an expression containing more than one exponentiation operator? For example, for the expression "10-2^2^0.5": "10-(2^2) ^0.5"= 8 or "10-2^ (2^0.5)" = 7.33485585731?
The result differs across languages and (possible) interpreters. However, most of them uses right-associative rule.
In Lua print(10-2^2^0.5) returns 7.3348 and in Visual Basic, Console.WriteLine(10-2^2^0.5) returns 8.
The fact that different systems uses different rules suggest me that there is no defined rule for that.
Related
I am writing a program for which I need terms in their prefix notation.
The point is to being able to parse mathematical expressions to prefix notation, while preserving the correct order of Operations. I then want to save the result in the database for later use (using assert), which includes translating to another language, which uses prefix notation. Prolog Operators do all have a fixed priority which is a feature I want to use, as I will be using all sorts of operators (including clp operators).
As among others I need to include complete mathematical expressions, such as the equality operator. Thus I cannot recursively use the Univ operator (=..), because it won't accept equality operators etc. Or can I somehow use =.. ?
Essentially I want to work with the internal representation of
N is 3*4+5 % just a random example
which would be
is(N,+(*(3,4),5))
Now, I do know that I can use, write_canonical(N is 3*4+5) to get the internal representation as seen above.
So is there a way to somehow get the internal representation as a term or a list, or something.
Would it be possible to bind the output of write_canonical to a variable?
I hope my question is clear enough.
Prolog terms can be despicted as trees. But, when writing a term, the way a term is displayed depends on the defined operators and write options. Consider:
?- (N is 3*4+5) = is(N,+(*(3,4),5)).
true.
?- (N is 3*4+5) = is(Variable, Expression).
N = Variable,
Expression = 3*4+5.
?- 3*4+5 = +(*(3,4),5).
true.
I.e. operators are syntactic sugar. They don't change how terms are represented, only how terms are displayed.
I'm trying to understand the difference in Z3 between equality testing and biconditional. My understanding is that = is used to express biconditional, but how is equality tested?
For example. I am trying to write something similar to the following (toy) statement in z3:
on_table(o, a) ↔ (in_hand(o) Λ a != pickup(o)) ∨ a = put_on_table(o)
Note: I am aware the above statement can be factored into a set of implications, but I am interested in expressing it as a single biconditional.
For the Bool type, equality and biconditional are the same operations. For any other type, biconditional doesn't really make sense.
All logics in SMT come equipped with the notion of equality, which is essentially term-level equality of objects. The standard explicitly states:
Version 2.6 of the SMT-LIB format adopts as its underlying logic a
version of many-sorted first-order logic with equality [Man93, Gal86,
End01].
See Section 2.2 of http://smtlib.cs.uiowa.edu/papers/smt-lib-reference-v2.6-r2017-07-18.pdf
The same document also says (Section 3.7.1):
Note the absence of a symbol for double implication. Such a connective
is superfluous because the equality symbol = can be used in its place.
I suspect though, perhaps, you are trying to ask for something else. Some further examples would definitely help.
After reading this answer on a CSS question, I wonder:
In Computer Science, is a single, constant value considered an expression?
In other words, is 7px an expression? What about just 7?
Quoting Wikipedia, emphasis mine:
An expression in a programming language is a combination of one or more explicit values, constants, variables, operators, and functions that the programming language interprets [...] and computes to produce [...] another value. This process, as for mathematical expressions, is called evaluation.
Quoting MS Docs, emphasis mine:
An expression is a sequence of one or more operands and zero or more operators that can be evaluated to a single value, object, method, or namespace. Expressions can consist of a literal value [...].
These both seems to indicate that values are expressions. However, one could argue that a value will not be evaluated, as it is already only a value, and therefore doesn't qualify.
Quoting Techopedia, emphasis mine:
[...] In terms of structure, experts point out that an expression inherently needs at least one 'operand’ or value that is acted on, and must have one or more operators. [...]
This suggests that even x does not qualify as expression as it is lacking one or more operators.
It depends on the exact definition of course, but under most definitions expressions are defined recursively with constants being one of the basis cases. So, yes, literal values are special cases of expressions.
You can look at grammars for various languages such as the one for Python
If you trace through the grammar you see that an expr can be an atom which includes number literals. The fact that number literals are Python expressions is also obvious when you consider productions like:
comparison: expr (comp_op expr)*
This is the production which captures expressions like x < 7, which wouldn't be captured if 7 isn't a valid expression.
In Computer Science, is a single, constant value considered an expression?
It depends entirely on the context. For example, FORTRAN, BASIC, and COBOL all have line numbers. Those are numeric constant values that are not expressions.
In other contexts (even within those languages) a numeric constant may be an expression.
I have a huge set (20000) of boolean expressions. They consist of AND, OR and NOT operators and a large number of boolean variables A1, A2, A3 ... (about 1000). Most expression contain only 5, maybe 20 of these variables.
Given an assignment of the variables (A1 = true, A2 = false, A3 = false ...) I have to find those expressions that evaluate to false.
The same set of expressions will be evaluated for multiple (10-100) assignments
For this purpose:
How should I store the expressions on disk so I can load and parse them fast (I currently have them either as some specialized DSL or as a more or less normalized (and dead slow) relational data structure, but I can change that)
Is there a fast algorithm / data structure for evaluating such expressions that I can use?
Do implementations on the JVM exist?
You may want to look at converting your expressions into Conjunctive Normal Form and combining like terms. You then can have a two-way mapping of an expression to a set of terms, any of which evaluating to false implies that the whole expression evaluates false. For each assignment of variables, start with a set of expressions, evaluate CNF terms until one evaluates to false. If that term is false, then all expressions involving that term will also be false, so those expressions can also be removed from the set.
Whether such an approach fits your case can't be said without looking at the expressions - with 1000 variables and 20000 expressions, it might not be that they have many CNF terms in common.
Outside of Java, and for much larger numbers of expressions, DNF is possibly more useful, since its implementation on the GPU is obvious.
The SOP answer to this is to store the expressions as strings in RPN (Reverse Polish Notation) and then write a simple Stack Machine parser to evaluate them.
Generally, an RPN string can be evaluated almost as fast as an already in-memory AST (Abstract Symbol Tree). And the stack machine parser is dead easy to write.
You seem attached to Java, but have you considered feeding these things to a language that has an eval() function? It would probably reduce the problem to saving an expression in a file and evaluating it. Note that if you don't trust the (source of the) expressions, this has security implications!
Jython comes to mind, but there are probably several that would make very short work of this.
If you're married to java, you could probably implement a recursive descent parser for boolean algebra. But that's quite a bit more involved.
UPDATE: The following site has code that might help.
Convert your list of expressions into source code for a function that when called with the value of the variables will evaluate all the functions and return an indication of which expressions evaluate to false. compile the function then call it for your different variable values.
I have done similar and used Python. The only parsing and interpretation I had to write was to translate the input boolean operators, '&', '|', '~' into their Python equivalents.
Your problem size seems quite OK for a Python solution.
You could build an index where for each variable you record two sets of expressions, those where the variable occurs positively and those where it occurs negatively. Depending on the values of the variables you collect those expressions which could become false due to this variable (positive occurrences if the variables is set to false and vice versa). Edit: These are just candidates, you still need to evaluate them to find out if they really become false.
Whether this helps compared to just evaluating all your expressions depends on the structure of your expressions and how many evaluate to false.
Try to convert them into CNF and use MiniSat to check whether the expression evaluates to true or false
I know how each of them can be converted to one another but never really understood what their applications are. The usual infix operation is quite readable, but where does it fail which led to inception of prefix and postfix notation
Infix notation is easy to read for humans, whereas pre-/postfix notation is easier to parse for a machine. The big advantage in pre-/postfix notation is that there never arise any questions like operator precedence.
For example, consider the infix expression 1 # 2 $ 3. Now, we don't know what those operators mean, so there are two possible corresponding postfix expressions: 1 2 # 3 $ and 1 2 3 $ #. Without knowing the rules governing the use of these operators, the infix expression is essentially worthless.
Or, to put it in more general terms: it is possible to restore the original (parse) tree from a pre-/postfix expression without any additional knowledge, but the same isn't true for infix expressions.
Postfix notation, also known as RPN, is very easy to process left-to-right. An operand is pushed onto a stack; an operator pops its operand(s) from the stack and pushes the result. Little or no parsing is necessary. It's used by Forth and by some calculators (HP calculators are noted for using RPN).
Prefix notation is nearly as easy to process; it's used in Lisp.
At least for the case of the prefix notation: The advantage of using a prefix operator is that syntactically, it reads as if the operator is a function call
Another aspect of prefix/postfix vs. infix is that the arity of the operator (how many arguments it is applied to) no longer has to be limited to exactly 2. It can be more, or sometimes less (0 or 1 when defaults are implied naturally, like zero for addition/subtraction, one for multiplication/division).