given an expression x'=x+1, I wish to rename x' to y. How to do using z3?
There are a number of API functions that let you modify terms and substitute new subterms for old ones. They are described under the Modifiers section that contain the modifiers Z3_update_term, z3_substitute, and Z3_substitute_vars (there is also Z3_translate to port terms between two contexts).
Here is the link:
http://research.microsoft.com/en-us/um/redmond/projects/z3/group__capi.html#gaa7497c70a827db2d61ba98889fe657b5
You can also traverse terms directly and write utilities to modify terms.
The display_ast example shows the main cases for recursively traversing terms:
http://research.microsoft.com/en-us/um/redmond/projects/z3/group__capi__ex.html#ga807b5fe0e26acdec09e52a77318208d0
Related
Most of the proofs suggested by Sledgehammer use this notation of number inside of parentheses:
by (smt (z3) ApplyAllResult.distinct(1)
ApplyResult.case(1)
ApplyResult.case(2)
ApplyResult.exhaust
applyInput.simps(1))
What does it mean for the fact to have such number?
Isabelle permits the use of fact lists indexed by a natural number starting with 1. Given a fact list fs and an index i, you can access an individual fact from the list by using the syntax fs(i). You can also select multiple facts from the list using multiple indexes (e.g., fs(1,3)), ranges (e.g. fs(2-5), fs(3-)) or a combination of both (e.g., fs(2,4-6)).
Examples of predefined fact lists are assms (which contains the assumptions of a theorem) and f.simps (which contains the equations defining function f).
I have created a grammar to read a file of equations then created AST nodes for each rule.My question is how can I do simplification or substitute vales on the equations that the parser is able to read correctly. in which stage? before creating AST nodes or after?
Please provide me with ideas or tutorials to follow.
Thank you.
I'm assuming you equations are something like simple polynomials over real-value variables, like X^2+3*Y^2
You ask for two different solutions to two different problems that start with having an AST for at least one equation:
How to "substitute values" into the equation and compute the resulting value, e.g, for X==3 and Y=2, substitute into the AST for the formula above and compute 3^2+3*2^2 --> 21
How to do simplification: I assume you mean algebraic simplification.
The first problem of substituting values is fairly easy if yuo already have the AST. (If not, parse the equation to produce the AST first!) Then all you have to do is walk the AST, replacing every leaf node containing a variable name with the corresponding value, and then doing arithmetic on any parent nodes whose children now happen to be numbers; you repeat this until no more nodes can be arithmetically evaluated. Basically you wire simple arithmetic into a tree evaluation scheme.
Sometimes your evaluation will reduce the tree to a single value as in the example, and you can print the numeric result My SO answer shows how do that in detail. You can easily implement this yourself in a small project, even using JavaCC/JJTree appropriately adapted.
Sometimes the formula will end up in a state where no further arithmetic on it is possible, e.g., 1+x+y with x==0 and nothing known about y; then the result of such a subsitution/arithmetic evaluation process will be 1+y. Unfortunately, you will only have this as an AST... now you need to print out the resulting AST in order for the user to see the result. This is harder; see my SO answer on how to prettyprint a tree. This is considerably more work; if you restrict your tree to just polynomials over expressions, you can still do this in small project. JavaCC will help you with parsing, but provides zero help with prettyprinting.
The second problem is much harder, because you must not only accomplish variable substitution and arithmetic evaluation as above, but you have to somehow encode knowledge of algebraic laws, and how to match those laws to complex trees. You might hardwire one or two algebraic laws (e.g., x+0 -> x; y-y -> 0) but hardwiring many laws this way will produce an impossible mess because of how they interact.
JavaCC might form part of such an answer, but only a small part; the rest of the solution is hard enough so you are better off looking for an alternative rather than trying to build it all on top of JavaCC.
You need a more organized approach for this: a Program Transformation System (PTS). A typical PTS will allow you specify
a grammar for an arbitrary language (in your case, simply polynomials),
automatically parses instance to ASTs and can regenerate valid text from the AST. A good PTS will let you write source-to-source transformation rules that the PTS will apply automatically the instance AST; in your case you'd write down the algebraic laws as source-to-source rules and then the PTS does all the work.
An example is too long to provide here. But here I describe how to define formulas suitable for early calculus classes, and how to define algebraic rules that simply such formulas including applying some class calculus derivative laws.
With sufficient/significant effort, you can build your own PTS on top of JavaCC/JJTree. This is likely to take a few man-years. Easier to get a PTS rather than repeat all that work.
Is there any difference between the following ways of expressing multiple necessary conditions in ontology (via Protege)
Each necessary condition expressed one by one inside the SubclassOf Section (for class A):
instrument some B
object some C
All of those stated at once (via the class expression editor)
instrument some B and object some C
Are 1 and 2 semantically the same?
Yes they are equivalent. The choice of which way to go is yours: which approach do you find more readable? That is the best choice.
I have implemented IEnumerable for a collection I built, and (although I have not tested them all) the Seq values appear to work correctly. Is it possible to override some Seq values, for instance "last", when the native performance of a value of my collection is better than using Seq's IEnumerable based function? I have not found any information on overriding Seq.
No -- the functions in the Seq module can't be overridden. However, some of them do try to optimize performance by checking their input value (the seq<'T> instance you pass them) to see if it's an instance of IList<'T> or 'T[]; if it is, the functions will take some optimized code path. For example, if you pass an array ('T[]) to Seq.length, it'll be able to quickly determine the length by using the .Length property of arrays.
If you're stuck on using the Seq module, the only performance optimization I can think of would be to have your collection also implement ICollection<'T> and/or IList<'T>. That may optimize some cases, but it won't be all cases.
As already said in the other answer, there is no way you can override the functions in the Seq module. If you're implementing a custom collection, then the best thing to do is to follow the standard pattern used by the core F# libraries.
The Seq module contains the most often used functions and functions that can be reasonably provided for any sequence.
Modules like Array or List provide more efficient implementations for a specific collection type and they add more functions (not available in Seq) that are specific to the collection (for example, functions List.tail and Array.get).
The best way when adding your own collection is to follow this pattern:
Implement IEnumerable<'T> so that the functions from Seq module work for your type
Create MyCollection module that contains efficient implementations of standard functions (at least those that matter to you) and adds more functionality that is specific to your collection.
I have what I imagine will be a fairly involved technical challenge: I want to be able to reliably alpha-rename identifiers in multiple languages (as many as possible). This will require special consideration for each language, and I'm asking for advice for how to minimize the amount of work I need to do by sharing code. Something like a unified parsing or abstract syntax framework that already has support for many languages would be great.
For example, here is some python code:
def foo(x):
def bar(y):
return x+y
return bar
An alpha renaming of x to y changes the x to a y and preserves semantics. So it would become:
def foo(y):
def bar(y1):
return y+y1
return bar
See how we needed to rename y to y1 in order to keep from breaking the code? That is why this is a hard problem. It seems like the program would have to have a pretty good knowledge of what constitutes a scope, rather than just doing, say, a string search and replace.
I would also like to preserve as much of the formatting as possible: comments, spacing, indentation. But that is not 100% necessary, it would just be nice.
Any tips?
To do this safely, you need to be able to to determine
all the identifiers (and those things that are not, e.g., the middle of a comment) in your code
the scopes of validity for each identifer
the ability to substitute a new identifier for an old one in the text
the ability to determine if renaming an identifier causes another name to be shadowed
To determine identifiers accurately, you need a least a langauge-accurate lexer. Identifiers in PHP look different than the do in COBOL.
To determine scopes of validity, you have to be determine program structure in practice, since most "scopes" are defined by such structure. This means you need a langauge-accurate parser; scopes in PHP are different than scopes in COBOL.
To determine which names are valid in which scopes, you need to know the language scoping rules. Your language may insist that the identifier X will refer to different Xes depending on the context in which X is found (consider object constructors named X with different arguments). Now you need to be able to traverse the scope structures according to the naming rules. Single inheritance, multiple inheritance, overloading, default types all will pretty much require you to build a model of the scopes for the programs, insert the identifiers and corresponding types into each scope, and then climb from the point of encounter of an identifier in the program text through the various scopes according to the language semantics. You will need symbol tables, inheritance linkages, ASTs, and the ability to navigage all of these. These structures are different from PHP and COBOL, but they share lots of common ideas so you likely need a library with the common concept support.
To rename an identifier, you have to modify the text. In a million lines of code, you need to point carefully. Modifying an AST node is one way to point carefully. Actually, you need to modify all the identifiers that correspond to the one being renamed; you have to climb over the tree to find them all, or record in the AST where all the references exist so they can be found easily. After modifyingy the tree you have to regenerate the source text after modifying the AST. That's a lot of machinery; see my SO answer on how to prettyprint ASTs preseriving all of the stuff you reasonably suggest should be preserved.
(Your other choice is to keep track in the AST of where the text for the string is,
and the read/patch/write the file.)
Before you update the file, you need to check that you haven't shadowed something. Consider this code:
{ local x;
x=1;
{local y;
y=2;
{local z;
z=y
print(x);
}
}
}
We agree this code prints "1". Now we decide to rename y to x.
We've broken the scoping, and now the print statement which referred
conceptually to the outer x refers to an x captured by the renamed y. The code now prints "2", so our rename broke it. This means that one must check all the other identifiers in scopes in which the renamed variable might be found, to see if the new name "captures" some name we weren't expecting. (This would be legal if the print statement printed z).
This is a lot of machinery.
Yes, there is a framework that has almost all of this as well as a number of robust language front ends. See our DMS Software Reengineering Toolkit. It has parsers producing ASTs, prettyprinters to produce text back from ASTs, generic symbol table management machinery (including support for multiple inheritance), AST visiting/modification machinery. Ithas prettyprinting machinery to turn ASTs back into text. It has front ends for C, C++, COBOL and Java that implement name and type resolution (e.g. instanting symbol table scopes and identifier to symbol table entry mappings); it has front ends for many other langauges that don't have scoping implemented yet.
We've just finished an exercise in implementing "rename" for Java. (All the above issues of course appeared). We about about to start one for C++.
You could try to create Xtext based implementations for the involved languages. The Xtext framework provides reliable infrastructure for cross language rename refactoring. However, you'll have to provide a grammar a at least a "good enough" scope resolution for each language.
Languages mostly guarantee tokens will be unique, whatever the context. A naive first approach (and this will break many, many pieces of code) would be:
cp file file.orig
sed -i 's/\b(newTokenName)\b/TEMPTOKEN/g' file
sed -i 's/\b(oldTokenName)\b/newTokenName/g' file
With GNU sed, this will break on PHP. Rewriting \b to a general token match, like ([^a-zA-Z~$-_][^a-zA-Z0-9~$-_]) would work on most C, Java, PHP, and Python, but not Perl (need to add # and % to the token characters. Beyond that, it would require a plugin architecture that works for any language you wanted to add. At some point, there will be two languages whose variable and function naming rules will be incompatible, and at that point, you'll need to do more and more in the plugin.