Is there an equivalent to define-const in the Z3 Python API? I am trying to rewrite the example below (similar to this example) in Python:
(define-const a String "hello")
(define-const b String "world")
(simplify (= a b))
(check-sat)
(get-model)
whose output is
false
sat
(
(define-fun b () String
"world")
(define-fun a () String
"hello")
)
I tried replacing define-const with a constant declaration followed by an assertion, but the result was not quite the same:
from z3 import *
a = String('a')
b = String('b')
s = Solver()
s.add(a == 'hello')
s.add(b == 'world')
print(simplify(a == b))
print(s.check())
print(s.model())
output:
a == b
sat
[a = "hello", b = "world"]
I also tried using StringVal() but the result was also not the same:
from z3 import *
a = StringVal('hello')
b = StringVal('world')
s = Solver()
s.add(a == 'hello')
s.add(b == 'world')
print(simplify(a == b))
print(s.check())
print(s.model())
output:
False
sat
[]
I have tried searching documentation and elsewhere but was unsuccessful.
This is a bit of a strange thing to do, since define-const is simply a macro. If you're using the Python API, you'd simply use a StringVal. But then, those variables don't show up in the solver, because there's no connection between the Python level object, and what gets asserted in the solver. This is what you are seeing in the last example you tried.
And this is perfectly fine: In SMTLib, constants live together with all the other variables; but in z3py you get two different worlds. A better question to ask is, perhaps, what exactly are you trying to achieve? Trying to simulate SMTLib exactly by the Python interface isn't an explicit goal. In particular, the Python interface allows for easier programming and access to z3 internals that are not available in the SMTLib world. Don't try to equate them!
Having said all this, the most "faithful" way to simulate the SMTLib you have in z3py would be:
from z3 import *
a = StringVal("hello")
b = StringVal("world")
print(simplify(a == b))
s = Solver()
print(s.check())
m = s.model()
print("a =", m.evaluate(a))
print("b =", m.evaluate(b))
This prints:
False
sat
a = "hello"
b = "world"
This matches the output you're expecting, but then again it's not really the best use of the API. Instead of trying to simulate SMTLib exactly, "what problem are you trying to solve" is a better goal to focus on.
Related
from z3 import *
p = Int('p')
q = Int('q')
solve(Or(p==1,p==2,p==3), Or(q==1,q==2), Not(p==q), q==1)
Gives me [p = 2, q = 1], but p could be 2 or 3. So the answer should be {2,3}. How can I tell z3 to inform me about multiple answers?
This question comes up often, and there are a few caveats. Essentially, you have to write a loop to "reject" earlier models and keep querying for satisfiability. For your particular problem, you can code it like this:
from z3 import *
p = Int('p')
q = Int('q')
s = Solver()
s.add(Or(p==1,p==2,p==3), Or(q==1,q==2), Not(p==q), q==1)
res = s.check()
while (res == sat):
m = s.model()
print(m)
block = []
for var in m:
block.append(var() != m[var])
s.add(Or(block))
res = s.check()
When I run this, I get:
[p = 2, q = 1]
[p = 3, q = 1]
Note that z3 doesn't always produce a "full" model: It'll stop producing assignments once it determines the problem is sat. So, you might have to track your variables and use model-completion explicitly. See Trying to find all solutions to a boolean formula using Z3 in python for details on how to do that.
I'm aware that Z3 has stack-based caching, where additional formulas can be added and cached. Is there a built-in way or extension that allows two Z3 caches to be combined?
Example (Z3 py)
from z3 import Solver
solver = Solver()
solver.push()
solver2 = Solver()
# solver.combine(solver2) ?
Not quite cleear what you mean by "combine." But, you can get the assertions from one and add it to the other:
from z3 import *
i = Int('x')
s1 = Solver()
s1.add(i == 3)
s1.push()
s2 = Solver()
s2.add(s1.assertions())
print s2.check()
print s2.model()
This prints:
sat
[x = 3]
You can use this trick to do your own combinations I suppose.
I have defined a type
data Expr =
Const Double
| Add Expr Expr
| Sub Expr Expr
and declared it as an instance of Eq typeclass:
instance Eq Expr where
(Add (Const a1) (Const a2)) == Const b = a1+a2 == b
(Add (Const a1) (Const a2)) == (Add (Const b1) (Const b2)) = a1+a2 == b1 + b2
Of course, the evaluation of the expression Sub (Const 1) (Const 1) == Const 0 will fail. How can I debug at runtime the pattern matching process to spot that it's failing? I would like to see how Haskell takes the arguments of == and walks through the patterns. Is it possible at all?
edit: providing a real answer to the question...
I find the easiest way to see what patterns are matching is to add trace statements, like so:
import Debug.Trace
instance Eq Expr where
(Add (Const a1) (Const a2)) == Const b = trace "Expr Eq pat 1" $ a1+a2 == b
(Add (Const a1) (Const a2)) == (Add (Const b1) (Const b2)) = trace "Expr Eq pat 2" $ a1+a2 == b1 + b2
-- catch any unmatched patterns
l == r = error $ "Expr Eq failed pattern match. \n l: " ++ show l ++ "\n r: " ++ show r
If you don't include a final statement to catch any otherwise unmatched patterns, you'll get a runtime exception, but I find it's more useful to see what data you're getting. Then it's usually simple to see why it doesn't match the previous patterns.
Of course you don't want to leave this in production code. I only insert traces as necessary then remove them when I've finished. You could also use CPP to leave them out of production builds.
I also want to say that I think pattern matching is the wrong way to go about this. You'll end up with a combinatorial explosion in the number of patterns, which quickly grows unmanageable. If you want to make a Float instance for Expr for example, you'll need several more primitive constructors.
Instead, you presumably have an interpreter function interpret :: Expr -> Double, or at least could write one. Then you can define
instance Eq Expr where
l == r = interpret l == interpret r
By pattern matching, you're essentially re-writing your interpret function in the Eq instance. If you want to make an Ord instance, you'll end up re-writing the interpret function yet again.
If you wish to get some examples on how the matching may fail, you could have a look at QuickCheck. There's an example on the manual (the size of test data) about generating and testing recursive data types that seems to perfectly suit your needs.
While the -Wall flag gives you a list of patterns non matched, a run of QuickCheck gives you examples of input data that lead your given proposition to failure.
For example, if I write a generator for your Expr and I give in input to quickCheck a proposition prop_expr_eq :: Expr -> Bool that checks if an Expr is equal to itself, I obtain very quickly Const 0.0 as a first example of non-matching input.
import Test.QuickCheck
import Control.Monad
data Expr =
Const Double
| Add Expr Expr
| Sub Expr Expr
deriving (Show)
instance Eq Expr where
(Add (Const a1) (Const a2)) == Const b = a1+a2 == b
(Add (Const a1) (Const a2)) == (Add (Const b1) (Const b2)) = a1+a2 == b1 + b2
instance Arbitrary Expr where
arbitrary = sized expr'
where
expr' 0 = liftM Const arbitrary
expr' n | n > 0 =
let subexpr = expr' (n `div` 2)
in oneof [liftM Const arbitrary,
liftM2 Add subexpr subexpr,
liftM2 Sub subexpr subexpr]
prop_expr_eq :: Expr -> Bool
prop_expr_eq e = e == e
As you see, running the test gives you a counterexample to prove that your equality test is wrong. I know this may be a little bit an overkill, but the advantage if you write things good is that you also get unit tests for your code that look at arbitrary properties, not only pattern matching exhaustiveness.
*Main> quickCheck prop_expr_eq
*** Failed! Exception: 'test.hs:(11,5)-(12,81): Non-exhaustive patterns in function ==' (after 1 test):
Const 0.0
PS: Another good reading about unit testing with QuickCheck is in the free book real world haskell.
You can break your complex pattern into simpler patterns and use trace to see what's going on. Something like this:
instance Eq Expr where
x1 == x2 | trace ("Top level: " ++ show (x, y1)) True,
Add x11 x12 <- x1,
trace ("First argument Add: " ++ show (x11, x12)) True,
Const a1 <- x11,
trace ("Matched first Const: " ++ show a1) True,
Const a2 <- x12,
trace ("Matched second Const: " ++ show a2) True,
Const b <- x2
trace ("Second argument Const: " ++ show b) True
= a1+a2 == b
It's a bit desperate, but desperate times calls for desperate measures. :)
As you get used to Haskell you rarely, if ever, need to do this.
I'm trying to learn Haskell and after an article in reddit about Markov text chains, I decided to implement Markov text generation first in Python and now in Haskell. However I noticed that my python implementation is way faster than the Haskell version, even Haskell is compiled to native code. I am wondering what I should do to make the Haskell code run faster and for now I believe it's so much slower because of using Data.Map instead of hashmaps, but I'm not sure
I'll post the Python code and Haskell as well. With the same data, Python takes around 3 seconds and Haskell is closer to 16 seconds.
It comes without saying that I'll take any constructive criticism :).
import random
import re
import cPickle
class Markov:
def __init__(self, filenames):
self.filenames = filenames
self.cache = self.train(self.readfiles())
picklefd = open("dump", "w")
cPickle.dump(self.cache, picklefd)
picklefd.close()
def train(self, text):
splitted = re.findall(r"(\w+|[.!?',])", text)
print "Total of %d splitted words" % (len(splitted))
cache = {}
for i in xrange(len(splitted)-2):
pair = (splitted[i], splitted[i+1])
followup = splitted[i+2]
if pair in cache:
if followup not in cache[pair]:
cache[pair][followup] = 1
else:
cache[pair][followup] += 1
else:
cache[pair] = {followup: 1}
return cache
def readfiles(self):
data = ""
for filename in self.filenames:
fd = open(filename)
data += fd.read()
fd.close()
return data
def concat(self, words):
sentence = ""
for word in words:
if word in "'\",?!:;.":
sentence = sentence[0:-1] + word + " "
else:
sentence += word + " "
return sentence
def pickword(self, words):
temp = [(k, words[k]) for k in words]
results = []
for (word, n) in temp:
results.append(word)
if n > 1:
for i in xrange(n-1):
results.append(word)
return random.choice(results)
def gentext(self, words):
allwords = [k for k in self.cache]
(first, second) = random.choice(filter(lambda (a,b): a.istitle(), [k for k in self.cache]))
sentence = [first, second]
while len(sentence) < words or sentence[-1] is not ".":
current = (sentence[-2], sentence[-1])
if current in self.cache:
followup = self.pickword(self.cache[current])
sentence.append(followup)
else:
print "Wasn't able to. Breaking"
break
print self.concat(sentence)
Markov(["76.txt"])
--
module Markov
( train
, fox
) where
import Debug.Trace
import qualified Data.Map as M
import qualified System.Random as R
import qualified Data.ByteString.Char8 as B
type Database = M.Map (B.ByteString, B.ByteString) (M.Map B.ByteString Int)
train :: [B.ByteString] -> Database
train (x:y:[]) = M.empty
train (x:y:z:xs) =
let l = train (y:z:xs)
in M.insertWith' (\new old -> M.insertWith' (+) z 1 old) (x, y) (M.singleton z 1) `seq` l
main = do
contents <- B.readFile "76.txt"
print $ train $ B.words contents
fox="The quick brown fox jumps over the brown fox who is slow jumps over the brown fox who is dead."
a) How are you compiling it? (ghc -O2 ?)
b) Which version of GHC?
c) Data.Map is pretty efficient, but you can be tricked into lazy updates -- use insertWith' , not insertWithKey.
d) Don't convert bytestrings to String. Keep them as bytestrings, and store those in the Map
Data.Map is designed under the assumption that the class Ord comparisons take constant time. For string keys this may not be the case—and when the strings are equal it is never the case. You may or may not be hitting this problem depending on how large your corpus is and how many words have common prefixes.
I'd be tempted to try a data structure that is designed to operate with sequence keys, such as for example a the bytestring-trie package kindly suggested by Don Stewart.
I tried to avoid doing anything fancy or subtle. These are just two approaches to doing the grouping; the first emphasizes pattern matching, the second doesn't.
import Data.List (foldl')
import qualified Data.Map as M
import qualified Data.ByteString.Char8 as B
type Database2 = M.Map (B.ByteString, B.ByteString) (M.Map B.ByteString Int)
train2 :: [B.ByteString] -> Database2
train2 words = go words M.empty
where go (x:y:[]) m = m
go (x:y:z:xs) m = let addWord Nothing = Just $ M.singleton z 1
addWord (Just m') = Just $ M.alter inc z m'
inc Nothing = Just 1
inc (Just cnt) = Just $ cnt + 1
in go (y:z:xs) $ M.alter addWord (x,y) m
train3 :: [B.ByteString] -> Database2
train3 words = foldl' update M.empty (zip3 words (drop 1 words) (drop 2 words))
where update m (x,y,z) = M.alter (addWord z) (x,y) m
addWord word = Just . maybe (M.singleton word 1) (M.alter inc word)
inc = Just . maybe 1 (+1)
main = do contents <- B.readFile "76.txt"
let db = train3 $ B.words contents
print $ "Built a DB of " ++ show (M.size db) ++ " words"
I think they are both faster than the original version, but admittedly I only tried them against the first reasonable corpus I found.
EDIT
As per Travis Brown's very valid point,
train4 :: [B.ByteString] -> Database2
train4 words = foldl' update M.empty (zip3 words (drop 1 words) (drop 2 words))
where update m (x,y,z) = M.insertWith (inc z) (x,y) (M.singleton z 1) m
inc k _ = M.insertWith (+) k 1
Here's a foldl'-based version that seems to be about twice as fast as your train:
train' :: [B.ByteString] -> Database
train' xs = foldl' (flip f) M.empty $ zip3 xs (tail xs) (tail $ tail xs)
where
f (a, b, c) = M.insertWith (M.unionWith (+)) (a, b) (M.singleton c 1)
I tried it on the Project Gutenberg Huckleberry Finn (which I assume is your 76.txt), and it produces the same output as your function. My timing comparison was very unscientific, but this approach is probably worth a look.
1) I'm not clear on your code.
a) You define "fox" but don't use it. Were you meaning for us to try to help you using "fox" instead of reading the file?
b) You declare this as "module Markov" then have a 'main' in the module.
c) System.Random isn't needed. It does help us help you if you clean code a bit before posting.
2) Use ByteStrings and some strict operations as Don said.
3) Compile with -O2 and use -fforce-recomp to be sure you actually recompiled the code.
4) Try this slight transformation, it works very fast (0.005 seconds). Obviously the input is absurdly small, so you'd need to provide your file or just test it yourself.
{-# LANGUAGE OverloadedStrings, BangPatterns #-}
module Main where
import qualified Data.Map as M
import qualified Data.ByteString.Lazy.Char8 as B
type Database = M.Map (B.ByteString, B.ByteString) (M.Map B.ByteString Int)
train :: [B.ByteString] -> Database
train xs = go xs M.empty
where
go :: [B.ByteString] -> Database -> Database
go (x:y:[]) !m = m
go (x:y:z:xs) !m =
let m' = M.insertWithKey' (\key new old -> M.insertWithKey' (\_ n o -> n + 1) z 1 old) (x, y) (M.singleton z 1) m
in go (y:z:xs) m'
main = print $ train $ B.words fox
fox="The quick brown fox jumps over the brown fox who is slow jumps over the brown fox who is dead."
As Don suggested, look into using the stricer versions o your functions: insertWithKey' (and M.insertWith' since you ignore the key param the second time anyway).
It looks like your code probably builds up a lot of thunks until it gets to the end of your [String].
Check out: http://book.realworldhaskell.org/read/profiling-and-optimization.html
...especially try graphing the heap (about halfway through the chapter). Interested to see what you figure out.
I'm learning more about Scala, and I'm having a little trouble understanding the example of anonymous functions in http://www.scala-lang.org/node/135. I've copied the entire code block below:
object CurryTest extends Application {
def filter(xs: List[Int], p: Int => Boolean): List[Int] =
if (xs.isEmpty) xs
else if (p(xs.head)) xs.head :: filter(xs.tail, p)
else filter(xs.tail, p)
def modN(n: Int)(x: Int) = ((x % n) == 0)
val nums = List(1, 2, 3, 4, 5, 6, 7, 8)
println(filter(nums, modN(2)))
println(filter(nums, modN(3)))
}
I'm confused with the application of the modN function
def modN(n: Int)(x: Int) = ((x % n) == 0)
In the example, it's called with one argument
modN(2) and modN(3)
What does the syntax of modN(n: Int)(x: Int) mean?
Since it's called with one argument, I'm assuming they're not both arguments, but I can't really figure out how the values from nums get used by the mod function.
This is a fun thing in functional programming called currying. Basically Moses Schönfinkel and latter Haskell Curry (Schonfinkeling would sound weird though...) came up with the idea that calling a function of multiple arguments, say f(x,y) is the same as the chain of calls {g(x)}(y) or g(x)(y) where g is a function that produces another function as its output.
As an example, take the function f(x: Int, y: Int) = x + y. A call to f(2,3) would produce 5, as expected. But what happens when we curry this function - redefine it as f(x:Int)(y: Int)and call it as f(2)(3). The first call, f(2) produces a function taking an integer y and adding 2 to it -> therefore f(2) has type Int => Int and is equivalent to the function g(y) = 2 + y. The second call f(2)(3) calls the newly produced function g with the argument 3, therefore evaluating to 5, as expected.
Another way to view it is by stepping through the reduction (functional programmers call this beta-reduction - it's like the functional way of stepping line by line) of the f(2)(3) call (note, the following is not really valid Scala syntax).
f(2)(3) // Same as x => {y => x + y}
|
{y => 2 + y}(3) // The x in f gets replaced by 2
|
2 + 3 // The y gets replaced by 3
|
5
So, after all this talk, f(x)(y) can be viewed as just the following lambda expression (x: Int) => {(y: Int) => x + y} - which is valid Scala.
I hope this all makes sense - I tried to give a bit of a background of why the modN(3) call makes sense :)
You are partially applying the ModN function. Partial function application is one of the main features of functional languages. For more information check out these articles on Currying and Pointfree style.
In that example, modN returns a function that mods by the particular N. It saves you from having to do this:
def mod2(x:Int): Boolean = (x%2) == 0
def mod3(x:Int): Boolean = (x%3) == 0
The two pairs of parens delimit where you can stop passing arguments to the method. Of course, you can also just use a placeholder to achieve the same thing even when the method only has a single argument list.
def modN(n: Int, x: Int): Boolean = (x % n) == 0
val nums = List(1, 2, 3, 4, 5)
println(nums.filter(modN(2, _)))
println(nums.filter(modN(3, _)))