We've seen two seemingly disparate examples of a monad and I will show you some more in the next tutorial. state monad in LINQ. This is especially painful when dealing with concurrency. Let's continue with our program to define a new monad. Here's the complete runnable version of the calculator that uses our Symbol Table Monad. The first choice would mean that the state never changes and, in fact, doesn't even have to be returned by the action. To make output easy, we will convert empty nodes of type Tree a to Node "_" Empty Empty. This is a great starting point from the point of composability. relabel :: Tree a -> State (Tree Int) relabel (Node l r) = do l’ <- relabel l r’ <- relabel r return (Node l’ r’) Reasoning about monads • How can we prove that the relabelling function is correct? But now, when you look at a do block, it looks very much like imperative code with hidden side effects. Now we can test our implementation of the glue code in one place, or still better, use the library code. Monads are Trees with Grafting This is an attempt to collect together, in tutorial form, a few of the things I've said about monads going back as far as my field guide. By running it! Well, how about exectuting its result? evalState (tagStep tree) 0 -- This is one monadic "step" of the calculation. The example is taken from Simon Thompson’s book Haskell - The Craft of Functional Programming, chapter 18. I promised to show you how another monad, the state monad, can eliminate explicit symbol-table threading. If Applicative is already present and flatten is well-behaved, extending the Applicative to a Monad is trivial. Now we have x of type a so we can call the continuation k: The continuation returns an action act' of the type State s b. Here is a picture of this tree. Note that the actual type definition of the generic transformer is hidden from us, so we must use only the publicly exported functions: get, put and apply (in addition to the monadic functions we get for free.) state Source. These effects are propagated when composing functions (as was the effect of modifying the symbol table, or being undefined for some values of arguments) and can be checked at compile time. Remember, bind is the glue that binds the output of one function to the input of another function -- the one we call a continuation. To understand the State Monad program, it may be best to start with Main, seeing how the various facilities are used, then backtrack through the code learning first the non-monadic tree labeler, starting with the function Label, then finally the monadic tree labeler, starting with the function MLabel. The State monad can help us by maintaining this value for us. They will be used to replace the two placeholders in the expression. We use monads to abstract the tedium of this translation. If the translation of a computation required adding input parameters to the original signature (passing the symbol table in, for instance), I used currying and turned the output type into a function type. from leaves to root). The basic premise of all programming is that you can decompose a complex computation into a set of simpler ones. Note that the actual type definition of the generic transformer is hidden from us, so we must use only the publicly exported functions: get, put and apply (in addition to the monadic functions we get for free.) The compiler may help with flagging explicit mismatches, but it can't check the implicit ones. Tree numbering is used as an example of using the State Monad in the Hackage page for Control.Monad.State.Lazy. In the future, we plan on making a post about using this to print out the tree in a, For pedagogical reasons, our method will allow us to demonstrate how to use the. Note that the actual type definition of the generic transformer is hidden from us, so we must use only the publicly exported functions: get, put and apply (in addition to the monadic functions we get for free.) relabel :: Tree a -> State (Tree Int) relabel (Node l r) = do l’ <- relabel l r’ <- relabel r return (Node l’ r’) Reasoning about monads • How can we prove that the relabelling function is correct? evalState (tagStep tree) 0 -- This is one monadic "step" of the calculation. Now let’s test our simple left to right print function on our example tree to make sure it is constructed properly. 1 Introduction Monads help structure functional programs. Ah right, the state monad is the triplet represented by S, Bind and Return, while MLabel and MLabel1 are just our logic to label a tree in a third way besides manually and functionally: monadically. In our case, this type constructor is of the form (SymTab -> (a, SymTab)), with the type parameter a nested inside the return type of an action. How can we extract x and symTab' from that action? import Control.Monad.State -- Function that numbers the nodes of a `Tree`. So to compute the width information, we will use the following function (no need to use the State monad yet): After calculating the width information for each node, we can calculate the horizontal position of each node. Since I'm new to haskell, and trying to get smart on monads, particularly the state monad, I'm beginning to believe the answer is no, you can't push the state monad down. Now, our strategy will be to do two passes of traversing the tree in the following order: We think of this problem recursively. Ex 2. For this reformatting of the tree, we don’t really need to use the type of the tree. The State type. Strict state monads, passing an updatable state through a computation. import Control.Monad.State -- Function that numbers the nodes of a `Tree`. Modifying an argument (when it's a reference). Sign in to view. It has the form: where Blob stands for the type constructor we are trying to monadize. Ex 1. Besides the mouthful, did I just say state monad? These functions encapsulate their results into Maybe or Either types. Note that the actual type definition of the generic transformer is hidden from us, so we must use only the publicly exported functions: get, put and apply (in addition to the monadic functions we get for free.) For each node, we will keep track of the following data types recording important size information: The field nodeWidth records the length of the string at the current node. So, now we can show the entire tree. State monad: Data type for modeling mutable state in a purely functional way; Either monad: Data type that contains either of two values that represent either a success or a failure; Monad transformers: Type constructors to combine multiple monads into one; The complete source code can be found on GitHub. So what have we gained in comparison to C? All common monads are members of it: class Monad m where (>>=):: m a-> (a-> m b)-> m b (>>):: m a-> m b-> m b return:: a-> m a fail:: String-> m … If we could only find a way to insert this returnS and then immediately cancel it. The StateT Monad structure is parameterized over two things: s - The state. The field leftWidth records the width of the array of String necessary to print the left child subtree. The return function leaves the state unchanged, while >>= uses the final state of the first computation as the initial state of the second. To accumulate a value without using it on the way, see Control.Monad.Trans.Writer. Consider the binary search tree for the words in the sentence “haskell is a cool language but state monads are hard”; the words are simply inserted in the order they appear in the sentence. The problem. For each node, we calculate the positions of the left and right children using the width information of each child. In the original code there were way too many opportunities to use the wrong symbol table for the wrong call. import Control.Monad.State -- Function that numbers the nodes of a `Tree`. This traversal is depth first. Symbol table passing is what "actions" are supposed to do; evaluate should only construct the tracks for the symbol-table train. The code is not only cleaner, but also less error prone. State is defined as a function that takes in some state value, and returns the next state, along with another value. data Tree a = L a | Tree a :+: Tree a. Bind must therefore define a lambda of the signature s -> (b, s) and pass it to state. Since I'm new to haskell, and trying to get smart on monads, particularly the state monad, I'm beginning to believe the answer is no, you can't push the state monad down. The great thing about it is that now all this additional information is visible to the compiler and the type checker. I will freely admit that C++ is not the best language for functional programming, and that monadic code in C++ looks overly complicated. The interesting thing is that this constructor is not exported from the library so you can't pattern match on it. Two things happened (not necessarily in that order) to change this situation: I tried to emphasize the same two steps when introducing monads. Here's the signature of >>= that is required by the Monad class as applied to State s: The first observation is that, in order to run the continuation k, we need a value of type a. To our great relief, this highly repetitive (and error-prone) glue code can be abstracted into just two functions: >>= and return (optionally >> and fail). We run. We might not see the hidden effects, but the compiler does. Still, hidden dependencies make composition fragile. A computation in the state monad State s a takes an initial state as its argument. The standard monad libraries de ne a number of \bread and butter" monads, in-cluding the State, Reader, Writer, List and Maybe monads. It assumes that -- it has access to the current counter value implicitly. Also note that since we have converted empty nodes of type Tree a to non-empty nodes, we have for empty nodes that WidthInfo {nodeWidth = 0, leftWidth = 0, rightWidth = 0}, however we will simply convert empty nodes to empty nodes. How can we turn this value into an evaluator? Even if you want to hold a reference to a chain for some reason, that's what delegates are for. State has one data constructor also called State. That is, the root will be at the left and the leaf nodes will be on the right. The generic monad is defined in the Control.Monad.State.Lazy and, mush like our definition takes two arguments: one for state and one for the result. Well, for the state monad, the monad is actually State s, so if that s was different, we'd be using >>= between two different monads. -- Accumulator function for folding over each level. The first one is used to represent the state (in our case that would be SymTab), and the second is the generic type parameter of every monad type constructor. If it type checks, it must be correct, right? By calling state with a function. To get the example given at the beginning of this post, we simply run. You just stick at every place the formal parameter appears in the body. There are only three places where you'll see explicit use of the symbol table: lookUp, addSymbol, and the main loop -- as it should be! Given the first dot, we have Dimension Zero, the singularity, or monad. The ST monad lets you use update-in-place, but is escapable (unlike IO). This enables you to dynamically replace or select between different (sub)algorithms at run-time. In case of action-returning functions, we need to run these actions, provide the additional parameters they need, and pass results to the next action. A state monad parameterized by the type s of the state to carry. As of March 2020, School of Haskell has been switched to read-only mode. The recipe for turning any functor into a monad is as follows: Create a generic discriminated union. If you use older versions of C#, the basic implementation of Maybe monad … A function signature may expose the effects of a function in addition to just turning input into output types. So now we can show entire tree where each node holds the string and its horizontal position. Here's our monadic UnaryNode evaluator: All references to the symbol table are magically gone. In the new scheme, the inner call to evaluate will no longer return a pair (x, symTab') but a function (SymTab -> (Double, SymTab)). tag :: Tree a -> Tree (a, Int) tag tree = -- tagStep is where the action happens. “The mineral “monad” is not an individuality latent, but an all-pervading Force which has for its present vehicle matter in its lowest and most concrete terrestrial state; in man the monad is fully developed, potential, and either passive or absolutely active, according to its vehicle, the five lower and more physical human principles. From #haskell (see 13:05:37 in the log): 1. For instance, in C you use a combination of functions and side effects. We managed to manipulate state -- the symbol table -- in a purely functional way. That's what monadic return is supposed to do. tag :: Tree a -> Tree (a, Int) tag tree = -- tagStep is where the action happens. In Haskell, all state information usually must be passed to functions explicitly as arguments. A parameterizable state monad where sis the type of the state to carry and ais the type of the return value. However, I have extensively revised and updated these notes as the basis of the new chapter on "Monads and more" in the second edition of my textbook "Programming in … There is even a name for this system in type theory: the effect system. Some computations may not require the full power of state transformers: For a read-only state, see Control.Monad.Trans.Reader. It's supposed to model computations that have access to some read-only environment. For now, a state is an // integer, pure and simple, and a label is a state. Using this data type, we want to define a function, which enumerates the leafs of a tree from left to right. See below for examples. Now you have seen with your own eyes that all this can be done with pure functions. 1, p. 21) “For him [i.e. So we will work with the general tree of type Tree a. The state might look like a global variable but it's not. data Tree a = L a | Tree a :+: Tree a. Control.Monad.Trans.State.Strict. The state, in the case of our expression monad, is a collection of two values — the arguments to the lambda. The state in our return value represents the position of the root node. Note that the actual type definition of the generic transformer is hidden from us, so we must use only the publicly exported functions: get, put and apply (in addition to the monadic functions we get for free.) A frequently asked question about F# is: what's the F# equivalent to an interface?There's no single answer to this question, because, as always, It Depends™. Its signature, again, is determined by the Monad class (I'm calling it returnS for now to avoid name conflicts): The implementation is a no-brainer, really. Use the State monad from Control.Monad.State to re-implement the evaluator. To provide evidence that a type belongs in the Monad type class, cats’ implementation requires us to provide an implementation of pure (which can be reused from Applicative) and flatMap.. We can use flatten to define flatMap: flatMap is just map followed by flatten. On first attempt we might think of the following lambda as our continuation: but it's the wrong type. So to compute the positions we will use the following function: Now we combine these positions with the original strings into a single tree. Let us use our generic state monad to rewrite the tree labeling function from above. First, we write an action that returns the next fresh integer. I recommend studying the complete code for the calculculator listed at the end of this tutorial, with special attention to those functions. Then, to actually use it, we have to interpret it somehow; since the tree is ultimately just a data structure, we can interpret it … Exercise 7: Verify the Monad laws, either abstractly (pencil and paper), or mechnically, via a program, for the state monad. In fact, the State monad would be unnecessary in this case. Here we have a choice: we can run it with the original st or with the new st'. Sign in to view. This is a very general pattern: Take a computation that takes input and produces output but does it in a non-functional way, and modify input and output types in such a way that the computation becomes functional. • Why not exploit monadic structure during the proof? There is a perfectly good monad built on this assumption: it's called the reader monad (see the exercise at the end of this tutorial). It depicts the emergence of the “Monads” from their state of absorption within the ONE; the earliest and highest stage in the formation of “Worlds,” the term Monad being one which may apply equally to the vastest Solar System or the tiniest atom.” (HPB, “The Secret Doctrine” Vol. Consider a function that extracts the schema of a JSON value. evalState (tagStep tree) 0 -- This is one monadic "step" of the calculation. preclude such errors, we show how the state monad may be used to carry the number implicitly as the tree is traversed. The second reason is that this form brings us closer to our goal of abstracting away the tedium of symbol-table passing. So let's familiarize ourselves with the Control.Monad.State version of the state monad (strictly speaking the state monad is defined using a monad transformer, so the actual code in the library may look a bit different from what I present). In imperative code such environment is often implemented as a global object. So this is lesson one: A computation can be turned into a function that encapsulates the originally non-functional bits into its modified (decorated, fancified, or whatever you call it) output data type. For this, we will traverse the tree in a depth first traversal and calculate the positions starting from the root, going downward to the leaves. The use of the Either monad helped us simplify error processing in the last tutorial. This just gets the ball -- rolling, with `0` as the initial counter value. That is, we will be printing the root at the top of the tree and the tree grows downward. tree in a left to right manner. I also showed you how to deal with mutable state by passing it as an additional parameter into and out of a function. "Ideally every term/node has to have its id associated with it at construction time," the poster said. Interestingly, this separation between creating an action and running it turned out to be quite useful in C++, as I showed in my old post Monads in C++. As we traverse each level of the tree, we store the next level in a list. We calculate the position of each node, going in the order of top to bottom (i.e. The field rightWidth is similar, but corresponds to the right child subtree. In Haskell you often see functions whose implementation is determined by their signatures. But we don't have any state yet. That's it. This traversal is also depth first. One is to implement a function that takes two values: The other is to implement a function that takes one argument and returns another function of one argument (parentheses added for emphasis): This might seem like a trivial transformation, but I'll show you how it can help us in coding the evaluator. The most important part of a monad is the bind function. However, they are not the only monads available to an enterprising Haskeller’s toolbox. Using monads, we can effectively hide some state information. Let us use our generic state monad to rewrite the tree labeling function from above. Recall the action that returns the next fresh integer. Monad is an abstraction of computation: you construct a monad chain, you execute it and then you garb the result of the computation. We called the new evaluate only to immediately execute the resulting action? Let us use our generic state monad to rewrite the tree labeling function from above. Our lambda, though, must return a pair of the type (b, s). First, we need to encapsulate our evaluator type in a newtype declaration. The great Brian Beckman demonstrates three ways of labeling a binary tree with unique integer node numbers: (1) by hand, (2) non-monadically, but functionally, by threading an updating counter state variable through function arguments, and (3) monadically, by using a partially generalized state-monad implementation to handle the threading via composition. For simplicity of exposition, let’s just manually define the instance of our tree: For starters and debugging purposes, let’s write a function for printing our After we have reformatted our tree into lists of levels of type [[(String, Position)]], we need to convert each level into a string. Yet proofs about monadic programs often start by expanding the definition of … Using this data type, we want to define a function, which enumerates the leafs of a tree from left to right. Monadic bind makes sure that the state is threaded from function to function. State is defined as a function that takes in some state value, and returns the next state, along with another value. But before I do that, let's have a short refresher on currying, since it's relevant to the construction of the state monad (there is actually some beautiful math behind the relationship of currying and the state monad). Monad instances. Note, that for aesthetic reasons, we wish to separate the node from the left child subtree by using one space. Why are these two functions so important? Our method will actually determine the positions of each node in the print out. For computing the width information at each node, we will do a depth first traversal of the tree, and we will compute the information starting at the leaves of the tree and going up to the root. Finally, before we can work on showing the entire original tree, we first need to convert the original empty nodes to nodes holding strings “_” representing them. As for the state monad, we just invented it, … Let us make a remark about the method we will use in this post. Copy link Quote reply halfAbee commented Apr 16, 2015. Let us use our generic state monad to rewrite the tree labeling function from above. What is its intended use? It is possible to do the horizontal printing above using a simple one pass recursion with functions that return [String]. Here’s the definition of our state — a list of integers (we really need only two, but I’m being sloppy… I mean general): type Args = [Int] Actions Imagine that we have a tree of strings (i.e. Let us use our generic state monad to rewrite the tree labeling function from above. First, we write an action that returns the next fresh integer. Values in the State monad are represented as transition functions from an initial state to a (value,newState) pair and a new type definition is provided to describe this construct: State s a is the type of a value of type a inside the State monad with state of type s. The type constructor State s is an instance of the Monad class. This muddles things a little, but is necessary if we want to use it in an instance declaration. Using monads, we can effectively hide some state information. This is a Java 8 porting of the C# code prepared by Brian Beckman for his lesson about the state monad. That is, a monad is a parameterised type m that supports return and >>= functions of the specified types. State monad is defined by a new type State, which is parameterized by two type variables. Note that the actual type definition of the generic transformer is hidden from us, so we must use only the publicly exported functions: get, put and apply (in addition to the monadic functions we get for free.) We can clearly see the two arguments to bind: one is act, the result of evaluate tree, and the other is the continuation k. The rest must be bind. root to leaves). A state monad parameterized by the type s of the state to carry. To do this, we will traverse the tree in a depth first manner and use the State monad to keep track of the indentation level as we go. The Hoare State Monad Proof Pearl Wouter Swierstra Chalmers University of Technology wouter@chalmers.se Abstract. The continuation is a function that returns an evaluator. Check this out: When you execute a lambda, you simply replace it with its body and replace the formal parameter with the actual argument. In functional languages we need to pass it as an argument to every function that might potentially need access to it. Let us use our generic state monad to rewrite the tree labeling function from above. It takes a function as an argument. First, I showed you how to translate partial computations into total functions. Let me call this function act for action. Copy link Quote reply halfAbee commented Apr 16, 2015. To convince yourself that this indeed works, first apply k to x -- this will just replace x' with x. What do they have in common, other than implementing the functions return and >>=? Recently, I've asked a question for building DFS tree from Graph in Stackoverflow and had learned that it can be simply implemented by using State Monad. But the network of functions depends only on the original parse tree. Now try using it with the pure State monad. Then why even bother with the intermediate step? Why do you need an interface in the first place? let state, tree = monad 0: printTree tree: Console.WriteLine() Console.WriteLine("DONE") Console.ReadLine() |> ignore: 0 // return an integer exit code: This comment has been minimized. Let us use our generic state monad to rewrite the tree labeling function from above. That is, a type a deriving Show, we have already converted a tree of type Tree a to Tree String. I've shown you how to extract the bind operator from state-threading code, but there is a more general derivation that's based on types. State Monad. We once again consider a data type for trees with values only in leaf positions. The following is the output for printing this tree … The fact that m must be a parameterised type, rather than just a type, is inferred from its use in the types for the two functions. An example of a monadic programming in Scheme that juxtaposes Haskell code for a particular state monad with the corresponding Scheme code. Construct a state monad computation from a function. The example involves traversing a binary tree with String leaves and substituting each String with an Integer, each time incremented by one. An explanation in Haskell-Cafe. Let us use our generic state monad to rewrite the tree labeling function from above. • Usual approach: expand definitions of return and bind, perform equational reasoning. According to Robin Waterfield’s translation of The Theology of Arithmetic, “[t]he monad is the non-spatial source of number… the monad is… linear and plane and solid.” In a purely latent state of potentiality, the monad … of type [[(String, Position)]]. One of the simplest monads that we can craft is a state-passing monad. Control.Monad.Trans.State.Strict. Let me remind you what the signature of the function evaluate was -- to make things simpler, let's consider the version from before the introduction of the Either monad: I'm going to parenthesize it the way that highlights the currying interpretation: Let's read this signature carefully: evaluate is a function that takes a Tree and returns a function, which takes a SymTab and returns a pair (Double, SymTab). The starting point of functional programming is that functions have no side effects whatsoever, so function composition is a straightforward matter of passing the results of one function as the input to the next. Monads can be viewed as a standard programming interface to various data or control structures, which is captured by the Monad class. Next, I showed you a way to do the same thing by modifying only the return types of the computation. It's time for some deeper insights. We can use do-notation and the standard monad functions with our DSL. Therefore, leftWidth is one more than the width necessary to print the left child subtree. It assumes that -- it has access to the current counter value implicitly. Sometimes, in OOP, an interface can be used for a Strategy. Note that the entire width necessary to print the current node and its subtrees is (nodeWidth widths) + (leftWidth widths) + (rightWidth widths). The monad will be a new type that contains the functor; you will not be turning the functor itself into a monad. It was, hopefully, a good learning experience, but one that shouldn't be repeated when writing production code. The return function leaves the state unchanged, while >>= uses the final state of the first computation as the initial state of the second.
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