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--- lfa:lab02-dfa [2020/10/19 12:27]
dmihai [1. Writing DFAs]
+++ lfa:lab02-dfa [2021/10/13 16:13] (current)
ioana.georgescu [Classes in Python]
@@ Line 1: / Line 1: @@
-====== Deterministic finite automata ======
+====== 2. Deterministic finite automata ======
-==== 1. Writing DFAs ====
+===== Classes in Python =====
-.1. Write a DFA which accepts the language $ L=\{w \in \{0,1\}^* \text{ | w contains an odd number of 1s} \} $
-Hint:
-  * in a DFA, delta is total!
-.2 Define a DFA which accepts arithmetic expressions. Consider the following definition for arithmetic expressions:
+Python supports a limited version of Object-Oriented programming, which includes **class definitions** and **inheritance**. The concept of **interface** and **interface implementation** is absent. Hence, when inheriting a function, it is the job of the programmer to make sure a method is overloaded correctly (otherwise it is just another definition of the class).
-<code>
+Below, we illustrate some examples of Python's object-oriented idiom:
-<expr> ::= <var> | <expr> + <expr> | <expr> * <expr>
-<var> ::= a | b
+<code python>
+class Example:
+   # the class constructor.
+   def __init__(self,param1,param2):
+      # the keyword self is similar to this from Java
+      # it is the only legal mode of initialising and referring class member variables
+      self.member1 = param1
+      # here member2 is a local variable, which is not visible outside of the constructor
+      member2 = param2
+   def fun(self):
+      # a member function must always refer self as shown here. Otherwise it is just a function
+      # defined in the scope of the class, not a member function.
+      return 0
+   def plus(self,x,y):
+      return x + y
+   # this is the equivalent of Java's toString method
+   def __str__(self):
+      string = ...
+      return string
+#global scope
+#class instantiation
+e = Example(1,2)
+#method calls:
+e.fun()
+print(e)
 </code>
-Hint:
+===== 2.1. The class Dfa (Python) =====
-  * how would you define the alphabet for the DFA?
-  * can <expr> be the empty string?
+**2.1.1.** Define the class ''Dfa'' which encodes deterministic finite automata. It must store:
+  * the alphabet
+  * the delta function
+  * the initial state
+  * the set of final states
+**Optional:** you may consider defining a class ''State'' to encode more general Dfas where states are not confined to integers. This will be useful later in your project. However, you will need to define a hash-function in order to use dictionaries over states. More details, google ''hashing in Python''.
+**2.1.2.** Define a constructor for Dfas, which takes a multi-line string of the following form:
-==== 2. Implementing DFAs ====
-Consider the following encoding of a DFA:
 <code>
-   <number_of_states>
+<initial_state>
-   <list_of_final_states>
+<state> <char> <state>
-   <state> <symbol> <state>
+...
+...
+<final_state_1> <final_state_2> ... <final_state_n>
 </code>
+where:
+  * the initial state is given on the first line of the input
+  * each of the subsequent lines encode transitions (states are integers)
+  * the last line encodes the set of final states.
 Example:
 <code>
-3
 a 1
 b 2
 a 0
+2
 </code>
-.1. Write a function which takes a DFA encoding as above and returns a DFA representation. Define a class "DFA".
+**2.1.3.** Implement a member function which takes a **configuration** (pair of state and rest of word) and returns the next configuration.
-.2. Add a method accept which takes a word and returns true if it is accepted by the DFA
+**2.1.4.** Implement a member function which verifies if a word is **accepted** by a Dfa.
-.3. Add a method step with takes a DFA configuration and returns the "next-step" configuration of the DFA. How is a configuration defined?
+===== 2.2. The Dfa Algebraic Datatype (Haskell) =====
-.4(*) Write a method which:
+During the lecture, Dfa states were encoded as integers. As we will soon see, we will need to provision our implementation, so that other types may encode states.
-  * takes a list of DFAs $ a_1, a_2, ..., a_n$
+  * Should the ADT ''DFA'' be **monomorphic** or **polymorphic**?
-  * takes a string $ s$. We know the string consists of a sequence of words, each accepted by some dfa in the list.
-  * returns a list of pairs $ (w_1,a_1), ...(w_i,a_i) ... (w_n,a_n)$ such that $ w_1w_2... w_n = s$ and the dfa $ a_i$ accepts word $ w_i$, for each i from 1 to n.
-Example:
+Also, we shall require a few constraints on what a proper state type should be. States should be: (i) **comparable** via equality, (ii) support an **ordering**. Later on we may add:
+  * new constraints
+  * **operations** which are specific to states only.
+How can we **group** all these constraints and support for future state operations?
+Finally, in order to encode transitions and sets, we need the ''Map'' and ''Set'' datatypes which are available in their own modules:
+<code haskell>
+{- We do import the data constructor Map and the infix function (!) as unqualified, to make them easier to use (Map.Map and Map.(!) is not very legible) -}
+import Data.Map (Map, (!))
+{-
+the keyword qualified forces us to prefix each function call from the module Data.Map with "Map." this is useful for two reasons:
+   - some function names overlap
+   - it makes the code more legible, by making the programmer aware of the module location of the called function
+-}
+import qualified Data.Map as Map
+import Data.Set (Set)
+import qualified Data.Set as Set
+</code>
+<blockquote>** How to choose between lists and sets during the implementation?**
+  * Do you need to make sure elements are unique? (go for sets)
+  * Do you need to iterate a lot over elements, and the collection size is not really big (go for lists)
+</blockquote>
+**2.2.1.** Implement the datatype ''DFA''. It must store:
+  * the alphabet
+  * the delta function
+  * the initial state
+  * the set of final states
+----
+**2.2.2.** Define a function which takes a string of the following form showed below, and returns a DFA **with states as integers**:
 <code>
-l = [a1, a2, a3]
+<initial_state>
- '''
+<state> <char> <state>
-    a1 accepts sequences of digits [0-9]
+...
-    a2 accepts sequences of lowercase symbols [a-z]
+...
-    a3 accepts operands (+ and *)
+<final_state_1> <final_state_2> ... <final_state_n>
- '''
+</code>
-s = "var+40*2300"
-our function returns:
+where:
-[("var",2), ("+",3), ("40",1), ("*",3), ("2300",1)]
+  * the initial state is given on the first line of the input
+  * each of the subsequent lines encode transitions (states are integers)
+  * the last line encodes the set of final states.
+Example:
+<code>
+
+a 1
+b 2
+a 0
+2
 </code>
+**Hint:** Use ''splitBy'' from PP.
+----
+**2.2.3.** Enroll the DFA type in class Show.
+**2.2.4.** Implement function which takes a DFA and a **configuration** (pair of state and rest of word) and returns the next configuration.
+**2.2.5.** Implement a function which verifies if a word is **accepted** by a Dfa. What kind of general list-operation best matches the accepting process?
+===== 2.3. Dfa practice =====
+Write Dfas and test them using your implementation, for the following languages:
+  * **2.3.1.** $ L=\{w \in \{0,1\}^* \text{ | w contains an odd number of 1s} \} $
+  * **2.3.2.** The language of binary words which contain **exactly** two ones
+  * **2.3.3.** The language of binary words which encode odd numbers (the last digit is least significative)
+  * **2.3.4.** (hard) The language of words which encode numbers divisible by 3.