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CD5560 FABER Formal Languages, Automata and Models of Computation Lecture 14

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## CD5560 FABER Formal Languages, Automata and Models of Computation Lecture 14

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**CD5560**FABER Formal Languages, Automata and Models of Computation Lecture 14 Mälardalen University 2010**ContentRecursive FunctionsPrimitive Recursion Ackermann**function & -recursive functions Relations Among Function Classes**Recursion**In computer programming, recursion is related to performing computations in a loop.**Recursion in Problem Modelling**Reducing the complexity by • breaking up computational sequences into its simplest forms. • synthesizing components into more complex objects by replicating simple component sequences over and over again.**"A reduction is a way of converting one problem into another**problem in such a way that a solution to the second problem can be used to solve the first problem." Michael Sipser, Introduction to the Theory of Computation**Recursion can be seen as concept of well-defined**self-reference. We use recursion frequently. Consider, for example, the following hypothetical “definition of a Jew”. I found it on web, as a joke. “Somebody is a Jew if she is Abraham's wife Sarah, or if his or her mother is a Jew.” (My digression: I wonder what about Abraham?)**So if I want to know if I am a Jew, I look at this**definition. I'm not Sarah, so I need to know whether my mother is a Jew. How do I know about my mother? I look at the definition again. She isn't Sarah either, so I ask about her mother. I keep going back through the generations - recursively.**Self-referential definitions can be dangerous if we're not**careful to avoid circularity. The definition ''A rose is a rose'‘* just doesn't cut it. This is why our definition of recursion includes the word well-defined. *Know Gertrude Stein? '' A rose is a rose is a rose''**Yet another recursive definition: an immigrant…**We can write pseudocode to determine whether somebody is an immigrant: FUNCTION isAnImmigrant(person): IF person immigrated herself, THEN: return true ELSE: return isAnImmigrant(person's parent) END IF This is a recursive function, since it uses itself to compute its own value. [According to some authors (Rudbeckius) Adam and Eve were Swedish.]**Functions**From math classes, we have seen many ways of defining and combining numerical functions. • Inverse f-1 • Composition f ◦g • Derivatives f´(x), f´´(x), … • Iteration f1(x), f2(x), f3(x), f4(x), … • …**Functions**Look at what happens when we use only some of these. • How can we define standard interesting functions? • How do these relate to e.g. TM computations? We have seen TMs as functions. They are cumbersome! As alternative, look at a more intuitive definition of functions.**Notation**For brevity, limit to functions on natural numbers N= {0,1,2,…} Notation will also use n-tuples of numbers (m1, …, mn)**Natural Numbers**Start with standard recursive definition of natural numbers (remember Peano?): • A natural number is either • 0, or • successor(n), where n is a natural number.**What is a recurrence?**A recurrence is a well-defined mathematical function written in terms of itself. It is a mathematical function defined recursively.**Fibonacci sequence**1, 1, 2, 3, 5, 8, 13, 21, 34, 55,... The first two numbers of the sequence are both 1, while each succeeding number is the sum of the two numbers before it. (We arrived at 55 as the tenth number, since it is the sum of 21 and 34, the eighth and ninth numbers.)**F(0) = 1 F(1) = 1 (base cases)**F(n) = F(n - 1) + F(n - 2) F is called a recurrence, since it is defined in terms of itself evaluated at other values.**Recursion & Recurrence**A recursive process is one in which objects are defined in terms of other objects of the same type. Using some sort of recurrence relation*, the entire class of objects can then be built up from a few initial values and a small number of rules. (*Recurrence is a mathematical function defined recursively.)**Computable Function**Any computable function can be programmed using while-loops (i.e., "while something is true, do something else"). For-loops (which have a fixed iteration limit) are a special case of while-loops. Computable functions could also be coded using a combination of for- and while-loops.**Primitive Recursive Function**Total Function A function which can be implemented using only for-loops. A function defined for all possible input values.**An example function**Domain Range**We need a set of basic functions.**We need a way to define functions.**Basic Primitive Recursive Functions**Zero function: Successor function: Projection functions:**Building functions**Composition**Composition, Generally**• Given • g1 : Nk N • . • . • . • gm : Nk N • f : Nm N • h(n1,…,nk) = f(g1(n1,…,nk), …, gm(n1,…,nk)) • h = f ◦(g1,…,gm) Alternate notation. • Create h : Nk N**Primitive Recursion “Template”**N.B. For primitive recursive functions recursionin only oneargument.**Any function built from**the basic primitive recursive functions is called Primitive Recursive Function.**Basic Primitive Zero function**(a constant function) Example**Basic Primitive Identity function**Recursive definition**Using Basic Primitive Zero function**and a Successor function we can construct Constant functions etc..**A Primitive Recursive Function**(projection) (successor function)**Predecessor**Predecessor is a primitive recursive function with no direct self-reference.**Primitive Recursion: Logic**A predicate (Boolean function) with output in the set {0,1} which is interpreted as {yes, no}, can be used to define standard functions. • Logical connectives , ,, , … • Numeric comparisons =, < ,, … • Bounded existential quantification in, f(i) • Bounded universal quantification in, f(i) • Bounded minimization min i in, f(i) where result = 0 if f(i) never true within bounds.**Example**Recursive predicates can combine into powerful functions. What does this compute? ???(n) = in,jn, ((i=1 j=n) (j=1 i=n) ijn) Tests primality.**ExampleAnother version of prime(n)**prime(n) = n2 i<n, (i1 mod(n,i) > 0) mod(m,n) = if n>0 then (min i im, div(m,n)n+i=m) else 0 div(m,n) = if n>0 then (min i im, (i+1)n>m) else 0**our construction**primitive recursive template