The WHILE Hierarchy of Program Schemes is Infinite

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1 The WHILE Hierarchy of Program Schemes is Infinite Can Adam Alayrak and Thomas Noll RWTH Aachen Ahornstr. 55, Aachen, Germany and fax: Astract. We exhiit a sequence S n (n 0) of while program schemes, i. e., while programs without interpretation, with the property that the while nesting depth of S n is n, and prove that any while program scheme which is scheme equivalent to S n, i. e., equivalent for all interpretations over aritrary domains, has while nesting depth at least n. This shows that the while nesting depth imposes a strict hierarchy (the while hierarchy) when programs are compared with respect to scheme equivalence and contrasts with Kleene s classical result that every program is equivalent to a program of while nesting depth 1 (when interpreted over a fixed domain with arithmetic on non negative integers). Our proof is ased on results from formal language theory; in particular, we make use of the notion of star height of regular languages. 1 Introduction When comparing programming languages, one often has a vague impression of one language eing more powerful than another. However, a asic result of the theory of computaility is that even simple models of computation like Turing machines, while programs (with arithmetic), and partial recursive functions are universal in the following sense: They descrie exactly the class of computale functions, according to Church s thesis. The proof uses encodings of functions y non negative integers with the help of zero and successor function. Thus, if the programming language under consideration supports arithmetic on non negative integers, then it is capale of simulating any effective control structure. A compiler implementing a programming language could in principle adopt this method. In general, such languages do not only specify computations over non negative integers ut they handle data types like floating point numers, character strings, and trees as well. Additionally, modern programming languages allow recursion as means for the description of algorithms. In principle, these extended capailities could e implemented (1) y emedding them in the setting of non negative integers using appropriate encodings, (2) y simulating their ehavior as a computale function, and (3) y translating the result

2 ack into the original context. However, it is clear that this approach is of purely theoretical interest; there is no hope for achieving good efficiency in this way. Instead these concepts are implemented directly: for example recursion is usually translated into iterative algorithms using a run time stack. Thus a comparison of the computational power of programming languages requires the distinction etween the control structures of a program and other aspects like the semantic domains involved in the computations. Therefore we use the approach to decompose a program into a program scheme and an interpretation (which comprises the semantic domain). We study only the scheme part as an astraction of the family of all programs represented y this scheme. In this generalized approach, two schemes are considered to e equivalent iff the concrete programs otained y addition of an interpretation are equivalent for all interpretations. It is well known that the schematic concept of recursion is more powerful than that of iteration [12], that recursion equals iteration + stack [2], and that iteration equals while + Boolean variales [1]. Unfortunately the question of scheme equivalence is undecidale in the general case [11]. The reason is not, as one might expect, the large numer of interpretations which one has to apply for deciding scheme equivalence it suffices to consider free interpretations (or Herrand interpretations) only. Instead, the undecidaility is caused y the structure of the state space of the program, more precisely the state space has too many components. If we astract from the state space we otain simple or monadic schemes for which the question of scheme equivalence ecomes decidale in most cases. In this paper we consider the class of Dijkstra schemes which are inductively uilt up from atomic statements y means of sequential composition, ranching instructions, and while loops. A characterization of scheme equivalence via regular languages is exploited: the star height of the regular language associated with a Dijkstra scheme yields a lower ound for the while nesting depth required. We exhiit a sequence S n (n 0) of Dijkstra schemes with the property that the while nesting depth of S n is n, and prove (via the correspondence to regular languages) that any Dijkstra scheme which is equivalent to S n has while nesting depth at least n. This shows that the while nesting depth of Dijkstra schemes imposes a strict hierarchy with respect to the computational power of the corresponding class of programs the while hierarchy. It contrasts with Kleene s classical result [10] that every program is equivalent to a program of while nesting depth 1 (eside some fixed numer of other loops) when interpreted over a fixed domain with arithmetic on non negative integers. 2 Dijkstra schemes Here we introduce the class of Dijkstra schemes. The only construction elements for Dijkstra schemes are sequential composition, ranching, and conditional it-

3 eration. Thus, Dijkstra schemes can e regarded as while programs without interpretations. Let Ω, Π e non empty, finite, and disjoint sets of unary function symols and unary predicate symols. (Ω, Π) is called a signature. The set BExp(Π) of Boolean expressions over Π is the smallest set which contains Π and which is closed under the Boolean operations (i. e., and ). The class Dij(Ω, Π) of Dijkstra schemes over Ω and Π is the smallest set which, for every S, S 1, S 2 Dij(Ω, Π) and BExp(Π), satisfies the following conditions: Ω Dij(Ω, Π) (S 1 ; S 2 ) Dij(Ω, Π) if then S 1 else S 2 fi Dij(Ω, Π) while do S done Dij(Ω, Π). We allow to omit races. Example 1. Let Ω := {f, g, h} and Π := {p}. Then is a Dijkstra scheme. f; while p do (g; h) done A (Ω, Π) interpretation, or interpretation for short, is a pair A := A; α where A is a non empty set, the domain of the interpretation, and α is a mapping which assigns a predicate α(p) : A {0, 1} to every symol p Π and a total function α(f) : A A to every symol f Ω. Instead of α(f) we write f A. The class of all (Ω, Π) interpretations is denoted y Int(Ω, Π). A pair (S, A) consisting of a Dijkstra scheme S Dij(Ω, Π) and an interpretation A Int(Ω, Π) is called a Dijkstra program. The semantics of (S, A) is the (partial) mapping [S ] A : A A, given as follows: f A (a), if S Ω [S 2 ] A ([S 1 ] A (a)), if S = (S 1 ; S 2 ) [S 1 ] A (a), if S = if then S 1 else S 2 fi and [] A (a) = 1 [S ] A (a) := [S 2 ] A (a), if S = if then S 1 else S 2 fi and [] A (a) = 0 [S ] k A (a), if S = while do S done and the while condition holds undefined else where [] A (a) is the truth value of on input a which is induced y the interpretation A and where the while condition depending on, S and a is given y i {0,..., k 1} : [] A ([S ] i A (a)) = 1 and [] A ([S ] k A (a)) = 0.

4 As usual [] A ([S ] i A (a)) = 1 means that the ith iteration [S ] i A of the mapping [S ] A applied to a is defined and that the results satisfies condition. Example 2. Let S the Dijkstra scheme in Example 1 and A := IN 2 ; α (IN is the set of all non negative integers) with f A (m, n) := (m, 1), g A (m, n) := (m, m n), h A (m, n) := (max{0, m 1}, n) and p A (m, n) = 1 m 0. Then [S ] A computes the factorial function, more precisely [S ] A (m, n) = (0, m!). As mentioned in the introduction we define: Two Dijkstra schemes S 1, S 2 Dij(Ω, Π) are (strongly) equivalent iff the equation [S 1 ] A = [S 2 ] A is valid for all interpretations A Int(Ω, Π). We write S 1 S 2 iff S 1 and S 2 are equivalent. Hence scheme equivalence comprises program equivalence which expresses that, under a fixed interpretation, oth programs compute the same function. 3 Characterization of Dijkstra scheme equivalence Now we give a characterization of scheme equivalence in terms of formal languages. The language L S which we associate with a given Dijkstra scheme S is a regular language capturing the full computation potential of S. To simulate the ehaviour of S under aritrary interpretations, we especially record the decisions which have een taken in the Boolean conditions. The languages we define use Boolean vectors for this protocol; a word of this language consists of function symols and Boolean vectors in alternation. The central point is the representation of scheme composition y a conditional product [9] of the corresponding languages. It allows two computations to e concatenated only if their adjacent Boolean vectors coincide. For a set Ω of unary function symols and a set Π = {p 1,..., p n } of predicate symols with n elements let B := {0, 1} n e the set of all Boolean vectors of length n. We associate with each Boolean expression BExp(Π) a set of Boolean vectors L B y induction: for p i Π let L pi := {(x 1,..., x i 1, 1, x i+1,..., x n ) x 1,..., x i 1, x i+1,..., x n {0, 1}}, and L 1 2 := L 1 L 2, L 1 2 := L 1 L 2, and L := B \ L. Now we are ready to specify the Dijkstra scheme language L S of an aritrary scheme

5 S Dij(Ω, Π). It is given y the following inductive definition: L f :=B {f} B L (S1;S 2) :=L S1 L S2 L if then S1 else S 2 fi :=(L L S1 ) ((B \ L ) L S2 ) ( ) L while do S done := (L L S ) i (B \ L ) i IN where L 1 L 2 is the conditional product defined y L 1 L 2 := {wβv β B, w (BΩ), v (ΩB), wβ L 1 and βv L 2 } and L 0 := B and L i+1 := L i L for every i IN. The class of all Dijkstra scheme languages over Ω and Π is denoted y L Dij (Ω, Π). Example 3. Let S e the Dijkstra scheme in Example 1, and let B := (0 + 1). Then L S is the language denoted y the regular expression Bf(1gBh) 0. Proposition 4. (Characterization of Dijkstra scheme equivalence) For any two Dijkstra schemes S 1, S 2 Dij(Ω, Π) the following condition holds: S 1 S 2 L S1 = L S2. Proof. It is well known that every Dijkstra scheme is translatale into an equivalent Ianov scheme, which can e considered as an uninterpreted (monadic) flowchart (see [7], [6], [8], and [13] for further details). For this class of schemes, I. I. Ianov gave a language theoretic description of equivalence y assigning to every scheme a deterministic finite automaton whose recognized language characterizes the scheme equivalence. The comination of oth techniques yields our proof: y induction on the syntactic structure of S Dij(Ω, Π) it is possile to show that the language associated with the equivalent Ianov scheme trans(s) and the language L S assigned to S coincide. Example 5. Let S once again e the Dijkstra scheme in Example 1 and S e the Dijkstra scheme f; if ( p) then while p do (g; h) done else (g; h); while p do (g; h) done fi Let B := (0+1). Then the Dijkstra scheme language L S is the language denoted y the regular expression (Bf0 ) + ( Bf1gBh(1gBh) 0 ). Since it is the same language as the Dijkstra scheme language for S we can deduce y Proposition 4 that S and S are equivalent.

6 4 The star height of regular languages In order to prove the main theorem we use the concept of star height of regular languages. After presenting some known facts concerning the star height, we show that there exists an infinite family of regular languages (L n ) n IN such that every language L n of this family has star height n. This knowledge will e exploited in the next section. We use to denote the empty language, ε to denote the language which consists of the empty word and L(α) for the language denoted y the regular expression α. The set of all regular expressions over a finite alphaet Σ is denoted y RE(Σ). The star height of a regular expression is the maximal numer of nested stars which appear in this expression, and the star height of a regular language is the minimal star height of all regular expressions denoting this language, more formally one defines for a finite alphaet Σ sh( ) = sh(ε) = sh(a) = 0 for all symols a Σ sh(αβ) = sh(α + β) = max{sh(α),sh(β)} for α, β RE(Σ) sh(α ) = sh(α) + 1 for α RE(Σ), and for a regular language L Σ is called the star height of L. sh(l) := min{sh(α) α RE(Σ) and L(α) = L} Example 6. (Star height [3]) Let Σ := {a, }. The regular expression (a ) has star height 2 ut the language L((a ) ) denoted y this regular expression has at most star height 1, ecause L(ε + (a + ) ) = L((a ) ) and sh(ε + (a + ) ) = 1. Furthermore it is easy to show that a regular language is finite iff it has zero star height. So we get sh(l((a ) )) = 1. In 1963 L. C. Eggan has raised up the question whether there are languages of aritrary star height over a two letter alphaet [5]. F. Dejean and M. P. Schützenerger gave a positive answer to this question y showing that for every n IN \ { 0 } the language L n over the alphaet {a, } which is recognized y the deterministic finite automaton A n

7 a q 1 a q 0 a q 2n 1 a q 2 q 2 n 2 a a q 3 a with 2 n states has star height n. We pick up the technique which has een used in [4] for showing that in a special suclass of regular languages, the class L Dij (Ω, Π) of all Dijkstra scheme languages, there also exist languages of aritrary star height. The following well known lemma, which we need in the next section, is easy to prove. Lemma 7. (Star height of homomorphic images) Let Σ e a finite alphaet and h : Σ Σ e a homomorphism on Σ. Then for every regular language L Σ : sh(h(l)) sh(l). The next lemma presents the regular language family y which we are going to estalish the connection to Dijkstra schemes. Lemma 8. (Star height of a certain family of regular languages) Let (α n ) n IN e a family of regular expressions over the alphaet Σ := {f, g}, defined inductively y α 0 α 1 := ε := (fg) α n+1 := (f 2n α n fgg 2n α n fg) (for n IN \ { 0 } ). (1) Then for all n IN it holds that sh(l(α n )) = n. Proof. To identify the star height of a language given y a regular expression, one has to prove the nonexistence of equivalent expressions of lower star height. Here we are forced to give a proof for every parameter n IN. The technique applied in [4] (cf. also [14] for a similar approach) can e used to otain this

8 result. Here we only sketch the proof. For every n IN \ { 0 }, let K n e a class of regular languages which satisfies the following three conditions (a), (), and (c): (a) For every language L in K n z w L : w f w g = z, where w f and w g denote the numer of occurrences of f and g, respectively, in the word w. () For m, n IN \ { 0 } let w (n,m) {f, g} e given y w (1,m) := fg ( ) mfgg ( ) w (n+1,m) := f 2n 2 n mfg w(n,m) w(n,m). For every n IN \ { 0 }, the n suword index set of a language L is T L n := {m IN \ { 0 } (w (n,m) ) m is a suword of a word in L}. The cardinality of this set, which is called n suword index of L, must e infinite for each L in K n : T L n =, i. e. there are (for every index n of K n ) infinitely many suwords of the form (w (n,m) ) m in L. (c) Every element L K n is minimal with respect to the star height among all languages which satisfy the conditions (a) and (), i. e. for all regular languages L over {f, g} which also fulfil conditions (a) and () it holds that sh(l) sh(l ). Thus all languages in K n have the same star height. It is easy to see that, for every n IN\ { 0 }, L(α n ) (cf. (1)) has properties (a) and (). Hence, sh(l) n for every L K n. The proof of the reverse inequation is shown y induction on n. For the case where n = 1 this follows from the fact that every infinite regular language has a star height of at least 1. For the inductive step we consider a decomposition of L K n+1 in a finite union of expressions of the form γ 0 γ 1γ 2 γ 3... γ 2k 1 γ 2k with sh(γ i ) < sh(l) and verify that there is an index i 0 such that γ i0 meets (a) and () for the parameter n. With the inductive assumption we conclude sh(l) n + 1.

9 5 Nested WHILE loops in Dijkstra schemes We now consider Dijkstra schemes with nested while loops. We want to know whether it is possile to restrict the numer of nested while loops if we do not use coding mechanisms like in recursion theory, and if we do not require any special data structures. We will show that such a limit does not exist in general. To this aim we exploit our characterization of Dijkstra scheme equivalence y formal languages and the star height property of regular languages. According to our preliminary definitions, the proof must e founded on a fixed finite signature of function and predicate symols. Before studying this situation we consider the simpler case where the set of predicate symols may ecome aritrarily large. In this case it suffices to consider the value language val(s) of a Dijkstra scheme S to estalish the connection to formal language theory. It collects all execution paths of S, represented y the sequence of function symols as they are applied, and is defined as the homomorphic image of the Dijkstra scheme language L S under the homomorphism which erases all Boolean vectors. Proposition 9. (Value language of a Dijkstra scheme) Let Ω e a set of unary function symols and R Ω e an aritrary non empty regular language over Ω. Then there exist a set Π R of predicate symols and a Dijkstra scheme S R Dij(Ω, Π R ) such that val(s R ) = R. Proof. The proof is an easy induction on the set RE(Ω) of all regular expressions over Ω, where for the inductive step we assume that the sets of predicate symols of the constituent schemes are disjoint and where we otain one of the schemes if p then S R1 else S R2 fi with a new predicate symol p for the case R 1 R 2 (S R1 ; S R2 ) for R 1 R 2 while p do S R1 done with a new predicate symol p for the case R 1. Note that the while nesting depth of the resulting scheme coincides with the star height of the regular expression representing R. Corollary 10. (Star height of Dijkstra scheme languages with infinite signatures) Let Ω e a set of function symols with at least two elements and Π e an aritrary large set of predicate symols. Then for every n IN there exists a Dijkstra scheme S n Dij(Ω, Π) such that sh(l Sn ) = n, i. e. the star height of Dijkstra scheme languages over infinite signatures is unounded.

10 Proof. We use the following result, cited in Section 4: In the class of regular languages over an alphaet with at least two elements there exists, for every numer n IN, a regular language L n such that sh(l n ) = n. Let n IN, and let α e a regular expression with L(α) = L n and sh(α) = n. According to Proposition 9, there exists (a set Π of predicate symols and) a Dijkstra schema S with val(s) = L n, constructed inductively on the structure of α. Because its value language val(s) is a homomorphic image of the scheme language L S, Lemma 7 yields sh(l S ) sh(val(s)) = n. (2) As mentioned aove, since S has een uilt up according to α, it contains at most n nested while loops. On the other hand, only while loops yield a contriution to the star height of the scheme language L S. Thus we otain sh(l S ) n and hence, y (2), sh(l S ) = n. The question arises whether it is really necessary to introduce new predicate symols, as in the proof of Theorem 9. If it was possile to reuse them, then our proof could e ased on a fixed signature. The following example illustrates the difficulties. Example 11. Let Π := {p} and Ω := {f, g, h}. We consider the Dijkstra schemes S 1 and S 2 over this signature where and S 1 := if p then f else g fi S 2 := h. Then we get L S1 = {1f0, 1f1, 0g0, 0g1} and L S1 = {0h0, 0h1, 1h0, 1h1} and therefore val(s 1 ) = {f, g} and val(s 2 ) = {h}. If in the case R 1 R 2 of the aove construction we would not introduce a new predicate symol then we would otain the Dijkstra scheme S = if p then (if p then f else g fi) else h fi, which has the scheme language L S = {1f0, 1f1, 0h0, 0h1} and thus the value language val(s) = {f, h}. But then val(s) = {f, h} {f, g, h} = val(s 1 ) val(s 2 ). The reason for this is simply that g ecomes never applied in any interpretation ecause of the repeated use of the predicate symol p S is not free.

11 Now we present our proof of the hierarchy result with a fixed signature. We assume that we have at least one predicate symol p and at least two function symols. In the discussion at the end of the paper we will explain why we cannot extend our proof technique to signatures where we have one function symol only. The set of predicate symols which we need in the proof of Theorem 9 must contain at least as many symols as the numer of occurrences of + and in the regular expression where we start from. To restrict the numer of predicate symols we should sparingly use the symol +, and we should reuse predicate symols. The aove example shows that such a reuse can at est e accomplished y employing free Dijkstra schemes, i. e. Dijkstra schemes where etween two condition evaluations a computation (function application) must take place. This can e achieved y appending function symols after while loops. An appropriate family (S n ) n IN of Dijkstra schemes over Ω = {f, g} and Π := {p} is given as follows. For every n IN, let f 2n := f;... ; f }{{} 2 n times (g 2n analogously). Then (S n ) n IN is defined as S 0 S 1 :=while (p p) do f; g done :=while p do f; g done; S n+1 :=while p do f 2n ; S n ; f; g; g 2n ; S n ; f; g; done; (3) (where n 1). Now the following theorem holds: Theorem 12. (Star height of Dijkstra scheme languages over a fixed signature) Let Ω := {f, g} and Π := {p}. For n IN let S n Dij(Ω, Π) e the Dijkstra schemes defined in (3). The Dijkstra scheme language L Sn has the following property: sh(l Sn ) = n. Proof. An easy induction over n IN shows for the value language val(s n ): val(s n ) = L(α n ), where α n is the regular expression defined in Lemma 8. According to Proposition 8 we get sh(val(s n )) = n (for every n IN). As in the proof of Corollary 10, Lemma 7 on the star height of homomorphic images and the oservation that only while loops can contriute to the star height of a Dijkstra scheme language yield n (8) = sh(val(s n )) (7) sh(l Sn ) n, which implies that sh(l Sn ) = n.

12 From this theorem we can deduce a corollary which shows clearly the effect of the different notions of equivalence (program equivalence, scheme equivalence) and of the encodings y means of special data structures. While from the standpoint of recursion theory the numer of nested loops can e ounded, such a limit does not exist from the standpoint of program scheme theory (ecause otherwise there would exist a limit on the star height of Dijkstra scheme languages). We express the main result of this section: Corollary 13. (The WHILE Hierarchy of Dijkstra schemes) The hierarchy of nested while loops in Dijkstra schemes is strict, i. e. for every n IN there exists a Dijkstra scheme S n+1 such that S n+1 uses n + 1 nested while loops and S n+1 cannot e equivalent to any Dijkstra scheme with less than n + 1 nested while loops. 6 Conclusion and Discussion Conclusion: By comining two well known techniques we characterized the equivalence of Dijkstra schemes which respect to the inductive structure of the class of Dijkstra schemes. We have shown, y considering the star height of Dijkstra scheme languages, that the renounce of coding mechanism and special data structures leads to an infinite hierarchy concerning the numer of nested while loops. Discussion: Unfortunately Theorem 12 does not express anything aout minimal signatures, i. e. signatures with one predicate symol and one function symol only. Since value languages of a Dijkstra scheme over such a signature are regular languages over a one letter alphaet, the star height of the value language can only e 0 or 1. So the technique we used in our proof can not e extended to such a signature, ecause the inequality sh(val(s n )) sh(l Sn ) degenerates to 0 sh(l Sn ) or 1 sh(l Sn ), respectively. Thus the question is still open in this setting. Since it suffices to identify languages of aritrary star height in the homomorphic images of the scheme languages, a possile approach might e a homomorphism which erases the function symols instead of the Boolean vectors, yielding a regular language over the two letter alphaet of truth values. Acknowledgements: We would like to thank Klaus Indermark for the premise of this work, as well as Markus Mohnen and Thomas Wilke for the effort of reading a draft version of this paper. References 1. Corrado Böhm and Giuseppe Jacopini. Flow diagrams, Turing machines and languages with only two formation rules. Communications of the ACM, 9(5): , 1966.

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