Greibach Normal Form for ω -Algebraic Systems and Weighted Simple ω -Pushdown Automata
GGreibach Normal Form for ω -Algebraic Systems andWeighted Simple ω -Pushdown Automata Manfred Droste a , Sven Dziadek a,1 , Werner Kuich b,2 a Institut f¨ur Informatik, Universit¨at Leipzig, Germany b Institut f¨ur Diskrete Mathematik und Geometrie, Technische Unversit¨at Wien, Austria
Abstract
In weighted automata theory, many classical results on formal languages have been extended into a quantitative setting.Here, we investigate weighted context-free languages of infinite words, a generalization of ω -context-free languages(Cohen, Gold 1977) and an extension of weighted context-free languages of finite words (Chomsky, Sch¨utzenberger1963). As in the theory of formal grammars, these weighted context-free languages, or ω -algebraic series, can berepresented as solutions of mixed ω -algebraic systems of equations and by weighted ω -pushdown automata.In our first main result, we show that (mixed) ω -algebraic systems can be transformed into Greibach normalform . We use the Greibach normal form in our second main result to prove that simple ω -reset pushdown automata recognize all ω -algebraic series. Simple ω -reset automata do not use (cid:15) -transitions and can change the stack only by atmost one symbol. These results generalize fundamental properties of context-free languages to weighted context-freelanguages. Keywords:
Greibach normal form, weighted automata, omega-pushdown automata, omega context-free languages
1. Introduction
Context-free languages provide a fundamental concept for programming languages in computer science. In orderto model quantitative properties, already in 1963, Chomsky and Sch¨utzenberger [3] introduced weighted context-freelanguages. The theory of weighted pushdown automata has been extensively studied; for background, we refer thereader to the survey [21] and the books [22, 18, 11]. In 1977, Cohen and Gold [4] investigated context-free languagesof infinite words. Weighted ω -context-free languages, i.e., ω -algebraic series were studied by ´Esik and Kuich [16].The goal of this paper is the investigation of weighted context-free languages and weighted pushdown automataon infinite words. As in [18], the weighted context-free languages of infinite words are described by solutions of ω -algebraic systems and mixed ω -algebraic systems of equations. In our first main result, we show that these systemscan be transformed into a Greibach normal form. In the literature, Greibach normal forms, central for the theory ofcontext-free languages of finite words, have been established for ω -context-free languages (of infinite words), see [4],and also for algebraic systems of equations for series over finite words [22, 18]; this latter result is employed in ourproof. Hence here we extend these classical results to a weighted version for infinite words.Recently, Droste, ´Esik and Kuich introduced weighted ω -pushdown automata in [9, 10]. In our second mainresult, we consider weighted simple ω -pushdown automata that we call simple ω -reset pushdown automata here.These automata do not use (cid:15) -transitions and utilize only three simple stack commands: popping a symbol, pushinga symbol or leaving the stack unaltered; moreover, it is only possible to read the topmost stack symbol by poppingit. Observe that together with the restriction of not allowing (cid:15) -transitions, restrictions for the actions on the stack are Email addresses: [email protected] (Manfred Droste), [email protected] (Sven Dziadek), [email protected] (Werner Kuich) Supported by Deutsche Forschungsgemeinschaft (DFG), Graduiertenkolleg 1763 (QuantLA) Partially supported by Austrian Science Fund (FWF): grant no. I1661 N25 a r X i v : . [ c s . F L ] J u l on-trivial. In our second main result we show that these simple ω -reset pushdown automata recognize all weighted ω -context-free languages. For our proof, we use that ω -algebraic systems can be brought into Greibach normal form byour present first main result. Our construction of simple ω -reset pushdown automata is deduced from the constructionused in a recent corresponding result [6] that states that simple reset pushdown automata on finite words recognize allalgebraic series.We believe the model of simple ω -reset pushdown automata to be very natural. Similar expressivity equivalenceresults in the unweighted case hold for context-free languages of finite words, hidden in a proof by Blass and Gure-vich [1], and also for ω -context-free languages, see [8].After the Preliminaries, in Section 3, we characterize ω -algebraic series by a series of equivalent statements.In Section 4, we define mixed ω -algebraic systems and their canonical solutions; l th canonical solutions are bydefinition unique. This allows us to perform the following proof method in Section 7: The proof that each of twoexpressions is the m th component of the l th canonical solution implies the equality of these two expressions. (Comparethis with the proof method in continuous semirings: The proof that each of two expressions is the m th component ofthe least solution of an algebraic system implies the equality of these two expressions.) The main result of Section 4states that each ω -algebraic series is a component of a canonical solution of a mixed ω -algebraic system in Greibachnormal form.In Section 5 we specialize the main result of Section 4: now each ω -algebraic series is a component of a canonicalsolution of an ω -algebraic system in Greibach normal form.We consider simple reset pushdown automata in Section 6 and recall the result of [6] that for each algebraic series r there exists a simple reset pushdown automaton with behavior r .Simple ω -reset pushdown automata are introduced in Section 7. The main result of this section and of the wholepaper is that for each ω -algebraic series r it is possible to construct a simple ω -reset pushdown automaton with behav-ior r . The proof of this main result is performed by our proof method described above. We consider an ω -algebraicseries that is the m th component of the l th canonical solution of an ω -algebraic system in Greibach normal form andconstruct a simple ω -reset pushdown automaton whose moves depend only on the coe ffi cients of this Greibach nor-mal form. We prove that the behavior of this simple ω -reset pushdown automaton equals the m th component of the l th canonical solution of this Greibach normal form. Hence, we conclude that for for each ω -algebraic series r we canconstruct a simple ω -reset pushdown automaton with behavior r .A preliminary version of this paper appeared in [7]. In this version, we strengthen the first main result by provingthat already ω -algebraic systems can be transformed into Greibach normal form. In [7], we only showed the existenceof the Greibach normal form for mixed ω -algebraic systems. The stronger result in this work allows us to generalizethe second main result: weighted simple ω -pushdown recognize all ω -algebraic series. For this, we needed to adaptthe construction such that our simple ω -reset pushdown automata behave exactly like the canonical solutions of ω -algebraic systems. Furthermore, we add a result (see Theorem 2) describing ω -powers of matrices consideringB¨uchi-acceptance. We give complete arguments and further examples for our results.
2. Preliminaries
For the convenience of the reader, we recall definitions and results from ´Esik, Kuich [18].A monoid (cid:104) S , + , (cid:105) is called complete if it is equipped with sum operations (cid:80) I for all families ( a i | i ∈ I ) ofelements of S , where I is an arbitrary index set, such that the following conditions are satisfied (see Conway [5],Eilenberg [14], Kuich [21]):(i) (cid:88) i ∈∅ a i = , (cid:88) i ∈{ j } a i = a j , (cid:88) i ∈{ j , k } a i = a j + a k for j (cid:44) k , (ii) (cid:88) j ∈ J (cid:0) (cid:88) i ∈ I j a i (cid:1) = (cid:88) i ∈ I a i , if (cid:91) j ∈ J I j = I and I j ∩ I j (cid:48) = ∅ for j (cid:44) j (cid:48) .Furthermore, a semiring (cid:104) S , + , · , , (cid:105) is called complete if (cid:104) S , + , (cid:105) is a complete monoid and if we additionally have(iii) (cid:88) i ∈ I ( c · a i ) = c · (cid:0) (cid:88) i ∈ I a i (cid:1) , (cid:88) i ∈ I ( a i · c ) = (cid:0) (cid:88) i ∈ I a i (cid:1) · c .2his means that a semiring S is complete if it has “infinite sums” (i) that are an extension of the finite sums, (ii)that are associative and commutative and (iii) that satisfy the distributivity laws.A semiring S equipped with an additional unary star operation ∗ : S → S is called a starsemiring . In completesemirings for each element a , the star a ∗ of a is defined by a ∗ = (cid:88) j ≥ a j . Hence, each complete semiring is a starsemiring, called a complete starsemiring .Starsemirings allow us to generalize the star operation to matrices. Let M ∈ S n × n , then we define M ∗ ∈ S n × n inductively as in ´Esik, Kuich [18], pp. 14–15 as follows. For n = M = ( a ), for a ∈ S , we let M ∗ = ( a ∗ ). Now,for n >
1, we partition M into submatrices, called blocks , M = (cid:32) a bc d (cid:33) , (1)with a ∈ S × , b ∈ S × ( n − , c ∈ S ( n − × , d ∈ S ( n − × ( n − , and we define M ∗ = (cid:32) ( a + bd ∗ c ) ∗ ( a + bd ∗ c ) ∗ bd ∗ ( d + ca ∗ b ) ∗ ca ∗ ( d + ca ∗ b ) ∗ (cid:33) . (2)Whenever we use a matrix M as defined in (1), the corresponding automaton can be illustrated as follows:1 2 bca d A semiring is called continuous if it is ordered, each directed subset has a least upper bound and addition andmultiplication preserve the least upper bound of directed sets. Any continuous semiring is complete. See ´Esik,Kuich [18] for background.Suppose that S is a semiring and V is a commutative monoid written additively. We call V a (left) S -semimoduleif V is equipped with a (left) action S × V → V ( s , v ) (cid:55)→ sv subject to the following rules: s ( s (cid:48) v ) = ( ss (cid:48) ) v , ( s + s (cid:48) ) v = sv + s (cid:48) v , s ( v + v (cid:48) ) = sv + sv (cid:48) , v = v , v = , s = , for all s , s (cid:48) ∈ S and v , v (cid:48) ∈ V . If V is an S -semimodule, we call ( S , V ) a semiring-semimodule pair .Suppose that ( S , V ) is a semiring-semimodule pair such that S is a starsemiring and S and V are equipped with anomega operation ω : S → V . Then we call ( S , V ) a starsemiring-omegasemimodule pair .´Esik, Kuich [19] define a complete semiring-semimodule pair to be a semiring-semimodule pair ( S , V ) such that S is a complete semiring and V is a complete monoid with s (cid:16)(cid:88) i ∈ I v i (cid:17) = (cid:88) i ∈ I sv i and (cid:16)(cid:88) i ∈ I s i (cid:17) v = (cid:88) i ∈ I s i v , for all s ∈ S , v ∈ V , and for all families ( s i ) i ∈ I over S and ( v i ) i ∈ I over V ; moreover, it is required that an infinite productoperationS ω (cid:51) ( s , s , . . . ) (cid:55)→ (cid:89) j ≥ s j ∈ V
3s given mapping infinite sequences over S to V subject to the following three conditions:(i) (cid:89) i ≥ s i = (cid:89) i ≥ ( s n i − + · · · · · s n i ) ,(ii) s · (cid:89) i ≥ s i + = (cid:89) i ≥ s i ,(iii) (cid:89) j ≥ (cid:88) i j ∈ I j s i j = (cid:88) ( i , i ,... ) ∈ I × I × ... (cid:89) j ≥ s i j ,where in the first equation 0 = n ≤ n ≤ n ≤ . . . and I , I , . . . are arbitrary index sets. This means that the leftaction of the semimodule is distributive and it is required that it has “infinite products” mapping infinite sequencesover S to V such that the product (i) can be partitioned (an infinite form of associativity), (ii) can be extended fromthe left and (iii) satisfies an infinite distributivity law.Suppose that ( S , V ) is complete. Then we define s ∗ = (cid:88) i ≥ s i and s ω = (cid:89) i ≥ s , for all s ∈ S . This turns ( S , V ) into a starsemiring-omegasemimodule pair. Observe that, if ( S , V ) is a completesemiring-semimodule pair, then 0 ω = star-omega semiring is a semiring S equipped with unary operations ∗ and ω : S → S . A star-omega semiring S is called complete if ( S , S ) is a complete semiring-semimodule pair, i.e., if S is complete and is equipped with aninfinite product operation that satisfies the three conditions stated above. A complete star-omega semiring S is called continuous if the semiring S is continuous. Example 1.
Formal languages are covered by our model. Let (cid:104) B , + , · , , (cid:105) be the Boolean semiring. Then let ∗ = ∗ = and take infima as infinite products. This makes B a continuous star-omega and commutative semiring. It thenfollows that B (cid:104)(cid:104) Σ ∗ (cid:105)(cid:105)× B (cid:104)(cid:104) Σ ω (cid:105)(cid:105) is isomorphic to formal languages of finite and infinite words with the usual operations.The semiring (cid:104) N ∞ , + , · , , (cid:105) with N ∞ = N ∪ {∞} and the natural infinite product operation of numbers is acontinuous star-omega and commutative semiring.The tropical semiring (cid:104) N ∞ , min , + , ∞ , (cid:105) with the usual infinite sum operation as infinite product is a commutativesemiring and a continuous star-omega semiring.Analogously, the arctic semiring (cid:104) ¯ N , max , + , −∞ , (cid:105) with ¯ N = N ∪ {−∞ , ∞} and the infinite sum operation asinfinite product is a commutative semiring and a continuous star-omega semiring. A Conway semiring (see Conway [5], Bloom, ´Esik [2]) is a starsemiring S satisfying the sum star identity ( a + b ) ∗ = a ∗ ( ba ∗ ) ∗ and the product star identity ( ab ) ∗ = + a ( ba ) ∗ b for all a , b ∈ S . Observe that by ´Esik, Kuich [18], Theorem 1.2.24, each complete starsemiring is a Conway semiring.Note that from the identities in Conway semirings, it follows a ∗ = + aa ∗ = + a ∗ a , a ( ba ) ∗ = ( ab ) ∗ a , (3)for all a , b ∈ S .If S is a Conway semiring then so is S n × n . Let M ∈ S n × n . Assume that n > M as in (1). Applying theidentities of Conway semirings, we get an equivalent definition (cf. Conway [5], pp. 27–28) to (2): M ∗ = (cid:32) ( a + bd ∗ c ) ∗ a ∗ b ( d + ca ∗ b ) ∗ d ∗ c ( a + bd ∗ c ) ∗ ( d + ca ∗ b ) ∗ (cid:33) . (4)4ollowing Bloom, ´Esik [2], we call a starsemiring-omegasemimodule pair ( S , V ) a Conway semiring-semimodulepair if S is a Conway semiring and if the omega operation satisfies the sum omega identity and the product omegaidentity :( a + b ) ω = ( a ∗ b ) ω + ( a ∗ b ) ∗ a ω and ( ab ) ω = a ( ba ) ω ,for all a , b ∈ S . By ´Esik, Kuich [19] each complete semiring-semimodule pair is a Conway semiring-semimodulepair.Observe that the omega fixed-point equation holds, i.e. aa ω = a ω ,for all a ∈ S .Consider a starsemiring-omegasemimodule pair ( S , V ). Following Bloom, ´Esik [2], we define a matrix operation ω : S n × n → V n × on a starsemiring-omegasemimodule pair ( S , V ) as follows. If n = M ω is the unique element of V , and if n =
1, so that M = ( a ), for some a ∈ S , M ω = ( a ω ). Assume now that n > M as in (1). Then M ω = (cid:32) ( a + bd ∗ c ) ω + ( a + bd ∗ c ) ∗ bd ω ( d + ca ∗ b ) ω + ( d + ca ∗ b ) ∗ ca ω (cid:33) .Additionally, the matrix star identity is valid for Conway semirings and states that the star of a matrix is independentof the partitioning of the matrix. The matrix omega identity is valid for Conway semiring-semimodule pairs and statesthat the operation ω is independent of the partitioning of the matrix, i.e., the blocks of (1) can have arbitrary sizes: a ∈ S n × n , b ∈ S n × n , c ∈ S n × n , d ∈ S n × n for n + n = n . If ( S , V ) is a Conway semiring-semimodule pair, thenso is ( S n × n , V n ). See also ´Esik, Kuich [18], page 106.Following ´Esik, Kuich [17], we define matrix operations ω, t : S n × n → V n × for 0 ≤ t ≤ n as follows. Assume that M ∈ S n × n is decomposed into blocks a , b , c , d as in (1), but with a of dimension t × t and d of dimension ( n − t ) × ( n − t ).Then M ω, t = (cid:32) ( a + bd ∗ c ) ω d ∗ c ( a + bd ∗ c ) ω (cid:33) . (5)Observe that M ω, = M ω, n = M ω . Intuitively, M can be interpreted as an adjacency matrix and M ω, t are infinitepaths where the first t states are repeated states, i.e., states that are B¨uchi-accepting.The next theorem states that, in case of a Conway semiring, M ω, t , for 0 ≤ t ≤ n , can be computed also in a waydi ff erent from its definition and, with certain limits, is independent of the partitioning of the matrix M . Theorem 2.
Let S be a Conway semiring and ≤ t ≤ k ≤ n. Assume M ∈ S n × n is decomposed into blocksM = (cid:32) a bc d (cid:33) with block a being of dimension k × k and block d of dimension ( n − k ) × ( n − k ) .Then we have,M ω, t = (cid:32) ( a + bd ∗ c ) ω, t d ∗ c ( a + bd ∗ c ) ω, t (cid:33) . (6) Proof.
The proof resembles the proof of the matrix omega identity (cf. [18], Theorem 5.3.13). Assume M ∈ S n × n isdecomposed into nine blocks M = f g hi a bj c d f ∈ S t × t , a ∈ S ( k − t ) × ( k − t ) and d ∈ S ( n − k ) × ( n − k ) . Consider the following two partitionings: M = f g hij a bc d M (cid:48) = f gi a hbj c d Now we need to show that M ω, t , calculated as in (5) M ω, t = α (cid:32) a bc d (cid:33) ∗ (cid:32) ij (cid:33) α ,where α = (cid:32) f + (cid:16) g h (cid:17) (cid:32) a bc d (cid:33) ∗ (cid:32) ij (cid:33)(cid:33) ω is equal to M (cid:48) ω, t , calculated as in (6) M (cid:48) ω, t = (cid:32) µ d ∗ (cid:16) j c (cid:17) µ (cid:33) ,where µ = (cid:32)(cid:32) f gi a (cid:33) + (cid:32) hb (cid:33) d ∗ (cid:16) j c (cid:17)(cid:33) ω, t .In the case t = k , we have M = M (cid:48) = (cid:32) f hj d (cid:33) .It follows that α = ( f + hd ∗ j ) ω = ( f + hd ∗ j ) ω, t = µ ,where the second equality is due to t being the full dimension of f + hd ∗ j . The second components of M ω, t and M (cid:48) ω, t then both reduce to d ∗ j ( f + hd ∗ j ) ω .If k = n , we have M = M (cid:48) = (cid:32) f gi a (cid:33) .Now, the second component of M (cid:48) ω, t and the second summand of µ have dimension 0 and thus M (cid:48) ω, t = (cid:32) f gi a (cid:33) ω, t = M ω, t Hence, in the following, we can assume t < k < n .First, we compute M ω, t . We denote the blocks of M ω, t by ( M ω, t ) i for 1 ≤ i ≤
3. Then we have( M ω, t ) = α = (cid:32) f + (cid:16) g h (cid:17) (cid:32) a bc d (cid:33) ∗ (cid:32) ij (cid:33)(cid:33) ω = (cid:32) f + (cid:16) g h (cid:17) (cid:32) ( a + bd ∗ c ) ∗ a ∗ b ( d + ca ∗ b ) ∗ d ∗ c ( a + bd ∗ c ) ∗ ( d + ca ∗ b ) ∗ (cid:33) (cid:32) ij (cid:33)(cid:33) ω = (cid:32) f + (cid:16) g h (cid:17) (cid:32) ( a + bd ∗ c ) ∗ i + a ∗ b ( d + ca ∗ b ) ∗ jd ∗ c ( a + bd ∗ c ) ∗ i + ( d + ca ∗ b ) ∗ j (cid:33)(cid:33) ω = (cid:0) f + g ( a + bd ∗ c ) ∗ i + ga ∗ b ( d + ca ∗ b ) ∗ j + hd ∗ c ( a + bd ∗ c ) ∗ i + h ( d + ca ∗ b ) ∗ j (cid:1) ω .6ere, we used the star of a matrix in the form shown in (4). We will now compute the other two blocks by using thestar of a matrix as in (2): (cid:32) ( M ω, t ) ( M ω, t ) (cid:33) = (cid:32) a bc d (cid:33) ∗ (cid:32) ij (cid:33) α = (cid:32) ( a + bd ∗ c ) ∗ ( a + bd ∗ c ) ∗ bd ∗ ( d + ca ∗ b ) ∗ ca ∗ ( d + ca ∗ b ) ∗ (cid:33) (cid:32) ij (cid:33) α = (cid:32) ( a + bd ∗ c ) ∗ i + ( a + bd ∗ c ) ∗ bd ∗ j ( d + ca ∗ b ) ∗ ca ∗ i + ( d + ca ∗ b ) ∗ j (cid:33) α = (cid:32)(cid:0) ( a + bd ∗ c ) ∗ i + ( a + bd ∗ c ) ∗ bd ∗ j (cid:1) α (cid:0) ( d + ca ∗ b ) ∗ ca ∗ i + ( d + ca ∗ b ) ∗ j (cid:1) α (cid:33) Now, we compute M (cid:48) ω, t . We denote the blocks of M (cid:48) ω, t by ( M (cid:48) ω, t ) i for 1 ≤ i ≤
3. Then we have (cid:32) ( M (cid:48) ω, t ) ( M (cid:48) ω, t ) (cid:33) = µ = (cid:32)(cid:32) f gi a (cid:33) + (cid:32) hb (cid:33) d ∗ (cid:16) j c (cid:17)(cid:33) ω, t = (cid:32)(cid:32) f gi a (cid:33) + (cid:32) hd ∗ j hd ∗ cbd ∗ j bd ∗ c (cid:33)(cid:33) ω, t = (cid:32) f + hd ∗ j g + hd ∗ ci + bd ∗ j a + bd ∗ c (cid:33) ω, t = (cid:32) δ ( a + bd ∗ c ) ∗ ( i + bd ∗ j ) δ (cid:33) ,where δ = (cid:0) f + hd ∗ j + ( g + hd ∗ c )( a + bd ∗ c ) ∗ ( i + bd ∗ j ) (cid:1) ω .It remains to calculate( M (cid:48) ω, t ) = d ∗ (cid:16) j c (cid:17) µ = d ∗ (cid:0) j + c ( a + bd ∗ c ) ∗ ( i + bd ∗ j ) (cid:1) δ .The last step is to verify the three equalities ( M ω, t ) i = ( M (cid:48) ω, t ) i for 1 ≤ i ≤
3. The first equality follows basicallyfrom Lemma 1.2.16 of [18]. We will mark the use of Lemma 1.2.16 by ♦ and obtain( M ω, t ) = α = (cid:0) f + g ( a + bd ∗ c ) ∗ i + ga ∗ b ( d + ca ∗ b ) ∗ j + hd ∗ c ( a + bd ∗ c ) ∗ i + h ( d + ca ∗ b ) ∗ j (cid:1) ω ♦ = (cid:0) f + hd ∗ j + g ( a + bd ∗ c ) ∗ i + g ( a + bd ∗ c ) ∗ bd ∗ j + hd ∗ c ( a + bd ∗ c ) ∗ i + hd ∗ c ( a + bd ∗ c ) ∗ bd ∗ j (cid:1) ω = (cid:0) f + hd ∗ j + g ( a + bd ∗ c ) ∗ ( i + bd ∗ j ) + hd ∗ c ( a + bd ∗ c ) ∗ ( i + bd ∗ j ) (cid:1) ω = (cid:0) f + hd ∗ j + ( g + hd ∗ c )( a + bd ∗ c ) ∗ ( i + bd ∗ j ) (cid:1) ω = δ = ( M (cid:48) ω, t ) For the second equality, we have( M ω, t ) = (cid:0) ( a + bd ∗ c ) ∗ i + ( a + bd ∗ c ) ∗ bd ∗ j (cid:1) α = (cid:0) ( a + bd ∗ c ) ∗ ( i + bd ∗ j ) (cid:1) δ = ( M (cid:48) ω, t ) . 7ow, for the third equality, it su ffi ces to prove( d + ca ∗ b ) ∗ ca ∗ i + ( d + ca ∗ b ) ∗ j = d ∗ (cid:0) j + c ( a + bd ∗ c ) ∗ ( i + bd ∗ j ) (cid:1) .We have d ∗ (cid:0) j + c ( a + bd ∗ c ) ∗ ( i + bd ∗ j ) (cid:1) = d ∗ j + d ∗ c ( a + bd ∗ c ) ∗ ( i + bd ∗ j ) = d ∗ j + d ∗ c ( a + bd ∗ c ) ∗ i + d ∗ c ( a + bd ∗ c ) ∗ bd ∗ j = d ∗ j + d ∗ c ( a ∗ bd ∗ c ) ∗ a ∗ i + d ∗ c ( a ∗ bd ∗ c ) ∗ a ∗ bd ∗ j = d ∗ j + ( d ∗ ca ∗ b ) ∗ d ∗ ca ∗ i + ( d ∗ ca ∗ b ) ∗ d ∗ ca ∗ bd ∗ j = ( d ∗ ca ∗ b ) ∗ d ∗ ca ∗ i + ( d ∗ ca ∗ b ) ∗ d ∗ ca ∗ bd ∗ j + d ∗ j = ( d ∗ ca ∗ b ) ∗ d ∗ ca ∗ i + (cid:0) ( d ∗ ca ∗ b ) ∗ d ∗ ca ∗ b + (cid:1) d ∗ j = ( d ∗ ca ∗ b ) ∗ d ∗ ca ∗ i + ( d ∗ ca ∗ b ) ∗ d ∗ j = ( d + ca ∗ b ) ∗ ca ∗ i + ( d + ca ∗ b ) ∗ j .Note that for this calculation, we rely heavily on commutativity of addition, distributivity and the sum star identity andthe product star identity of Conway semirings together with their derived identities (3). This completes the proof.For a complete definition of quemirings, we refer the reader to [18], page 110. Here we note that a quemiring T isisomorphic to a quemiring S × V determined by the semiring-semimodule pair ( S , V ); it follows that we can identifyevery element t of a quemiring T by a pair ( s , v ) of a semiring-semimodule pair ( S , V ). A quemiring is an algebraicstructure with an addition given componentwise, i.e.,( s , v ) + ( s (cid:48) , v (cid:48) ) = ( s + s (cid:48) , v + v (cid:48) ) ,a semidirect product type multiplication (using that S acts on V ), i.e.,( s , v ) · ( s (cid:48) , v (cid:48) ) = ( ss (cid:48) , v + sv (cid:48) ) ,and two constants 0 = (0 ,
0) and 1 = (1 ,
0) (and a unary operation ¶ , but we will not use it here). A quemiring S × V satisfies a set of axioms inherited from semiring-semimodule pairs; those axioms make a quemiring quasi a semiring (cf. Elgot [15], ´Esik, Kuich [18], page 109; in fact, a quemiring is not necessarily distributive from the left and 0 onlybehaves like a zero from the left). Also, one can define a natural star operation on S × V , i.e.,( s , v ) ⊗ = ( s ∗ , s ω + s ∗ v ) ,making it a generalized starquemiring , see [18].For an alphabet Σ , we call mappings r of Σ ∗ into S series . The collection of all such series r is denoted by S (cid:104)(cid:104) Σ ∗ (cid:105)(cid:105) . We call the set supp( r ) = { w | ( r , w ) (cid:44) } the support of a series r . The set of series with finite support S (cid:104) Σ ∗ (cid:105) = { s ∈ S (cid:104)(cid:104) Σ ∗ (cid:105)(cid:105) | supp( s ) is finite } is called the set of polynomials . We denote by S (cid:104) Σ (cid:105) , S (cid:104){ (cid:15) }(cid:105) and S (cid:104) Σ ∪ { (cid:15) }(cid:105) the series with support in Σ , { (cid:15) } and Σ ∪ { (cid:15) } , respectively. Series s with | supp( s ) | ≤ Σ ω into S are called ω -series and their collection is denoted by S (cid:104)(cid:104) Σ ω (cid:105)(cid:105) . See [22, 18] for moreinformation. Examples of monomials in S (cid:104) Σ ∗ (cid:105) for a semiring (cid:104) S , + , · , , (cid:105) are 0, w , sw for s ∈ S and w ∈ Σ ∗ , definedby (0 , w ) = w , ( w , w ) = w , w (cid:48) ) = w (cid:44) w (cid:48) , ( sw , w ) = s and ( sw , w (cid:48) ) = w (cid:44) w (cid:48) . ω -Algebraic Systems This and the next two sections describe the Greibach normal form for (mixed) ω -algebraic series.8or this section and the next two sections, Sections 3, 4 and 5, S is a continuous, and therefore complete, star-omega semiring . If we consider S (cid:104)(cid:104) Σ ∗ (cid:105)(cid:105) or S (cid:104)(cid:104) Σ ω (cid:105)(cid:105) , then we assume additionally that the underlying semiring S iscommutative . Let further Σ denote an alphabet.By Theorem 5.5.5 of ´Esik, Kuich [18], ( S (cid:104)(cid:104) Σ ∗ (cid:105)(cid:105) , S (cid:104)(cid:104) Σ ω (cid:105)(cid:105) ) is a complete semiring-semimodule pair, hence a Con-way semiring-semimodule pair, satisfying (cid:15) ω =
0. Hence, S (cid:104)(cid:104) Σ ∗ (cid:105)(cid:105) × S (cid:104)(cid:104) Σ ω (cid:105)(cid:105) is a generalized starquemiring.In the sequel, x , y and z denote vectors of dimension n , i.e., x = ( x , . . . , x n ), y = ( y , . . . , y n ) and z = ( z , . . . , z n ).Later, we will also use z of dimension m . It is clear by the context whether they are used as row or as column vectors.Similar conventions hold for vectors p , σ , ω and τ . Moreover, X denotes the set of variables { x , . . . , x n } for S (cid:104)(cid:104) Σ ∗ (cid:105)(cid:105) ,while { z , . . . , z n } is the set of variables for S (cid:104)(cid:104) Σ ω (cid:105)(cid:105) . The set Y denotes the set of variables { y i , . . . , y n } for the quemiring S (cid:104)(cid:104) Σ ∗ (cid:105)(cid:105)× S (cid:104)(cid:104) Σ ω (cid:105)(cid:105) . We will be working with two di ff erent generalizations of ω -context-free grammars, the ω -algebraicsystems and the mixed ω -algebraic systems.An ω -algebraic system over the quemiring S (cid:104)(cid:104) Σ ∗ (cid:105)(cid:105)× S (cid:104)(cid:104) Σ ω (cid:105)(cid:105) consists of an algebraic system over S (cid:104)(cid:104) Σ ∗ (cid:105)(cid:105)× S (cid:104)(cid:104) Σ ω (cid:105)(cid:105) y = p ( y ) , p ∈ ( S (cid:104) ( Σ ∪ Y ) ∗ (cid:105) ) n × .The vector of quemiring elements τ ∈ ( S (cid:104)(cid:104) Σ ∗ (cid:105)(cid:105) × S (cid:104)(cid:104) Σ ω (cid:105)(cid:105) ) n is a solution of the ω -algebraic system y = p ( y ) ,if τ = p ( τ ) .Note that every p i is a polynomial, i.e., a finite sum of monomials in S (cid:104) ( Σ ∪ Y ) ∗ (cid:105) . Let y i = ( x i , z i ), for 1 ≤ i ≤ n .Now, we can apply the quemiring addition and multiplication to p .Consider a monomial t ( y , . . . , y n ) = sw y i w . . . w k − y i k w k ,where s ∈ S and w i ∈ Σ ∗ for 1 ≤ i ≤ k . Note that from the quemiring operations, we have t (( x , z ) , . . . , ( x n , z n )) = ( sw x i w . . . w k − x i k w k , sw z i + sw x i w z i + . . . + sw x i w · · · w k − x i k − w k − z i k ) .Therefore, following ´Esik, Kuich [18], p. 138, we define t x ( x , . . . , x n , z , . . . , z n ) = sw z i + sw x i w z i + . . . + sw x i w · · · w k − x i k − w k − z i k ,and for a polynomial p ( y , . . . , y n ) = (cid:80) ≤ j ≤ m t j ( y , . . . , y n ), we let p x ( x , . . . , x n , z , . . . , z n ) = (cid:88) ≤ j ≤ m ( t j ) x ( x , . . . , x n , z , . . . , z n ) .For an ω -algebraic system y = p ( y ) over S (cid:104)(cid:104) Σ ∗ (cid:105)(cid:105) × S (cid:104)(cid:104) Σ ω (cid:105)(cid:105) , we call x = p ( x ), z = p x ( x , z ) the mixed ω -algebraicsystem over S (cid:104)(cid:104) Σ ∗ (cid:105)(cid:105) × S (cid:104)(cid:104) Σ ω (cid:105)(cid:105) induced by y = p ( y ).In general, a mixed ω -algebraic system over the quemiring S (cid:104)(cid:104) Σ ∗ (cid:105)(cid:105) × S (cid:104)(cid:104) Σ ω (cid:105)(cid:105) consists of an algebraic system over S (cid:104)(cid:104) Σ ∗ (cid:105)(cid:105) x = p ( x ) , p ∈ ( S (cid:104) ( Σ ∪ X ) ∗ (cid:105) ) n × and a linear system over S (cid:104)(cid:104) Σ ω (cid:105)(cid:105) z = (cid:37) ( x ) z , (cid:37) ∈ ( S (cid:104) ( Σ ∪ X ) ∗ (cid:105) ) m × m .The pair ( σ, ω ) ∈ ( S (cid:104)(cid:104) Σ ∗ (cid:105)(cid:105) ) n × ( S (cid:104)(cid:104) Σ ω (cid:105)(cid:105) ) m is a solution of the mixed ω -algebraic system x = p ( x ) , z = (cid:37) ( x ) z , 9f σ = p ( σ ) , ω = (cid:37) ( σ ) ω .Observe that, by Theorem 5.5.1 of ´Esik, Kuich [18], ω ( k ) = (cid:37) ( σ ) ω, k for each 1 ≤ k ≤ n , is solution for the linearsystem z = (cid:37) ( σ ) z .A solution ( σ , . . . , σ n ) of the algebraic system x = p ( x ) is termed least solution if σ i ≤ τ i , for each 1 ≤ i ≤ n , for all solutions ( τ , . . . , τ n ) of x = p ( x ).If σ is the least solution of x = p ( x ), then z = (cid:37) ( σ ) z is an S alg (cid:104)(cid:104) Σ ∗ (cid:105)(cid:105) -linear system and ( σ, ω ( k ) ) = ( σ, (cid:37) ( σ ) ω, k ),where k ∈ { , , . . . , m } , is called k th canonical solution of x = p ( x ), z = (cid:37) ( x ) z . Observe that the k th canonical solutionis unique by definition. A solution ( σ, ω ) is called canonical , if there exists a k such that ( σ, ω ) is the k th canonicalsolution. The k th canonical solution of an ω -algebraic system y = p ( y ) is defined to be the k th canonical solution ofthe mixed ω -algebraic system x = p ( x ), z = p x ( x , z ) induced by y = p ( y ).Recall that S alg (cid:104)(cid:104) Σ ∗ (cid:105)(cid:105) is the collection of all components of least solutions of algebraic systems x i = p i where p i ∈ S (cid:104) ( Σ ∪ X ) ∗ (cid:105) for 1 ≤ i ≤ n .We define S alg (cid:104)(cid:104) Σ ω (cid:105)(cid:105) to be the collection of all components of vectors M ω, k , where M ∈ ( S alg (cid:104)(cid:104) Σ ∗ (cid:105)(cid:105) ) n × n , n ≥
1, and k ∈{ , . . . , n } . Moreover, we define ω - Rat ( S alg (cid:104)(cid:104) Σ ∗ (cid:105)(cid:105) ) to be the ω -Kleene closure of (i.e., the generalized starquemiringgenerated by) S alg (cid:104)(cid:104) Σ ∗ (cid:105)(cid:105) . Example 3.
We consider the following ω -algebraic system over the quemiring B (cid:104)(cid:104) Σ ∗ (cid:105)(cid:105) × B (cid:104)(cid:104) Σ ω (cid:105)(cid:105) for the Booleansemiring (cid:104) B , + , · , , (cid:105) y = y y + (cid:15) y = ay b + (cid:15) ,where a , b ∈ Σ . This induces the following mixed ω -algebraic systemx = x x + (cid:15) z = z + x z x = ax b + (cid:15) z = az .Then for the algebraic system x = p ( x ) over B (cid:104)(cid:104) Σ ∗ (cid:105)(cid:105) , we get the least solution σ = (cid:80) n ≥ a n b n and therefore σ = ( (cid:80) n ≥ a n b n ) ∗ . For the semimodule part, we can consider the first canonical solution where only z is B¨uchi-accepting and the second canonical solution where both z and z are B¨uchi-accepting. The first canonical solutionof the mixed ω -algebraic system x = p ( x ) , z = p x ( x , z ) over B (cid:104)(cid:104) Σ ∗ (cid:105)(cid:105) × B (cid:104)(cid:104) Σ ω (cid:105)(cid:105) is then ( σ , σ ; ( (cid:80) n ≥ a n b n ) ω , . Thesecond canonical solution would be ( σ , σ ; ( (cid:80) n ≥ a n b n ) ω + ( (cid:80) n ≥ a n b n ) ∗ a ω , a ω ) . Example 4.
We consider the following mixed ω -algebraic system over the quemiring N ∞ (cid:104)(cid:104) Σ ∗ (cid:105)(cid:105) × N ∞ (cid:104)(cid:104) Σ ω (cid:105)(cid:105) for thetropical semiring (cid:104) N ∞ , min , + , ∞ , (cid:105) x = ax b + ab z = cz z = x z + z where a , b , c ∈ Σ and using the natural number .Then for the algebraic system x = p ( x ) over N ∞ (cid:104)(cid:104) Σ ∗ (cid:105)(cid:105) , we get the least solution σ = a n b n (cid:55)→ n. The first canonicalsolution of the mixed ω -algebraic system x = p ( x ) , z = (cid:37) ( x ) z over N ∞ (cid:104)(cid:104) Σ ∗ (cid:105)(cid:105) × N ∞ (cid:104)(cid:104) Σ ω (cid:105)(cid:105) is then ( σ, c ω (cid:55)→ , a n b n c ω (cid:55)→ n ) . Hence the series a n b n c ω (cid:55)→ n is ω -algebraic but it is clearly not recognizable by a weighted automaton withoutstack. ω -algebraic series. Theorem 5.
Let S be a continuous complete star-omega semiring with the underlying semiring S being commutativeand let Σ be an alphabet. Then the following statements are equivalent for ( s , υ ) ∈ S (cid:104)(cid:104) Σ ∗ (cid:105)(cid:105) × S (cid:104)(cid:104) Σ ω (cid:105)(cid:105) : (i) ( s , υ ) ∈ S alg (cid:104)(cid:104) Σ ∗ (cid:105)(cid:105) × S alg (cid:104)(cid:104) Σ ω (cid:105)(cid:105) , (ii) ( s , υ ) ∈ ω - Rat ( S alg (cid:104)(cid:104) Σ ∗ (cid:105)(cid:105) ) , (iii) ( s , υ ) = (cid:107) A (cid:107) , where A is a finite S alg (cid:104)(cid:104) Σ ∗ (cid:105)(cid:105) -automaton over S (cid:104)(cid:104) Σ ∗ (cid:105)(cid:105) × S (cid:104)(cid:104) Σ ω (cid:105)(cid:105) , (iv) s ∈ S alg (cid:104)(cid:104) Σ ∗ (cid:105)(cid:105) and υ = (cid:80) ≤ j ≤ l s j t ω j for some l ≥ , where s j , t j ∈ S alg (cid:104)(cid:104) Σ ∗ (cid:105)(cid:105) , (v) ( s , υ ) is a component of a canonical solution of a mixed ω -algebraic system over S (cid:104)(cid:104) Σ ∗ (cid:105)(cid:105) × S (cid:104)(cid:104) Σ ω (cid:105)(cid:105) .Proof. The statements (ii), (iii) and (iv) are equivalent by Theorem 5.4.9 (see also Theorem 5.6.6) of ´Esik, Kuich [18].(iii) ⇒ (v): Assume that ( s , υ ) = (cid:107) A (cid:107) , where A = ( n , I , M , P , k ) is a finite S alg (cid:104)(cid:104) Σ ∗ (cid:105)(cid:105) -automaton. Without loss of gen-erality A is normalized by Theorem 5.4.2 of ´Esik, Kuich [18]; i.e., I = e i for some i . Hence, ( s , υ ) = (( M ∗ P ) i , ( M ω, k ) i )is a component of the k th canonical solution of the mixed ω -algebraic system x = Mx + P , z = Mz .(v) ⇒ (i): Assume there exists a mixed ω -algebraic system x = p ( x ) , z = (cid:37) ( x ) z , with canonical solution ( σ, (cid:37) ( σ ) ω, k )such that ( s , υ ) = ( σ i , ( (cid:37) ( σ ) ω, k ) j ) for some i and j . Since the entries of σ and (cid:37) ( σ ) are in S alg (cid:104)(cid:104) Σ ∗ (cid:105)(cid:105) , ( s , υ ) is in S alg (cid:104)(cid:104) Σ ∗ (cid:105)(cid:105) × S alg (cid:104)(cid:104) Σ ω (cid:105)(cid:105) .(i) ⇒ (iv): Now assume s ∈ S alg (cid:104)(cid:104) Σ ∗ (cid:105)(cid:105) and υ = ( M ω, k ) i for some M ∈ ( S alg (cid:104)(cid:104) Σ ∗ (cid:105)(cid:105) ) n × n , n ≥
1, and i , k ∈ { , . . . , n } .By the definition of M ω, k , each entry of M ω, k is of the form (cid:80) ≤ j ≤ l s j t ω j for some l ≥
0, where s j , t j ∈ S alg (cid:104)(cid:104) Σ ∗ (cid:105)(cid:105) for1 ≤ j ≤ l .
4. Greibach Normal Form for Mixed ω -Algebraic Systems In this section we show that for any element ( s , υ ) of S alg (cid:104)(cid:104) Σ ∗ (cid:105)(cid:105) × S alg (cid:104)(cid:104) Σ ω (cid:105)(cid:105) there exists a mixed ω -algebraicsystem in Greibach normal form such that ( s , υ ) is a component of a solution of this ω -algebraic system.Similar to the definition for algebraic systems on finite words (cf. also Greibach [20]), a mixed ω -algebraic system x = p ( x ) , z = (cid:37) ( x ) z is in Greibach normal form ifsupp( p i ( x )) ⊆ { (cid:15) } ∪ Σ ∪ Σ X ∪ Σ XX , for all 1 ≤ i ≤ n , andsupp( (cid:37) i j ( x )) ⊆ Σ ∪ Σ X , for all 1 ≤ i , j ≤ m .For the construction of the Greibach normal form we need a corollary to Theorem 5. Corollary 6.
The following statement for ( s , υ ) ∈ S (cid:104)(cid:104) Σ ∗ (cid:105)(cid:105) × S (cid:104)(cid:104) Σ ω (cid:105)(cid:105) is equivalent to the statements (i) to (v) ofTheorem 5:s ∈ S alg (cid:104)(cid:104) Σ ∗ (cid:105)(cid:105) and υ = (cid:80) ≤ j ≤ l s j t ω j for some l ≥ , where s j , t j ∈ S alg (cid:104)(cid:104) Σ ∗ (cid:105)(cid:105) with ( t j , (cid:15) ) = ; moreover ( s j , (cid:15) ) = ors j = ( s j , (cid:15) ) (cid:15) .Proof. Assume ( s j , (cid:15) ) (cid:44)
0. Then s j = ( s j , (cid:15) ) (cid:15) + s (cid:48) j where ( s (cid:48) j , (cid:15) ) =
0, and s j t ω j = ( s j , (cid:15) ) t ω j + s (cid:48) j t ω j .Assume ( t j , (cid:15) ) (cid:44)
0. Then t j = ( t j , (cid:15) ) (cid:15) + t (cid:48) j , where ( t (cid:48) j , (cid:15) ) =
0. Since ( S (cid:104)(cid:104) Σ ∗ (cid:105)(cid:105) , S (cid:104)(cid:104) Σ ω (cid:105)(cid:105) ) is a Conway semiring-semimodule pair satisfying (cid:15) ω =
0, we obtain t ω j = (( t j , (cid:15) ) ∗ (cid:15) ∗ t (cid:48) j ) ω with ( t j , (cid:15) ) ∗ (cid:15) ∗ t (cid:48) j ∈ S alg (cid:104)(cid:104) Σ ∗ (cid:105)(cid:105) , since (( t j , (cid:15) ) (cid:15) ) ω = ( t j , (cid:15) ) ω (cid:15) ω = s , υ ) ∈ S alg (cid:104)(cid:104) Σ ∗ (cid:105)(cid:105)× S (cid:104)(cid:104) Σ ω (cid:105)(cid:105) is given in the form of Corollary 6 with l =
1. By Theorem 2.4.10of ´Esik, Kuich [18], there exist algebraic systems in Greibach normal form whose first component of their leastsolutions equals s , t . 11irstly, we deal with the case ( s , (cid:15) ) =
0. Let x i = p i ( x ) + (cid:88) ≤ j ≤ n p i j ( x ) x j , for each 1 ≤ i ≤ n , ( ∗ )where supp( p i ( x )) ⊆ Σ ∪ Σ X , supp( p i j ( x )) ⊆ Σ X , be the algebraic system in Greibach normal form for s and x (cid:48) i = p (cid:48) i ( x (cid:48) ) + (cid:88) ≤ j ≤ m p (cid:48) i j ( x (cid:48) ) x (cid:48) j , for each 1 ≤ i ≤ m , ( ∗∗ )where supp( p (cid:48) i ( x (cid:48) )) ⊆ Σ ∪ Σ X (cid:48) , supp( p i j ( x (cid:48) )) ⊆ Σ X (cid:48) , be the algebraic system in Greibach normal form for t . Let σ and σ (cid:48) with σ = s and σ (cid:48) = t be the least solutions of ( ∗ ) and ( ∗∗ ), respectively.Consider now the mixed ω -algebraic system consisting of the algebraic system ( ∗ ), ( ∗∗ ) over S (cid:104)(cid:104) Σ ∗ (cid:105)(cid:105) and thelinear system over S (cid:104)(cid:104) Σ ω (cid:105)(cid:105) z (cid:48)(cid:48) = p (cid:48) ( x (cid:48) ) z (cid:48)(cid:48) + (cid:88) ≤ j ≤ m p (cid:48) j ( x (cid:48) ) z (cid:48) j , z (cid:48) i = p (cid:48) i ( x (cid:48) ) z (cid:48)(cid:48) + (cid:88) ≤ j ≤ m p (cid:48) i j ( x (cid:48) ) z (cid:48) j , for 1 ≤ i ≤ m , z i = p i ( x ) z (cid:48)(cid:48) + (cid:88) ≤ j ≤ n p i j ( x ) z j , for 1 ≤ i ≤ n . ( ∗ ∗ ∗ )Observe that the mixed ω -algebraic system is in Greibach normal form. We then order the variables of the mixed ω -algebraic system ( ∗ ), ( ∗∗ ), ( ∗ ∗ ∗ ) as x , . . . , x n ; x (cid:48) , . . . , x (cid:48) m ; z (cid:48)(cid:48) ; z (cid:48) , . . . , z (cid:48) m ; z , . . . , z n . After an example, we will provethat ( σ , . . . , σ n ; σ (cid:48) , . . . , σ (cid:48) m ; σ (cid:48) σ (cid:48) ω ; σ (cid:48) σ (cid:48) ω , . . . , σ (cid:48) m σ (cid:48) ω ; σ σ (cid:48) ω , . . . , σ n σ (cid:48) ω ) (7)is a solution of ( ∗ ), ( ∗∗ ), ( ∗ ∗ ∗ ). Observe that σ (cid:48) σ (cid:48) ω = σ (cid:48) ω . Example 7.
Consider the quemiring N ∞ (cid:104)(cid:104) Σ ∗ (cid:105)(cid:105) × N ∞ (cid:104)(cid:104) Σ ω (cid:105)(cid:105) for the tropical semiring (cid:104) N ∞ , min , + , ∞ , (cid:105) . Note thatsubsequently, stands for the natural number and the neutral element of the semiring multiplication is = .We now define algebraic systems in Greibach normal form for s = a n b n (cid:55)→ n and t = (( dd ) ∗ c ) (cid:55)→ . Letx = ax + ax x x (cid:48) = c + dx (cid:48) x (cid:48) x = b x (cid:48) = dHere, x is the start variable for s and x (cid:48) is the start variable for t. In the proof, these two systems are called ( ∗ ) and ( ∗∗ ) . Now, we construct a mixed ω -algebraic system:z (cid:48)(cid:48) = cz (cid:48)(cid:48) + dx (cid:48) z (cid:48) z (cid:48) = cz (cid:48)(cid:48) + dx (cid:48) z (cid:48) z (cid:48) = dz (cid:48)(cid:48) z = ax z (cid:48)(cid:48) + ax z z = bz (cid:48)(cid:48) In the new system (corresponding to ( ∗ ∗ ∗ ) ), variable z (cid:48)(cid:48) is B¨uchi-accepting and variable z acts as the start variable,i.e., we consider the forth component (with the ordering z (cid:48)(cid:48) , z (cid:48) , z (cid:48) , z , z ) of the first canonical solution. The semimodulepart of the solution is st ω = a n b n (( dd ) ∗ c ) ω (cid:55)→ n. Note that the equation for z (cid:48)(cid:48) is needed in this example because z (cid:48) isnot allowed to be B¨uchi-accepting to prevent ( dd ) ω as part of the canonical solution. Lemma 8.
The mixed ω -algebraic system ( ∗ ), ( ∗∗ ), ( ∗ ∗ ∗ ) has solution ( σ , . . . , σ n ; σ (cid:48) , . . . , σ (cid:48) m ; σ (cid:48) σ (cid:48) ω ; σ (cid:48) σ (cid:48) ω , . . . , σ (cid:48) m σ (cid:48) ω ; σ σ (cid:48) ω , . . . , σ n σ (cid:48) ω )12 roof. Again, observe that σ (cid:48) ω = σ (cid:48) σ (cid:48) ω . We obtain, for the first equation, p (cid:48) ( σ (cid:48) ) σ (cid:48) ω + (cid:88) ≤ j ≤ m p (cid:48) j ( σ (cid:48) ) σ (cid:48) j σ (cid:48) ω = (cid:0) p (cid:48) ( σ (cid:48) ) + (cid:88) ≤ j ≤ m p (cid:48) j ( σ (cid:48) ) σ (cid:48) j (cid:1) σ (cid:48) ω = σ (cid:48) σ (cid:48) ω ,then, for 1 ≤ i ≤ m , and the second equation, p (cid:48) i ( σ (cid:48) ) σ (cid:48) ω + (cid:88) ≤ j ≤ m p (cid:48) i j ( σ (cid:48) ) σ (cid:48) j σ (cid:48) ω = (cid:0) p (cid:48) i ( σ (cid:48) ) + (cid:88) ≤ j ≤ m p (cid:48) i j ( σ (cid:48) ) σ (cid:48) j (cid:1) σ (cid:48) ω = σ (cid:48) i σ (cid:48) ω ,and, for 1 ≤ i ≤ n , and the third equation, p i ( σ ) σ (cid:48) ω + (cid:88) ≤ j ≤ n p i j ( σ ) σ j σ (cid:48) ω = (cid:0) p i ( σ ) + (cid:88) ≤ j ≤ n p i j ( σ ) σ j (cid:1) σ (cid:48) ω = σ i σ (cid:48) ω .But we need more: We will prove that this solution is the first canonical solution of ( ∗ ), ( ∗∗ ), ( ∗ ∗ ∗ ). Lemma 9.
The solution (7) is the first canonical solution of the mixed ω -algebraic system ( ∗ ), ( ∗∗ ), ( ∗ ∗ ∗ ).Proof. Let P (cid:48) m ( x (cid:48) ) = (cid:16) p (cid:48) ( x (cid:48) ) · · · p (cid:48) m ( x (cid:48) ) (cid:17) , P (cid:48) m ( x (cid:48) ) = p (cid:48) ( x (cid:48) ) ... p (cid:48) m ( x (cid:48) ) , P (cid:48) mm ( x (cid:48) ) = p (cid:48) ( x (cid:48) ) . . . p (cid:48) m ( x (cid:48) ) ... ... p (cid:48) m ( x (cid:48) ) . . . p (cid:48) mm ( x (cid:48) ) , P n ( x ) = p ( x ) ... p n ( x ) , P nn ( x ) = p ( x ) . . . p n ( x ) ... ... p n ( x ) . . . p nn ( x ) , z = z ... z n , z (cid:48) = z (cid:48) ... z (cid:48) m , and M ( x , x (cid:48) ) = p (cid:48) ( x (cid:48) ) P (cid:48) m ( x (cid:48) ) 0 P (cid:48) m ( x (cid:48) ) P (cid:48) mm ( x (cid:48) ) 0 P n ( x ) 0 P nn ( x ) .Then the linear system ( ∗ ∗ ∗ ) can be written in the form z (cid:48)(cid:48) z (cid:48) z = M ( x , x (cid:48) ) z (cid:48)(cid:48) z (cid:48) z .Hence, the first canonical solution of ( ∗ ), ( ∗∗ ), ( ∗∗∗ ) is ( σ, σ (cid:48) , M ( σ, σ (cid:48) ) ω, ). Before we prove our lemma, we provethree identities.The system ( ∗ ) can be written in the form x = P n ( x ) + P nn ( x ) x , for x = ( x , . . . , x n ) T . 13y the diagonal identity (see Proposition 2.2.11 of ´Esik, Kuich [18]) the system x = P n ( σ ) + P nn ( σ ) x has the same least solution as ( ∗ ). Hence, σ = P nn ( σ ) ∗ P n ( σ ) . (8)The system ( ∗∗ ) can be written in the form x (cid:48) = P (cid:48) m ( x (cid:48) ) + P (cid:48) mm ( x (cid:48) ) x (cid:48) , for x (cid:48) = ( x (cid:48) , . . . , x (cid:48) m ) T .Again, by the diagonal identity (see Proposition 2.2.11 of ´Esik, Kuich [18]) the system x (cid:48) = P (cid:48) m ( σ (cid:48) ) + P (cid:48) mm ( σ (cid:48) ) x (cid:48) has the same solution. Hence σ (cid:48) = P (cid:48) mm ( σ (cid:48) ) ∗ P (cid:48) m ( σ (cid:48) ) . (9)It follows for the first component σ (cid:48) = (cid:16) P (cid:48) mm ( σ (cid:48) ) ∗ P (cid:48) m ( σ (cid:48) ) (cid:17) = (cid:16) P (cid:48) m ( σ (cid:48) ) + P (cid:48) mm ( σ (cid:48) ) + P (cid:48) m ( σ (cid:48) ) (cid:17) = (cid:16) P (cid:48) m ( σ (cid:48) ) + P (cid:48) mm ( σ (cid:48) ) P (cid:48) mm ( σ (cid:48) ) ∗ P (cid:48) m ( σ (cid:48) ) (cid:17) = p (cid:48) ( σ (cid:48) ) + P (cid:48) m ( σ (cid:48) ) P (cid:48) mm ( σ (cid:48) ) ∗ P (cid:48) m ( σ (cid:48) ) . (10)We now compute( M ω, ( σ, σ (cid:48) )) z (cid:48)(cid:48) = (cid:34) p (cid:48) ( σ (cid:48) ) + (cid:16) P (cid:48) m ( σ (cid:48) ) 0 (cid:17) (cid:32) P (cid:48) mm ( σ (cid:48) ) 00 P nn ( σ ) (cid:33) ∗ (cid:32) P (cid:48) m ( σ (cid:48) ) P n ( σ ) (cid:33)(cid:35) ω = (cid:34) p (cid:48) ( σ (cid:48) ) + (cid:16) P (cid:48) m ( σ (cid:48) ) 0 (cid:17) (cid:32) P (cid:48) mm ( σ (cid:48) ) ∗ P nn ( σ ) ∗ (cid:33) (cid:32) P (cid:48) m ( σ (cid:48) ) P n ( σ ) (cid:33)(cid:35) ω = (cid:104) p (cid:48) ( σ (cid:48) ) + P (cid:48) m ( σ (cid:48) ) P (cid:48) mm ( σ (cid:48) ) ∗ P (cid:48) m ( σ (cid:48) ) (cid:105) ω = σ (cid:48) ω .The last equality is by (10).When starting with another variable z i or z (cid:48) j for 1 ≤ i ≤ n and 1 ≤ j ≤ m , we get( M ω, ( σ, σ (cid:48) )) ( z (cid:48) , z ) = (cid:32) P (cid:48) mm ( σ (cid:48) ) 00 P nn ( σ ) (cid:33) ∗ (cid:32) P (cid:48) m ( σ (cid:48) ) P n ( σ ) (cid:33) ( M ω, ( σ, σ (cid:48) )) z (cid:48)(cid:48) = (cid:32) P (cid:48) mm ( σ (cid:48) ) ∗ P nn ( σ ) ∗ (cid:33) (cid:32) P (cid:48) m ( σ (cid:48) ) P n ( σ ) (cid:33) σ (cid:48) ω = (cid:32) P (cid:48) mm ( σ (cid:48) ) ∗ P (cid:48) m ( σ (cid:48) ) P nn ( σ ) ∗ P n ( σ ) (cid:33) σ (cid:48) ω Thus, by (9), we have, for 1 ≤ i ≤ m ,( M ω, ( σ, σ (cid:48) )) z (cid:48) i = (cid:104) P (cid:48) mm ( σ (cid:48) ) ∗ P (cid:48) m ( σ (cid:48) ) (cid:105) i σ (cid:48) ω = σ (cid:48) i σ (cid:48) ω ,and, by (8), we have, for 1 ≤ i ≤ n ,( M ω, ( σ, σ (cid:48) )) z i = (cid:2) P nn ( σ ) ∗ P n ( σ ) (cid:3) i σ (cid:48) ω = σ i σ (cid:48) ω .This completes the proof. 14econdly, we deal with the case s = ( s , (cid:15) ) (cid:15) . Consider now the mixed ω -algebraic system consisting of ( ∗∗ ) andthe linear system over S (cid:104)(cid:104) Σ ω (cid:105)(cid:105) z (cid:48)(cid:48) = p (cid:48) ( x (cid:48) ) z (cid:48)(cid:48) + (cid:88) ≤ j ≤ m p (cid:48) j ( x (cid:48) ) z (cid:48) j , z (cid:48) i = p (cid:48) i ( x (cid:48) ) z (cid:48)(cid:48) + (cid:88) ≤ j ≤ m p (cid:48) i j ( x (cid:48) ) z (cid:48) j , ≤ i ≤ m , z = ( s , (cid:15) ) p (cid:48) ( x (cid:48) ) z (cid:48)(cid:48) + ( s , (cid:15) ) (cid:88) ≤ j ≤ m p (cid:48) j ( x (cid:48) ) z (cid:48) j . ( ∗∗∗∗ ) Lemma 10.
The mixed ω -algebraic system ( ∗∗ ), ( ∗∗∗∗ ) has solution ( σ (cid:48) , . . . , σ (cid:48) m ; σ (cid:48) σ (cid:48) ω ; σ (cid:48) σ (cid:48) ω , . . . , σ (cid:48) m σ (cid:48) ω ; ( s , (cid:15) ) σ (cid:48) ω ) . (11) Proof.
As in the proof of Lemma 8, we obtain that ( σ (cid:48) σ (cid:48) ω ; σ (cid:48) σ (cid:48) ω , . . . , σ (cid:48) m σ (cid:48) ω ) is solution of the z (cid:48)(cid:48) - and z (cid:48) i -equations,1 ≤ i ≤ m . For the right side of the z -equation, we obtain( s , (cid:15) ) (cid:16) p (cid:48) ( σ (cid:48) ) σ (cid:48) ω + (cid:88) ≤ j ≤ m p (cid:48) j ( σ (cid:48) ) σ (cid:48) j σ (cid:48) ω (cid:17) = ( s , (cid:15) ) σ (cid:48) σ (cid:48) ω = ( s , (cid:15) ) σ (cid:48) ω .But again we need more: We will prove that this solution is the first canonical solution of ( ∗∗ ), ( ∗∗∗∗ ). Lemma 11.
The solution (11) is the first canonical solution of the mixed algebraic system ( ∗∗ ), ( ∗∗∗∗ ).Proof. Let M (cid:15) ( x (cid:48) ) = p (cid:48) ( x (cid:48) ) P (cid:48) m ( x (cid:48) ) 0 P (cid:48) m ( x (cid:48) ) P (cid:48) mm ( x (cid:48) ) 0( s , (cid:15) ) p (cid:48) ( x (cid:48) ) ( s , (cid:15) ) P (cid:48) m ( x (cid:48) ) 0 .Then the linear system ( ∗∗∗∗ ) can be written in the form z (cid:48)(cid:48) z (cid:48) z = M (cid:15) ( x (cid:48) ) z (cid:48)(cid:48) z (cid:48) z .Hence, the first canonical solution of ( ∗∗ ), ( ∗∗∗∗ ) is ( σ (cid:48) , M (cid:15) ( σ (cid:48) ) ω, ). We now compute( M ω, (cid:15) ( σ (cid:48) )) z (cid:48)(cid:48) = (cid:34) p (cid:48) ( σ (cid:48) ) + (cid:16) P (cid:48) m ( σ (cid:48) ) 0 (cid:17) (cid:32) P (cid:48) mm ( σ (cid:48) ) 0( s , (cid:15) ) P (cid:48) m ( σ (cid:48) ) 0 (cid:33) ∗ (cid:32) P (cid:48) m ( σ (cid:48) )( s , (cid:15) ) p (cid:48) ( σ (cid:48) ) (cid:33)(cid:35) ω = (cid:34) p (cid:48) ( σ (cid:48) ) + (cid:16) P (cid:48) m ( σ (cid:48) ) 0 (cid:17) (cid:32) P (cid:48) mm ( σ (cid:48) ) ∗ s , (cid:15) ) P (cid:48) m ( σ (cid:48) ) P (cid:48) mm ( σ (cid:48) ) ∗ (cid:33) (cid:32) P (cid:48) m ( σ (cid:48) )( s , (cid:15) ) p (cid:48) ( σ (cid:48) ) (cid:33)(cid:35) ω = (cid:104) p (cid:48) ( σ (cid:48) ) + P (cid:48) m ( σ (cid:48) ) P (cid:48) mm ( σ (cid:48) ) ∗ P (cid:48) m ( σ (cid:48) ) (cid:105) ω = σ (cid:48) ω .The last equality is by (10).When starting with another variable z (cid:48) i or z for 1 ≤ i ≤ m , we get( M ω, (cid:15) ( σ (cid:48) )) ( z (cid:48) , z ) = (cid:32) P (cid:48) mm ( σ (cid:48) ) 0( s , (cid:15) ) P (cid:48) m ( σ (cid:48) ) 0 (cid:33) ∗ (cid:32) P (cid:48) m ( σ (cid:48) )( s , (cid:15) ) p (cid:48) ( σ (cid:48) ) (cid:33) ( M ω, (cid:15) ( σ (cid:48) )) z (cid:48)(cid:48) = (cid:32) P (cid:48) mm ( σ (cid:48) ) ∗ s , (cid:15) ) P (cid:48) m ( σ (cid:48) ) P (cid:48) mm ( σ (cid:48) ) ∗ (cid:33) (cid:32) P (cid:48) m ( σ (cid:48) )( s , (cid:15) ) p (cid:48) ( σ (cid:48) ) (cid:33) σ (cid:48) ω = (cid:32) P (cid:48) mm ( σ (cid:48) ) ∗ P (cid:48) m ( σ (cid:48) )( s , (cid:15) ) P (cid:48) m ( σ (cid:48) ) P (cid:48) mm ( σ (cid:48) ) ∗ P (cid:48) m ( σ (cid:48) ) + ( s , (cid:15) ) p (cid:48) ( σ (cid:48) ) (cid:33) σ (cid:48) ω ≤ i ≤ m ,( M ω, (cid:15) ( σ (cid:48) )) z (cid:48) i = (cid:104) P (cid:48) mm ( σ (cid:48) ) ∗ P (cid:48) m ( σ (cid:48) ) (cid:105) i σ (cid:48) ω = σ (cid:48) i σ (cid:48) ω ,and, by (10), we have( M ω, (cid:15) ( σ (cid:48) )) z = (cid:0) ( s , (cid:15) ) P (cid:48) m ( σ (cid:48) ) P (cid:48) mm ( σ (cid:48) ) ∗ P (cid:48) m ( σ (cid:48) ) + ( s , (cid:15) ) p (cid:48) ( σ (cid:48) ) (cid:1) σ (cid:48) ω = ( s , (cid:15) ) (cid:0) P (cid:48) m ( σ (cid:48) ) P (cid:48) mm ( σ (cid:48) ) ∗ P (cid:48) m ( σ (cid:48) ) + p (cid:48) ( σ (cid:48) ) (cid:1) σ (cid:48) ω = ( s , (cid:15) ) σ (cid:48) σ (cid:48) ω = ( s , (cid:15) ) σ (cid:48) ω .We now consider general sums of series of the above form. The next lemma shows how to construct a mixed ω -algebraic system whose canonical solution is the sum of the canonical solutions of multiple mixed ω -algebraicsystems as given in Lemmas 9 and 11. Lemma 12.
Let ( s , υ ) ∈ S alg (cid:104)(cid:104) Σ ∗ (cid:105)(cid:105) × S (cid:104)(cid:104) Σ ω (cid:105)(cid:105) be given in the form of Corollary 6. Then there exists a mixed ω -algebraic system in Greibach normal form such that υ is a component of its l th canonical solution.Proof. Let υ = (cid:80) ≤ i ≤ l s i t ω i as in the statement of Corollary 6 and let l ≥
1. By Lemmas 9 and 11, for 1 ≤ i ≤ l , thereexist mixed ω -algebraic systems x i = p i ( x i ) , ( (cid:93) ) (cid:32) z i ¯ z i (cid:33) = M i ( x i ) (cid:32) z i ¯ z i (cid:33) , in Greibach normal form with M i ( x i ) = (cid:32) a i b i c i d i (cid:33) , where a i ∈ ( S (cid:104) ( Σ ∪ X ) ∗ (cid:105) ) × , b i ∈ ( S (cid:104) ( Σ ∪ X ) ∗ (cid:105) ) × ( n i − , c i ∈ ( S (cid:104) ( Σ ∪ X ) ∗ (cid:105) ) ( n i − × , d i ∈ ( S (cid:104) ( Σ ∪ X ) ∗ (cid:105) ) ( n i − × ( n i − , such that s i t ω i is a component of the first canonical solution of the i th system. We will assume without loss of generalitythat s i t ω i is the first component of variable ¯ z i , i.e., s i t ω i = (cid:104) ( M ω, i ) ¯ z i (cid:105) = (cid:2) ( d ∗ i c i )( a i + b i d ∗ i c i ) ω (cid:3) . (12)Similarly to the case of summation in Theorem 5.4.4 of ´Esik, Kuich [18], we consider now the mixed ω -algebraicsystem consisting of the algebraic systems ( (cid:93) ) over S (cid:104)(cid:104) Σ ∗ (cid:105)(cid:105) and the linear system over S (cid:104)(cid:104) Σ ω (cid:105)(cid:105) ˆ z = M ˆ z , ( (cid:93)(cid:93) )with M = a . . . a l b . . . b l c . . . c l d . . . d l (cid:16) c · · · c l (cid:17) (cid:16) d · · · d l (cid:17) , ˆ z = z ... z l ¯ z ... ¯ z l z (cid:48) .16ote that this system (cid:93)(cid:93) is still in Greibach normal form.We order the variables of the mixed ω -algebraic system ( (cid:93) ), ( (cid:93)(cid:93) ) as z , . . . , z l ; ¯ z , . . . , ¯ z l ; z (cid:48) . We now compute the l th canonical solution, starting with variable z = ( z , . . . , z l ) T . Then( M ω, l ) z = a . . . a l + b . . . b l d . . . d l (cid:16) d · · · d l (cid:17) ∗ c . . . c l (cid:16) c · · · c l (cid:17) ω = a . . . a l + b . . . b l d . . . d l ∗ (cid:16) d · · · d l (cid:17) d . . . d l ∗ c . . . c l (cid:16) c · · · c l (cid:17) ω = a . . . a l + b . . . b l d . . . d l ∗ c . . . c l ω = ( a + b d ∗ c ) ω ... ( a l + b l d ∗ l c l ) ω .When starting with the new variable z (cid:48) , we get a sum of the original solutions:( M ω, l ) z (cid:48) = d . . . d l (cid:16) d · · · d l (cid:17) ∗ c . . . c l (cid:16) c · · · c l (cid:17) ( M ω, l ) z l + = d . . . d l ∗ (cid:16) d · · · d l (cid:17) d . . . d l ∗ c . . . c l (cid:16) c · · · c l (cid:17) ( M ω, l ) z l + = d . . . d l ∗ c . . . c l (cid:16) d d ∗ · · · d l d ∗ l (cid:17) c . . . c l + (cid:16) c · · · c l (cid:17) ( M ω, l ) z l + = d ∗ c . . . d ∗ l c l (cid:16) d d ∗ c + c · · · d l d ∗ l c l + c l (cid:17) ( a + b d ∗ c ) ω ... ( a l + b l d ∗ l c l ) ω l + d ∗ c ( a + b d ∗ c ) ω ... d ∗ l c l ( a l + b l d ∗ l c l ) ω (cid:80) ≤ i ≤ l ( d i d ∗ i c i + c i )( a i + b i d ∗ i c i ) ω l + = (cid:88) ≤ i ≤ l ( d i d ∗ i c i + c i )( a i + b i d ∗ i c i ) ω = (cid:88) ≤ i ≤ l ( d ∗ i c i )( a i + b i d ∗ i c i ) ω Thus, the first component is (by identity (12)) (cid:104) ( M ω, l ) z (cid:48) (cid:105) = (cid:88) ≤ i ≤ l ( d ∗ i c i )( a i + b i d ∗ i c i ) ω = (cid:88) ≤ i ≤ l (cid:2) ( d ∗ i c i )( a i + b i d ∗ i c i ) ω (cid:3) = (cid:88) ≤ i ≤ l s i t ω i = υ .We can now conclude the following theorem. Theorem 13.
The following statement for ( s , υ ) ∈ S (cid:104)(cid:104) Σ ∗ (cid:105)(cid:105) × S (cid:104)(cid:104) Σ ω (cid:105)(cid:105) is equivalent to the statements of Theorem 5: ( s , υ ) is component of a canonical solution of a mixed ω -algebraic system over S (cid:104)(cid:104) Σ ∗ (cid:105)(cid:105) × S (cid:104)(cid:104) Σ ω (cid:105)(cid:105) in Greibach normalform.Proof. The above statement trivially implies statement (v) of Theorem 5. By Corollary 6 and Lemma 12, the state-ments of Theorem 5 imply the above statement.
5. Greibach Normal Form for ω -Algebraic Systems For the following sections, we need the Greibach normal form not only for mixed ω -algebraic systems but alsofor ω -algebraic systems. So we show in this section a specialization of Theorem 13 for ω -algebraic systems.Similar to the definition for mixed ω -algebraic systems, an ω -algebraic system y = p ( y )where { y , . . . , y n } is a set of variables for the quemiring S alg (cid:104)(cid:104) Σ ∗ (cid:105)(cid:105) × S alg (cid:104)(cid:104) Σ ω (cid:105)(cid:105) , is in Greibach normal form ifsupp( p i ( y )) ⊆ { (cid:15) } ∪ Σ ∪ Σ Y ∪ Σ YY , for all 1 ≤ i ≤ n .Our first main result is the following. Theorem 14.
The following statement for ( s , υ ) ∈ S (cid:104)(cid:104) Σ ∗ (cid:105)(cid:105) × S (cid:104)(cid:104) Σ ω (cid:105)(cid:105) is equivalent to the statements of Theorem 5: ( s , υ ) is component of a canonical solution of an ω -algebraic system over S (cid:104)(cid:104) Σ ∗ (cid:105)(cid:105) × S (cid:104)(cid:104) Σ ω (cid:105)(cid:105) in Greibach normal form.Proof. By Theorem 13, we can assume that ( s , υ ) is component of the t th canonical solution of a mixed ω -algebraicsystem over S (cid:104)(cid:104) Σ ∗ (cid:105)(cid:105) × S (cid:104)(cid:104) Σ ω (cid:105)(cid:105) in Greibach normal form for a t ∈ N . Let the mixed ω -algebraic system be given inthe following form: x i = p i + (cid:88) ≤ j ≤ n ( p i j x + q i j ) x j , for 1 ≤ i ≤ n , ( (cid:62) ) z i = (cid:88) ≤ j ≤ m ( p (cid:48) i j x + q (cid:48) i j ) z j , for 1 ≤ i ≤ m , ( (cid:62)(cid:62) )where p i j ∈ S (cid:104) Σ (cid:105) × n , for 1 ≤ i , j ≤ n , p (cid:48) i j ∈ S (cid:104) Σ (cid:105) × n , for 1 ≤ i , j ≤ m , 18nd supp( p i ) ⊆ { (cid:15) } ∪ Σ , supp( p i j x ) ⊆ Σ X , supp( q i j ) ⊆ Σ ,supp( p (cid:48) i j x ) ⊆ Σ X , supp( q (cid:48) i j ) ⊆ Σ .Note that p i j x = (cid:88) ≤ k ≤ n ( p i j ) k x k ;we decided for this notation because of brevity, important especially in matrices.For the remainder of the proof, consider integers k and l to be fixed such that the t th canonical solution of ( (cid:62) ),( (cid:62)(cid:62) ) is ( σ, ω ) with σ k = s and ω l = υ .We will later need a simple implication: We can write the linear system ( (cid:62)(cid:62) ) as z = P (cid:48) mm ( x ) z ,where P (cid:48) mm ( x ) = p (cid:48) x + q (cid:48) · · · p (cid:48) m x + q (cid:48) m ... . . . ... p (cid:48) m x + q (cid:48) m · · · p (cid:48) mm x + q (cid:48) mm .Note that t ≤ m . It follows that ω = P (cid:48) mm ( σ ) ω, t . (13)Now, we construct from ( (cid:62) ), ( (cid:62)(cid:62) ) an ω -algebraic system ( (cid:62)(cid:62)(cid:62) ) where the variables x are substituted by ¯ y and z by ˆ y . Additionally, we add a new equation and a new variable ˙ y to combine the k th component of the semiring partand the l th component of the semimodule part:ˆ y i = (cid:88) ≤ j ≤ m ( p (cid:48) i j ¯ y + q (cid:48) i j )ˆ y j , for 1 ≤ i ≤ m ,¯ y i = p i + (cid:88) ≤ j ≤ n ( p i j ¯ y + q i j )¯ y j , for 1 ≤ i ≤ n ,˙ y = p k + (cid:88) ≤ j ≤ n ( p k j ¯ y + q k j )¯ y j + (cid:88) ≤ j ≤ m ( p (cid:48) l j ¯ y + q (cid:48) l j )ˆ y j . ( (cid:62)(cid:62)(cid:62) )Note that ( (cid:62)(cid:62)(cid:62) ) is in Greibach normal form. Moreover, note that we order the equations such that the first equationsare those corresponding to the old equations of variables z i . This ensures that the t th canonical solution still considersthe correct variables as B¨uchi-accepting. Claim : The ( m + n + th component of the t th canonical solution of ( (cid:62)(cid:62)(cid:62) ) is ( σ k , ω l ) = ( s , υ ).We now compute this solution. The t th canonical solution of the ω -algebraic system ( (cid:62)(cid:62)(cid:62) ) is defined to bethe t th canonical solution of the mixed ω -algebraic system induced by ( (cid:62)(cid:62)(cid:62) ). The corresponding induced mixed ω -algebraic system is given by the algebraic system over S alg (cid:104)(cid:104) Σ ∗ (cid:105)(cid:105) ˆ x i = (cid:88) ≤ j ≤ m ( p (cid:48) i j ¯ x + q (cid:48) i j ) ˆ x j , for 1 ≤ i ≤ m ,¯ x i = p i + (cid:88) ≤ j ≤ n ( p i j ¯ x + q i j ) ¯ x j , for 1 ≤ i ≤ n ,˙ x = p k + (cid:88) ≤ j ≤ n ( p k j ¯ x + q k j ) ¯ x j + (cid:88) ≤ j ≤ m ( p (cid:48) l j ¯ x + q (cid:48) l j ) ˆ x j , ( S alg (cid:104)(cid:104) Σ ω (cid:105)(cid:105) ˆ z i = (cid:88) ≤ j ≤ m ( p (cid:48) i j ¯ x + q (cid:48) i j )ˆ z j + p (cid:48) i j ¯ z , for 1 ≤ i ≤ m ,¯ z i = (cid:88) ≤ j ≤ n ( p i j ¯ x + q i j )¯ z j + p i j ¯ z , for 1 ≤ i ≤ n ,˙ z = (cid:88) ≤ j ≤ n ( p k j ¯ x + q k j )¯ z j + p k j ¯ z + (cid:88) ≤ j ≤ m ( p (cid:48) l j ¯ x + q (cid:48) l j )ˆ z j + p (cid:48) l j ¯ z . ( Claim : (0 , . . . , σ ; σ k ) is the least solution of ( m equations, and for 1 ≤ i ≤ m , (cid:88) ≤ j ≤ m ( p (cid:48) i j σ + q (cid:48) i j )0 = ≤ i ≤ n , p i + (cid:88) ≤ j ≤ n ( p i j σ + q i j ) σ j = σ i ;because σ is a solution of ( (cid:62) ). Finally, we obtain by the same reason, for the last equation, p k + (cid:88) ≤ j ≤ n ( p k j σ + q k j ) σ j + (cid:88) ≤ j ≤ m ( p (cid:48) l j σ + q (cid:48) l j )0 j = p k + (cid:88) ≤ j ≤ n ( p k j σ + q k j ) σ j + = σ k .The algebraic system ( , . . . , σ ; σ k ) is also the leastsolution. This proves the claim.Now consider the linear system ( P (cid:48) mm ( ¯ x ) be defined as above and let further P nn ( ¯ x ) = p ¯ x + q · · · p n ¯ x + q n ... . . . ... p n ¯ x + q n · · · p nn ¯ x + q nn , R nn = (cid:80) ≤ j ≤ n ( p j ) · · · (cid:80) ≤ j ≤ n ( p j ) n ... . . . ... (cid:80) ≤ j ≤ n ( p n j ) · · · (cid:80) ≤ j ≤ n ( p n j ) n , R (cid:48) mn = (cid:80) ≤ j ≤ m ( p (cid:48) j ) · · · (cid:80) ≤ j ≤ m ( p (cid:48) j ) n ... . . . ... (cid:80) ≤ j ≤ m ( p (cid:48) m j ) · · · (cid:80) ≤ j ≤ m ( p (cid:48) m j ) n .Note that for ( ≤ i ≤ m , we have (cid:88) ≤ j ≤ m p (cid:48) i j ¯ z = (cid:88) ≤ j ≤ m (cid:88) ≤ k ≤ n ( p (cid:48) i j ) k ¯ z k = (cid:88) ≤ k ≤ n (cid:88) ≤ j ≤ m ( p (cid:48) i j ) k ¯ z k = (cid:16) (cid:88) ≤ j ≤ m ( p (cid:48) i j ) , · · · , (cid:88) ≤ j ≤ m ( p (cid:48) i j ) n (cid:17) ¯ z = ( R (cid:48) mn ) i ¯ z . 20nalogously, we can prove (cid:80) ≤ j ≤ n p i j ¯ z = ( R nn ) i ¯ z . We let M ( ˆ x , ¯ x , x ) = P (cid:48) mm ( ¯ x ) R (cid:48) mn P nn ( ¯ x ) + R nn P (cid:48) mm ( ¯ x )) l ( P nn ( ¯ x )) k + ( R nn ) k + ( R (cid:48) mn ) l ,then the linear system ( ˆ z ¯ zz = M ( ˆ x , ¯ x , x ) ˆ z ¯ zz .Now, we can plug the semiring part (0 , σ, σ k ) of the solution into M . By Theorem 2, the semimodule part of thecanonical solution of ( M (0 , σ, σ k ) ω, t = ξ ω, t (cid:32) P nn ( σ ) + R nn χ (cid:33) ∗ (cid:32) P (cid:48) mm ( σ )) l (cid:33) ξ ω, t with χ = ( P nn ( σ )) k + ( R nn ) k + ( R (cid:48) mn ) l and ξ = P (cid:48) mm ( σ ) + (cid:16) R (cid:48) mn (cid:17) (cid:32) P nn ( σ ) + R nn χ (cid:33) ∗ (cid:32) P (cid:48) mm ( σ )) l (cid:33) = P (cid:48) mm ( σ ) + (cid:16) R (cid:48) mn (cid:17) (cid:32) ( P nn ( σ ) + R nn ) ∗ χ ( P nn ( σ ) + R nn ) ∗ (cid:33) (cid:32) P (cid:48) mm ( σ )) l (cid:33) = P (cid:48) mm ( σ ) + (cid:16) R (cid:48) mn ( P nn ( σ ) + R nn ) ∗ (cid:17) (cid:32) P (cid:48) mm ( σ )) l (cid:33) = P (cid:48) mm ( σ ) + = P (cid:48) mm ( σ ) .It follows that M (0 , σ, σ k ) ω, t = P (cid:48) mm ( σ ) ω, t (cid:32) P nn ( σ ) + R nn χ (cid:33) ∗ (cid:32) P (cid:48) mm ( σ )) l (cid:33) P (cid:48) mm ( σ ) ω, t = P (cid:48) mm ( σ ) ω, t (cid:32) ( P nn ( σ ) + R nn ) ∗ χ ( P nn ( σ ) + R nn ) ∗ (cid:33) (cid:32) P (cid:48) mm ( σ )) l (cid:33) P (cid:48) mm ( σ ) ω, t = P (cid:48) mm ( σ ) ω, t (cid:32) P (cid:48) mm ( σ )) l (cid:33) P (cid:48) mm ( σ ) ω, t = P (cid:48) mm ( σ ) ω, t P (cid:48) mm ( σ )) l P (cid:48) mm ( σ ) ω, t .Now, we have for the last component (cid:16) M (0 , σ, σ k ) ω, t (cid:17) m + n + = ( P (cid:48) mm ( σ )) l P (cid:48) mm ( σ ) ω, t = (cid:16) P (cid:48) mm ( σ ) P (cid:48) mm ( σ ) ω, t (cid:17) l = (cid:16) P (cid:48) mm ( σ ) ω, t (cid:17) l = ω l , 21here the third equality is by Theorem 5.5.1 of [18] and the last equality is by (13). In summary, the ( n + m + th component of the t th canonical solution of ( σ k , ω l ) = ( s , υ ). As defined for ω -algebraic systems, it thenfollows that also the t th canonical solution of ( (cid:62)(cid:62)(cid:62) ) is ( s , υ ).As the mixed ω -algebraic system in the preceding proof does not depend on the previous discussion and since weproved that we can construct the Greibach normal form when needed, we infer the following. Corollary 15.
Let ( s , υ ) be a component of a canonical solution of a mixed ω -algebraic system over S (cid:104)(cid:104) Σ ∗ (cid:105)(cid:105)× S (cid:104)(cid:104) Σ ω (cid:105)(cid:105) .Then we can construct an ω -algebraic system over S (cid:104)(cid:104) Σ ∗ (cid:105)(cid:105) × S (cid:104)(cid:104) Σ ω (cid:105)(cid:105) (in Greibach normal form) where ( s , υ ) is acomponent of a canonical solution.
6. Simple Reset Pushdown Automata
Now that we have proved the existence of the Greibach normal form for every ω -algebraic system and everymixed ω -algebraic system, we want to use it in the second part of the paper to show that each ω -algebraic series canbe represented as the behavior of a simple ω -reset pushdown automaton. The next section will prove that result. Forthe proof, we will need the corresponding result for finite words as an intermediate step. We have shown in [6], thatfor every algebraic series r (of finite words), there exists a simple reset pushdown automaton with behavior r . Werecall the construction of the simple reset pushdown automata here for the convenience of the reader, as variants ofthese automata will be used in Section 7 for ω -algebraic series.Following Kuich, Salomaa [22] and Kuich [21], we introduce pushdown transitions matrices. These matrices canbe considered as adjacency matrices of graphs representing automata. A special form, the reset pushdown matrices, isused for pushdown automata starting with an empty stack and allowing the automaton to push onto the empty stack.Here, we are interested in simple reset pushdown matrices, introduced in [6]. This simple form allows the automatononly to push one symbol, to pop one symbol or to ignore the stack. The corresponding automata, the simple resetpushdown automata are a generalization of the unweighted automata used in [8]. They do not use (cid:15) -transitions and donot allow the inspection of the topmost stack symbol.A matrix M ∈ ( S n × n ) Γ ∗ × Γ ∗ is called row-finite if { π (cid:48) | M π,π (cid:48) (cid:44) } is finite for all π ∈ Γ ∗ .Let Γ be an alphabet, called pushdown alphabet and let n ≥
1. A matrix ¯ M ∈ ( S n × n ) Γ ∗ × Γ ∗ is called a pushdownmatrix (with pushdown alphabet Γ and stateset { , . . . , n } ) if(i) ¯ M is row-finite;(ii) for all π , π ∈ Γ ∗ ,¯ M π ,π = ¯ M p ,π , if there exist p ∈ Γ , π, π (cid:48) ∈ Γ ∗ with π = p π (cid:48) and π = ππ (cid:48) , , otherwise.Intuitively, here (ii) means that the infinite pushdown matrix ¯ M is fully represented already by the blocks ¯ M p ,π where p ∈ Γ , π ∈ Γ ∗ , and (i) means that only finitely many such blocks are nonzero.Let Γ be a pushdown alphabet and { , . . . , n } , n ≥
1, be a set of states. A reset matrix M R ∈ ( S n × n ) Γ ∗ × Γ ∗ is arow-finite matrix such that( M R ) π ,π = π , π ∈ Γ ∗ with π (cid:44) (cid:15) .A reset pushdown matrix M ∈ ( S n × n ) Γ ∗ × Γ ∗ is the sum of a reset matrix M R and a pushdown matrix ¯ M , M = M R + ¯ M .Intuitively, a reset pushdown matrix is similar to a pushdown matrix with the additional possibility to push onto theempty stack, i.e., M (cid:15),π is allowed to be nonzero. Note that the entries of reset pushdown matrices are determined byfinitely many values because it is row-finite and property (ii) of pushdown matrices ensures that the value of M p π (cid:48) ,ππ (cid:48) is equal to (and therefore can be derived from) M p ,π . 22 reset pushdown matrix M is called simple if, M ∈ (cid:0) ( S (cid:104) Σ (cid:105) ) n × n (cid:1) Γ ∗ × Γ ∗ for some n ≥
1, and for all p , p ∈ Γ , M p ,(cid:15) , M p , p = M (cid:15),(cid:15) and M p , p p = M (cid:15), p , are the only blocks M π,π (cid:48) , where π ∈ { (cid:15), p } and π (cid:48) ∈ Γ ∗ , that may be unequal to the zero matrix 0.Hence, a simple reset pushdown matrix M is defined by its blocks M (cid:15),(cid:15) and M p ,(cid:15) , M (cid:15), p ( p ∈ Γ ). Intuitively, theautomata will only be allowed to ignore the stack (modeled by M (cid:15),(cid:15) ), pop one symbol ( M p ,(cid:15) ) or push one symbol( M (cid:15), p ). Note also that the matrix M ∈ (( S (cid:104) Σ (cid:105) ) n × n ) Γ ∗ × Γ ∗ forbids (cid:15) -transitions. Moreover, the equalities M p , p = M (cid:15),(cid:15) and M p , p p = M (cid:15), p imply that the next transition does not depend on the topmost symbol of the stack except whenpopping it (modeled by M p ,(cid:15) ).A reset pushdown automaton (with input alphabet Σ ) A = ( n , Γ , I , M , P ) is given by • a set of states { , . . . , n } , n ≥ • a pushdown alphabet Γ , • a reset pushdown matrix M ∈ (( S (cid:104) Σ ∪ { (cid:15) }(cid:105) ) n × n ) Γ ∗ × Γ ∗ called transition matrix , • a row vector I ∈ ( S (cid:104){ (cid:15) }(cid:105) ) × n , called initial state vector , • a column vector P ∈ ( S (cid:104){ (cid:15) }(cid:105) ) n × , called final state vector .The behavior (cid:107) A (cid:107) of a reset pushdown automaton A is defined by (cid:107) A (cid:107) = I ( M ∗ ) (cid:15),(cid:15) P .A reset pushdown automaton A = ( n , Γ , I , M , P ) is called simple if M is a simple reset pushdown matrix.Example 18 will show a simple reset pushdown automaton and the corresponding simple reset pushdown matrix.Given a series r ∈ S alg (cid:104)(cid:104) Σ ∗ (cid:105)(cid:105) , we want to construct a simple reset pushdown automaton with behavior r . ByTheorems 5.10 and 5.4 of [21], r is a component of the unique solution of a strict algebraic system in Greibach normalform.We only consider the algebraic series r with ( r , (cid:15) ) =
0; cf. [6] for the other case. So we assume without loss ofgenerality that r is the x -component of the unique solution of the algebraic system (14) with variables x , . . . , x n x i = p i , ≤ i ≤ n , of the form x i = (cid:88) ≤ j , k ≤ n (cid:88) a ∈ Σ ( p i , ax j x k ) ax j x k + (cid:88) ≤ j ≤ n (cid:88) a ∈ Σ ( p i , ax j ) ax j + (cid:88) a ∈ Σ ( p i , a ) a . (14)As in [6], we now construct the simple reset pushdown automaton A m = ( n + , Γ , I m , M , P ), 1 ≤ m ≤ n , with r = (cid:107) A (cid:107) as follows:We let Γ = { x , . . . , x n } ; we also denote the state n + f ; the entries of M of the form ( M x k , x k ) i , j , ( M x k ,(cid:15) ) i , j , ( M (cid:15), x k ) i , j ,( M (cid:15),(cid:15) ) i , j , ( M (cid:15),(cid:15) ) i , f , 1 ≤ i , j , k ≤ n , that may be unequal to 0 are( M (cid:15), x k ) i , j = (cid:88) a ∈ Σ ( p i , ax j x k ) a ,( M x k , x k ) i , j = ( M (cid:15),(cid:15) ) i , j = (cid:88) a ∈ Σ ( p i , ax j ) a ,( M x k ,(cid:15) ) i , k = ( M x k , x k ) i , f = ( M (cid:15),(cid:15) ) i , f = (cid:88) a ∈ Σ ( p i , a ) a ; 23 MN TB Fa ↓ Sb ↑ S a ↓ T a a ↓ Ta b ↑ Tb ↑ S b ↑ Tb ↑ B b ↑ B b b a ↓ B a ↓ B b ↑ B Figure 1: Example 18: Simple reset pushdown automaton, where ↓ X means push symbol X , ↑ X means pop X , and a and pushing B onto the stack that haveweight 1. All other possible transitions have weight −∞ . we further put ( I m ) m = (cid:15), ( I m ) i = ≤ i ≤ m − m + ≤ i ≤ n +
1; finally let P f = (cid:15) and P j = ≤ j ≤ n ;The following motivation will be essential for our later construction for ω -pushdown automata. Intuitively, thevariables in the algebraic system are simulated by states in the simple reset pushdown automaton A m . By the Greibachnormal form, only two variables on the right-hand side are allowed. The first is modeled directly by changing thestate, the second is pushed to the pushdown tape and the state is changed to it later when the variable is popped again.The special final state f will only be used as the last state.Note that ( M x k , x k ) i , f allows the automaton to change to the final state with a non-empty pushdown tape. This is anartificial addition to fit the definition of simple reset pushdown matrices. If the simple reset automaton is not popping asymbol from the pushdown tape, it cannot distinguish between di ff erent pushdown states. Even though the automatoncan enter the final state too early, it can not continue from there as it is a sink.Observe that (cid:107) A m (cid:107) = (( M ∗ ) (cid:15),(cid:15) ) s , f for all 1 ≤ m ≤ n .This simple reset pushdown matrix M is called the simple pushdown matrix induced by the Greibach normalform (14). The simple reset pushdown automata A m , 1 ≤ m ≤ n , are called the simple reset pushdown automata induced by the Greibach normal form (14).The following (main) theorem of [6] states that the behavior of the simple reset pushdown automata induced bythe Greibach normal form (14) is the unique solution of the original algebraic system (14). Theorem 16 (Theorem 11 of [6]) . The unique solution of the algebraic system (14) is ( (cid:107) A (cid:107) , . . . , (cid:107) A n (cid:107) ) = ((( M ∗ ) (cid:15),(cid:15) ) , f , . . . , (( M ∗ ) (cid:15),(cid:15) ) n , f ) . Corollary 17 (Corollary 12 of [6]) . Let r ∈ S alg (cid:104)(cid:104) Σ ∗ (cid:105)(cid:105) . Then there exists a simple reset pushdown automaton withbehavior r. Example 18.
Consider the semiring ¯ N (cid:104)(cid:104) Σ ∗ (cid:105)(cid:105) for the arctic semiring (cid:104) ¯ N , max , + , −∞ , (cid:105) with ¯ N = N ∪ {−∞ , ∞} .Analogously to Example 7, we let = −∞ and = and we note that in the following, stands for the naturalnumber .We define the algebraic systemS = aMS + aNT + aN T = aMT + aMM = b + aMB N = b + aNBB = bwith the variables S , T , M , N , B. These variables facilitate reading the equations, but for comparison with equa-tion (14) , consider the variable mapping x = T , x = S , x = N , x = M , x = B. ow, the variable M derives a string a n b n + for n ∈ N . The variable N does the same but at the same time producesthe weight n. The variables S and T add another a.Let L = { a n b n | n ≥ } . In total, the second component (i.e., with S being the start variable) of the least solution isu with ( u , a n b n a n b n . . . a n k b n k ) = max n i for k ≥ and ( u , w ) = −∞ for w (cid:60) L + .From this, we can construct a simple reset pushdown automaton A = ( n , Γ , I , M , P ) as shown in Figure 1. Thus,we have n = , Γ = { T , S , N , M , B } . The initial state vector is I = (cid:15) and I i = for i (cid:44) . The final state vector isP = (cid:15) and P i = for i (cid:44) . The simple reset pushdown matrix is defined asM = M (cid:15),(cid:15) M (cid:15), T M (cid:15), S M (cid:15), N M (cid:15), M M (cid:15), B · · · M T ,(cid:15) M (cid:15),(cid:15) · · · M S ,(cid:15) M (cid:15),(cid:15) · · · M N ,(cid:15) M (cid:15),(cid:15) · · · M M ,(cid:15) M (cid:15),(cid:15) · · · M B ,(cid:15) M (cid:15),(cid:15) · · · ... ... ... ... ... ... . . . ,with, for instanceM (cid:15),(cid:15) = a a b b and M (cid:15), B = a a .The rest of the matrix M can be inferred by the rules of pushdown matrices. The behavior (cid:107) A (cid:107) is equal to the secondcomponent of the least solution of the algebraic system above.
7. Simple ω -Reset Pushdown Automata In this section, we will prove that for every ω -algebraic series r , there exists a simple ω -reset pushdown automatonwith behavior r . We first prove some results for infinite applications of simple reset pushdown matrices. Then weintroduce simple ω -reset pushdown automata, and the main theorem will show that they can recognize all ω -algebraicseries.In the sequel, ( S , V ) is a complete semiring-semimodule pair .We will use sets P l comprising infinite sequences over { , . . . , n } as defined in [9]: P l = { ( j , j , . . . ) ∈ { , . . . , n } ω | j t ≤ l for infinitely many t ≥ } .We obtain, for a reset pushdown matrix M ∈ ( S n × n ) Γ ∗ × Γ ∗ , π ∈ Γ + and for 1 ≤ j ≤ n ,(( M ω, l ) π ) j = (cid:88) π ,π , ···∈ Γ ∗ (cid:88) ( j , j ,... ) ∈ P l ( M π,π ) j , j ( M π ,π ) j , j ( M π ,π ) j , j · · · . (15)Observe the following summation identity: Assume that M , M , . . . are matrices in S n × n . Then for 0 ≤ l ≤ n ,1 ≤ j ≤ n , and m ≥
1, we have (cid:88) ( j , j ,... ) ∈ P l ( M ) j , j ( M ) j , j · · · = (cid:88) ≤ j ,..., j m ≤ n ( M ) j , j · · · ( M m ) j m − , j m (cid:88) ( j m + , j m + ,... ) ∈ P l ( M m + ) j m , j m + · · · .By Theorem 5.5.1 of ´Esik, Kuich [18] we obtain, for a finite matrix M and for 0 ≤ l ≤ n , the equality MM ω, l = M ω, l . By Theorem 6 of Droste, ´Esik, Kuich [9], we have a similar result for pushdown matrices. We will now showthe same equality for a reset pushdown matrix M . 25 heorem 19. Let ( S , V ) be a complete semiring-semimodule pair and let further M ∈ ( S n × n ) Γ ∗ × Γ ∗ be a reset pushdowntransition matrix. Then, for ≤ l ≤ n,M ω, l = MM ω, l .Proof. We obtain for π ∈ Γ ∗ and 1 ≤ j ≤ n ,(( MM ω, l ) π ) j = (cid:88) π ∈ Γ ∗ (cid:88) ≤ j ≤ n ( M π ,π ) j , j (cid:88) π ,π ,... ∈ Γ ∗ (cid:88) ( j , j ... ) ∈ P l ( M π,π ) j , j ( M π ,π ) j , j . . . = (cid:88) π,π ,π ... ∈ Γ ∗ (cid:88) ( j , j , j ,... ) ∈ P l ( M π ,π ) j , j ( M π,π ) j , j ( M π ,π ) j , j . . . = (( M ω, l ) π ) j . The following two theorems compare reset pushdown matrices to pushdown matrices in the course of an infiniteapplication. They state that either the topmost stack symbol p is popped or the reset pushdown matrix behaves similarto pushdown matrices. Theorem 20.
Let ( S , V ) be a complete semiring-semimodule pair. Let M be a reset pushdown matrix. Then ( M ω ) p = ( ¯ M ω ) p + ( ¯ M ∗ ) p ,(cid:15) ( M ω ) (cid:15) , for any p ∈ Γ .Proof. We obtain, for p ∈ Γ ,( M ω ) p = (cid:88) π ,π , ···∈ Γ + M p ,π M π ,π · · · + (cid:88) t ≥ (cid:88) π ,...,π t − ∈ Γ + M p ,π · · · M π t − ,(cid:15) ( M ω ) (cid:15) = ( ¯ M ω ) p + (cid:88) t ≥ ( ¯ M t ) p ,(cid:15) ( M ω ) (cid:15) = ( ¯ M ω ) p + ( ¯ M ∗ ) p ,(cid:15) ( M ω ) (cid:15) . Theorem 21.
Let ( S , V ) be a complete semiring-semimodule pair. Let M be a reset pushdown matrix and ≤ l ≤ n.Then ( M ω, l ) p = ( ¯ M ω, l ) p + ( ¯ M ∗ ) p ,(cid:15) ( M ω, l ) (cid:15) , for any p ∈ Γ .Proof. We use the proof of Theorem 20. We obtain, for p ∈ Γ , 0 ≤ l ≤ n and 1 ≤ j ≤ n ,(( M ω, l ) p ) j = (cid:88) π ,π , ···∈ Γ ∗ (cid:88) ( j , j ,... ) ∈ P l ( M p ,π ) j , j ( M π ,π ) j , j · · · = (cid:88) π ,π , ···∈ Γ + (cid:88) ( j , j ,... ) ∈ P l ( M p ,π ) j , j ( M π ,π ) j , j · · · + (cid:88) t ≥ (cid:88) π ,...,π t − ∈ Γ + π t + ,π t + , ···∈ Γ ∗ (cid:88) ( j , j ,... ) ∈ P l ( M p ,π ) j , j · · · ( M π t − ,(cid:15) ) j t − , j t · ( M (cid:15),π t + ) j t , j t + ( M π t + ,π t + ) j t + , j t + · · · = (( ¯ M ω, l ) p ) j + (cid:88) t ≥ (cid:88) π ,...,π t − ∈ Γ + (cid:88) ≤ j ,..., j t ≤ n ( M p ,π ) j , j · · · ( M π t − ,(cid:15) ) j t − , j t · (cid:88) π t + ,π t + , ···∈ Γ ∗ (cid:88) ( j t + , j t + ,... ) ∈ P l ( M (cid:15),π t + ) j t , j t + ( M π t + ,π t + ) j t + , j t + · · · = (( ¯ M ω, l ) p ) j + (cid:88) t ≥ (cid:88) ≤ j (cid:48) ≤ n (( ¯ M t ) p ,(cid:15) ) j , j (cid:48) (( M ω, l ) (cid:15) ) j (cid:48) = (( ¯ M ω, l ) p + ( ¯ M ∗ ) p ,(cid:15) ( M ω, l ) (cid:15) ) j .For simple reset pushdown matrices, the following two lemmas state that infinite paths starting with symbol p onthe pushdown tape can either ignore that symbol or pop it and then continue with an infinite path from the empty tape.26 emma 22. Let ( S , V ) be a complete semiring-semimodule pair. Let M be a simple reset pushdown matrix. Then, forp ∈ Γ , ( M ω ) p = ( M ω ) (cid:15) + ( M ∗ ) (cid:15),(cid:15) M p ,(cid:15) ( M ω ) (cid:15) .Proof. We obtain, for p ∈ Γ ,( M ω ) p = (cid:88) π ,π , ···∈ Γ ∗ M p ,π M π ,π · · · = (cid:88) π ,π , ···∈ Γ ∗ M p ,π p M π p ,π p · · · + (cid:88) t ≥ (cid:88) π ,...,π t − ∈ Γ ∗ M p ,π p · · · M π t − p , p M p ,(cid:15) ( M ω ) (cid:15) = ( M ω ) (cid:15) + (cid:0) (cid:88) t ≥ (cid:88) π ,...,π t − ∈ Γ ∗ M (cid:15),π · · · M π t − ,(cid:15) (cid:1) M p ,(cid:15) ( M ω ) (cid:15) = ( M ω ) (cid:15) + (cid:88) t ≥ ( M t ) (cid:15),(cid:15) M p ,(cid:15) ( M ω ) (cid:15) = ( M ω ) (cid:15) + ( M ∗ ) (cid:15),(cid:15) M p ,(cid:15) ( M ω ) (cid:15) . Lemma 23.
Let ( S , V ) be a complete semiring-semimodule pair. Let M be a simple reset pushdown matrix. Then, forp ∈ Γ and ≤ l ≤ n, ( M ω, l ) p = ( M ω, l ) (cid:15) + ( M ∗ ) (cid:15),(cid:15) M p ,(cid:15) ( M ω, l ) (cid:15) .Proof. We use the proof of Lemma 22. We obtain, for p ∈ Γ , 0 ≤ l ≤ n and 1 ≤ j ≤ n ,(( M ω, l ) p ) j = (cid:88) π ,π , ···∈ Γ ∗ (cid:88) ( j , j ,... ) ∈ P l ( M p ,π ) j , j ( M π ,π ) j , j · · · = (cid:88) π ,π , ···∈ Γ ∗ (cid:88) ( j , j ,... ) ∈ P l ( M p ,π p ) j , j ( M π p ,π p ) j , j · · · + (cid:88) t ≥ (cid:88) π ,π , ···∈ Γ ∗ (cid:88) ( j , j ,... ) ∈ P l ( M p ,π p ) j , j · · · ( M π t − p , p ) j t − , j t · ( M p ,(cid:15) ) j t , j t + ( M (cid:15),π t + ) j t + , j t + ( M π t + ,π t + ) j t + , j t + · · · = (( M ω, l ) (cid:15) ) j + (cid:88) t ≥ (cid:88) π ,...,π t − ∈ Γ ∗ (cid:88) ≤ j ,..., j t + ≤ n ( M p ,π p ) j , j · · · ( M π t − p , p ) j t − , j t ( M p ,(cid:15) ) j t , j t + · (cid:88) π t + ,π t + , ···∈ Γ ∗ (cid:88) ( j t + , j t + ,... ) ∈ P l ( M (cid:15),π t + ) j t + , j t + ( M π t + ,π t + ) j t + , j t + · · · = (( M ω, l ) (cid:15) ) j + (cid:88) t ≥ (cid:88) π ,...,π t − ∈ Γ ∗ (cid:88) ≤ j ,..., j t + ≤ n ( M (cid:15),π ) j , j · · · ( M π t − ,(cid:15) ) j t − , j t · ( M p ,(cid:15) ) j t , j t + (( M ω, l ) (cid:15) ) j t + = (( M ω ) (cid:15) ) j + (cid:88) t ≥ (cid:88) ≤ j (cid:48) , j (cid:48)(cid:48) ≤ n (( M t ) (cid:15),(cid:15) ) j , j (cid:48) ( M p ,(cid:15) ) j (cid:48) , j (cid:48)(cid:48) (( M ω, l ) (cid:15) ) j (cid:48)(cid:48) = (( M ω ) (cid:15) + ( M ∗ ) (cid:15),(cid:15) M p ,(cid:15) ( M ω, l ) (cid:15) ) j .Next, an ω -reset pushdown automaton A = ( n , Γ , I , M , P , l )is given by a reset pushdown automaton ( n , Γ , I , M , P ) and an integer l with 0 ≤ l ≤ n , which indicates that 1 , . . . , l arethe repeated states of A . The behavior (cid:107) A (cid:107) of this ω -reset pushdown automaton A is defined by (cid:107) A (cid:107) = I ( M ∗ ) (cid:15),(cid:15) P + I ( M ω, l ) (cid:15) .The ω -reset pushdown automaton A = ( n , Γ , I , M , P , l ) is called simple if M is a simple reset pushdown matrix.27 a , ( ↓ , Z ) : 1 b , ( ↑ , X ) b , ( ↑ , Z ) a , ( ↓ , X ) : 1 b , ( ↑ , X ) c , c , b , ( ↑ , Z ) Figure 2: Example 24: Simple ω -reset pushdown automaton, where ( ↓ , X ) means push symbol X , ( ↑ , X ) means pop X , and a and pushing a symbol ontothe stack that have weight 1. All other possible transitions have weight ∞ . Example 24.
Figure 2 shows a simple ω -reset pushdown automaton A = (4 , Γ , I , M , P , over the quemiring N ∞ (cid:104)(cid:104) Σ ∗ (cid:105)(cid:105) × N ∞ (cid:104)(cid:104) Σ ω (cid:105)(cid:105) for the tropical semiring (cid:104) N ∞ , min , + , = ∞ , = (cid:105) with Σ = { a , b , c } , Γ = { Z , X } , I = ,I i = ∞ for i (cid:44) and P i = ∞ for all ≤ i ≤ . Then the adjacency matrix M of the automaton shown in Figure 2 is asimple reset pushdown matrix. As an indication, M is defined with ( M (cid:15),(cid:15) ) , = c, ( M (cid:15),(cid:15) ) , = c, ( M (cid:15), Z ) , = a, etc.,resulting in e.g.,M (cid:15),(cid:15) = c c and finally M = M (cid:15),(cid:15) M (cid:15), Z M (cid:15), X · · · M Z ,(cid:15) M (cid:15),(cid:15) · · · M X ,(cid:15) M (cid:15),(cid:15) · · · ... ... ... . . . ,where the excluded part of M can be derived from the rules of pushdown and simple reset pushdown matrices. Theautomaton A has the behavior a n b n c ω (cid:55)→ n, similar to the mixed ω -algebraic system in Example 4. Example 25.
Reconsider Example 18. We define the simple ω -reset pushdown automaton A = (6 , Γ , I , M , P , wherewe define the state ordering T , S , N , M , B , F to make state T B¨uchi-accepting. The behavior in the semiring part isequal to before; the behavior in the semimodule part is u with ( u , a n b n a n b n . . . ) = max n i and ( u , w ) = −∞ forw (cid:60) { a n b n | n ≥ } ω . Example 26.
Consider the ω -algebraic systemy = a + cy y = ay y + ay . (16) We will consider the second component of the first canonical solution, i.e., variable y is B¨uchi-accepting and variabley is the start variable.The ω -algebraic system induces the following mixed ω -algebraic systemx = a + cx z = cz x = ax x + ax z = az + ax z . (17) The least solution of x = p ( x ) is σ = (cid:32) c ∗ a ( ac ∗ a ) + (cid:33) .Now, we write the linear system z = (cid:37) ( σ ) z in the matrix form and compute the first canonical solution. (cid:37) ( σ ) ω, = (cid:32) c a a ( c ∗ a ) (cid:33) ω, = (cid:32) ( c + aa ) ∗ a ) ω ( ac ∗ a ) ∗ a ( c + aa ) ∗ a ) ω (cid:33) = (cid:32) c ω ( ac ∗ a ) ∗ ac ω (cid:33) (cid:67) (cid:32) ω (1)1 ω (1)2 (cid:33) = ω (1) ote that the second component, ω (1)2 , does not contain the ω -words ( ac ∗ a ) ω even though for an unweighted ω -context-free grammar corresponding to (16) , the derivationy → ay y → ( aa ) y → ( aa ) ay y → ( aa ) y → ω a ω would be successful even with only y B¨uchi-accepting. The di ff erence is due to the fact that y is not significant in the ω -algebraic system above because it is exchanged by x in the derivation (for more information, see [18] pp. 140 ff .).Now, we look at the simple ω -reset pushdown automaton induced by ω -algebraic system (16) :2 1 fa , a , ( ↓ , y ) a , ( ↑ , y ) a , c , The behavior of this automaton is ((( M ∗ ) (cid:15),(cid:15) ) , f , (( M ∗ ) (cid:15),(cid:15) ) , f ; (( M ω, ) (cid:15) ) , (( M ω, ) (cid:15) ) ) = ( c ∗ a , ( ac ∗ a ) + ; c ω , ( ac ∗ a ) ∗ ac ω + ( ac ∗ a ) ω ) Here, the first two components are equal to σ , as desired. But the last component di ff ers from ω (1)2 ; the last componentis however equal to the behavior of unweighted ω -context-free grammars.Note that the desired component ω (1)2 = ( ac ∗ a ) ∗ ac ω is not recognized by this automaton, even when changing theB¨uchi-accepting states. If no states are B¨uchi-accepting, the behavior is 0, if all of them are B¨uchi-accepting, we havethe same behavior as above. If only state 2 is B¨uchi-accepting (can be achieved by renaming), we only recognize ( ac ∗ a ) ω .We now propose a di ff erent construction; this new construction models exactly the canonical solutions of mixed ω -algebraic systems. The following is the simple ω -reset pushdown automaton induced by the mixed ω -algebraicsystem (17) ; this new construction will be defined after the example. Basically, the construction is similar to the oldconstruction but it di ff erentiates between variables x and z; it therefore uses the states x , . . . , x n , z , . . . , z n :z z x x fa , a , ( ↓ , X ) a , ( ↑ , X ) a , c , a , a , ( ↓ , Z ) a , ( ↑ , Z ) c , This simple ω -reset pushdown automaton has exactly the behavior ( σ, ω (1) ) . This means, if only z is B¨uchi-accepting,then the automaton does not allow the run ( ac ∗ a ) ω .The rest of the paper will show that in general, the l th canonical solution of a mixed ω -algebraic system x = p ( x ) , z = (cid:37) ( x ) z is exactly the behavior of the simple ω -reset pushdown automaton induced by x = p ( x ) , z = (cid:37) ( x ) z. Given a series r ∈ S alg (cid:104)(cid:104) Σ ∗ (cid:105)(cid:105) × S alg (cid:104)(cid:104) Σ ω (cid:105)(cid:105) , we want to construct a simple ω -reset pushdown automaton withbehavior r . By Theorem 14 and Theorem 5, r is a component of a canonical solution of an ω -algebraic system (18)29compare this to the algebraic system (14)) in Greibach normal form over the quemiring S (cid:104)(cid:104) Σ ∗ (cid:105)(cid:105) × S (cid:104)(cid:104) Σ ω (cid:105)(cid:105) , y i = (cid:88) ≤ j , k ≤ n (cid:88) a ∈ Σ ( p i , ay j y k ) ay j y k + (cid:88) ≤ j ≤ n (cid:88) a ∈ Σ ( p i , ay j ) ay j + (cid:88) a ∈ Σ ( p i , a ) a . (18)The variables of this system are y i , (1 ≤ i ≤ n ); they are variables for S (cid:104)(cid:104) Σ ∗ (cid:105)(cid:105) × S (cid:104)(cid:104) Σ ω (cid:105)(cid:105) . The system (18) inducesthe following mixed ω -algebraic system: x i = (cid:88) ≤ j , k ≤ n (cid:88) a ∈ Σ ( p i , ay j y k ) ax j x k + (cid:88) ≤ j ≤ n (cid:88) a ∈ Σ ( p i , ay j ) ax j + (cid:88) a ∈ Σ ( p i , a ) a . (14)and z i = (cid:88) ≤ j , k ≤ n (cid:88) a ∈ Σ ( p i , ay j y k ) a ( z j + x j z k ) + (cid:88) ≤ j ≤ n (cid:88) a ∈ Σ ( p i , ay j ) az j (19)But this system hides information, for instance, y j y k will never be derived by two consecutive variables z j z k of S (cid:104)(cid:104) Σ ω (cid:105)(cid:105) . Our new construction is therefore based on the following mixed ω -algebraic system that can be gained fromthe last system (14), (19) by renaming: x i = (cid:88) ≤ j , k ≤ n (cid:88) a ∈ Σ ( p i , ax j x k ) ax j x k + (cid:88) ≤ j ≤ n (cid:88) a ∈ Σ ( p i , ax j ) ax j + (cid:88) a ∈ Σ ( p i , a ) a . (14)and z i = (cid:88) ≤ j , k ≤ n (cid:88) a ∈ Σ ( p i , ax j z k ) ax j z k + (cid:88) ≤ j ≤ n (cid:88) a ∈ Σ ( p i , az j ) az j (20)We now want to construct a simple ω -reset pushdown automaton. Here, we introduce our new construction. Let A lm = (2 n + , Γ , I m , M , P , l ), 1 ≤ m ≤ n , 0 ≤ l ≤ n , be defined as follows:We let Γ = { X , . . . , X n , Z , . . . , Z n } ; we denote the states 1 , . . . , n + z , . . . , z n , x , . . . , x n , f ; the entries of M ofthe form ( M π,π (cid:48) ) v , v (cid:48) for 1 ≤ v , v (cid:48) ≤ n + π, π (cid:48) ∈ Γ ∗ with | π | , | π (cid:48) | ≤ M (cid:15), X k ) x i , x j = (cid:88) a ∈ Σ ( p i , ax j x k ) a ,( M Z k , Z k ) x i , x j = ( M X k , X k ) x i , x j = ( M (cid:15),(cid:15) ) x i , x j = (cid:88) a ∈ Σ ( p i , ax j ) a ,( M Z k ,(cid:15) ) x i , z k = ( M X k ,(cid:15) ) x i , x k = ( M Z k , Z k ) x i , f = ( M X k , X k ) x i , f = ( M (cid:15),(cid:15) ) x i , f = (cid:88) a ∈ Σ ( p i , a ) a ,( M Z k , Z k ) z i , z j = ( M X k , X k ) z i , z j = ( M (cid:15),(cid:15) ) z i , z j = (cid:88) a ∈ Σ ( p i , az j ) a ,( M (cid:15), Z k ) z i , x j = (cid:88) a ∈ Σ ( p i , ax j z k ) a ,30or 1 ≤ i , j , k ≤ n ; we further put ( I m ) x m = ( I m ) z m = (cid:15) , and ( I m ) x i = ( I m ) z i = ≤ i ≤ m − m + ≤ i ≤ n and( I m ) f =
0; finally let P f = (cid:15) and P j = ≤ j ≤ n ;In the following, we assume that r ∈ S alg (cid:104)(cid:104) Σ ∗ (cid:105)(cid:105) × S alg (cid:104)(cid:104) Σ ω (cid:105)(cid:105) is the m th component of the l th canonical solutionof (18). We want to show that for the l th canonical solution τ = ( σ, ω ) of (14), (20), and therefore also of (18), wehave τ m = σ m + ω m = (cid:107) A lm (cid:107) .This simple reset pushdown matrix M is called the simple reset pushdown matrix induced by the Greibach normalform (14), (20). The simple ω -reset pushdown automata A lm (1 ≤ m ≤ n , 0 ≤ l ≤ n ) are called the simple ω -resetpushdown automata induced by the Greibach normal form (14), (20).For the rest of the paper, we will use the following notation (cf. [22], page 179). Note that M ∈ ( S k × k ) Γ ∗ × Γ ∗ for k = n +
1. By isomorphism, we can transform this into (cid:98) M ∈ ( S Γ ∗ × Γ ∗ ) k × k . We then have ( M π,π (cid:48) ) v , v (cid:48) = ( (cid:98) M v , v (cid:48) ) π,π (cid:48) for π, π (cid:48) ∈ Γ ∗ and 1 ≤ v , v (cid:48) ≤ n +
1. (By the notation 1 ≤ v ≤ n +
1, we mean v can be any of the states z , . . . , z n , x , . . . , x n , f .) Example 27.
This notation allows us to add up matrices with suitable pushdown indexes while still keeping theinformation of the states. For instance, note that (cid:88) ≤ k ≤ n (cid:88) π ∈ Γ ∗ ( (cid:98) M z i , x k ) (cid:15),π ( (cid:98) M x k , z j ) π,(cid:15) = (cid:88) ≤ k ≤ n (cid:0) (cid:98) M z i , x k (cid:98) M x k , z j (cid:1) (cid:15),(cid:15) .Now consider the term (cid:88) ≤ k ≤ n (cid:88) π ∈ Γ ∗ ( M (cid:15),π ) z i , x k ( M π,(cid:15) ) x k , z j ,which cannot be simplified because (cid:80) π ∈ Γ ∗ ( M (cid:15),π M π,(cid:15) ) z i , z j does no longer hold the information that the path passesonly through states x i , i.e., it contains also the path ( M (cid:15),π ) z i , z k ( M π,(cid:15) ) z k , z j (for all ≤ k ≤ n). In the proofs below, wewill specifically need to distinguish paths that pass through states x i and those that pass through states z i as in themixed ω -algebraic system, we also distinguish between variables x i for finite derivations and variables z i for infinitederivations. Lemma 28.
Let M ∈ ( S k × k ) Γ ∗ × Γ ∗ be a reset pushdown matrix. Then, (cid:99) M ∗ = (cid:98) M ∗ .Proof. For 1 ≤ v , v (cid:48) ≤ k and for π, π (cid:48) ∈ Γ ∗ , we obtain(( (cid:99) M ∗ ) v , v (cid:48) ) π,π (cid:48) = (( M ∗ ) π,π (cid:48) ) v , v (cid:48) = (cid:88) n ≥ (( M n ) π,π (cid:48) ) v , v (cid:48) = (cid:88) n ≥ (( (cid:98) M n ) v , v (cid:48) ) π,π (cid:48) = (( (cid:98) M ∗ ) v , v (cid:48) ) π,π (cid:48) .Similarly, we need the above result for another operator. Lemma 29.
Let M ∈ ( S k × k ) Γ ∗ × Γ ∗ be a reset pushdown matrix. Then, for ≤ l ≤ k, (cid:100) M ω, l = (cid:98) M ω, l .Proof. For 1 ≤ v ≤ k and for π ∈ Γ ∗ , we obtain(( (cid:100) M ω, l ) v ) π = (( M ω, l ) π ) v = (cid:88) π ,π ,... ∈ Γ ∗ (cid:88) ( v , v ,... ) ∈ P l ( M π,π ) v , v ( M π ,π ) v , v ( M π ,π ) v , v · · · = (cid:88) π ,π ,... ∈ Γ ∗ (cid:88) ( v , v ,... ) ∈ P l ( (cid:98) M v , v ) π,π ( (cid:98) M v , v ) π ,π ( (cid:98) M v , v ) π ,π · · · = (( (cid:98) M ω, l ) v ) π . 31et M be a simple reset pushdown matrix induced by the Greibach normal form (14), (20). We define some blocksof the matrix (cid:98) M to make the following argumentation easier. We take the idea of the above-mentioned isomorphismand divide (cid:98) M like (cid:98) M = (cid:98) M z , z (cid:98) M z , x (cid:98) M x , z (cid:98) M x , x (cid:98) M x , f , (21)where the respective blocks are defined as (cid:98) M z , z = (cid:98) M z , z · · · (cid:98) M z , z n ... . . . ... (cid:98) M z n , z · · · (cid:98) M z n , z n , (cid:98) M z , x = (cid:98) M z , x · · · (cid:98) M z , x n ... . . . ... (cid:98) M z n , x · · · (cid:98) M z n , x n , (cid:98) M x , z = (cid:98) M x , z · · · (cid:98) M x , z n ... . . . ... (cid:98) M x n , z · · · (cid:98) M x n , z n , (cid:98) M x , x = (cid:98) M x , x · · · (cid:98) M x , x n ... . . . ... (cid:98) M x n , x · · · (cid:98) M x n , x n , (cid:98) M x , f = (cid:98) M x , f ... (cid:98) M x n , f ,and where each (cid:98) M v , v (cid:48) ∈ S Γ ∗ × Γ ∗ for 1 ≤ v , v (cid:48) ≤ n +
1. For notational convenience, we also set (cid:98) M z i , x = (cid:16) (cid:98) M z i , x · · · (cid:98) M z i , x n (cid:17) , (cid:98) M x , z i = (cid:98) M x , z i ... (cid:98) M x n , z i .Note that we have not defined the blocks (cid:98) M z , f , (cid:98) M f , z , (cid:98) M f , x and (cid:98) M f , f as they would all be zero by our constructionfor simple reset pushdown matrices induced by the Greibach normal form (14), (20).Analogously, let M z , z , M z , x , M x , z , M x , x , M x , f ∈ ( S (2 n + × (2 n + ) Γ ∗ × Γ ∗ be the isomorphic copy of (cid:98) M z , z , (cid:98) M z , x , (cid:98) M x , z , (cid:98) M x , x , (cid:98) M x , f , respectively. Then, for u , v ∈ { x , z } and for π, π (cid:48) ∈ Γ ∗ , the matrix ( M u , v ) π,π (cid:48) is M π,π (cid:48) restricted to the variables u i , v j (for 1 ≤ i , j ≤ n ). Similarly, M x , f is M restricted to variables x i , f (for 1 ≤ i ≤ n ). For instance, ( (cid:98) M x , x ) ∗ andequally ( M x , x ) ∗ consider only paths passing through states x i and no paths through y i or f (for 1 ≤ i ≤ n ). Their onlydi ff erence is the order of indexes.The following theorem computes the behavior of induced simple ω -reset pushdown automata. Theorem 30.
Let M be a simple reset pushdown matrix induced by the Greibach normal form (14) , (20) . Then, forall ≤ i ≤ n and ≤ l ≤ n, (cid:0) ( M ω, l ) (cid:15) (cid:1) x i = (cid:0) ( M ω, l ) (cid:15) (cid:1) f = ,and (cid:0) ( M ω, l ) (cid:15) (cid:1) z i = (cid:16)(cid:0)(cid:0) M z , z + M z , x ( M ∗ ) x , x M x , z (cid:1) ω, l (cid:1) (cid:15) (cid:17) i .Proof. For the matrix M , we have, by above notation (21), (cid:98) M = (cid:98) M z , z (cid:98) M z , x (cid:98) M x , z (cid:98) M x , x (cid:98) M x , f = (cid:98) M z , z (cid:98) M z , x (cid:98) M x , z (cid:98) M x , x (cid:98) M x , f .Thus, by Theorem 2 and Lemma 29, we obtain M ω, l = α ω, l (cid:32) M x , x M x , f (cid:33) ∗ (cid:32) M x , z (cid:33) α ω, l , 32here α = M z , z + (cid:16) M z , x (cid:17) (cid:32) M x , x M x , f (cid:33) ∗ (cid:32) M x , z (cid:33) = M z , z + (cid:16) M z , x (cid:17) (cid:32) ( M x , x ) ∗ ( M x , x ) ∗ M x , f (cid:33) (cid:32) M x , z (cid:33) = M z , z + (cid:16) M z , x ( M x , x ) ∗ M z , x ( M x , x ) ∗ M x , f (cid:17) (cid:32) M x , z (cid:33) = M z , z + M z , x ( M x , x ) ∗ M x , z . (22)Now, we continue with the term from before and get M ω, l = α ω, l (cid:32) M x , x M x , f (cid:33) ∗ (cid:32) M x , z (cid:33) α ω, l = ( M z , z + M z , x ( M x , x ) ∗ M x , z ) ω, l (cid:32) M x , x M x , f (cid:33) ∗ (cid:32) M x , z (cid:33) ( M z , z + M z , x ( M x , x ) ∗ M x , z ) ω, l = ( M z , z + M z , x ( M x , x ) ∗ M x , z ) ω, l (cid:32) ( M x , x ) ∗ ( M x , x ) ∗ M x , f (cid:33) (cid:32) M x , z (cid:33) ( M z , z + M z , x ( M x , x ) ∗ M x , z ) ω, l = ( M z , z + M z , x ( M x , x ) ∗ M x , z ) ω, l (cid:32) ( M x , x ) ∗ M x , z (cid:33) ( M z , z + M z , x ( M x , x ) ∗ M x , z ) ω, l = (cid:0) M z , z + M z , x ( M x , x ) ∗ M x , z (cid:1) ω, l (cid:0) ( M x , x ) ∗ M x , z (cid:1)(cid:0) M z , z + M z , x ( M x , x ) ∗ M x , z (cid:1) ω, l .Then, we start the run of the automaton with an empty stack and get( M ω, l ) (cid:15) = (cid:0) M z , z + M z , x ( M x , x ) ∗ M x , z (cid:1) ω, l (cid:0) ( M x , x ) ∗ M x , z (cid:1)(cid:0) M z , z + M z , x ( M x , x ) ∗ M x , z (cid:1) ω, l (cid:15) = (cid:0)(cid:0) M z , z + M z , x ( M x , x ) ∗ M x , z (cid:1) ω, l (cid:1) (cid:15) (cid:16)(cid:0) ( M x , x ) ∗ M x , z (cid:1)(cid:0) M z , z + M z , x ( M x , x ) ∗ M x , z (cid:1) ω, l (cid:17) (cid:15) = (cid:0)(cid:0) M z , z + M z , x ( M x , x ) ∗ M x , z (cid:1) ω, l (cid:1) (cid:15) (cid:80) π ∈ Γ ∗ (cid:0) ( M x , x ) ∗ M x , z (cid:1) (cid:15),π (cid:0)(cid:0) M z , z + M z , x ( M x , x ) ∗ M x , z (cid:1) ω, l (cid:1) π = (cid:0)(cid:0) M z , z + M z , x ( M x , x ) ∗ M x , z (cid:1) ω, l (cid:1) (cid:15) (cid:80) π ∈ Γ ∗ (cid:0)(cid:0) M z , z + M z , x ( M x , x ) ∗ M x , z (cid:1) ω, l (cid:1) π = (cid:0)(cid:0) M z , z + M z , x ( M x , x ) ∗ M x , z (cid:1) ω, l (cid:1) (cid:15) where the forth equality uses the fact that (( M x , x ) ∗ M x , z ) (cid:15),π = M x , z ) π,π (cid:48) = π (cid:44) Z k π (cid:48)(cid:48) (1 ≤ k ≤ n and π (cid:48)(cid:48) ∈ Γ ∗ ) and at the same time, (( M x , x ) ∗ ) (cid:15), Z k π (cid:48)(cid:48) = M z , x ) (cid:15), Z k (cid:44) M ω, l ) (cid:15) is indexed by z , . . . , z n , x , . . . , x n , f , thus completing the proof.33e want to apply the results from Section 6. The following three lemmas investigate the star operation appliedto simple reset pushdown matrices M induced by the Greibach normal form (14), (20). The lemmas state that in acomputation ( M ∗ ) (cid:15),(cid:15) , the new states y k are never reached when starting in a state x i and therefore, these computationsare equivalent to the computations ( M (cid:48)∗ ) (cid:15),(cid:15) for M (cid:48) being induced by the Greibach normal form (14), i.e., for M (cid:48) builtby the old construction. Lemma 31.
Let M be a simple reset pushdown matrix induced by the Greibach normal form (14) , (20) . Then, for all ≤ i , k ≤ n, (( M ∗ ) (cid:15),(cid:15) ) x k , x i = ((( M x , x ) ∗ ) (cid:15),(cid:15) ) x k , x i .Proof. Let ∆ = { X , . . . , X n } . We have(( M ∗ ) (cid:15),(cid:15) ) x k , x i = (cid:88) t ≥ (( M t ) (cid:15),(cid:15) ) x k , x i = (cid:88) t ≥ (cid:88) π ,...,π t − ∈ Γ ∗ (cid:16) M (cid:15),π M π ,π · · · M π t − ,(cid:15) (cid:17) x k , x i = (cid:88) t ≥ (cid:88) π ∈ ∆ ∗ π ,...,π t − ∈ Γ ∗ (cid:88) ≤ j ≤ n ( M (cid:15),π ) x k , x j (cid:16) M π ,π · · · M π t − ,(cid:15) (cid:17) x k , x i = (cid:88) t ≥ (cid:88) π ,...,π t − ∈ ∆ ∗ (cid:88) ≤ j ,..., j t − ≤ n ( M (cid:15),π ) x k , x j ( M π ,π ) x j , x j · · · ( M π t − ,(cid:15) ) x jt − , x i = (cid:0)(cid:0)(cid:88) t ≥ ( M x , x ) t (cid:1) (cid:15),(cid:15) (cid:1) x k , x i = ((( M x , x ) ∗ ) (cid:15),(cid:15) ) x k , x i ,where the third equality (and similarly the forth equality) is by definition of induced pushdown matrices; the blocks( M (cid:15), X k ) x i , x j , ( M X k , X k ) x i , x j and ( M (cid:15),(cid:15) ) x i , x j are the only non-null blocks that describe a step in the matrix starting from astate x i and having (cid:15) or X k as the topmost stack symbol. Lemma 32.
Let M be a simple reset pushdown matrix induced by the Greibach normal form (14) , (20) . Then, wehave (( M x , x + M x , z ( M z , z ) ∗ M z , x ) ∗ ) (cid:15),(cid:15) = (( M x , x ) ∗ ) (cid:15),(cid:15) .Proof. Let ∆ = { X , . . . , X n } . In some sense similar to the proof of Lemma 31, we have(( M x , x + M x , z ( M z , z ) ∗ M z , x ) ∗ ) (cid:15),(cid:15) = (cid:16)(cid:88) t ≥ ( M x , x + M x , z ( M z , z ) ∗ M z , x ) t (cid:17) (cid:15),(cid:15) = (cid:88) t ≥ (cid:88) π ,...,π t − ∈ Γ ∗ ( M x , x + M x , z ( M z , z ) ∗ M z , x ) (cid:15),π · · · ( M x , x + M x , z ( M z , z ) ∗ M z , x ) π t − ,(cid:15) = (cid:88) t ≥ (cid:88) π ,...,π t − ∈ Γ ∗ (cid:16) ( M x , x ) (cid:15),π + (cid:0) (cid:88) π,π (cid:48) ∈ Γ ∗ ( M x , z ) (cid:15),π (( M z , z ) ∗ ) π,π (cid:48) ( M z , x ) π,π (cid:1)(cid:17) · · · ( M x , x + M x , z ( M z , z ) ∗ M z , x ) π t − ,(cid:15) = (cid:88) t ≥ (cid:88) π ∈ ∆ ∗ π ,...,π t − ∈ Γ ∗ ( M x , x ) (cid:15),π ( M x , x + M x , z ( M z , z ) ∗ M z , x ) π ,π · · · ( M x , x + M x , z ( M z , z ) ∗ M z , x ) π t − ,(cid:15) = (cid:88) t ≥ (cid:88) π ∈ ∆ ∗ π ,...,π t − ∈ Γ ∗ ( M x , x ) (cid:15),π (cid:16) ( M x , x ) π ,π + (cid:0) (cid:88) π,π (cid:48) ∈ Γ ∗ ( M x , z ) π ,π (( M z , z ) ∗ ) π,π (cid:48) ( M z , x ) π,π (cid:1)(cid:17) · · · ( M x , x + M x , z ( M z , z ) ∗ M z , x ) π t − ,(cid:15) = (cid:88) t ≥ (cid:88) π ,...,π t − ∈ ∆ ∗ ( M x , x ) (cid:15),π ( M x , x ) π ,π · · · ( M x , x ) π t − ,(cid:15) = (( M x , x ) ∗ ) (cid:15),(cid:15) ,where the forth equality is because ( M x , z ) (cid:15),π = π ∈ Γ ∗ . Similarly, for the sixth equality, we use the fact that( M x , z ) π i ,π = π i ∈ ∆ ∗ (and π ∈ Γ ∗ ). 34 emma 33. Let M be a simple reset pushdown matrix induced by the Greibach normal form (14) , (20) and M (cid:48) beinduced by the Greibach normal form (14) . Then, for all ≤ i ≤ n, (( M ∗ ) (cid:15),(cid:15) ) x i , f = (( M (cid:48)∗ ) (cid:15),(cid:15) ) i , f .Proof. Note that by construction, we have (cid:98) M (cid:48) = (cid:32) (cid:98) M x , x (cid:98) M x , f (cid:33) .By applying Lemma 28, we infer M (cid:48)∗ = (cid:32) ( M x , x ) ∗ ( M x , x ) ∗ M x , f (cid:33) ,and we get(( M (cid:48)∗ ) (cid:15),(cid:15) ) i , f = (cid:0) (( M x , x ) ∗ M x , f ) (cid:15),(cid:15) (cid:1) i . (23)At the same time, we have (cid:98) M = (cid:98) M z , z (cid:98) M z , x (cid:98) M x , z (cid:98) M x , x (cid:98) M x , f = (cid:98) M z , z (cid:98) M z , x (cid:98) M x , z (cid:98) M x , x (cid:98) M x , f .By Lemma 28, we obtain M ∗ = α ∗ α ∗ (cid:16) M z , x (cid:17) β ∗ (cid:32) M x , z (cid:33) β ∗ ,with α = M z , z + (cid:16) M z , x (cid:17) (cid:32) M x , x M x , f (cid:33) ∗ (cid:32) M x , z (cid:33) = M z , z + M z , x ( M x , x ) ∗ M x , z ,by (22) in the proof of Theorem 30 and β ∗ = (cid:32)(cid:32) M x , x M x , f (cid:33) + (cid:32) M x , z (cid:33) ( M z , z ) ∗ (cid:16) M z , x (cid:17)(cid:33) ∗ = (cid:32)(cid:32) M x , x M x , f (cid:33) + (cid:32) M x , z ( M z , z ) ∗ M z , x
00 0 (cid:33)(cid:33) ∗ = (cid:32) M x , x + M x , z ( M z , z ) ∗ M z , x M x , f (cid:33) ∗ = (cid:32) ( M x , x + M x , z ( M z , z ) ∗ M z , x ) ∗ ( M x , x + M x , z ( M z , z ) ∗ M z , x ) ∗ M x , f (cid:33) . (24)35e deduce that(( M ∗ ) (cid:15),(cid:15) ) x i , f = (cid:0)(cid:0) ( M x , x + M x , z ( M z , z ) ∗ M z , x ) ∗ M x , f (cid:1) (cid:15),(cid:15) (cid:1) i = (cid:0) (( M x , x + M x , z ( M z , z ) ∗ M z , x ) ∗ ) (cid:15),(cid:15) ( M x , f ) (cid:15),(cid:15) (cid:1) i = (cid:0) (( M x , x ) ∗ ) (cid:15),(cid:15) ( M x , f ) (cid:15),(cid:15) (cid:1) i = (cid:0) (( M x , x ) ∗ M x , f ) (cid:15),(cid:15) (cid:1) i = (( M (cid:48)∗ ) (cid:15),(cid:15) ) i , f ,where the third equality is by Lemma 32 and the last equality is by (23). This concludes the proof.The following lemma investigates the final state f in infinite paths. It states that a finite run of induced simple ω -reset pushdown automata is equivalent to another path only through states x and with symbol Z j initially on thepushdown tape and ending in state z j with an empty pushdown tape. Lemma 34.
Let M be a simple reset pushdown matrix induced by the Greibach normal form (14) , (20) . Then, for all ≤ j , k ≤ n, (( M ∗ ) (cid:15),(cid:15) ) x k , f = (cid:0) (( M x , x ) ∗ ) Z j , Z j M Z j ,(cid:15) (cid:1) x k , z j .Proof. The beginning of the proof is similar to the proof of Lemma 10 of [6]. We obtain(( M ∗ ) (cid:15),(cid:15) ) x k , f = (( M + ) (cid:15),(cid:15) ) x k , f = (( M ∗ M ) (cid:15),(cid:15) ) x k , f = (cid:88) ≤ v ≤ n + (( M ∗ ) (cid:15),(cid:15) ) x k , v ( M (cid:15),(cid:15) ) v , f + (cid:88) ≤ v ≤ n + (cid:88) P ∈ Γ (( M ∗ ) (cid:15), P ) x k , v ( M P ,(cid:15) ) v , f = (cid:88) ≤ v ≤ n + (( M ∗ ) (cid:15),(cid:15) ) x k , v ( M (cid:15),(cid:15) ) v , f = (cid:88) ≤ i ≤ n (( M ∗ ) (cid:15),(cid:15) ) x k , x i ( M (cid:15),(cid:15) ) x i , f = (cid:88) ≤ i ≤ n ((( M x , x ) ∗ ) (cid:15),(cid:15) ) x k , x i ( M (cid:15),(cid:15) ) x i , f = (cid:88) ≤ i ≤ n ((( M x , x ) ∗ ) Z j , Z j ) x k , x i ( M Z j ,(cid:15) ) x i , z j = ((( M x , x ) ∗ ) Z j , Z j M Z j ,(cid:15) ) x k , z j ,where the forth equality is since ( M P ,(cid:15) ) v , f = ≤ v ≤ n + P ∈ Γ by our construction. In the fifthequality, we use the fact that ( M (cid:15),(cid:15) ) v , f = v (cid:44) x i (1 ≤ i ≤ n ). The sixth equality is by Lemma 31. The seventhequality is also by construction and by the definition of pushdown matrices.The following lemma treats a case similar to the previous lemma. It states that an infinite path starting withsymbol Z k on the pushdown tape is equivalent to a finite run starting with an empty pushdown tape and ending in state f followed by an infinite run that starts in state z k with an empty pushdown tape. Lemma 35.
Let M be a simple reset pushdown matrix induced by the Greibach normal form (14) , (20) . Then, for all ≤ j , k ≤ n and ≤ l ≤ n, (( M ω, l ) Z k ) x j = (( M ∗ ) (cid:15),(cid:15) ) x j , f (( M ω, l ) (cid:15) ) z k .Proof. By Lemma 23, we have(( M ω, l ) Z k ) x j = (cid:2) ( M ω, l ) (cid:15) + ( M ∗ ) (cid:15),(cid:15) M Z k ,(cid:15) ( M ω, l ) (cid:15) (cid:3) x j = (cid:0) ( M ω, l ) (cid:15) (cid:1) x j + (cid:0) ( M ∗ ) (cid:15),(cid:15) M Z k ,(cid:15) ( M ω, l ) (cid:15) (cid:1) x j .Consider the first summand. By Theorem 30, we know that(( M ω, l ) (cid:15) ) x j = ≤ j , k ≤ n , we have (cid:0) ( M ∗ ) (cid:15),(cid:15) M Z k ,(cid:15) ( M ω, l ) (cid:15) (cid:1) x j = (cid:88) ≤ v , v ≤ n + (( M ∗ ) (cid:15),(cid:15) ) x j , v ( M Z k ,(cid:15) ) v , v (( M ω, l ) (cid:15) ) v = (cid:88) ≤ v ≤ n + (( M ∗ ) (cid:15),(cid:15) ) x j , v ( M Z k ,(cid:15) ) v , z k (( M ω, l ) (cid:15) ) z k = (cid:88) ≤ v ≤ n + (( M ∗ ) (cid:15),(cid:15) ) x j , v ( M (cid:15),(cid:15) ) v , f (( M ω, l ) (cid:15) ) z k = (( M ∗ ) (cid:15),(cid:15) M (cid:15),(cid:15) ) x j , f (( M ω, l ) (cid:15) ) z k = (( M ∗ ) (cid:15),(cid:15) ) x j , f (( M ω, l ) (cid:15) ) z k ,where the second equality holds because we defined ( M Z k ,(cid:15) ) v , v = v (cid:44) z k and the third equality is because wehave ( M Z k ,(cid:15) ) v , z k = ( M (cid:15),(cid:15) ) v , f for induced simple pushdown matrices. The result follows.We now discuss the behaviors of our constructed simple ω -reset pushdown automata. Lemma 36.
Let the simple ω -reset pushdown automata A lm = (2 n + , Γ , I m , M , P , l ) , for ≤ m ≤ n and ≤ l ≤ n, beinduced by the Greibach normal form (14) , (20) . We then have (cid:107) A lm (cid:107) = (( M ∗ ) (cid:15),(cid:15) ) x m , f + (( M ω, l ) (cid:15) ) z m .Proof. Let 1 ≤ m ≤ n and 0 ≤ l ≤ n . We obtain (cid:107) A lm (cid:107) = I ( M ∗ ) (cid:15),(cid:15) P + I ( M ω, l ) (cid:15) = (( M ∗ ) (cid:15),(cid:15) ) x m , f + (( M ∗ ) (cid:15),(cid:15) ) z m , f + (( M ω, l ) (cid:15) ) x m + (( M ω, l ) (cid:15) ) z m , = (( M ∗ ) (cid:15),(cid:15) ) x m , f + (( M ∗ ) (cid:15),(cid:15) ) z m , f + (( M ω, l ) (cid:15) ) z m .where the last equality is by Theorem 30.It remains to show that (( M ∗ ) (cid:15),(cid:15) ) z m , f =
0. We have (cid:98) M = (cid:98) M z , z (cid:98) M z , x (cid:98) M x , z (cid:98) M x , x (cid:98) M x , f .Now let M ∗ = (cid:32) α βγ δ (cid:33) ,where we are only interested in the second component of β . By lemma 28 and by (24) in the proof of Lemma 33, wehave β = ( M z , z ) ∗ (cid:16) M z , x (cid:17) (cid:34)(cid:32) M x , x M x , f (cid:33) + (cid:32) M x , z (cid:33) ( M z , z ) ∗ (cid:16) M z , x (cid:17)(cid:35) ∗ = (cid:16) ( M z , z ) ∗ M z , x (cid:17) (cid:32) ( M x , x + M x , z ( M z , z ) ∗ M z , x ) ∗ ( M x , x + M x , z ( M z , z ) ∗ M z , x ) ∗ M x , f (cid:33) = (cid:16) ( M z , z ) ∗ M z , x ( M x , x + M x , z ( M z , z ) ∗ M z , x ) ∗ , ( M z , z ) ∗ M z , x ( M x , x + M x , z ( M z , z ) ∗ M z , x ) ∗ M x , f (cid:17) .Now, we obtain(( M ∗ ) (cid:15),(cid:15) ) z m , f = (cid:16)(cid:16) ( M z , z ) ∗ M z , x ( M x , x + M x , z ( M z , z ) ∗ M z , x ) ∗ M x , f (cid:17) (cid:15),(cid:15) (cid:17) m = (cid:16)(cid:0) ( M z , z ) ∗ M z , x ( M x , x + M x , z ( M z , z ) ∗ M z , x ) ∗ (cid:1) (cid:15),(cid:15) ( M x , f ) (cid:15),(cid:15) (cid:17) m = (cid:16) (( M z , z ) ∗ ) (cid:15),(cid:15) (cid:0) M z , x ( M x , x + M x , z ( M z , z ) ∗ M z , x ) ∗ (cid:1) (cid:15),(cid:15) ( M x , f ) (cid:15),(cid:15) (cid:17) m = (cid:88) ≤ i ≤ n (cid:16) (( M z , z ) ∗ ) (cid:15),(cid:15) ( M z , x ) (cid:15), Z i (cid:0) ( M x , x + M x , z ( M z , z ) ∗ M z , x ) ∗ (cid:1) Z i ,(cid:15) ( M x , f ) (cid:15),(cid:15) (cid:17) m , (25)37here in the second equality, we have ( M x , f ) π,(cid:15) = π (cid:44) (cid:15) . The third equality uses that ( M z , z ) ∗ ) (cid:15),π = π (cid:44) (cid:15) . Inthe forth equality, we have ( M z , x ) (cid:15),π = π (cid:60) { Z i | ≤ i ≤ n } .We concentrate on the factor in the center, where we have (cid:0) ( M x , x + M x , z ( M z , z ) ∗ M z , x ) ∗ (cid:1) Z i ,(cid:15) = (cid:88) t ≥ (cid:88) π ,...,π t − ∈ Γ ∗ ( M x , x + M x , z ( M z , z ) ∗ M z , x ) Z i ,π · · · ( M x , x + M x , z ( M z , z ) ∗ M z , x ) π t − ,(cid:15) = (cid:88) t ≥ (cid:88) π ,...,π t − ∈ Γ ∗ ( M x , x + M x , z ( M z , z ) ∗ M z , x ) Z i ,π · · · ( M x , x + M x , z ( M z , z ) ∗ M z , x ) π t − ,(cid:15) ( M x , x ) (cid:15),(cid:15) = (cid:88) t ≥ ( M x , x ) Z i ,(cid:15) · · · ( M x , x ) (cid:15),(cid:15) ( M x , x ) (cid:15),(cid:15) = M x , x + M x , z ( M z , z ) ∗ M z , x ) π t − ,(cid:15) = ( M x , x ) (cid:15),(cid:15) because( M x , x ) π t − ,(cid:15) = π t − (cid:44) (cid:15) and because( M x , z ( M z , z ) ∗ M z , x ) π t − ,(cid:15) = (cid:88) π,π (cid:48) ∈ Γ ∗ ( M x , z ) π t − ,π (( M z , z ) ∗ ) π,π (cid:48) ( M z , x ) π (cid:48) ,(cid:15) = M z , x ) π (cid:48) ,(cid:15) = π (cid:48) . In the last equality, ( M x , x ) Z i ,(cid:15) = M ∗ ) (cid:15),(cid:15) ) z m , f = (cid:88) ≤ i ≤ n (cid:16) (( M z , z ) ∗ ) (cid:15),(cid:15) ( M z , x ) (cid:15), Z i (cid:0) ( M x , x + M x , z ( M z , z ) ∗ M z , x ) ∗ (cid:1) Z i ,(cid:15) ( M x , f ) (cid:15),(cid:15) (cid:17) m = (cid:88) ≤ i ≤ n (cid:16) (( M z , z ) ∗ ) (cid:15),(cid:15) ( M z , x ) (cid:15), Z i M x , f ) (cid:15),(cid:15) (cid:17) m = ω -reset pushdown automata with the solutionsof system (18). Theorem 37.
Let ( S , V ) be a complete semiring-semimodule pair. Let the simple ω -reset pushdown automata A lm , for ≤ m ≤ n and ≤ l ≤ n, be induced by the Greibach normal form (14) , (20) .Then, for ≤ l ≤ n, ( (cid:107) A l (cid:107) , . . . , (cid:107) A ln (cid:107) ) = (cid:0) (( M ∗ ) (cid:15),(cid:15) ) x , f + (( M ω, l ) (cid:15) ) z , . . . , (( M ∗ ) (cid:15),(cid:15) ) x n , f + (( M ω, l ) (cid:15) ) z n (cid:1) is a solution of (18) .Proof. We show that((( M ∗ ) (cid:15),(cid:15) ) x , f , . . . , (( M ∗ ) (cid:15),(cid:15) ) x n , f ) and ((( M ω, l ) (cid:15) ) z , . . . , (( M ω, l ) (cid:15) ) z n )are solutions of the mixed ω -algebraic system (14), (20).Let now M (cid:48) be induced by the Greibach normal form (14). By Theorem 16, ((( M (cid:48)∗ ) (cid:15),(cid:15) ) , f , . . . , (( M (cid:48)∗ ) (cid:15),(cid:15) ) n , f ) is asolution of (14). By Lemma 33, we deduce that ((( M ∗ ) (cid:15),(cid:15) ) x , f , . . . , (( M ∗ ) (cid:15),(cid:15) ) x n , f ) is also a solution of (14).We now show that ((( M ω, l ) (cid:15) ) z , . . . , (( M ω, l ) (cid:15) ) z n ) is a solution of (20) and substitute it into the right sides of (20): (cid:88) ≤ j , k ≤ n (cid:88) a ∈ Σ ( p i , ax j z k ) a σ j ω k + (cid:88) ≤ j ≤ n (cid:88) a ∈ Σ ( p i , az j ) a ω j = (cid:88) ≤ j , k ≤ n ( M (cid:15), Z k ) z i , x j (( M ∗ ) (cid:15),(cid:15) ) x j , f (( M ω, l ) (cid:15) ) z k + (cid:88) ≤ j ≤ n ( M (cid:15),(cid:15) ) z i , z j (( M ω, l ) (cid:15) ) z j = (cid:88) ≤ j , k ≤ n ( M (cid:15), Z k ) z i , x j (( M ω, l ) Z k ) x j + (cid:88) ≤ j ≤ n ( M (cid:15),(cid:15) ) z i , z j (( M ω, l ) (cid:15) ) z j = (cid:88) ≤ k ≤ n ( M (cid:15), Z k ( M ω, l ) Z k ) z i + ( M (cid:15),(cid:15) ( M ω, l ) (cid:15) ) z i = (cid:0) MM ω, l ) (cid:15) (cid:1) z i = (( M ω, l ) (cid:15) ) z i , for each 1 ≤ i ≤ n ,38here the second equality is by Lemma 35, the last equality is by Theorem 19.The theorem above, Theorem 37, is not su ffi cient for our main result. The following theorem extends the previoustheorem by stating that ( (cid:107) A l (cid:107) , . . . , (cid:107) A ln (cid:107) ) is a canonical solution of (18). Theorem 38.
Let ( S , V ) be a complete semiring-semimodule pair. Let the simple ω -reset pushdown automata A lm , for ≤ m ≤ n and ≤ l ≤ n, be induced by the Greibach normal form (14) , (20) .Then, for ≤ l ≤ n, ( (cid:107) A l (cid:107) , . . . , (cid:107) A ln (cid:107) ) = (cid:0) (( M ∗ ) (cid:15),(cid:15) ) x , f + (( M ω, l ) (cid:15) ) z , . . . , (( M ∗ ) (cid:15),(cid:15) ) x n , f + (( M ω, l ) (cid:15) ) z n (cid:1) is the l th canonical solution of (18) .Proof. We show that((( M ∗ ) (cid:15),(cid:15) ) x , f , . . . , (( M ∗ ) (cid:15),(cid:15) ) x n , f ) and ((( M ω, l ) (cid:15) ) z , . . . , (( M ω, l ) (cid:15) ) z n )is the l th canonical solution of the mixed ω -algebraic system (14), (20).Let M (cid:48) be induced by the Greibach normal form (14). Then, by Theorem 16, ((( M (cid:48)∗ ) (cid:15),(cid:15) ) , f , . . . , (( M (cid:48)∗ ) (cid:15),(cid:15) ) n , f ) is theunique (and therefore least) solution of (14). By Lemma 33, we can conclude that σ = ((( M ∗ ) (cid:15),(cid:15) ) x , f , . . . , (( M ∗ ) (cid:15),(cid:15) ) x n , f )is also the least solution of (14).Fix l with 1 ≤ l ≤ n for the remainder of the proof. It remains to show that for the system (20), written as z = (cid:37) ( x ) z ,we have (cid:37) ( σ ) ω, l = ((( M ω, l ) (cid:15) ) z , . . . , (( M ω, l ) (cid:15) ) z n )We start with the right side of equation (20). We have, for 1 ≤ i ≤ n , (cid:37) ( σ ) i z = (cid:88) ≤ j , k ≤ n (cid:88) a ∈ Σ ( p i , ax j z k ) a σ j z k + (cid:88) ≤ j ≤ n (cid:88) a ∈ Σ ( p i , az j ) az j = (cid:88) ≤ j , k ≤ n (cid:88) a ∈ Σ ( p i , ax k z j ) a σ k z j + (cid:88) ≤ j ≤ n (cid:88) a ∈ Σ ( p i , az j ) az j = (cid:88) ≤ j ≤ n (cid:16) (cid:88) ≤ k ≤ n (cid:88) a ∈ Σ ( p i , ax k z j ) a σ k + (cid:88) a ∈ Σ ( p i , az j ) a (cid:17) z j = (cid:88) ≤ j ≤ n (cid:16) (cid:88) ≤ k ≤ n ( M (cid:15), Z j ) z i , x k (( M ∗ ) (cid:15),(cid:15) ) x k , f + ( M (cid:15),(cid:15) ) z i , z j (cid:17) z j = (cid:88) ≤ j ≤ n (cid:16) (cid:88) ≤ k ≤ n ( M (cid:15), Z j ) z i , x k ((( M x , x ) ∗ ) Z j , Z j M Z j ,(cid:15) ) x k , z j + ( M (cid:15),(cid:15) ) z i , z j (cid:17) z j = (cid:88) ≤ j ≤ n (cid:16) (cid:88) ≤ k , k (cid:48) ≤ n ( M (cid:15), Z j ) z i , x k ((( M x , x ) ∗ ) Z j , Z j ) x k , x k (cid:48) ( M Z j ,(cid:15) ) x k (cid:48) , z j + ( M (cid:15),(cid:15) ) z i , z j (cid:17) z j = (cid:88) ≤ j ≤ n (cid:16) (cid:88) ≤ k , k (cid:48) ≤ n ( (cid:98) M z i , x k ) (cid:15), Z j ((( (cid:98) M x , x ) ∗ ) x k , x k (cid:48) ) Z j , Z j ( (cid:98) M x k (cid:48) , z j ) Z j ,(cid:15) + ( (cid:98) M z i , z j ) (cid:15),(cid:15) (cid:17) z j = (cid:88) ≤ j ≤ n (cid:16) (cid:88) ≤ k , k (cid:48) ≤ n (cid:88) P ∈ Γ ( (cid:98) M z i , x k ) (cid:15), P ((( (cid:98) M x , x ) ∗ ) x k , x k (cid:48) ) P , P ( (cid:98) M x k (cid:48) , z j ) P ,(cid:15) + ( (cid:98) M z i , z j ) (cid:15),(cid:15) (cid:17) z j = (cid:88) ≤ j ≤ n (cid:16) (cid:88) ≤ k , k (cid:48) ≤ n (cid:0) (cid:98) M z i , x k (( (cid:98) M x , x ) ∗ ) x k , x k (cid:48) (cid:98) M x k (cid:48) , z j (cid:1) (cid:15),(cid:15) + ( (cid:98) M z i , z j ) (cid:15),(cid:15) (cid:17) z j = (cid:88) ≤ j ≤ n (cid:16) (cid:98) M z i , x ( (cid:98) M x , x ) ∗ (cid:98) M x , z j + (cid:98) M z i , z j (cid:17) (cid:15),(cid:15) z j ,where the fifth equality is by Lemma 34. The eighth equality is because for P (cid:44) Z j , we have ( (cid:98) M x k (cid:48) , z j ) P ,(cid:15) = (cid:37) of the system z = (cid:37) ( x ) z , we obtain (cid:37) ( σ ) = (cid:0) (cid:98) M z , x ( (cid:98) M x , x ) ∗ (cid:98) M x , z + (cid:98) M z , z (cid:1) (cid:15),(cid:15) · · · (cid:0) (cid:98) M z , x ( (cid:98) M x , x ) ∗ (cid:98) M x , z n + (cid:98) M z , z n (cid:1) (cid:15),(cid:15) ... . . . ... (cid:0) (cid:98) M z n , x ( (cid:98) M x , x ) ∗ (cid:98) M x , z + (cid:98) M z n , z (cid:1) (cid:15),(cid:15) · · · (cid:0) (cid:98) M z n , x ( (cid:98) M x , x ) ∗ (cid:98) M x , z n + (cid:98) M z n , z n (cid:1) (cid:15),(cid:15) = (cid:16) M z , x ( M x , x ) ∗ M x , z + M z , z (cid:17) (cid:15),(cid:15) . Then, we have (cid:0) (cid:37) ( σ ) ω, l (cid:1) j = (cid:16)(cid:0)(cid:0) M z , x ( M x , x ) ∗ M x , z + M z , z (cid:1) (cid:15),(cid:15) (cid:1) ω, l (cid:17) j = (cid:88) ( j , j ,... ) ∈ P l (cid:0) (cid:98) M z j , x ( (cid:98) M x , x ) ∗ (cid:98) M x , z j + (cid:98) M z j , z j (cid:1) (cid:15),(cid:15) (cid:0) (cid:98) M z j , x ( (cid:98) M x , x ) ∗ (cid:98) M x , z j + (cid:98) M z j , z j (cid:1) (cid:15),(cid:15) · · · = (cid:88) ( j , j ,... ) ∈ P l (cid:88) π ,π ,... ∈ Γ ∗ (cid:0) (cid:98) M z j , x ( (cid:98) M x , x ) ∗ (cid:98) M x , z j + (cid:98) M z j , z j (cid:1) (cid:15),π (cid:0) (cid:98) M z j , x ( (cid:98) M x , x ) ∗ (cid:98) M x , z j + (cid:98) M z j , z j (cid:1) π ,π · · · = (cid:16)(cid:0) ( M z , x ( M x , x ) ∗ M x , z + M z , z ) ω, l (cid:1) (cid:15) (cid:17) j , (26)where the forth equality uses the fact that (cid:0) (cid:98) M z i , x ( (cid:98) M x , x ) ∗ (cid:98) M x , z j + (cid:98) M z i , z j (cid:1) (cid:15),π = π (cid:44) (cid:15) which is because ( (cid:98) M z i , z j ) (cid:15),π = π (cid:44) (cid:15) by definition and because, by our construction, we have M z , x ( M x , x ) ∗ M x , z = (cid:88) ≤ j ≤ n ( M z , x ) (cid:15), Z j (( M x , x ) ∗ ) Z j , Z j ( M x , z ) Z j ,(cid:15) .Inductively, the above argument can be applied to all factors (cid:0) (cid:98) M z ji , x ( (cid:98) M x , x ) ∗ (cid:98) M x , z ji + + (cid:98) M z ji , z ji + (cid:1) π i ,π i + because we learnfrom the preceding factor that π i = (cid:15) .Now, we proceed from the other direction. From Theorem 30, we know that for the simple ω -reset pushdownautomaton A lm and a variable z j , we have(( M ω, l ) (cid:15) ) z j = (cid:16)(cid:0)(cid:0) M z , z + M z , x ( M x , x ) ∗ M x , z (cid:1) ω, l (cid:1) (cid:15) (cid:17) j = (cid:37) ( σ ) ω, lj ,where the last equality is by (26). This completes the proof.We now combine our previous discussion and Theorem 38 to get our second main result. Corollary 39.
Let S be a continuous star-omega semiring with the underlying semiring S being commutative and letr ∈ S alg (cid:104)(cid:104) Σ ∗ (cid:105)(cid:105) × S alg (cid:104)(cid:104) Σ ω (cid:105)(cid:105) .Then there exists a simple ω -reset pushdown automaton with behavior r.Proof. Let r ∈ S alg (cid:104)(cid:104) Σ ∗ (cid:105)(cid:105) × S alg (cid:104)(cid:104) Σ ω (cid:105)(cid:105) . As discussed on page 29, by Theorem 14 (and Theorem 5), r is a componentof a canonical solution of an ω -algebraic system in Greibach normal form over S (cid:104)(cid:104) Σ ∗ (cid:105)(cid:105) × S (cid:104)(cid:104) Σ ω (cid:105)(cid:105) . Let (18) be sucha system and assume that the m th component of the l th canonical solution of (18) is r , i.e., assume τ m = r for the l th canonical solution τ .Now, we can construct the simple ω -reset pushdown automata A lm induced by the Greibach normal form (14),(20), for which, by Theorem 38, ( (cid:107) A l (cid:107) , . . . , (cid:107) A ln (cid:107) ) is the l th canonical solution of (18). As the l th canonical solution isunique, we can conclude that (cid:107) A lm (cid:107) = τ m = r . 40 . Discussion We have extended the characterization of ω -algebraic series so that we can use the ω -Kleene closure to transferthe property of Greibach normal form from algebraic systems to mixed ω -algebraic systems. This generalizes afundamental property from context-free languages.We believe that the same technique can be used to transfer other properties of algebraic systems to infinite words.Cohen, Gold [4] use this technique also for the elimination of chain rules, for the Chomsky normal form and fore ff ective decision methods of emptiness, finiteness and infiniteness.The second part of this paper applies the Greibach normal form for the construction of ω -pushdown automata.Simple ω -reset pushdown automata do not use (cid:15) -transitions; in the literature, this is also called a realtime pushdownautomaton. Realtime pushdown automata read a symbol of the input word in every transition - exactly like context-free grammars in Greibach normal form generate a letter in every derivation step. Additionally, each derivation stepof context-free grammars in Greibach normal form increases the number of non-terminals in the sentential formby at most one. We showed that for realtime pushdown automata it su ffi ces to handle at most one stack symbolper transition. Here the Greibach normal form provides exactly the properties needed to construct simple ω -resetpushdown automata.The model of simple ω -reset pushdown automata seems to be very natural. They occur when applying generalhomomorphisms to nested-word automata [1, 8]. Their unweighted counterparts have been used for a B¨uchi-typelogical characterization of timed pushdown languages of finite words [12] and ω -context-free languages [8]. Alsoin the weighted setting, simple reset pushdown automata of finite words have been used in [13]. A B¨uchi-typelogical characterization for weighted ω -context-free languages is currently in development and uses the simple ω -reset pushdown automata introduced here. References [1] Blass, A., & Gurevich, Y. (2006). A note on nested words.
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