Composition Theorem for Generalized Sum - Semantic Scholar
Composition Theorem for Generalized Sum Alexander Rabinovich Dept. of CS, Tel Aviv University [email protected], September 19, 2001
1 Introduction Composition Theorems are tools which reduce sentences about some compound structure to sentences about its parts. A seminal example of such a result is the Feferman-Vaught Theorem [2] which reduces the first-order theory of generalized products to the first order theory of its factors and the monadic second-order theory of index structure. Shelah [19] defined the notion of generalized sum and stated the composition theorem which reduces the monadic second-order theory of the generalized sum to the monadic second-order theories of the summands and of the index structures. An important example of generalized sums is the ordinal sum of linearly ordered sets. In [19] the composition theorem for linear orders was one of the main tools for obtaining very strong decidability result for the monadic theory of linear orders. Shelah [19] hinted that the composition theorem for generalized sums can be obtained by Ehrenfeucht-Fraiss´e games. Gurevich in [4] proved a finite version of Feferman-Vaught composition theorem and then derived the composition theorem for linear orders. In [6] several composition theorems for monadic-second order logic over trees were given and were used to show undefinability of choice functions and the failure of uniformization in the monadic theory of the binary tree. The composition theorems for linear orders and trees were used in [4, 6, 7, 8, 9, 10, 14, 15]. The composition theorem for linear orders was described in a survey exposition by Gurevich [5] and Thomas [22]; it was illustrated there for the decidability of monadic logic of order, as an alternative to the automata-theoretic approach popularized by B¨uchi. The composition method, despite its success, is largely ignored by the theoretical computer science community. W. Thomas surveys in [22] Shelah’s composition theorem for linear orders and writes: “Although the subject was exposed in Gurevich’s concise survey [5], it did not attract much attention among theoretical computer scientists. Preference was (and is still) given to automata theoretic method . . . because it does not involve frightening logical technicalities as one finds them in [19]. Thus there is a tendency that the merits of model theoretic approach are overlooked. This is unfortunate, because it excludes some interesting applications.” There is no doubt that many applications of this method can be found in finite model theory and descriptive complexity, because (1) the composition methods are among few fundamental theorems which are valid on finite models; (2) many constructions in descriptive complexity are based on assembling a model from components (see e.g., Immerman [12]) and first-order reductions. The main technical contribution of our paper is (1) a definition of a generalized sum of structure and (2) a composition theorem for first-order logic over the generalized sum. After this work had been completed, Saharon Shelah pointed to us that the generalized sum is a degenerate case of the distorted sum construct [20]. Moreover, a variant of the composition theorem for the generalized sum was stated without proof (and with some misprints) as Theorem 5.3 of [20]. Consequently, our paper can be considered as an explanation of this degenerate case of the distorted sum. The paper is organized as follows. Section 2 - Preliminaries. The notion of the generalized sum is very natural. However, it requires a considerable amount of notations. In Section 3 we present two generic examples which are instances of the generalized sum. Then explains the generalized sum construct and illustrates it by many examples. Section 4 provides the composition theorem for first-order logic. Section 5 contains consequences of the composition theorem. In Section 6 we illustrate applications of the composition theorem to definability and replace sometimes tricky game (or inductive) arguments by transparent reductions. Our paper aims at popularizing Composition Theorems. Therefore, Section 7 discusses the Feferman-Vaught [2] composition theorem for generalized product, explains Shelah’s definition of generalized sum [19] and the Shelah composition theorem for monadic second-order logic. 1
2 Preliminaries In this section we present the definitions and notations used throughout the paper. All the languages considered here are either first-order or monadic second-order. We assu me that our languages contain as logical symbols: equality, True and False with their standard interpretations. To specify a logical language in the paper, we need only to give its set of relational and functional symbols and their arity (the signature of the language) and indicate whether the logic is first-order or monadic second-order. No restriction on the cardinality of the signature is assumed. However, if not stated otherwise, we assume that the signatures do not contain functional symbols. We use ti for first-order variables. Often we consider several languages. In such cases we will use ti as first-order variables of one language and vi as first-order variables of another. We use ti for tuples of variables. We use Zi for monadic predicate variables. Similarly Zi is used for tuples of monadic predicate variables. Sometimes for the sake of readability we will be liberal about naming the variables. A first-order monadic extension of a language over a signature is the first-order language over the signature [ fXi i 2 Natg, where Xi are new monadic predicate symbols. As usual if ' is a formula we may write ' t1 ; : : : tk X1 : : : Xm to indicate that the free variables in ' are among t1 ; : : : ; tk and the monadic predicate names are among X1 ; : : : ; Xm . For formula ' of the first-order language L, the quantifier rank of ' (notation qr ' ) is defined by the structural induction as usual. An alternation type is a finite sequence of positive natural numbers. We define below when a formula ' has an alternation type hn1 ; : : : nk i. However, unlike the quantifier rank, a formula has many alternation types.
:
(
;
)
()
2. If n is an alternation type of '1 and of '2 then n is an alternation type of :'1 ,of '1 ^ '2 and of '1 _ '2.
1. If n is an alternation type and ' is atomic then n is an alternation type of '.
3. If hn1 ; : : : nk i is an alternation type of ' then hn0 ; n1 ; : : : nk i is an alternation type of 8u1 : : : 8um ' and of 9u1 : : : 9um' for n0 m.
23 11111
221
For example if ' is in prenex normal form with the quantifier prefix 82 93 then h ; i, h ; ; ; ; i and h ; ; i are alternation types of ', but h ; ; i is not one of its alternation type. Finally, we say that ' has an alternation rank k if ' has an alternation type of length k. Let A be a structure for a signature . We use j A j for the universe of A and RA for the interpretation of the relational symbol R in A. However, whenever there is no confusion we will also use A for the universe of A; sometimes we use “a 2 A” instead of “a 2j A j”. Let ' t1 ; : : : tk X1 : : : Xm be a formula in the extension of a language L by monadic predicate names X1 : : : Xm . In order to check if the formula ' t1 ; : : : tk X1 : : : Xm is true we need to specify which model A of L is intended and what are the elements a1 : : : ak in A and the predicates (subsets) P1 Pm over A which are assigned to the variables and predicate names t1 : : : tk ; X1 Xm . Hence, the notation will usually be hA; a1 ; : : : ak P1 ; Pm i j ' t1 ; tk X1 : : : Xm which we also abbreviate to A j ' a1 : : : ak P1 : : : Pm or to A j ' a; P where the bar denotes a tuple of the appropriate length; whenever a and P are clear from a context we even will use A j '.
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3 Generalized Sum 3.1 Generic Examples The notion of a generalized sum is very natural. However, it requires a considerable amount of notations. Therefore, we start this section with two examples which are instances of the generalized sum - disjoint union of structures and ordinal sum of ordered sets. Example 3.1 (Disjoint Union) Let Lsummand be a language and let Ai (for i 2 Ind) be a set of disjoint structures for Lsummand . The disjoint union of Ai is the following structure A for the language Lresult Lsummand .
=
= [ j Ai j.
The universe: j A j
For n-ary relational symbol R 2 Lsummand and a tuple a1 ; : : : ; an of elements of A
ha1 ; : : : an i 2 RA iff there is i 2j Ind j such that ha1 ; : : : ; an i 2 RAi 2
Hence, if a tuple of elements satisfies in A an atomic Lresult formula then all the elements of the tuple are in the same summand structure Ai . Example 3.2 (Ordinal sum of ordered sets) Let Ind = (j Ind j; m no sentence of quantifier rank at most n distinguishes
1. Between Cir n and Cir k . 2. Between Cir n and Cir k
[ Cir k .
=
=
=
Proof (1) Consider the following rule for the languages Lsummand ;, Lind f
1 Introduction Composition Theorems are tools which reduce sentences about some compound structure to sentences about its parts. A seminal example of such a result is the Feferman-Vaught Theorem [2] which reduces the first-order theory of generalized products to the first order theory of its factors and the monadic second-order theory of index structure. Shelah [19] defined the notion of generalized sum and stated the composition theorem which reduces the monadic second-order theory of the generalized sum to the monadic second-order theories of the summands and of the index structures. An important example of generalized sums is the ordinal sum of linearly ordered sets. In [19] the composition theorem for linear orders was one of the main tools for obtaining very strong decidability result for the monadic theory of linear orders. Shelah [19] hinted that the composition theorem for generalized sums can be obtained by Ehrenfeucht-Fraiss´e games. Gurevich in [4] proved a finite version of Feferman-Vaught composition theorem and then derived the composition theorem for linear orders. In [6] several composition theorems for monadic-second order logic over trees were given and were used to show undefinability of choice functions and the failure of uniformization in the monadic theory of the binary tree. The composition theorems for linear orders and trees were used in [4, 6, 7, 8, 9, 10, 14, 15]. The composition theorem for linear orders was described in a survey exposition by Gurevich [5] and Thomas [22]; it was illustrated there for the decidability of monadic logic of order, as an alternative to the automata-theoretic approach popularized by B¨uchi. The composition method, despite its success, is largely ignored by the theoretical computer science community. W. Thomas surveys in [22] Shelah’s composition theorem for linear orders and writes: “Although the subject was exposed in Gurevich’s concise survey [5], it did not attract much attention among theoretical computer scientists. Preference was (and is still) given to automata theoretic method . . . because it does not involve frightening logical technicalities as one finds them in [19]. Thus there is a tendency that the merits of model theoretic approach are overlooked. This is unfortunate, because it excludes some interesting applications.” There is no doubt that many applications of this method can be found in finite model theory and descriptive complexity, because (1) the composition methods are among few fundamental theorems which are valid on finite models; (2) many constructions in descriptive complexity are based on assembling a model from components (see e.g., Immerman [12]) and first-order reductions. The main technical contribution of our paper is (1) a definition of a generalized sum of structure and (2) a composition theorem for first-order logic over the generalized sum. After this work had been completed, Saharon Shelah pointed to us that the generalized sum is a degenerate case of the distorted sum construct [20]. Moreover, a variant of the composition theorem for the generalized sum was stated without proof (and with some misprints) as Theorem 5.3 of [20]. Consequently, our paper can be considered as an explanation of this degenerate case of the distorted sum. The paper is organized as follows. Section 2 - Preliminaries. The notion of the generalized sum is very natural. However, it requires a considerable amount of notations. In Section 3 we present two generic examples which are instances of the generalized sum. Then explains the generalized sum construct and illustrates it by many examples. Section 4 provides the composition theorem for first-order logic. Section 5 contains consequences of the composition theorem. In Section 6 we illustrate applications of the composition theorem to definability and replace sometimes tricky game (or inductive) arguments by transparent reductions. Our paper aims at popularizing Composition Theorems. Therefore, Section 7 discusses the Feferman-Vaught [2] composition theorem for generalized product, explains Shelah’s definition of generalized sum [19] and the Shelah composition theorem for monadic second-order logic. 1
2 Preliminaries In this section we present the definitions and notations used throughout the paper. All the languages considered here are either first-order or monadic second-order. We assu me that our languages contain as logical symbols: equality, True and False with their standard interpretations. To specify a logical language in the paper, we need only to give its set of relational and functional symbols and their arity (the signature of the language) and indicate whether the logic is first-order or monadic second-order. No restriction on the cardinality of the signature is assumed. However, if not stated otherwise, we assume that the signatures do not contain functional symbols. We use ti for first-order variables. Often we consider several languages. In such cases we will use ti as first-order variables of one language and vi as first-order variables of another. We use ti for tuples of variables. We use Zi for monadic predicate variables. Similarly Zi is used for tuples of monadic predicate variables. Sometimes for the sake of readability we will be liberal about naming the variables. A first-order monadic extension of a language over a signature is the first-order language over the signature [ fXi i 2 Natg, where Xi are new monadic predicate symbols. As usual if ' is a formula we may write ' t1 ; : : : tk X1 : : : Xm to indicate that the free variables in ' are among t1 ; : : : ; tk and the monadic predicate names are among X1 ; : : : ; Xm . For formula ' of the first-order language L, the quantifier rank of ' (notation qr ' ) is defined by the structural induction as usual. An alternation type is a finite sequence of positive natural numbers. We define below when a formula ' has an alternation type hn1 ; : : : nk i. However, unlike the quantifier rank, a formula has many alternation types.
:
(
;
)
()
2. If n is an alternation type of '1 and of '2 then n is an alternation type of :'1 ,of '1 ^ '2 and of '1 _ '2.
1. If n is an alternation type and ' is atomic then n is an alternation type of '.
3. If hn1 ; : : : nk i is an alternation type of ' then hn0 ; n1 ; : : : nk i is an alternation type of 8u1 : : : 8um ' and of 9u1 : : : 9um' for n0 m.
23 11111
221
For example if ' is in prenex normal form with the quantifier prefix 82 93 then h ; i, h ; ; ; ; i and h ; ; i are alternation types of ', but h ; ; i is not one of its alternation type. Finally, we say that ' has an alternation rank k if ' has an alternation type of length k. Let A be a structure for a signature . We use j A j for the universe of A and RA for the interpretation of the relational symbol R in A. However, whenever there is no confusion we will also use A for the universe of A; sometimes we use “a 2 A” instead of “a 2j A j”. Let ' t1 ; : : : tk X1 : : : Xm be a formula in the extension of a language L by monadic predicate names X1 : : : Xm . In order to check if the formula ' t1 ; : : : tk X1 : : : Xm is true we need to specify which model A of L is intended and what are the elements a1 : : : ak in A and the predicates (subsets) P1 Pm over A which are assigned to the variables and predicate names t1 : : : tk ; X1 Xm . Hence, the notation will usually be hA; a1 ; : : : ak P1 ; Pm i j ' t1 ; tk X1 : : : Xm which we also abbreviate to A j ' a1 : : : ak P1 : : : Pm or to A j ' a; P where the bar denotes a tuple of the appropriate length; whenever a and P are clear from a context we even will use A j '.
131
(
; = [
;
)
= ( =
℄
(
;
;
)
)
= [
;
℄
3 Generalized Sum 3.1 Generic Examples The notion of a generalized sum is very natural. However, it requires a considerable amount of notations. Therefore, we start this section with two examples which are instances of the generalized sum - disjoint union of structures and ordinal sum of ordered sets. Example 3.1 (Disjoint Union) Let Lsummand be a language and let Ai (for i 2 Ind) be a set of disjoint structures for Lsummand . The disjoint union of Ai is the following structure A for the language Lresult Lsummand .
=
= [ j Ai j.
The universe: j A j
For n-ary relational symbol R 2 Lsummand and a tuple a1 ; : : : ; an of elements of A
ha1 ; : : : an i 2 RA iff there is i 2j Ind j such that ha1 ; : : : ; an i 2 RAi 2
Hence, if a tuple of elements satisfies in A an atomic Lresult formula then all the elements of the tuple are in the same summand structure Ai . Example 3.2 (Ordinal sum of ordered sets) Let Ind = (j Ind j; m no sentence of quantifier rank at most n distinguishes
1. Between Cir n and Cir k . 2. Between Cir n and Cir k
[ Cir k .
=
=
=
Proof (1) Consider the following rule for the languages Lsummand ;, Lind f
