P. A. Whigham. A schema theorem for context-free grammars. In 1995 IEEE Conference on Evolutionary Computation, pages 178--181. 230  Home/Search   Document Details and Download   Summary   Related Articles   Check

....comparatively little work has been devoted to a theory of GP. However such a theory is of great impor tance to the field as a whole. Work so far has been principly split between extending the Schema Theorem [Holland, 1992] from Genetic Algorithms to GP [Koza, 1992; O Reilly and Oppacher, 1995; Whigham, 1995; Poli and Langdon, 1997] studying benchmark problems (e.g. Royal Trees [Tackett, 1995; Punch et al. 1996] and applying population genetics to GP [Altenberg, 1994; Altenberg, 1995] In this paper we combine the later two approaches by applying Price s Theorem from theoretical biology, plus ....

P. A. Whigham. A schema theorem for context-free grammars. In 1995 IEEE Conference on Evolutionary Computation, pages 178-181. 9

....GP under the general framework of inductive learning. There was no direct connection with EHW. Whigham [15] 16] used similar crossover and mutation operators to those used by Hemmi et al. 11] A schema theorem under the derivation tree representation and the crossover and mutation was given [18]. The idea of evolving the grammar itself was mentioned but not tested. Whigham s work provides a different view toward the evolution of grammars. Such evolution can be regarded as the evolution of biases; i.e. certain knowledge or heuristics about what kind of circuits should be evolved. Such ....

P. Whigham, "A schema theorem for context-free grammars," in Proc. 1995 IEEE Int. Conf. Evolutionary Computat. (ICEC'95). Piscataway, NJ: IEEE Press, vol. 1, pp. 178--182.

....results on GP using the idea of schema is that the definition of schema is much less straightforward than for GAs and a few alternative definitions have been proposed in the literature. All of them define schemata as composed of one or multiple trees or fragments of trees. In some definitions [23, 24, 25, 26, 27] schema components are non rooted and, therefore, a schema can be present multiple times within the same program. This, together with the variability of the size and shape of the programs matching the same schema, leads to considerable complications in the calculations necessary to formulate ....

P. A. Whigham, "A schema theorem for context-free grammars," in 1995.

....the purely macroscopic ones. This distinction is important because the latter are simpler and easier to analyse than the former. The theory of schemata in genetic programming has had a difficult childhood. After some excellent early efforts leading to different worst case scenario schema theorems [6, 1, 11, 24, 17, 21], only very recently exact schema theories have become available [14, 12] which give exact formulations (rather than lower bounds) for the expected number of instances of a schema at the next generation. These exact theories are applicable to GP with one point crossover [16, 17, 18] No exact ....

....schema theories normally provide a lower bound for or, equivalently, for . Obtaining theoretical results for GP using the idea of schema is much less straightforward than for GAs. A few alternative definitions of schema have been proposed in the literature [6, 1, 11, 24, 17, 21], but for brevity here we will describe only the definition introduced in [17, 18] This is used in the rest of this paper. We will refer to this kind of schemata as fixed size and shape schemata. Syntactically a GP fixed size and shape schema is a tree composed of functions from the set ....

P. A. Whigham. A schema theorem for context-free grammars. In

....by adding up the contributions given by each possible member or each possible pair of members of the population, like in [2] In the case of GP theory the space of models is signi cantly less densely populated than the space of GA models. Before the present work only approximate schema theorems [15, 24, 58, 33, 44], which we will review in the next section, and one exact 4 microscopic model presented in [1] see Section 2.2) were available. These models are represented by the gray blobs in Figure 1. This paper presents the rst theoretical results on GP schemata which provide an exact macroscopic ....

....on GP using the idea of schema is that the de nition of schema is much less straightforward than for GAs and a few alternative de nitions have been proposed in the literature. Syntactically all of them de ne schemata as composed of one or multiple trees or fragments of trees. In some de nitions [15, 1, 24, 58, 59] schema components are non rooted and a schema is seen as a set of components (subtrees) that can be present multiple times within the same program. The focus in these theories is to predict how the number or the frequency of such components vary over time. However, the variability of the size and ....

P. A. Whigham. A schema theorem for context-free grammars. In

....One way of creating a theory for GP is to define a concept of schema for parse trees and to extend the GA schema theorem. Unfortunately, until very recently the efforts in this direction have given limited results. In the last few years alternative definitions of schema have been proposed [6, 12, 20]. All these definitions are based on the idea that a schema is composed of one or more trees or fragments of trees and that each schema represents all the programs in which such trees or tree fragments are present. These notions of schema have led to some theoretical results which, however, have a ....

P. A. Whigham. A schema theorem for context-free grammars. In

....and node reference systems introduced in other recent research. This theory generalises and refines previous work in GP and GA theory. 1 Introduction Genetic programming theory has had a difficult childhood. After some excellent early efforts leading to different approximate schema theorems [1, 2, 3, 4, 5, 6, 7], only very recently have schema theories become available which give exact formulations (rather than lower bounds) for the expected number of instances of a schema at the next generation. These exact theories are applicable to GP with one point crossover [8, 9, 10] standard crossover and other ....

....equivalently, for # . One of the difficulties in obtaining theoretical results on GP using the idea of schema is that finding a workable definition of a schema is much less straightforward than for GAs. Several alternative definitions have been proposed in the literature [1, 2, 3, 4, 6, 7, 5]. For brevity here we will describe only the definition introduced in [6, 7] since this is what is used in the rest of this paper. We will refer to this kind of schemata as fixed size and shape schemata. Syntactically a GP fixed size and shape schema is a tree composed of functions from the set ....

P. A. Whigham, "A schema theorem for context-free grammars, " in

....the purely macroscopic ones. This distinction is important because the latter are simpler and easier to analyse than the former. The theory of schemata in genetic programming has had a difficult childhood. After some excellent early efforts leading to different worst case scenario schema theorems [6, 1, 11, 24, 17, 21], only very recently exact schema theories have become available [14, 12] which give exact formulations (rather than lower bounds) for the expected number of instances of a schema at the next generation. These exact theories are applicable to GP with one point crossover [16, 17, 18] No exact ....

....worst casescenario schema theories normally provide a lower bound for (H; t) or, equivalently, for E[m(H; t 1) Obtaining theoretical results for GP using the idea of schema is much less straightforward than for GAs. A few alternative definitions of schema have been proposed in the literature [6, 1, 11, 24, 17, 21], but for brevity here we will describe only the definition introduced in [17, 18] This is used in the rest of this paper. We will refer to this kind of schemata as fixed size and shape schemata. Syntactically a GP fixed size and shape schema is a tree composed of functions from the set F [ f=g ....

P. A. Whigham. A schema theorem for context-free grammars. In

....are drawn in Section 6. 2 BACKGROUND 2.1 GP SCHEMA THEORIES Several alternative definitions of GP schema have been proposed in the literature (see [6, 4] for more details) All of them define schemata as templates composed of one or multiple trees or fragments of trees. In some definitions [12, 13, 14] schema components are non rooted and, therefore, a schema can be present multiple times within the same program. This leads to considerable mathematical difficulties. In more recent definitions [3, 5] schemata are represented by rooted trees or tree fragments, which x = 2 = Fixed Size and ....

P. A. Whigham, "A schema theorem for context-free grammars, " in

....macroscopic ones. The paper gives examples which show how the theory can be specialised to specific operators. 1 Introduction The theory of schemata in genetic programming has had a difficult childhood. After some excellent early efforts leading to different worst case scenario schema theorems [1, 2, 3, 4, 5, 6, 7], exact schema theories have become available only very recently [8, 9, 10, 11] These new theories give exact formulations (rather than lower bounds) for the expected number of instances of a schema at the next generation, and are applicable to GP with various types of subtree crossover. No exact ....

....normally provide a lower bound for (H; t) or, equivalently, for E[m(H; t 1) One of the difficulties in obtaining theoretical results on GP using the idea of schema is that its definition is much less straightforward than for GAs. Various definitions have been proposed in the literature [1, 2, 3, 4, 5, 7], but for brevity here we will describe only the definition of fixed size and shape schema introduced in [5, 6] which is what is used in this paper and in other recent work [8, 9, 10, 11, 16] 2.1 GP Schemata Syntactically a GP fixed size and shape schema (or just schema for simplicity) is a ....

P. A. Whigham. A schema theorem for context-free grammars. In

....macroscopic ones. The paper gives examples which show how the theory can be specialised to specific operators. 1 Introduction The theory of schemata in genetic programming has had a difficult childhood. After some excellent early efforts leading to different worst case scenario schema theorems [1, 2, 3, 4, 5, 6, 7], exact schema theories have become available only very recently [8, 9, 10, 11] These new theories give exact formulations (rather than lower bounds) for the expected number of instances of a schema at the next generation, and are applicable to GP with various types of subtree crossover. No exact ....

....provide a lower bound for 4 or, equivalently, for 42050 . One of the difficulties in obtaining theoretical results on GP using the idea of schema is that its definition is much less straightforward than for GAs. Various definitions have been proposed in the literature [1, 2, 3, 4, 5, 7], but for brevity here we will describe only the definition of fixed size and shape schema introduced in [5, 6] which is what is used in this paper and in other recent work [8, 9, 10, 11, 16] 2.1 GP Schemata Syntactically a GP fixed size and shape schema (or just schema for simplicity) is a ....

P. A. Whigham. A schema theorem for context-free grammars. In

....and to extend Holland s schema theorem. One of the difficulties in obtaining theoretical results using the idea of schema is that the definition of schema for GP is much less straightforward than for GAs and alternative definitions have been proposed [Koza, 1992, O Reilly and Oppacher, 1995, Whigham, 1995] These define schemata as composed of one or multiple fragments of a tree. These definitions allow a schema to be present E mail: R.Poli,W.B.Langdon cs.bham.ac.uk multiple times within the same program. This, together with the variability of the size and shape of the programs matching ....

....blocks in GP and, therefore, experimental studies are necessary to corroborate theoretical results on schemata. In the framework of his GP system based on context free grammars (CFG GP) Whigham produced a very general concept of schema for context free grammars and the related schema theorem [Whigham, 1995]. In CFG GP programs are the result of applying a set of rewrite rules to a starting symbol. The process of creation of a program can be represented with a derivation tree whose internal nodes are rewrite rules and whose terminals are the functions and terminals used in the program. In CFG GP the ....

Whigham, P. A. A schema theorem for context-free grammars. In 1995 IEEE Conference on Evolutionary Computation, volume 1, pages 178--181, Perth, Australia. IEEE Press.

....and node reference systems introduced in other recent research. This theory generalises and refines previous work in GP and GA theory. 1 Introduction Genetic programming theory has had a difficult childhood. After some excellent early efforts leading to different approximate schema theorems [1, 2, 3, 4, 5, 6, 7], only very recently have schema theories become available which give exact formulations (rather than lower bounds) for the expected number of instances of a schema at the next generation. These exact theories are applicable to GP with one point crossover [8, 9, 10] standard crossover and other ....

.... (H; t) or, equivalently, for E[m(H; t 1) One of the difficulties in obtaining theoretical results on GP using the idea of schema is that finding a workable definition of a schema is much less straightforward than for GAs. Several alternative definitions have been proposed in the literature [1, 2, 3, 4, 6, 7, 5]. For brevity here we will describe only the definition introduced in [6, 7] since this is what is used in the rest of this paper. We will refer to this kind of schemata as fixed size and shape schemata. Syntactically a GP fixed size and shape schema is a tree composed of functions from the set ....

P. A. Whigham, "A schema theorem for context-free grammars, " in

....In the last year or so the theory of schemata has made considerable progress, both for GAs and GP. This includes several new schema theorems which give exact formulations (rather than the lower bounds previously presented in the literature (Koza 1992, Altenberg 1994, O Reilly and Oppacher 1995, Whigham 1995, Poli and Langdon 1997, Rosca 1997, Poli and Langdon 1998b) for the expected number of instances of a schema at the next generation. These exact theories are applicable to GP with one point crossover (Poli 2000a, Poli 2000b, Poli 2001a) standard and other subtree swapping crossovers (Poli ....

Whigham, P. A. (1995). A schema theorem for context-free grammars. In: 1995 IEEE Conference on Evolutionary Computation. Vol. 1. IEEE Press. Perth, Australia. pp. 178--181.

....This theory generalises and refines previous work in GP and GA theory. 1 Introduction The theory for genetic programming has had a difficult childhood. After some excellent early efforts leading to different approximate schema theorems (Koza 1992, Altenberg 1994, O Reilly and Oppacher 1995, Whigham 1995, Poli and Langdon 1997, Poli and Langdon 1998b, Rosca 1997) only very recently have schema theories become available which give exact formulations (rather than lower bounds) for the expected number of instances of a schema at the next generation. These exact theories are applicable to GP with ....

....the difficulties in obtaining theoretical results on GP using the idea of schema is that finding a workable definition of a schema is much less straightforward than for GAs. Several alternative definitions have been proposed in the literature (Koza 1992, Altenberg 1994, O Reilly and Oppacher 1995, Whigham 1995, Poli and Langdon 1997, Poli and Langdon 1998b, Rosca 1997) For brevity here we will describe only the definition introduced in (Poli and Langdon 1997, Poli and Langdon 1998b) since this is what is used in the rest of this paper. We will refer to this kind of schemata as fixed size and shape ....

Whigham, P. A. (1995). A schema theorem for context-free grammars. In: 1995 IEEE Conference on Evolutionary Computation. Vol. 1. IEEE Press. Perth, Australia. pp. 178--181.

....1997, Stephens and Waelbroeck 1999) then don t have to be so. 1 The theory of schemata in genetic programming has had a difficult childhood. After some excellent early efforts leading to different worst case scenario schema theorems (Koza 1992, Altenberg 1994, O Reilly and Oppacher 1995, Whigham 1995, Poli and Langdon 1997b, Rosca 1997) only very recently exact schema theories have become available (Poli 2000a, Poli 2000b) which give exact formulations (rather than lower bounds) for the expected number of instances of a schema at the next generation. 1 These exact theories are applicable ....

....is much less straightforward than for GAs. A few alternative definitions have been proposed in the literature. Syntactically all of them define schemata as composed of one or multiple trees or fragments of trees. In some definitions (Koza 1992, Altenberg 1994, O Reilly and Oppacher 1995, Whigham 1995, Whigham 1996) schema components are non rooted and a schema in seen as a set of subtrees that can be present multiple times within the same program. The focus in these theories is to predict how the number or the frequency of such subtrees vary over time. In more recent definitions (Poli and ....

Whigham, P. A. (1995). A schema theorem for context-free grammars. In: 1995 IEEE Conference on Evolutionary Computation. Vol. 1. IEEE Press. Perth, Australia. pp. 178--181.

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P. A. Whigham. A schema theorem for context-free grammars. In 1995 IEEE Conference on Evolutionary Computation, pages 178--181. 230

No context found.

P. A. Whigham. A schema theorem for context-free grammars. In 1995.

No context found.

Whigham, P. A. (1995). A schema theorem for context-free grammars. In 1995 IEEE Conference on Evolutionary Computation, volume 1, pages 178--181, Perth, Australia. IEEE Press.

No context found.

Whigham, P. A. A schema theorem for context-free grammars. In 1995 IEEE Conference on Evolutionary Computation, volume 1, pages 178--181, Perth, Australia. IEEE Press.

No context found.

P. A. Whigham, "A schema theorem for context-free grammars, " in 1995.

No context found.

P. A. Whigham, "A schema theorem for context-free grammars ", in 1995.

No context found.

Whigham, P. A. (1995). A schema theorem for context-free grammars. In 1995 IEEE Conference on Evolutionary Computation, volume 1, pages 178--181, Perth, Australia. IEEE Press.

No context found.

P. A. Whigham, "A schema theorem for context-free grammars, " in 1995.

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P. A. Whigham. A schema theorem for context-free grammars. In 1995.

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