Corradini, A. and F. Gadducci, An algebraic presentation of term graphs, via gs-monoidal categories, Applied Categorical Structures 7 (1999), pp. 299--331.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....is proposed in [CG97] It is already known that term rewriting systems can be faithfully described by a Cartesian 2 category, where horizontal arrows represent terms, and cells represent rewriting sequences. A similar, original 2 categorical presentation is presented for term graph rewriting. In [CG98a] the main result 19 is a categorical characterization of term graphs that parallels the well known characterization of terms as arrows of the algebraic theory of a given signature. In particular, it is shown that term graphs over a signature Sigma are one to one with the arrows of the free ....

A. Corradini and F. Gadducci. An algebraic presentation of term graphs, via gs-monoidal categories. APCS, 1998. To appear.

....[24] The theory under the translation approach is based on a new link between double categories and 2 categories. Of special interest is the study of data structures to be used for representing con gurations and observations. Term graphs have been studied in detail in a categorical framework [27]. Moreover, a speci c analysis [22] has shown that quite a few classes of graphs, including all those which have been actually used in the case studies, can be suitably axiomatized and equipped with standard forms. In addition, these classes can be represented eciently in the SRI rewriting logic ....

....We think that a key issue in this area is the development of speci c algebras of graphs and graph rewriting techniques, providing operators which can be applied consistently not only to the static but also to the dynamic description of system components. We will try to apply our results in e.g. [27, 22]. Also, tiles naturally integrate graph descriptions at both the static and the dynamic level, e.g. actor systems and interaction diagrams. In addition, tiles can be drawn employing suggestive, wire and box diagrams which can be composed in a na ve fashion. A visual speci cation language based on ....

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A. Corradini, and F. Gadducci, An Algebraic Presentation of Term Graphs, via GS-Monoidal Categories. To appear in APCS, 1999.

.... ; rm = rn ; m = n Table 1. Axiomatization of Th[ Term graphs [9] are in some sense situated between linear and cartesian terms, because they allow for explicit sharing and discharging, in such a way that this information is preserved by composition. In fact, it has been shown in [6] that by abandoning the naturality of r and , one obtains a gs monoidal category GS[ which is isomorphic to the category of (ranked) term graphs on . For example, in GS[ the composition [t 1 =x 1 ] C[x 1 ] can be written as let x 1 = t 1 in C[x 1 ] with the convention that it evaluates to ....

A. Corradini and F. Gadducci. An algebraic presentation of term graphs, via gs-monoidal categories. Applied Categorical Structures, 7(4):299-331, 1999.

....idn naturality ( m;l = n;k ; rm = rn ; m = n (a) n s ## a 1 1 an ## 1 a ## n t ## 1 fx i a i y i j i 2 Ig C[x1 ; xn ] a D[y1 ; yn ] b) Figure 4. this information is preserved by composition. In fact, Corradini and Gadducci [8] have shown that by abandoning the naturality of r and in the construction of Th[ one obtains a category (called gs monoidal, from graph substitution) GS[ which is isomorphic to the category of (ranked) term graphs on . For example, in GS[ the composition [t 1 =x 1 ] C[x 1 ] can be ....

A. Corradini and F. Gadducci. An algebraic presentation of term graphs, via gs-monoidal categories. Applied Categorical Structures, 7(4):299-331, 1999.

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Corradini, A. and F. Gadducci, An algebraic presentation of term graphs, via gs-monoidal categories, Applied Categorical Structures 7 (1999), pp. 299--331.

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A. Corradini and F. Gadducci. An algebraic presentation of term graphs, via gs-monoidal categories. Applied Categorical Structures, 7:299-331, 1999.

.... axioms: r a b = r a r b ) a a;b b) a b = a b r e = e = id e r a ; r a a) r a ; a r a ) r a ; a;a = r a r a ; a a) a The previous situation is sometimes referred to in the literature by saying that each object is equipped with a comonoid structure [11] while, after [7], we denote such a structured category as gs monoidal. If duplicators and dischargers are natural (i.e. if they satisfy f ; r b = r a ; f f) and f ; b = a for all f : a b) then the category is actually cartesian. Similarly, co duplicators a : a a a and co dischargers a : e a must ....

A. Corradini and F. Gadducci. An algebraic presentation of term graphs, via gs-monoidal categories. Applied Categorical Structures, 7:299-331, 1999.

....from the ordinary tree like presentation of terms. We often use such theories in our paper, but we do not aim at any completeness, and we do not give any account of similar structures that have been presented in the literature (see e.g. 24,34,42,43,72] we refer the interested reader to [9,20]. 2 On the structure of terms 2.1 Some thoughts on name sharing We suggest an informal wire and box notation for giving an intuitive understanding of the name sharing mechanism, and more generally of the role played by the auxiliary structure in the ordinary representation of terms. In this ....

....differ only for two axioms, which represent the difference between implicit (as in the ordinary description of terms) and explicit (as it is the case instead in formalisms like term graphs 2 , usually presented with a set theoretical flavour) sharing of subterms. For more details, we refer to [20] and to a recent joint work of the authors with Roberto Bruni [9] in the latter we propose a schema for describing normal forms for this kind of structures, obtaining a framework where each structure finds a standard representation. 2.2 On theories We recall here some basic definitions, which ....

A. Corradini and F. Gadducci. An algebraic presentation of term graphs, via gs-monoidal categories. Applied Categorical Structures, 7:299--331, 1999.

....can also be presented in a categorical way similar to the 2 categorical presentation of term rewriting. In fact, ranked, acyclic) term graphs over a signature Sigma are in one to one correspondence with the arrows of the free gs monoidal category generated by Sigma (this result is presented in [15], where we also explain the origin of the acronym gs , which stays for graph substitution) And acyclic term graph rewriting sequences over a rewriting system R (according to the definition in [3] the most widely accepted in the literature) are faithfully represented by the cells of the free ....

....of R. REWRITING ON CYCLIC STRUCTURES 3 The gs monoidal categories are symmetric strict monoidal categories equipped with two transformations, the duplicator r (read nabla) and the discharger (bang) from which cartesian categories can be recovered requiring their naturality. As shown in [15], the non naturality of r is related to the fact that term graphs with different degree of sharing are distinct, while that of allows for the presence of garbage in a term graph (i.e. nodes not reachable from the roots) The main contribution of this paper is the generalisation of [13] to the ....

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A. Corradini and F. Gadducci. An algebraic presentation of term graphs, via gs-monoidal categories. Applied Categorical Structures, 1999. To appear.

....the theorem states that, for all processes p 1 ; p 2 , whenever CN (p 1 ) CN (p 2 ) is verified, also [p 1 ] p 2 ] holds. The proof is carried out first by finding a suitable normal form for the arrows of the subcategory CN (CP(N ) and then by induction on the length of normal forms, as in [11, 26, 9]. 5 Conclusions In this paper we provided a categorical characterization of the behaviour of contextual P T nets. We first defined the class of contextual processes of a CP T net (see Definition 10) then, we showed how these processes can be modeled as arrows of a suitable category, via a ....

A. Corradini and F. Gadducci. An algebraic presentation of term graphs, via gs-monoidal categories. Applied Categorical Structures, 1998. To appear.

.... the bicategories of processes of Walters [22,23] to the pre monoidal categories of Power and Robinson [34] to the action structures of Milner [30] to the interaction categories of Abramsky [1] to the sharing graphs of Hasegawa [20] and to the gs monoidal categories of Corradini and Gadducci [10,11], just to mention a few (see also [12,16,18,35] It is noteworthy that all these structures can be seen as enrichments of symmetric monoidal categories, which give the basis for the description of a distributed environment in terms of a wire and box diagram. We propose a schema for describing ....

....in this paper, via an implementation of their normal form representation as spaces. 2 A categorical view for di erent formalisms We recall here a few categorical de nitions. They represent suitable enrichments of monoidal categories and, as it is argued by various authors (as surveyed in e.g. [11,18]) these structures allow to recast the usual notion of term over a signature in a more general setting. Moreover, the progressive enrichment of a basic theory with di erent, additional constructors generates a great variety of di erent model classes, where the usual notions of relation, partial ....

[Article contains additional citation context not shown here]

A. Corradini and F. Gadducci. An algebraic presentation of term graphs, via gs-monoidal categories. Applied Categorical Structures, 7:299-331, 1999.

....specified also for closed (i.e. without variables) terms. 1 The distinction is made precise by the axiomatization of algebraic theories: Terms and term graphs differ for two axioms, representing, in a categorical setting, the naturality of transformations for copying and discharging arguments [8]. Many other mathematical structures have been proposed, for expressing formalisms different from the ordinary tree like presentation of terms. They range from the flownomial calculus of Stefanescu [6, 34] to the bicategories of processes of Walters [18, 19] to the pre monoidal categories of ....

.... the bicategories of processes of Walters [18, 19] to the pre monoidal categories of Power and Robinson [28] to the action structures of Milner [24] to the interaction categories of Abramsky [1] to the sharing graphs of Hasegawa [16] and to the gs monoidal categories of Corradini and Gadducci [7, 8], just to mention a few (see also [9, 11, 15, 29] All these structures can be seen as enrichments of symmetric monoidal categories, which give the basis for the description of a distributed environment in terms of a wire and box diagram. We propose a schema for describing normal forms for this ....

[Article contains additional citation context not shown here]

A. Corradini and F. Gadducci. An algebraic presentation of term graphs, via gs-monoidal categories. Applied Categorical Structures, 1998. To appear.

....(decomposition of graphs) Every graph can be obtained as the value of an expression containing only atomic graphs as constants and composition and union as operators. ut Space limitations force us to omit the proof. Nevertheless, a similar one for (acyclic) term graph can be actually found in [9], carried out by induction on G id 1 ffl 1 G ae 1 2 ffl ffl 1 2 Ge 1 ffl ffl 1 G ; Gr 1 ffl 1 2 G 1 ffl ; G Delta 1 2 ffl 1 G ; ffl 1 Fig. 2. Atomic graphs. the number of nodes of a given graph. In this version, the main technical tool is the use of a suitable ....

....ffl = Gc 1 ffl 1 In order to restrict e.g. the right interface, we may compose with G : Then G c ; G yields the (0; 1) ranked loop. ut 3 Graphs as Terms The aim of this section is to present a complete axiomatization for the category DG of ranked graphs, as done for (cyclic) term graphs in [9]. We first introduce the notion of dgs monoidal categories; we then show a finitary encoding of traced monoidal categories into the dgs monoidal structures, from which the completeness result can be inferred via a folklore characterization of hyper graphs [33, 21] 3.1 On (d)gs monoidal ....

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A. Corradini and F. Gadducci. An algebraic presentation of term graphs, via gsmonoidal categories. Applied Categorical Structures, 1998. To appear. Available at http://www.di.unipi.it/gadducci/papers/aptg.ps.

....and that, correspondingly, conditioned or generalized Kleene (strong) equations [6] are the natural notion of equation, among the many existing ones. Finally, for multi algebras we show that a concrete syntax for derived relations is given by term graphs , thanks to the main result of [8] which shows that they are in one to one correspondence with arrows of the gs monoidal theory. After a short summary on multi algebras, in Section 3 we discuss why a functorial semantics of multi algebras could not be based on the standard notion of algebraic theory. Simple counterexamples will ....

....F (b) such that F ( a ) e = 0 F (a) and F (r a ) a;a = r 0 F (a) it is strict if e and the components of are identities. The category of small strict gs monoidal categories and their strict functors is denoted by GSM Cat. Strict gs monoidal categories have been introduced in [8] to provide an algebraic characterization of term graphs as arrows of a suitable theory, in the same way as terms over a signature are represented by arrows of its algebraic theory. The pre x gs stands indeed for graph substitution, and it is justi ed by the main characterization result of [8] ....

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A. Corradini and F. Gadducci. An algebraic presentation of term graphs, via gs-monoidal categories. Applied Categorical Structures, 7:299-331, 1999.

....a ffl ## G G G G c 1 j g ffl a ffl ; w w w w c 2 j a ffl : g ffl Fig. 2: Example of explicit sharing. not give any account of similar structures that have been presented in the literature (see e.g. 24,34,42,43,72] we refer the interested reader to [9,20]. 2 On the structure of terms 2.1 Some thoughts on name sharing We suggest an informal wire and box notation for giving an intuitive understanding of the name sharing mechanism, and more generally of the role played by the auxiliary structure in the ordinary representation of terms. In this ....

....differ only for two axioms, which represent the difference between implicit (as in the ordinary description of terms) and explicit (as it is the case instead in formalisms like term graphs 2 , usually presented with a set theoretical flavour) sharing of sub terms. For more details, we refer to [20] and to a recent joint work of the authors with Roberto Bruni [9] in the latter we propose a schema for describing normal forms for this kind of structures, obtaining a framework where each structure finds a standard representation. 2.2 On theories We recall here some basic definitions, which ....

A. Corradini and F. Gadducci. An algebraic presentation of term graphs, via gs-monoidal categories. Applied Categorical Structures, 7:299--331, 1999. 42

....a term rewriting system can be faithfully described by a cartesian 2 category, where horizontal arrows represent terms, and cells represent rewriting sequences. In this paper we propose a similar, original 2 categorical presentation for term graph rewriting . Building on a result presented in [8], which shows that term graphs over a given signature are in one to one correspondence with arrows of a gs monoidal category freely generated from the signature, we associate with a term graph rewriting system a gs monoidal 2 category, and show that cells faithfully represent its rewriting ....

.... have been defined that, for example, extract from a term graph the term it represents (via some unraveling ) or, viceversa, that from a given term return a term graph representing it (and exhibiting some minimal or maximal sharing) The goal of this paper (together with its predecessor [8]) is to fill the gap described above, by proposing an original, clean categorical description of term graph rewriting. More precisely, while [8] has shown that term graphs over a signature correspond one to one to arrows of a gs monoidal category, building on that result we show here that term ....

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A. Corradini and F. Gadducci. An algebraic presentation of term graphs, via ps-monoidal categories. Submitted for publication. Available at http://www.di.unipi.it/gadducci/papers/aptg.ps, 1997.

.... F (a) Omega 0 F (b) such that F ( a ) OE e = 0 F (a) and F (r a ) OE = r 0 F (a) it is strict if OE and OE e are identities. The category of small strict gs monoidal categories and their strict functors is denoted by GSM Cat. Strict gs monoidal categories have been introduced in [2] for reasons apparently quite unrelated to the topic of the present paper, namely to provide an algebraic characterization of term graphs as arrows of a suitable theory, in the same way terms over a signature are represented by arrows of its algebraic theory. The prefix gs stays indeed for graph ....

....namely to provide an algebraic characterization of term graphs as arrows of a suitable theory, in the same way terms over a signature are represented by arrows of its algebraic theory. The prefix gs stays indeed for graph substitution, and is justified by the main characterization result of [2] (analogous correspondence results for other graph like structures, like cyclic term graphs and directed graphs, can be found in [3, 5] The non strict version is presented in [4] and it is based on non strict monoidal categories, where associativity and unit only hold up to natural isomorphism. ....

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A. Corradini and F. Gadducci. An algebraic presentation of term graphs, via gs-monoidal categories. Applied Categorical Structures, 1998. To appear.

....of acyclic term graphs can also be presented in a categorical way similar to the 2 categorical presentation of term rewriting. In fact, ranked, acyclic) term graphs over a signature Sigma are in one to one correspondence with the arrows of the free gsmonoidal category generated by Sigma [8]. And term graph rewriting sequences over a rewriting system R (according to the definition in [3] the most widely accepted in the literature) are faithfully represented by the cells of the free gs monoidal 2 category generated by a suitable representation of the rules in R. The gs monoidal ....

....generated by a suitable representation of the rules in R. The gs monoidal categories are symmetric strict monoidal categories equipped with two transformations, the duplicator r and the discharger , from which cartesian categories can be recovered requiring their naturality. As shown in [8], but already argued in [28] the non naturality of r is related to the fact that term graphs with different degree of sharing are distinct, while that of allows for the presence of garbage in a term graph (i.e. nodes not reachable from the roots) The main contribution of this paper is the ....

[Article contains additional citation context not shown here]

A. Corradini and F. Gadducci. An algebraic presentation of term graphs, via gsmonoidal categories. Applied Categorical Structures, 1998. To appear. Available at http://www.di.unipi.it/gadducci/papers/aptg.ps.

....graphs we are going to introduce are de nitely more complex than terms, we disagree with the other points. The next two sections will show that, comparing the functorial presentations of algebras and multi algebras (the former based on the Lawvere theory of , the latter on the gs monoidal theory [CG99a] term graphs play the same r ole for multi algebras that standard terms play for total algebras. Furthermore, the theory of term graphs and of term graph rewriting is nowadays well developed (see the book [SPvE93] and the references therein) and term graph rewriting is used, as a matter of ....

.... graph rewriting is nowadays well developed (see the book [SPvE93] and the references therein) and term graph rewriting is used, as a matter of fact, in many implementations of functional languages [PvE93] 4 Term Graphs This section introduces term graphs and some operations on them following [CG99a] The presentation departs slightly from the standard de nition (see e.g. BvEG 87] because we are more interested in algebraic properties of term graphs. In the following we assume that is a standard one sorted signature. De nition 4 (term graphs and morphisms) A directed acyclic graph ....

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A. Corradini and F. Gadducci. An algebraic presentation of term graphs, via gsmonoidal categories. Applied Categorical Structures, 1999. To appear.

....introducing the ownomial calculus [13,14] proposed as a calculus for owgraphs. Similar considerations arose independently also in di erent contexts, such as the allegories of circuits [3] based on the relational language ruby for circuit design [29] or the languages for (term) graph rewriting [10,11,20] and contextual net processes [22] or the calculi for concurrent and distributed processes [21,24,28] mostly arising from the work on pre monoidal categories [36] On the other hand, these structures have been used as suitable semantic models, that is, as suitable algebras for modeling the ....

....structure that could be used as a semantic domain, still retaining suitable properties of freeness. In our opinion, categories of spans [2,7] o er such a domain. Our view is supported by their use as models for circuits and predicate transformers [23,30,31] and as a syntax for nets and graphs [5,10,22]. More generally, it seems to us that these (co)span categories arise naturally in the description of distributed systems with interfaces , lifting and suitably weakening the properties of the underlying categories over which they are built. In the paper we provide a rst analysis of these ....

[Article contains additional citation context not shown here]

A. Corradini and F. Gadducci. An algebraic presentation of term graphs, via gs-monoidal categories. Applied Categorical Structures, 7:299-331, 1999.

.... the bicategories of processes of Walters [22,23] to the pre monoidal categories of Power and Robinson [34] to the action structures of Milner [30] to the interaction categories of Abramsky [1] to the sharing graphs of Hasegawa [20] and to the gs monoidal categories of Corradini and Gadducci [10,11], just to mention a few (see also [12,16,18,35] It is noteworthy that all these structures can be seen as enrichments of symmetric monoidal categories, which give the basis for the description of distributed environments in terms of wire and box diagrams. We propose a schema for describing ....

....in this paper, via an implementation of their normal form representation as spaces. 2 A categorical view for di erent formalisms We recall here a few categorical de nitions. They represent suitable enrichments of monoidal categories and, as it is argued by various authors (as surveyed in e.g. [11,18]) these structures allow to recast the usual notion of term over a signature in a more general setting. Moreover, the progressive enrichment of a basic theory with di erent, additional constructors generates a great variety of di erent model classes, where the notions of relation, partial order, ....

[Article contains additional citation context not shown here]

A. Corradini and F. Gadducci. An algebraic presentation of term graphs, via gs-monoidal categories. Applied Categorical Structures, 7:299-331, 1999.

....specified also for closed (i.e. without variables) terms. 1 The distinction is made precise by the axiomatization of algebraic theories: Terms and term graphs differ for two axioms, representing, in a categorical setting, the naturality of transformations for copying and discharging arguments [9]. Many other mathematical structures have been proposed, for expressing formalisms different from the ordinary tree like presentation of terms. They range from the flownomial calculus of Stefanescu [13,12] to the bicategories of processes of Walters [24,25] to the pre monoidal categories of ....

.... the bicategories of processes of Walters [24,25] to the pre monoidal categories of Power and Robinson [36] to the action structures of Milner [31] to the interaction categories of Abramsky [1] to the sharing graphs of Hasegawa [22] and to the gs monoidal categories of Corradini and Gadducci [8,9], just to mention a few (see also [10,18,20,38] All these structures can be seen as enrichments of symmetric monoidal categories, which give the basis for the description of a distributed environment in terms of a wire and box diagram. 1 Terms may share variables, but shared sub terms of a ....

[Article contains additional citation context not shown here]

A. Corradini and F. Gadducci. An algebraic presentation of term graphs, via gs-monoidal categories. Applied Categorical Structures, 1999. To appear.

....specified also for closed (i.e. without variables) terms. 1 The distinction is made precise by the axiomatization of algebraic theories: Terms and term graphs differ for two axioms, representing, in a categorical setting, the naturality of transformations for copying and discharging arguments [8]. Many other mathematical structures have been proposed, for expressing formalisms different from the ordinary tree like presentation of terms. They range from the flownomial calculus of Stefanescu [6, 34] to the bicategories of processes of Walters [18, 19] to the pre monoidal categories of ....

.... the bicategories of processes of Walters [18, 19] to the pre monoidal categories of Power and Robinson [28] to the action structures of Milner [24] to the interaction categories of Abramsky [1] to the sharing graphs of Hasegawa [16] and to the gs monoidal categories of Corradini and Gadducci [7, 8], just to mention a few (see also [9, 11, 15, 29] All these structures can be seen as enrichments of symmetric monoidal categories, which give the basis for the description of a distributed environment in terms of a wire and box diagram. We propose a schema for describing normal forms for this ....

[Article contains additional citation context not shown here]

A. Corradini and F. Gadducci. An algebraic presentation of term graphs, via gs-monoidal categories. Applied Categorical Structures, 1998. To appear.

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A. Corradini and F. Gadducci. An algebraic presentation of term graphs, via gs-monoidal categories. Applied Categorical Structures, 7:299-331, 1999.

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