Graphical Calculus Research Articles

A Port Graph Calculus for Autonomic Computing and Invariant Verification

From our previous work on biochemical applications, the structure of port graph (or multigraph with ports) and a rewriting calculus have proved to be well-suited formalisms for modeling interactions between proteins. Then port graphs have been proposed as a formal model for distributed resources and grid infrastructures, where each resource is modeled by a node with ports. The lack of global information and the autonomous and distributed behavior of components are modeled by a multiset of port graphs and rewrite rules which are applied locally, concurrently, and non-deterministically. Some computations take place wherever it is possible and in parallel, while others may be controlled by strategies. In this paper, we first introduce port graphs as graphs with multiple edges and loops, with nodes having explicit connection points, called ports, and edges attaching to ports of nodes. We then define an abstract biochemical calculus that instantiates to a rewrite calculus on these graphs. Rules and strategies are themselves port graphs, i.e. first-class objects of the calculus. As a consequence, they can be rewritten as well, and rules can create new rules, providing a way of modeling adaptive systems. This approach also provides a formal framework to reason about computations and to verify useful properties. We show how structural properties of a modeled system can be expressed as strategies and checked for satisfiability at each step of the computation. This provides a way to ensure invariant properties of a system. This work is a contribution to the formal specification and verification of adaptive systems and to theoretical foundations of autonomic computing.

An Intriguing Exponential Inequality

An algebra problem with a graphical solution challenges precalculus and calculus students to determine an analytical solution.

SOME NON-BRAIDED FUSION CATEGORIES OF RANK THREE

We classify all fusion categories for a given set of fusion rules with three simple object types. If a conjecture of Ostrik is true, our classification completes the classification of fusion categories with three simple object types. To facilitate the discussion, we describe a convenient, concrete and useful variation of graphical calculus for fusion categories, discuss pivotality and sphericity in this framework, and give a short and elementary re-proof of the fact that the quadruple dual functor is naturally isomorphic to the identity.

On graph reasoning

In this paper, we study the (positive) graph relational calculus. The basis for this calculus was introduced by Curtis and Lowe in 1996 and some variants, motivated by their applications to semantics of programs and foundations of mathematics, appear scattered in the literature. No proper treatment of these ideas as a logical system seems to have been presented. Here, we give a formal presentation of the system, with precise formulation of syntax, semantics, and derivation rules. We show that the set of rules is sound and complete for the valid inclusions, and prove a finite model result as well as decidability. We also prove that the graph relational language has the same expressive power as a first-order positive fragment (both languages define the same binary relations), so our calculus may be regarded as a notational variant of the positive existential first-order logic of binary relations. The graph calculus, however, has a playful aspect, with rules easy to grasp and use. This opens a wide range of applications which we illustrate by applying our calculus to the positive relational calculus (whose set of valid inclusions is not finitely axiomatizable), obtaining an algorithm for deciding the valid inclusions and equalities of the latter.

New Graph Calculi for Planar Non-3-Colorable Graphs

The Hajos calculus is a nondeterministic procedure which generates the class of non-3-colorable graphs. If all non-3-colorable graphs can be constructed in polynomial steps by the calculus, then NP = co-NP holds. Up to date, however, it remains open whether there exists a family of graphs that cannot be generated in polynomial steps. To attack this problem, we propose two graph calculi PHC and PHC* that generate non-3-colorable planar graphs, where intermediate graphs in the calculi are also restricted to be planar. Then we prove that PHC and PHC* are sound and complete. We also show that PHC* can polynomially simulate PHC.

A note on proofs with graphs

Curtis and Lowe (S. Curtis, G. Lowe, Proofs with graphs, Sci. Comput. Program. 26 (1996) 197–216) presented a graphical calculus to prove inclusions between relations, leaving completeness of the system as an open question. Here we recast their system, excluding complementation and union, by presenting a low complexity decision algorithm to decide its valid inclusions. Completeness follows as a corollary.

A diagrammatic reasoning system for the description logic [formula omitted

Diagrammatic reasoning is a tradition of visual logic that allows sentences that are equivalent to first order logic to be written in a visual or structural form: usually for improved usability. A calculus for the diagram can then be defined that allows well-formed formulas to be derived. This calculus is intended in the analog of logical inference. Description logics (DLs) have become a popular knowledge representation and processing language. DLs correspond to decidable fragments of first order logic; their notation is in the style of symbolic, variable-free formulas. Moreover, DLs are equipped with table au theorem provers that are proven to be sound and complete. Although DLs have roots in diagrammatic languages (such as semantic networks), they are elaborated in a purely symbolic manner. This paper discusses how DLs can be equivalently represented in terms of a diagrammatic reasoning system. First, existing diagrammatic reasoning systems, namely spider- and constraint diagrams, as well as existential and conceptual graphs, are investigated to determine if they are compatible with DLs. It turns out that Peirce's existential graphs are better suited for this purpose than the alternatives we examine. The paper then redevelops the DL ALC , which is the smallest propositional DL, by means of labeled trees, and provides a diagrammatic representation for these trees in the style of Peircean graphs. We provide a calculus based on C.S. Peirce's calculus for existential graphs and prove the soundness and completeness of the calculus. The calculus acts on labeled trees, but can be best understood as a diagrammatic calculus whose rules modify the Peircean-style representation of ALC .

Invariant tensors for the spin representation of (7)

AbstractWe give a graphical calculus for the invariant tensors of the eight dimensional spin representation of the quantum groupUq(B3). This leads to a finite confluent presentation of the centraliser algebras of the tensor powers of this representation and a construction of a cellular basis.

A rewriting calculus for cyclic higher-order term graphs

The Rewriting Calculus (ρ-calculus, for short) was introduced at the end of the 1990s and fully integrates term-rewriting and λ-calculus. The rewrite rules, acting as elaborated abstractions, their application and the structured results obtained are first class objects of the calculus. The evaluation mechanism, which is a generalisation of beta-reduction, relies strongly on term matching in various theories.In this paper we propose an extension of the ρ-calculus, called ρg-calculus, that handles structures with cycles and sharing rather than simple terms. This is obtained by using recursion constraints in addition to the standard ρ-calculus matching constraints, which leads to a term-graph representation in an equational style. Like in the ρ-calculus, the transformations are performed by explicit application of rewrite rules as first-class entities. The possibility of expressing sharing and cycles allows one to represent and compute over regular infinite entities.We show that the ρg-calculus, under suitable linearity conditions, is confluent. The proof of this result is quite elaborate, due to the non-termination of the system and the fact that ρg-calculus-terms are considered modulo an equational theory. We also show that the ρg-calculus is expressive enough to simulate first-order (equational) left-linear term-graph rewriting and α-calculus with explicit recursion (modelled using a letrec-like construct).

A definition of descendants at one point in graph calculus

We study the genus expansion of Barannikov-Kontsevich solutions of the WDVV equation. In terms of the related graph calculus we give a definition of descendants at one point and prove that this definition satisfies the topological recursion relations in genera 0, 1, and 2, string and dilaton equations, and the pull-back formula.

Reasoning with Graphs

In this paper we study the (positive) graph relational calculus. The basis for this calculus was introduced by S. Curtis and G. Lowe in 1996 and some variants, motivated by their applications to semantics of programs and foundations of mathematics, appear scattered in the literature. No proper treatment of these ideas as a logical system seems to have been presented. Here, we give a formal presentation of the system, with precise formulation of syntax, semantics, and derivation rules. We show that the set of rules is sound and complete for the valid inclusions, and prove a finite model result as well as decidability. We also prove that the graph relational language has the same expressive power as a first-order positive fragment (both languages define the same binary relations), so our calculus may be regarded as a notational variant of the positive existential first-order logic of binary relations. The graph calculus, however, has a playful aspect, with rules easier to grasp and use. This opens a wide range of applications which we illustrate by applying our calculus to the positive relational calculus (whose set of valid inclusions is not finitely axiomatizable), obtaining an algorithm for deciding the valid inclusions and equalities of the latter.

Graph Complexes in Deformation Quantization

Kontsevich's formality theorem and the consequent star-product formula rely on the construction of an $L_\infty$-morphism between the DGLA of polyvector fields and the DGLA of polydifferential operators. This construction uses a version of graphical calculus. In this article we present the details of this graphical calculus with emphasis on its algebraic features. It is a morphism of differential graded Lie algebras between the Kontsevich DGLA of admissible graphs and the Chevalley-Eilenberg DGLA of linear homomorphisms between polyvector fields and polydifferential operators. Kontsevich's proof of the formality morphism is reexamined in this light and an algebraic framework for discussing the tree-level reduction of Kontsevich's star-product is described.

A Rewriting Calculus for Cyclic Higher-order Term Graphs

Introduced at the end of the nineties, the Rewriting Calculus (ρ-calculus, for short) is a simple calculus that fully integrates term-rewriting and λ-calculus. The rewrite rules, acting as elaborated abstractions, their application and the obtained structured results are first class objects of the calculus. The evaluation mechanism, generalizing beta-reduction, strongly relies on term matching in various theories.In this paper we propose an extension of the ρ-calculus, handling graph like structures rather than simple terms. The transformations are performed by explicit application of rewrite rules as first class entities. The possibility of expressing sharing and cycles allows one to represent and compute over regular infinite entities.The calculus over terms is naturally generalized by using unification constraints in addition to the standard ρ-calculus matching constraints. This therefore provides us with the basics for a natural extension of an explicit substitution calculus to term graphs. Several examples illustrating the introduced concepts are given.

Feynman graph representation of the perturbation series for general functional measures

A representation of the perturbation series of a general functional measure is given in terms of generalized Feynman graphs and rules. The graphical calculus is applied to certain functional measures of Lévy type. A graphical notion of Wick ordering is introduced and is compared with orthogonal decompositions of the Wiener–Itô–Segal type. It is also shown that the linked cluster theorem for Feynman graphs extends to generalized Feynman graphs. We perturbatively prove existence of the thermodynamic limit for the free energy density and the moment functions. The results are applied to the gas of charged microscopic or mesoscopic particles—neutral in average—in d = 2 dimensions generating a static field φ with quadratic energy density giving rise to a pair interaction. The pressure function for this system is calculated up to fourth order. We also discuss the subtraction of logarithmically divergent self-energy terms for a gas of only one particle type by a local counterterm of first order.

Clover calculus for homology 3-spheres via basic algebraic topology

We present an alternative definition for the Goussarov--Habiro filtration of the Z-module freely generated by oriented integral homology 3-spheres, by means of Lagrangian-preserving homology handlebody replacements (LP-surgeries). Garoufalidis, Goussarov and Polyak proved that the graded space (G_n)_n associated to this filtration is generated by Jacobi diagrams. Here, we express elements associated to LP-surgeries as explicit combinations of these Jacobi diagrams in (G_n)_n. The obtained coefficient in front of a Jacobi diagram is computed like its weight system with respect to a Lie algebra equipped with a non-degenerate invariant bilinear form, where cup products in 3-manifolds play the role of the Lie bracket and the linking number replaces the invariant form. In particular, this article provides an algebraic version of the graphical clover calculus developed by Garoufalidis, Goussarov, Habiro and Polyak. This version induces splitting formulae for all finite type invariants of homology 3-spheres.

A diagrammatic approach to link invariants of finite degree

In [5] M. Polyak and O. Viro developed a graphical calculus of diagrammatic formulas for Vassiliev link invariants, and presented several explicit formulas for low degree invariants. M. Goussarov [2] proved that this arrow diagram calculus provides formulas for all Vassiliev knot invariants. The original note [5] contained no proofs, and it also contained some minor inaccuracies. This paper fills the gap in literature by presenting the material of [5] with all proofs and details, in a self-contained form. Furthermore, a compatible coalgebra structure, related to the connected sum of knots, is introduced on the algebra of based arrow diagrams with one circle.

Quantum Clifford Hopf gebra for quantum field theory

We give arguments for the necessity to employ quantum Clifford Hopf gebras in quantum field theory. The role of the antipode is examined, Feynman diagrams are re-interpreted as tangles of graphical calculus. Regularization due to the design of convolution Hopf gebras is given as a program for future research.

Quiver varieties and fusion products for [formula omitted

We construct the fusion product of finite-dimensional sl_2-modules in the homology of (or in the space of constructible functions on) a certain subvariety L_l(w_1, ..., w_r) of Nakajima's tensor product variety L(w_1,..., w_r). We also give a combinatorial description of the irreducible components of this subvariety using the notions of graphical calculus and crossingless matches for sl_2.

FEYNMAN DIAGRAMS VIA GRAPHICAL CALCULUS

We use Reshetikhin-Turaev graphical calculus to define Feynman diagrams and prove that asymptotic expansions of Gaussian integrals can be written as a sum over a suitable family of graphs. We discuss how different kinds of interactions give rise to different families of graphs. In particular, we show how symmetric and cyclic interactions lead to "ordinary" and "ribbon" graphs respectively. As an example, the 't Hooft-Kontsevich model for 2D quantum gravity is treated in some detail.

Topological Constructions in the o–Graph Calculus

Benedetti and Petronio developed a so called o–Graph Calculus, where a compact oriented 3–manifold with nonempty boundary could be described by a quadrivalent graph together with some extra structure. In this paper, we will show how topological constructions such as puncturing, connected sums, attaching handles, closing boundary components and product and mapping tori constructions can be translated into the o–graph calculus.