## Polygraphic programs and polynomial-time functions

Citations: | 5 - 0 self |

### BibTeX

@MISC{Bonfante_polygraphicprograms,

author = {Guillaume Bonfante and Inria Loria and Yves Guiraud and Inria Loria},

title = {Polygraphic programs and polynomial-time functions},

year = {}

}

### OpenURL

### Abstract

Abstract – We study the computational model of polygraphs. For that, we consider polygraphic programs, a subclass of these objects, as a formal description of first-order functional programs. We explain their semantics and prove that they form a Turing-complete computational model. Their algebraic structure is used by analysis tools, called polygraphic interpretations, for complexity analysis. In particular, we delineate a subclass of polygraphic programs that compute exactly the functions that are Turingcomputable in polynomial time.

### Citations

1372 |
Quantum Computation and Quantum Information
- Nielsen, Chuang
- 2000
(Show Context)
Citation Context ...ne should be able to compute, for example, algebraic cooperations, such as the ones found in Jean-Louis Loday’s generalized bigebras [33], or automorphisms of C n , such as the universal Deutsch gate =-=[37]-=- of quantum circuits. Going further, at this step, there will be no reason anymore to consider constructor 2-cells with one output only or values with no output. This way, one could consider algorithm... |

564 |
A Decision Method for Elementary Algebra and Geometry
- Tarski
- 1948
(Show Context)
Citation Context ...ead of additive ones. However, some studies are much more promising. First, to turn to polynomials over reals give some procedures to build interpretations (see [11]) via Alfred Tarski’s decidability =-=[41]-=-. Second, we plan to consider differential interpretations with values in multisets (instead of natural numbers), to characterize polynomial-space computations. 35sREFERENCES For each generalisation o... |

528 |
Theory of Self-reproducing Automata
- Neumann
- 1966
(Show Context)
Citation Context ...cuits: instead of computing on syntactical terms, polygraphs make use of a net of cells, which individually behave according to some local transition rules, as do John von Neumann’s cellular automata =-=[43]-=- and Yves Lafont’s interaction nets [26]. Following Niel Jones’ thesis that programming languages and semantics have strong connexions with complexity theory [24], we think that the syntactic features... |

163 |
Interaction nets
- Lafont
- 1990
(Show Context)
Citation Context ...al terms, polygraphs make use of a net of cells, which individually behave according to some local transition rules, as do John von Neumann’s cellular automata [43] and Yves Lafont’s interaction nets =-=[26]-=-. Following Niel Jones’ thesis that programming languages and semantics have strong connexions with complexity theory [24], we think that the syntactic features offered by polygraphs, with respect to ... |

140 |
P.J.: Introduction to Higher-Order Categorical Logic
- Lambek, Scott
- 1986
(Show Context)
Citation Context ...the polygraphic setting. Removing duplication and erasure from the standard definition means that one moves from a cartesian setting to a monoidal one. According to a variant of André Joyal’s paradox =-=[29]-=-, this is necessary to describe functions such as linear maps on finite-dimensional vector spaces. Thus, one should be able to compute, for example, algebraic cooperations, such as the ones found in J... |

114 |
On proving term rewrite systems are noetherian
- Lankford
- 1979
(Show Context)
Citation Context ...m. Complexity analysis of polygraphic programs – Here we use tools inspired by polynomial interpretations, which have been introduced by Dallas Lankford to prove termination of term rewriting systems =-=[30]-=-. They associate to each term a polynomial with natural numbers as coefficients, in a way that is naturally compatible with contexts and substitutions. When, for each rule, the interpretation of the l... |

38 |
Lambda calculus characterizations of poly-time
- Leivant, Marion
(Show Context)
Citation Context ..., in the field of implicit computational complexity, the notion of stratification has shown to be a fundamental tool of the discipline. This has been developped by Daniel Leivant and Jean-Yves Marion =-=[31, 32]-=- and by Stephen Bellantoni and Stephen Cook [6] to delineate FPTIME. Other characterizations include Neil Jones’ "Life without cons" WHILE programs [25] and Karl-Heinz Niggl and Henning Wunderlich’s c... |

34 |
Limits indexed by category-valued 2-functors
- Street
- 1976
(Show Context)
Citation Context ...er 19, 2006. Last modification – July 15, 2008. Polygraphs as a computational model – Polygraphs (or computads) are presentations by "generators" and "relations" of some higher-dimensional categories =-=[41, 12, 42, 43]-=-. Albert Burroni has proved that they provide an algebraic structure to equational theories [12]. Yves Lafont and the second author have explored some of the computational properties of these objects,... |

23 | What is an algorithm
- Moschovakis
- 2001
(Show Context)
Citation Context ...algorithm (up to the test ≤ on the natural numbers p and q) really mimics the "mechanics" of the fusion sort algorithm and, actually, we rediscover the complexity bound as given by Yannis Moschovakis =-=[36]-=-. Why don’t we internalize the comparision of numbers within the polygraphic program? This comes from the fact that the if-then-else construction implicitely involves an evaluation strategy: one first... |

21 | Resource analysis by sup-interpretation
- Marion, Péchoux
(Show Context)
Citation Context ...gly conjectured that the preliminary results developped in this paper can be used for other characterisations. In particular, the current interpretations can be seen as sup-interpretations, following =-=[35]-=-: this means that values have polynomial size. Coming back to FPTIME, in the field of implicit computational complexity, the notion of stratification has shown to be a fundamental tool of the discipli... |

21 |
Certifying Polynomial Time and Linear/Polynomial Space for Imperative Programs
- Niggl, Wunderlich
(Show Context)
Citation Context ...Cook [6] to delineate FPTIME. Other characterizations include Neil Jones’ "Life without cons" WHILE programs [25] and Karl-Heinz Niggl and Henning Wunderlich’s characterization of imperative programs =-=[38]-=-. There is also a logical approach to implicit computational complexity, based on a linear type discipline, in the seminal works of Jean-Yves Girard on light linear logic [16], Yves Lafont on soft lin... |

13 |
Term graph rewriting. Handbook of graph grammars and computing by graph transformation: vol. 2: applications, languages, and tools
- Plump
- 1999
(Show Context)
Citation Context ...n the polygraphic language: ⇛ ⇛ With these rules, one can actually "see" how the computation is made, by "unzipping" lists. Also, one can internalize in polygraphs the sharing operation of termgraphs =-=[39]-=-, described as an explicit and local duplication. As a consequence, the rules generating computations become linear: the operations for pointers management can be "seen" within the rules. Actually, in... |

9 |
A finiteness condition for rewriting systems, Theoret
- Squier, Otto, et al.
- 1994
(Show Context)
Citation Context ...riginal definition [18]. As pointed earlier, polygraphs are higher dimensional-categories. Philippe Malbos and the second author are currently adapting the finite derivation criterion of Craig Squier =-=[40]-=- to them, as was done before for 1-categories [34]. We think that this will lead us a computable sufficient condition to ensure that a function does not admit a finite, convergent polygraphic program ... |

7 |
an algebraic theory of boolean circuits
- Towards
(Show Context)
Citation Context ...tional theories [12]. Yves Lafont and the second author have explored some of the computational properties of these objects, mainly termination, confluence and their links with term rewriting systems =-=[27, 18]-=-. The present study, extending notions and results presented earlier by the same authors [9], concerns the complexity analysis of polygraphs. On a first approach, one can think of these objects as rew... |

6 |
Generalized bialgebras and triples of operads, preprint
- Loday
- 2006
(Show Context)
Citation Context ...rk. Polygraphs provide a uniform, algebraic and graphical description of objects coming from different domains: abstract, string and term rewriting systems [27, 17, 18], abstract algebraic structures =-=[12, 17, 33]-=-, Feynman and Penrose diagrams [4], braids, knots and tangles diagrams equipped with the Reidemeister moves [1, 17], Petri nets [20] and propositional proofs of classical and linear logics [19]. 1.2 P... |

3 |
linear logic and polynomial time, Theoretical Computer Science 318
- Soft
- 2004
(Show Context)
Citation Context ... also a logical approach to implicit computational complexity, based on a linear type discipline, in the seminal works of Jean-Yves Girard on light linear logic [16], Yves Lafont on soft linear logic =-=[28]-=- or Patrick Baillot and Kazushige Terui [5]. The second part of this document is devoted to general results about polygraphic interpretations of polygraphs. There, we explore the pieces of information... |

3 |
A foundational delineation of computational feasability
- Leivant
- 1991
(Show Context)
Citation Context ..., in the field of implicit computational complexity, the notion of stratification has shown to be a fundamental tool of the discipline. This has been developped by Daniel Leivant and Jean-Yves Marion =-=[31, 32]-=- and by Stephen Bellantoni and Stephen Cook [6] to delineate FPTIME. Other characterizations include Neil Jones’ "Life without cons" WHILE programs [25] and Karl-Heinz Niggl and Henning Wunderlich’s c... |

3 |
categories, strings, cubes and simplex equations
- Higher
- 1995
(Show Context)
Citation Context ...er 19, 2006. Last modification – July 15, 2008. Polygraphs as a computational model – Polygraphs (or computads) are presentations by "generators" and "relations" of some higher-dimensional categories =-=[41, 12, 42, 43]-=-. Albert Burroni has proved that they provide an algebraic structure to equational theories [12]. Yves Lafont and the second author have explored some of the computational properties of these objects,... |

2 |
For string rewriting systems the homotopical and homological finiteness conditions coincide, preprint
- Malbos
- 2007
(Show Context)
Citation Context ...raphs are higher dimensional-categories. Philippe Malbos and the second author are currently adapting the finite derivation criterion of Craig Squier [40] to them, as was done before for 1-categories =-=[34]-=-. We think that this will lead us a computable sufficient condition to ensure that a function does not admit a finite, convergent polygraphic program that computes it. The same collaboration has more ... |