## Proofs Without Syntax

### Cached

### Download Links

Venue: | Annals of Mathematics |

Citations: | 6 - 0 self |

### BibTeX

@ARTICLE{Hughes_proofswithout,

author = {J. D. Hughes},

title = {Proofs Without Syntax},

journal = {Annals of Mathematics},

year = {},

pages = {2006}

}

### OpenURL

### Abstract

[M]athematicians care no more for logic than logicians for mathematics. Augustus de Morgan, 1868 Proofs are traditionally syntactic, inductively generated objects. This paper presents an abstract mathematical formulation of propositional calculus (propositional logic) in which proofs are combinatorial (graph-theoretic), rather than syntactic. It defines a combinatorial proof of a proposition φ as a graph homomorphism h: C → G(φ), where G(φ) is a graph associated with φ and C is a coloured graph. The main theorem is soundness and completeness: φ is true if and only if there exists a combinatorial proof h: C → G(φ). 1.

### Citations

567 |
Untersuchungen über das logische Schließen
- Gentzen
- 1935
(Show Context)
Citation Context ...ormal proof system [CR79] since correctness can be checked in polynomial time. 3 There is a polynomial-time computable function taking a propositional sequent calculus proof of φ with n ≥ 0 cut rules =-=[Gen35]-=- to a combinatorial proof of φ with n cuts: a combinatorial proof of φ ∨ (θ1 ∧ ¬θ1) ∨ · · · ∨ (θn ∧ ¬θn) for propositions θi. In the example of a combinatorial proof drawn in the Introduction, observe... |

508 |
Matching Theory
- Lovász, Plummer
- 1986
(Show Context)
Citation Context ...n C whose union induces a matching in C: for each CiCj ∈ M ′ \ M pick a colour class {v, w} with v ∈ V (Ci) and w ∈ V (Cj). Since G has a unique perfect matching, some CkCk+1 ∈ M is a bridge [Kot59], =-=[LP86]-=-, i.e., ( V (G), E(G) \ CkCk+1 )=X ∨ Y with Ck ∈ V (X) and Ck+1 ∈ V (Y ). Let W be the union of all colour classes of C coincident with any Ci in X, and let W ′ = V (C) \ W . Then C[W ] and C[W ′ ] ar... |

328 | The relative efficiency of propositional proof systems
- Cook, Reckhow
- 1979
(Show Context)
Citation Context ...ass is labelled appropriately, and (3) h is a skew fibration, a lax form of graph fibration. Each condition can be checked in polynomial time, so combinatorial proofs constitute a formal proof system =-=[CR79]-=-. Acknowledgements. Nil Demirçubuk, Vaughan Pratt, Julien Basch, Rajat Bhattacharjee, Rob van Glabbeek, Sol Feferman, Grisha Mints and Don Knuth. Stanford grant 1DMA644. 2 Notation and terminology Gra... |

291 |
Graph classes: a survey
- Brandstädt, Le, et al.
- 1999
(Show Context)
Citation Context ...h with its path category). 3 The skew fibration and axiomatic conditions are clearly polynomial. Checking that a graph G is a cograph is polynomial by constructing its modular decomposition tree T(G) =-=[BLS99]-=-, and checking that G is nicely coloured is a simple breadth-first search on T(G).s5 Proof of Theorem 2 The diagram right shows the dependency between the Lemmas (1–9) and Theorems (T1–T4) in this pap... |

210 |
Closed categories
- Eilenberg, Kelly
- 1966
(Show Context)
Citation Context ...ely pairing an occurrence of a variable p with an occurrence of its dual p. The idea of pairing dual variable occurrences has arisen in the study of various forms of syntax, such as closed categories =-=[KM71]-=-, contraction-free predicate calculus [KW84] and linear logic [Gir87]. Combinatorial proofs relate only superficially to the connection/matrix method [Dav71, Bib74, And81]; the latter fails to provide... |

160 |
A new constructive logic: Classical Logic
- Girard
- 1991
(Show Context)
Citation Context ... connection/matrix method [Dav71, Bib74, And81]; the latter fails to provide a proof system [CR79]. A partially combinatorial notion of proof for classical logic, called a proof net, was presented in =-=[Gir91]-=-, though promptly dismissed by the author as overly syntactic: a proof net of a proposition φ has an underlying syntax tree containing not only ∧’s and ∨’s from φ, but also auxiliary syntactic connect... |

146 |
Elements of Homotopy Theory
- Whitehead
- 1978
(Show Context)
Citation Context ... E) and (V ′ , E ′ ) are isomorphic if there exists a bijection h : V → V ′ with vw ∈ E iff h(v)h(w) ∈ E ′ . 2 This is simply a convenient restatement of the familiar notions of fibration in topology =-=[Whi78]-=- and category theory [Gro59, Gra66]: a graph homomorphism is a graph fibration iff it satisfies the homotopy lifting property (when viewed as a continuous map by identifying each edge with a copy of t... |

120 |
Complement reducible graphs
- Corneil, Lerchs, et al.
- 1981
(Show Context)
Citation Context ...sjoint, the union G ∨ G ′ is (V ∪ V ′ , E ∪ E ′ ) and the join G∧G ′ is (V ∪V ′ , E ∪ E ′ ∪ {vv ′ : v ∈V, v ′ ∈V ′ }); colourings or labellings of G and G ′ are inherited. A graph (V, E) is a cograph =-=[CLS81]-=- if V is non-empty and for any distinct v, w, x, y ∈ V , the restriction of E to edges on {v, w, x, y} is not {vw, wx, xy}. A set W ⊆ V induces a matching if it is non-empty and for all w ∈W there is ... |

113 |
Theorem proving via general matings
- Andrews
- 1981
(Show Context)
Citation Context ...such as closed categories [KM71], contraction-free predicate calculus [KW84] and linear logic [Gir87]. Combinatorial proofs relate only superficially to the connection/matrix method [Dav71], [Bib74], =-=[And81]-=-; the latter fails to provide a proof system [CR79]. A partially combinatorial notion of proof for classical logic, called a proof net, was presented in [Gir91], though promptly dismissed by the autho... |

112 | Collected papers - Peirce - 1933 |

97 |
Technique de descente et théorème d’existence en géométrie algébrique
- Grothendieck
(Show Context)
Citation Context ...f there exists a bijection h : V → V with vw ∈ E if and only if h(v)h(w) ∈ E ′ . 2 This is simply a convenient restatement of the familiar notions of fibration in topology [Whi78] and category theory =-=[Gro59]-=-, [Gra66]: a graph homomorphism is a graph fibration if and only if it satisfies the homotopy lifting property (when viewed as a continuous map by identifying each edge with a copy of the unit interva... |

74 |
eine der arithmetischen nachgebildete Formelsprache des reinen Denkens
- Frege
- 1967
(Show Context)
Citation Context ...thematics undergraduates [Hil28], [Joh87]. Although the system itself is elegant (just three axiom schemata suffice), the syntactic proofs generated in it need not be. Other syntactic systems include =-=[Fr1879]-=-, [Gen35]. This paper presents an abstract mathematical formulation of propositional calculus (propositional logic) in which proofs are combinatorial (graphtheoretic), rather than syntactic. It define... |

50 | From Frege to Gödel. A Source Book - Heijenoort - 1967 |

29 | A decidable fragment of predicate calculus
- Ketonen
- 1984
(Show Context)
Citation Context ...th an occurrence of its dual p. The idea of pairing dual variable occurrences has arisen in the study of various forms of syntax, such as closed categories [KM71], contraction-free predicate calculus =-=[KW84]-=- and linear logic [Gir87]. Combinatorial proofs relate only superficially to the connection/matrix method [Dav71, Bib74, And81]; the latter fails to provide a proof system [CR79]. A partially combinat... |

28 | Fibred and cofibred categories - Gray - 1965 |

27 | Extremal Graph Theory - Bollabás - 2004 |

25 | Fibrations of graphs
- Boldi, Vigna
(Show Context)
Citation Context ...on of propositions is due to Peirce [Pei58, vol. 4:2], dating from the late 1800s. A skew fibration is a lax notion of graph fibration. A graph homomorphism h : G → G ′ is a graph fibration (see e.g. =-=[BV02]-=-) if for all v ∈ V (G) and h(v)w ∈ E(G ′ ) there is a unique v�w ∈ E(G) with h(�w) = w . 2 The definition of skew fibration drops uniqueness and relaxes h(�w)=w to ‘skewness’ h(�w)w �∈ E(G ′ ). Combin... |

23 |
Neubegründung der Mathematik”, Abhandlungen des Mathematischen Seminars Universität Hamburg 1, 157-177; English trans., “The New Grounding of Mathematics”, in Ewald
- Hilbert
- 1922
(Show Context)
Citation Context ...icians accustomed to beautiful symmetries, geometries, and rich layers of structure. Figure 1 (see Section 2) shows a syntactic proof in a standard Hilbert system taught to mathematics undergraduates =-=[Hil28]-=-, [Joh87]. Although the system itself is elegant (just three axiom schemata suffice), the syntactic proofs generated in it need not be. Other syntactic systems include [Fr1879], [Gen35]. This paper pr... |

22 |
Notes on Logic and Set Theory
- Johnstone
- 1987
(Show Context)
Citation Context ...p ⇒ q) ⇒ p) ⇒ p in a standard Hilbert system Figure 1 Below is a proof of Peirce’s law ((p ⇒ q) ⇒ p) ⇒ p in a standard Hilbert formulation of propositional logic, taught to mathematics undergraduates =-=[Joh87]-=-, with axiom schemata (a) x ⇒ (y ⇒ x) (b) (x ⇒ (y ⇒ z)) ⇒ ((x ⇒ y) ⇒ (x ⇒ z)) (c) ((x ⇒ ⊥) ⇒ ⊥) ⇒ x and where (m i j) marks modus ponens with hypotheses numbered i and j. Hilbert systems tend to empha... |

15 |
Handsome proof-nets: perfect matchings and cographs
- Retoré
(Show Context)
Citation Context ... syntactic connectives which are not even boolean operations (contraction and weakening). Nicely coloured cographs with two vertices in every colour class correspond to unlabelled chorded R&Bcographs =-=[Ret03]-=-. When labelled, the latter represent proof nets of mixed multiplicative linear logic [Gir87]. 4 Combinatorial propositions and truth A set W ⊆ V(G) is stable if vw �∈ E(G) for all v,w ∈ W. A clause i... |

10 | Heijenoort (ed.), From Frege to Gödel - A source book in mathematical logic, 1879–1931, 3rd printing ed - van - 1977 |

5 |
An approach to a systematic theorem proving procedure in first-order logic, Computing 12
- Bibel
- 1974
(Show Context)
Citation Context ... syntax, such as closed categories [KM71], contraction-free predicate calculus [KW84] and linear logic [Gir87]. Combinatorial proofs relate only superficially to the connection/matrix method [Dav71], =-=[Bib74]-=-, [And81]; the latter fails to provide a proof system [CR79]. A partially combinatorial notion of proof for classical logic, called a proof net, was presented in [Gir91], though promptly dismissed by ... |

5 |
On the theory of finite graphs with a linear factor
- Kotzig
- 1959
(Show Context)
Citation Context ...classes in C whose union induces a matching in C: for each CiCj ∈ M ′ \ M pick a colour class {v, w} with v ∈ V (Ci) and w ∈ V (Cj). Since G has a unique perfect matching, some CkCk+1 ∈ M is a bridge =-=[Kot59]-=-, [LP86], i.e., ( V (G), E(G) \ CkCk+1 )=X ∨ Y with Ck ∈ V (X) and Ck+1 ∈ V (Y ). Let W be the union of all colour classes of C coincident with any Ci in X, and let W ′ = V (C) \ W . Then C[W ] and C[... |

2 |
The synthesis of the resolution method and the inverse method, Zapiski Nauchnykh Seminarov Lomi 20
- Davydov
- 1971
(Show Context)
Citation Context ... forms of syntax, such as closed categories [KM71], contraction-free predicate calculus [KW84] and linear logic [Gir87]. Combinatorial proofs relate only superficially to the connection/matrix method =-=[Dav71]-=-, [Bib74], [And81]; the latter fails to provide a proof system [CR79]. A partially combinatorial notion of proof for classical logic, called a proof net, was presented in [Gir91], though promptly dism... |

2 | The Collected Papers of Gerhard - Gentzen - 1969 |

1 |
Fibred and cofibred categories, inProc
- Gray
- 1966
(Show Context)
Citation Context ...xists a bijection h : V → V with vw ∈ E if and only if h(v)h(w) ∈ E ′ . 2 This is simply a convenient restatement of the familiar notions of fibration in topology [Whi78] and category theory [Gro59], =-=[Gra66]-=-: a graph homomorphism is a graph fibration if and only if it satisfies the homotopy lifting property (when viewed as a continuous map by identifying each edge with a copy of the unit interval) if and... |

1 | Modern Graph Theory. Springer 2002. [BLS99 - BOLLABÁS - 1999 |

1 | Edited and introduced by - North-Holland - 1969 |

1 | A new constructive logic: Classical logic - Sci - 1987 |

1 | Casopis 9 & 10 - Fiz - 1959 |

1 | Algebra 1 1971 97–140; erratum, ibid - Appl |

1 | The Athenaeum 2 1868 - Hartshorne, Weiss, et al. |