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Fuzzy ChemistryAn Axiomatic Theory for General Chemistry
P. Amato(1), G. F. Cerofolini(2), and D. Narducci(2)
(1): DISCO(2): Scienze dei Materiali
January 8, 2009
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Introduction
Outline
1 IntroductionThe matter and its descriptionGeneral Chemistry (GC)
2 Formal chemistry (FC): Underlying ideasMolecules and graphsLoose metricReactionsLogical Framework
3 The axiomsStick-and-ballMolecular graphs
4 The natural interpretation5 Examples6 Conclusions
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Introduction The matter and its description
The Physical World: Hierarchical organization
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Introduction The matter and its description
Explanation and reduction
Different scientific theories are designed for explaining differentlevels of matterEach theory specializes in explaining its own particular set ofphenomenaObject studied at one level are composed of objects studied atlower levels
Does this mean that, in principle, the lowest level theory can subsumeal the higher-level science?
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Introduction The matter and its description
Reductionism and Emergence
the properties of a level are the statistical properties of the lowerlevel: hierarchythere exist properties which cannot be derived as statisticalproperties of the lower level: holism
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Introduction The matter and its description
The Physical World: Hierarchical description
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Introduction General Chemistry (GC)
General chemistry: the missing formal theory
chemistry may be viewed as a special chapter of physics devotedto the description of atomic assemblies (molecules, radicals,. . . )in terms of their constituting nuclei and electrons in mutualinteractionIt is generally believed that a description of molecules in terms ofproperties of their atomic constituents (General Chemistry (GC))is impossible.
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Introduction General Chemistry (GC)
GC basic fact
molecules are described as networks of atoms linked by bonds:stick-and-ball modelThough GC speaks using terms taken from physics (like atomicmass, bond length and energy, or formal charge, just to name afew), the properties of its fundamental characters (atoms andbonds) are autonomously postulated rather than being derivedfrom physicsGC has long reach and large predictive power, even using thelimited mathematical apparatus of college algebra
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Introduction General Chemistry (GC)
Relevance of GC formalization
Metatheoreticalto remove an anomaly in the hierarchy of description
PracticalIn its history, modern Chemistry went through two ages:
1 before quantum mechanics: concepts and rules defined bydescription and induction, based on empirical observation
2 after quantum mechanics: reductionistic approach, chemicalproperties explained by deduction from general principles.
Still Chemistry retains most of the concept that were developed beforeBohr’s atoms
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Introduction General Chemistry (GC)
Atoms in the hierarchical description of matter
Fortunately, many important properties of objects containing verymany atoms-such as the boiling and freezing of water-can be obtainedfrom simplified models of the structure of atoms and the lawsgoverning their interactions1.
1J. L. Lebowitz, Statistical mechanics: A selective review of two central issues, Rev.Mod. Phys. 71, 2 (1999) centenary issue
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Introduction General Chemistry (GC)
What is an atom in a molecule?
The derivation of the Hirshfeld atoms in molecules from informationtheory is clarified. The importance for chemistry of the concept ofatoms in molecules (AIM) is stressed, and it is argued that thisconcept, while highly useful, constitutes a noumenon in the sense ofKant2.
2Robert G. Parr, Paul W. Ayers, and Roman F. Nalewajski, What Is an Atom in aMolecule? J. Phys. Chem. A, 109 (17), 3957 -3959 ( 2005)
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Introduction General Chemistry (GC)
Non-reductionist Chemistry: Formal chemistry (FC)
Aimto recover and to dignify these concepts, which are still in their infancyfrom the viewpoint of their formalization.
ApproachTo develop an axiomatic theory Formal chemistry (FC): combining
Formal logicGraph theoryFuzzy sets theory
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Introduction General Chemistry (GC)
Hierarchical description: FC
atoms
molecules
phases
systems
Statistical Mechanics
Thermodynanics
Formal Chemistry
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Formal chemistry (FC): Underlying ideas
Outline
1 IntroductionThe matter and its descriptionGeneral Chemistry (GC)
2 Formal chemistry (FC): Underlying ideasMolecules and graphsLoose metricReactionsLogical Framework
3 The axiomsStick-and-ballMolecular graphs
4 The natural interpretation5 Examples6 Conclusions
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Formal chemistry (FC): Underlying ideas
Underlying ideas
(to some extent) a molecule may be regarded as a graphthe intrinsical uncertainty of GC can be included in a rigoroustheory by using characters taken from fuzzy logic
introduction of a “loose metric” for the description of atomic andmolecular properties
Chemical reactions preserve atoms but not bonds
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Formal chemistry (FC): Underlying ideas Molecules and graphs
Molecules as graphs
atoms: verticesbonds: edges
Figure: The molecule CH4 as graph (V ,E)
Note: V = {C,H,H,H,H}, and E = {C−H,C−H,C−H,C−H}. V and Eare bags (or multisets).
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Formal chemistry (FC): Underlying ideas Molecules and graphs
A molecules is more than a graph
A molecule, however, is in general much more complicated thana graph
vertex: intrinsic physical properties (like mass, self-energy, radius,electronegativity and group of belonging)edge: geometric and energetic properties (bond length and bondenergy)each vertex has a formal charge governed by the number ofbonds afferent to it
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Formal chemistry (FC): Underlying ideas Molecules and graphs
Example: H2O molecule
Figure: Water molecule: space-filled model (left), dimensions and geometricstructure (right)
Source: Water. (2008, December 20). In Wikipedia, The Free Encyclopedia. Retrieved 08:56, December 24, 2008, from
http://en.wikipedia.org/w/index.php?title=Water&oldid=259124804
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Formal chemistry (FC): Underlying ideas Loose metric
A (classical) probabilistic view is not suitable for thefoundation of general chemistry: an example
Example:character of theory: bond length l(a,b)
observable: internuclear distance d(a,b)
IssueIn general
|l(a,b)− d(a,b)| � measure error
Probabilistic solutiondefine l(a,b) as: E [all possible d(a,b)]but what “all possible” does mean?
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Formal chemistry (FC): Underlying ideas Loose metric
A fuzzy view is suitable for the foundation of generalchemistry
Rather than thinking of the characters of FC as real numbers, it issufficient to think of them as fuzzy numbers3
3after quantum mechanics we should be ready to accept any even morecomplicate assignation, provided it works.
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Formal chemistry (FC): Underlying ideas Loose metric
Fuzzy as a high theory
Fuzzy theory allows the separation axiom scheme of set theory tobe extended to predicates to which a precise truth value cannot beassignedIn this view, fuzzy theory is certainly a “high” theory allowing theextension of classical logic to fuzzy logicIn spite of that, however, fuzzy theory has had really few highapplications, its use being usually limited to cases where theuncertainty is due to a contingent poor knowledge of the situationunder analysis (think, for instance, of the fuzzy control of awashing machine).
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Formal chemistry (FC): Underlying ideas Reactions
Chemical reactions
Ultimate goal of chemistry:succeeding in elucidating notonly the properties ofmolecules, but also theirevolutionChemical evolution isconstrained by conservation ofmatter: the transformation of asubstance into another impliesthe consumption of the formerand the formation of the latter
Figure: The reactionCH4 + 2 O2 −−→ CO2 + 2 H2O
image source: http://www.chemistryland.com/
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Formal chemistry (FC): Underlying ideas Reactions
Linear Logic
Chemical evolution (the state transition associated with chemicalreaction) and finite resources can be tackled by using the features oflinear logic operators:
CH4 ⊗ O2 ⊗ O2 ( CO2 ⊗ H2O⊗ H2O,
where, use ⊗ (times) is the linear conjunction and ( (“lollipop”) thelinear implication
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Formal chemistry (FC): Underlying ideas Logical Framework
FC theory: formal framework
SyntaxMany-sorted linear predicate logic
Two sortsGraphs (sort GR)Numbers (sort NU)
Linear logiclinear conjunction ⊗ (times)linear implication ( (“lollipop”).
SemanticsNatural interpretation
(sub)set of elements of sort GR: moleculeselements of sort NU: fuzzy numbers
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The axioms
Outline
1 IntroductionThe matter and its descriptionGeneral Chemistry (GC)
2 Formal chemistry (FC): Underlying ideasMolecules and graphsLoose metricReactionsLogical Framework
3 The axiomsStick-and-ballMolecular graphs
4 The natural interpretation5 Examples6 Conclusions
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The axioms
Symbols
Besides the usual logical symbols4 (equality, conjunction,. . . ), FCrequires the following non-logical symbols5
N constant symbols c1, . . . , cN of sort GR (balls).The functional symbols e, l ,m, r , ξ, ζ,E ,M (GR 7→NU)The functional symbols Φ (NU 7→NU)The binary functional symbols e, l (GR×GR 7→NU)The predicate symbols Mol and ! (sort GR)All the needed constants (0,1) and functional symbols (+, . . .) forsort NU
4Here we consider linear logic as an extension of classical logic, and in this paperwe use both the symbols of usual and linear logics (each with its meaning).
5The concepts of graph, vertex, link, distance and angle are taken frommathematics and do not need to be defined in formal chemistry.
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The axioms Stick-and-ball
The sort GR
Definition (sort GR)Any 3-dimensional connected graph τ = (V ,E) where:
1 V is a bag of balls (constants of sort GR)2 each vertex is labeled by the value z ∈ N, referred to as formal
charge.3 each edge is labeled by the value n ∈ N, referred to as multiplicity.
is an element of sort GR.NOTE: from now on τ = (B(τ),S(τ)).
Definition (Sticks)A stick is an edge (unordered couple) defined on constants of sort GR
(balls).
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The axioms Stick-and-ball
Representing Balls
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The axioms Stick-and-ball
Representing Balls: a 3D view
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The axioms Stick-and-ball
Axiom (Four-dimensional ordering of balls)
The balls (costants of sort GR) of formal chemistry are arranged in a4-dimensional ordering. This order relation is described by quadruples(P,G,D,F )a of integer numbers with the following domains:
P = 1, · · · ,7 :P = 1 ⇒ D = 0 ∧ F = 0 ∧G = 1,8
P = 2,3 ⇒ D = 0 ∧ F = 0 ∧G = 1, · · · ,8
P = 4,5 ⇒{
F = 0 ∧ D = 0 =⇒ G = 1, · · · ,8F = 0 ∧G = 0⇒ D = 1, · · · ,10
P = 6,7 =⇒
(F = 0 ∧ D = 0)⇒ G = 1, · · · ,8F = 0 ∧G = 0⇒ D = 1, · · · ,10D = 2 ∧G = 0⇒ F = 1, · · · ,14
aIn chemistry the variable P is known as period, the variables G and D are knownas main group and transition group, respectively. The quadruples with G = 0 arecalled transition atoms; those with P = 6 and F 6= 0 form the lantanide series whereasthose with P = 7 and and F 6= 0 form the actinide series.
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The axioms Stick-and-ball
Definition (“Ballic” number)
Consider an arrangement of balls (constants of sort GR) in order ofincreasing P, inside each P of increasing G, with the insertion ofconstants with the ordering imposed by D in between G = 2 andG = 3, with the ordering imposed by F in between D = 1 and D = 2 forP = 6 and of the constants with the ordering imposed by F in betweenD = 1 and D = 2 for P = 7. The natural number giving the ordinal ofeach ball in this sequence is called the ballica number Z .
a“Ballic” is a neologism coined by the authors. Its meaning will be clarified atsemantic level.
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The axioms Stick-and-ball
4D ordering and ballic numbers
Figure: The 4-dimensional order (the P,G,D,F outside the table) and theballic numbers (the numbers inside the table) of the constants of sort GR,where N = 118
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The axioms Stick-and-ball
Balls: Physical properties
Axiom (Balls)
The balls are hard massive three-dimensional spheres, with radius(with functional symbol r ), mass (with functional symbol m), selfenergy(with functional symbol e) and electronegativity (with functional symbolξ) such that:
∀b (r(b) > 0 ∧ m(b) > 0 ∧ e(b) ≥ 0 ∧ ξ(b) > 0).
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The axioms Stick-and-ball
Sticks
Since each stick s is uniquely identified by a couple {a,b} of balls, inprinciple the functions on sticks take two arguments. The followingaxiom aims at reducing these functions to functions with just oneargument
Axiom (Sticks)
Let s = {a,b}
l(s) = r(a) + r(b) (stick length)e(s) = e(a) + e(b) + Φ(|ξ(a)− ξ(b)|) (stick energy)
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The axioms Molecular graphs
Imposing constraints on graphs
Graps have topological propertiesIn FC the metric properties do matter.To introduce the notion of molecular graph we need to define twokind of properties (topological and metric) that the graph termshave to satisfy.
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The axioms Molecular graphs
Definition (Topologically sound)A graph term is topologically sound if for any vertex b = (P,G,D,F )(ball) the following relationships are satisfied:
G − z + n − 8 = 0 if D = 0,F = 0 and G ≥ 3 (octet rule)G − z − n = 0 if D = 0,F = 0 and G ≤ 3 (group-valency rule)
where n denote the number of sticks (each counted for its multiplicity)connected to b, and z its formal charge.
Figure: Circles are balls and lines are sticks. The number iz inside a circleindicates the ball with ballic number i and formal charge z. The number overan edge is its multiplicity.
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The axioms Molecular graphs
Definition (Metrically sound)A graph term τ is metrically sound if there exist a 3D spatialarrangement such that:
for any stick {a,b} in S(τ) the distance between the centers of thespheres a and b is l(a,b)
for any couple of balls a,b in B(τ), the intersection of the spheresa and b is null.
Figure: A topologically and metrically sound graph. r(b1) = 25 pm,r(b8) = 60 pm
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The axioms Molecular graphs
The graphs relevant for formal chemistry
Axiom (Molecular graphs: predicate Mol)For any graph term τ , the predicate Mol(τ) is true if and only if τ istopologically and metrically sound. Each graph term τ such thatMol(τ) is true, is referred to as molecular graph. The set of allmolecular graphs, i.e. {τ : Mol(τ)}, is referred to as Γ set
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The axioms Molecular graphs
Axiom (Molecular graph properties)
For all γ ∈ Γ
M(γ) =∑
b∈B(γ)
m(b) (m.g. mass)
ζ(γ) =∑
b∈B(γ)
zb (m.g. charge)
E(γ) =∑
s∈S(γ)
e(s), (m.g. energy)
where γ = (B(γ),S(γ)) and zb is the formal charge of the ball b.
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The axioms Molecular graphs
Axiom (Allowed transition (!) predicate)Let L,R be bags of molecular graphs
L ! R ⇐⇒
∑γ∈L
B(γ) =∑γ′∈R
B(γ′) ∧∑γ∈L
ζ(γ) =∑γ′∈R
ζ(γ′)
The predicate {γ1, γ2}! γ3 is true when the “chemical” reaction thattransforms γ1 and γ2 in γ3 is possible. In the model this property isusually stated that in a chemical reaction matter (the bags of balls B)and charge (the formal charges zb) are conserved. On the contrary, ingeneral, bonds are not conserved in chemical reaction.
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The axioms Molecular graphs
Figure: Example of allowed transition: H2 + F2 −−→ 2 HF.
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The axioms Molecular graphs
Definition (Energy change (∆E))
Let L,R be bags of molecular graphs such that L ! R. Then:
∆E ,∑γ′∈R
E(γ′)−∑γ∈L
E(γ).
Axiom (Molecular graphs dynamics)
Let L,R be bags of molecular graphs such that F ! F ′
∆E > 0 =⇒
⊗γ∈L
React(γ) (⊗γ′∈R
Form(γ′)
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The natural interpretation
Outline
1 IntroductionThe matter and its descriptionGeneral Chemistry (GC)
2 Formal chemistry (FC): Underlying ideasMolecules and graphsLoose metricReactionsLogical Framework
3 The axiomsStick-and-ballMolecular graphs
4 The natural interpretation5 Examples6 Conclusions
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The natural interpretation
Models for FC
A model U for the two-sorted language of FC is an ordered triple〈G ,N , I〉, where G and N are nonempty domains, and I is andinterpretation function with domain the set of all constants, functionsand relations of the language such that:
if b is a ball (constant symbol of sort GR), then I(b) ∈ G
if n is a constant symbol of sort NU, then I(n) ∈ N
if f (δ=1...δm) is an m-placed function symbol, where δi ∈ {GR,NU},then I(f ) is an m-placed function whose i-th argument belongs toG (if δi =GR) or to N (if δi =NU)if R(δ1...δn) is an n-placed relation symbol, where δi ∈ {GR,NU},then I(R) is an n-placed relation whose i-th argument belongs toG (if δi =GR) or to N (if δi =NU)
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The natural interpretation
Natural interpretation
G is the set of adducts (i.e., any combination of bound atoms)the molecular graphs (the terms of sort GR satisfying the predicateM) are interpreted as moleculesN is the set F(R) of fuzzy numbers defined on Rthe symbols +, · · · , . . . are interpreted as the usual operators offuzzy arithmetic
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The natural interpretation
F(R) as N
Any function φ :GR 7→NU is interpreted as a function that associatesfuzzy quantities with molecules:
I(φ) : B 7→ N .
The shape of the membership function describing the fuzzy numberscan be different for different functions. In particular for some functionalsymbol (like m) it boils down to singleton characteristic function (i.e. tocrisp numbers).
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The natural interpretation
Summary of natural interpretation of FC theory
Syntactic object Natural Interpretation NoteBall Atom 3D sphereStick Bond Tangent spheres
Ballic formal charge Formal charge Integer, crispStick multiplicity Bond multiplicity Natural, crispBallic number Atomic number Natural, crisp
Mol “To be molecule” Crisp predicateMolecular graph Molecule 3D graph
m,M mass Real, crispr Atomics radius Real, fuzzyl internuclear distance Real, fuzzy
e,E Energy Real, fuzzyξ electronegativity Real, fuzzy
∆E Reaction enthalpy Real, fuzzy( Chemical reaction
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Examples
Outline
1 IntroductionThe matter and its descriptionGeneral Chemistry (GC)
2 Formal chemistry (FC): Underlying ideasMolecules and graphsLoose metricReactionsLogical Framework
3 The axiomsStick-and-ballMolecular graphs
4 The natural interpretation5 Examples6 Conclusions
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Examples
Example: heat generated by a chemical reaction
Chemical reaction:
H2 + F2 −−→ 2 HF. (1)
atoms involved: H (hydrogen) and F (fluorine)Balls: H = (1,1,0,0) and F = (2,7,0,0)
AtomAtomicnumber
Atomicweigth(g/mole)
Atomicradius(pm)
Elec-tronega-tivity
Self-energy(kJ/mole)
H 1 1.007 25 2.20 218F 9 18.998 50 3.98 79
Table: Properties of the atoms. The reported values are used as centers ofthe membership functions.
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Examples
Fuzzy radius and fuzzy bond length
Figure: Gaussian fuzzy numbers for atomic radius (up) and bond length(down).
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Examples
Bond properties
Bond Bond energy (kJ/mole)Bondlength (pm)
H–H 436 50F–F 158 100H–F 297 + 100(2.2− 3.98)2 = 613.84 75
Table: Properties of the bonds. The reported values are the centers of themembership functions.
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Examples
Reaction enthalpies
Bond Bond en-ergy (kJ/mole)
Number ofbonds
Right (+) H–F 613.84 2 1227.7Left (−) H–H 436 1 −436
F–F 157.8 1 −158Total 633.68
Substance Heat of re-action (kJ/mole)
Number ofsubstances
Right (+) HF 269 2 538Left (−) H2 0 1 0
F2 0 1 0Total 538
Table: Comparison of the reaction enthalpies for hydrogen fluoride, calculatedfrom bond energies (down) and from the heats of reaction (down)
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Examples
Figure: Gaussian fuzzy numbers for self-energy (up) bond energy (center)and energy change (down).
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Conclusions
Outline
1 IntroductionThe matter and its descriptionGeneral Chemistry (GC)
2 Formal chemistry (FC): Underlying ideasMolecules and graphsLoose metricReactionsLogical Framework
3 The axiomsStick-and-ballMolecular graphs
4 The natural interpretation5 Examples6 Conclusions
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Conclusions
Conclusions
To construct a formal theory of GC is possibleThis theory represents the reality only if its mathematical structureis based on fuzzy arithmeticsHave we really formulated a genuine theory of general chemistry?Of course no—the theory is both incomplete and inconsistent withsome experimental factsHowever, FC has two great advantages
it provides a rigorous axiomatic foundation for topics so dispersedthat can hardly be classified as a theory.it allows a description of matter where the properties of moleculesare given in terms of those of the constituting atoms and how theyare bonded
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Conclusions
Next steps
to deeply investigate fuzzy and interval arithmeticto set-up the machinery for fuzzy interpretation of FCto define operative tools based on FC to analyze molecules andreactionsTo develop algorithms on graphs to investigate charge flow insidemolecules
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