Skip to content

HTTPS clone URL

Subversion checkout URL

You can clone with
or
.
Download ZIP
Newer
Older
100644 532 lines (486 sloc) 22.745 kB
dd1ccae @teg Merged Formalism and Classical Logic chapters
authored
1 \chapter{Propositional Classical Logic}\label{chapter:PropositionalClassicalLogic}
beb7e3e @teg Defined a formalism
authored
2
531eaf8 @teg Informal comments
authored
3 The traditional formalism in deep inference is \emph{the calculus of structures} \cite{Gugl:06:A-System:kl}.
ed87c95 @teg Minor fixes
authored
4
9fd7aef @teg Submitted
authored
5 The idea of a new formalism, named \emph{formalism A} based on the calculus of structures, but where derivations contain less bureaucracy, was proposed by Guglielmi in \cite{Gugl:04:Formalis:am}, and later Br\"{u}nnler and Lengrand developed a term calculus around these ideas \cite{BrunLeng:05:On-Two-F:jf}.
fa2ca18 @teg Defined inference rule instance
authored
6
b6eadfb @teg Remove mention of bureaucracy
authored
7 In this chapter I define a formalism based on the ideas of formalism A and call it (as suggested by Fran\c{c}ois Lamarche) \emph{the functorial calculus}. The reason to introduce a new formalism is that it greatly simplifies the presentation of some of the more technical results in this thesis (in particular Section~\vref{subsection:ThresholdFormulae}).
280d92f @teg Informal text
authored
8
9 After presenting the functorial calculus we compare it briefly with the calculus of structures before we introduce the standard deductive system for classical logic in deep inference and show some preliminary results.
10
11 We now define `formulae' and `inference rules', which are in common between both the functorial calculus and the calculus of structures. Definitions~\vrefrange{definition:Formula}{definition:InferenceRuleInstance} are based on definitions given in \cite{BrusGugl:07:On-the-P:fk}. The focus of this thesis is classical propositional logic, and the following definitions reflect this. However, it is worth noting that the definitions can be generalised to other units and connectives, if one wants to present other logics.
f68726f @teg Finished comparison with CoS and finished definition of FuC
authored
12
8e3c317 @teg Added informal text and minor cleanups
authored
13 %---------------------------
0675709 @teg TODO's
authored
14 \newcommand{\fff}{\mathsf f}
15 \newcommand{\ttt}{\mathsf t}
16 \newcommand{\size}[1]{{\left\vert #1\right\vert}}\vlupdate\size
fa2ca18 @teg Defined inference rule instance
authored
17 \begin{definition}\label{definition:Formula}
8c3033c @teg Added index
authored
18 We define a set of \emph{formulae}\index{formulae}, denoted by $\alpha$, $\beta$, $\gamma$, $\delta$ to be:
fa2ca18 @teg Defined inference rule instance
authored
19 \begin{itemize}
f43d5b2 @teg Minor fixes
authored
20 \item \emph{atoms}, denoted by $a$, $b$, $c$, $d$ and $\bar a$, $\bar b$, $\bar c$, $\bar d$;
21 \item \emph{formula variables}, denoted by $A$, $B$, $C$, $D$;
22 \item \emph{units} $\fff$ (false) and $\ttt$ (true); and
23 \item the \emph{disjunction} and \emph{conjunction} of formulae $\alpha$ and $\beta$, denoted by $\vlsbr[\alpha.\beta]$ and $\vlsbr(\alpha.\beta)$, respectively.
fa2ca18 @teg Defined inference rule instance
authored
24 \end{itemize}
8c3033c @teg Added index
authored
25 A formula is \emph{ground}\index{formula!ground} if it contains no variables. We usually omit external brackets of formulae, and sometimes we omit dispensable brackets under associativity. We use $\equiv$ to denote literal equality of formulae. The \emph{size}\index{formula!size} $\size\alpha$ of a formula $\alpha$ is the number of unit, atom and variable occurrences appearing in it. On the set of atoms there is an involution $\bar\cdot$, called \emph{negation}\index{atoms!negation} (\emph{i.e.}, $\bar\cdot$ is a bijection from the set of atoms to itself such that $\bar{\bar a}\equiv a$); we require that $\bar a\not\equiv a$ for every $a$; when both $a$ and $\bar a$ appear in a formula, we mean that atom $a$ is mapped to by $\bar a$ by $\bar\cdot$. A \emph{context}\index{formula!context} is a formula where one \emph{hole}\index{hole} $\vlhole$ appears in the place of a subformula; for example, $\vls[a.(b.\vlhole)]$ is a context; the generic context is denoted by $\xi\vlhole$. The hole can be filled with formulae; for example, if $\xi\vlhole\equiv\vls(b.[\vlhole.c])$, then $\xi\{a\}\equiv\vls(b.[a.c])$, $\xi\{b\}\equiv\vls(b.[b.c])$ and $\xi\{\vls(a.b)\}\equiv\vls(b.[(a.b).c])$. The \emph{size}\index{formula!context!size} of $\xi\vlhole$ is defined as $\size{\xi\vlhole}=\size{\xi\{a\}}-1$.
fa2ca18 @teg Defined inference rule instance
authored
26 \end{definition}
8e3c317 @teg Added informal text and minor cleanups
authored
27 %---------------
fa2ca18 @teg Defined inference rule instance
authored
28
8e3c317 @teg Added informal text and minor cleanups
authored
29 %--------------------------------------------------------
fa2ca18 @teg Defined inference rule instance
authored
30 \begin{definition}\label{definition:RenamingSubstitution}
8c3033c @teg Added index
authored
31 A \emph{renaming}\index{atoms!renaming} is a map from the set of atoms to itself, and it is denoted by $\{a_1/b_1,a_2/b_2,\dots\}$. A renaming of $\alpha$ by $\{a_1/b_1,a_2/b_2,\dots\}$ is indicated by $\alpha\{a_1/b_1,a_2/b_2,\dots\}$ and is obtained by simultaneously substituting every occurrence of $a_i$ in $\alpha$ by $b_i$ and every occurrence of $\bar a_i$ by $\bar b_i$; for example, if $\alpha\equiv\vls(a.[b.(a.[\bar a.c])])$ then $\alpha\{a/\bar b,\bar b/c\}\equiv\vls(\bar b.[\bar c.(\bar b.[b.c])])$. A \emph{substitution}\index{substitution} is a map from the set of formula variables to the set of formulae, denoted by $\{A_1/\beta_1,A_2/\beta_2,\dots\}$. An \emph{instance}\index{formula!instance} of $\alpha$ by $\{A_1/\beta_1,A_2/\beta_2,\dots\}$ is indicated by $\alpha\{A_1/\beta_1,A_2/\beta_2,\dots\}$ and is obtained by simultaneously substituting every occurrence of variable $A_i$ in $\alpha$ by formula $\beta_i$; for example if $\alpha\equiv\vls[A.(b.c)]$ then $\alpha\{A/\vlsbr(c.\bar b)\}\equiv\vls[(c.\bar b).(b.c)]$.
fa2ca18 @teg Defined inference rule instance
authored
32 \end{definition}
8e3c317 @teg Added informal text and minor cleanups
authored
33 %---------------
fa2ca18 @teg Defined inference rule instance
authored
34
02b68f1 @teg Allow substitution on atom occurrences
authored
35 \begin{convention}
36 By the above definition, formula variables will only be used to define inference rules, and will never appear in derivations. However, when we perform normalisation we will sometimes single out atom occurrences (by decorating them) and substitute on them as if they were formula variables.
37 \end{convention}
0b6498c @teg Minor fixes
authored
38
8e3c317 @teg Added informal text and minor cleanups
authored
39 %---------------------------------------------------------
e3571b7 @teg Added labels and minor fixes
authored
40 \begin{definition}\label{definition:InferenceRuleInstance}
8c3033c @teg Added index
authored
41 An \emph{inference rule}\index{inference rule} $\rho$ is an expression $\vlinf{\rho}{}{\beta}{\alpha}$, where the formulae $\alpha$ and $\beta$ are called \emph{premiss}\index{inference rule!premiss} and \emph{conclusion}\index{inference rule!conclusion}, respectively. A (\emph{deductive}) \emph{system}\index{system} is a finite set of inference rules. An \emph{inference rule instance}\index{inference rule!instance} $\vlinf{\rho}{}{\delta}{\gamma}$ of $\vlinf{\rho}{}{\beta}{\alpha}$ is such that $\gamma$ and $\delta$ are ground, and $\gamma\equiv\alpha\{a_1/b_1,a_2/b_2,\dots\}\{A_1/\beta_1,A_2/\beta_2,\dots\}$ and $\delta\equiv\beta\{a_1/b_1,a_2/b_2,\dots\}\{A_1/\beta_1,A_2/\beta_2,\dots\}$, for some renaming $\{a_1/b_1,a_2/b_2,\dots\}$ and substiution $\{A_1/\beta_1,A_2/\beta_2,\dots\}$.
7d910ed @teg Defined inference rule and changed notation of composition of derivat…
authored
42 \end{definition}
8e3c317 @teg Added informal text and minor cleanups
authored
43 %---------------
44
45 %==================================================================
46 \section{The Functorial Calculus}\label{section:FunctorialCalculus}
7d910ed @teg Defined inference rule and changed notation of composition of derivat…
authored
47
280d92f @teg Informal text
authored
48 We now present the functorial calculus in the context of classical propositional logic and give some basic results.
49
be3798e @teg Informal comments
authored
50 The intuition behind derivations in the functorial calculus is that we can compose derivations by the same connectives we can compose formulae.
51
8e3c317 @teg Added informal text and minor cleanups
authored
52 %----------------------------------------------
e3571b7 @teg Added labels and minor fixes
authored
53 \begin{definition}\label{definition:Derivation}
8c3033c @teg Added index
authored
54 Given a deductive system $\mathcal S$, and formulae $\alpha$ and $\beta$; a (\emph{functorial calculus}) \emph{derivation $\Psi$ in $\mathcal S$ from $\alpha$ to $\beta$}\index{derivation!functorial calculus}, denoted $\vlder{\Psi}{\mathcal{S}}{\beta}{\alpha}$, is defined to be
beb7e3e @teg Defined a formalism
authored
55 \begin{enumerate}
0b6498c @teg Minor fixes
authored
56 \item\label{definition:Derivation:item:Formula} a formula: $\Psi\;=\;\alpha\equiv\beta$;
ebcb22a @teg Added TODO's
authored
57
8c3033c @teg Added index
authored
58 \item\label{definition:Derivation:item:Vertical} a \emph{vertical composition}\index{derivation!functorial calculus!vertical composition}:
beb7e3e @teg Defined a formalism
authored
59 \[
0b6498c @teg Minor fixes
authored
60 \Psi\;=\;
d8f02e0 @teg Defined size of derivations and minor fixes
authored
61 \vlderivation
2133672 @teg Remarked associativity of vertical composition of derivations
authored
62 {
d8f02e0 @teg Defined size of derivations and minor fixes
authored
63 \vlde{\Phi_2}{}
2133672 @teg Remarked associativity of vertical composition of derivations
authored
64 {
d8f02e0 @teg Defined size of derivations and minor fixes
authored
65 \beta
2133672 @teg Remarked associativity of vertical composition of derivations
authored
66 }
67 {
d8f02e0 @teg Defined size of derivations and minor fixes
authored
68 \vlin{\rho}{}
69 {
70 \alpha'
71 }
72 {
73 \vlde{\Phi_1}{}
74 {
75 \beta'
76 }
77 {
78 \vlhy
79 {
80 \alpha
81 }
82 }
83 }
2133672 @teg Remarked associativity of vertical composition of derivations
authored
84 }
85 }
86 \quad,
beb7e3e @teg Defined a formalism
authored
87 \]
0b6498c @teg Minor fixes
authored
88 where $\vlinf{\rho}{}{\alpha'}{\beta'}$ is an instance of an inference rule from $\mathcal{S}$, and $\vlder{\Phi_1}{\mathcal S}{\beta'}{\alpha}$ and $\vlder{\Phi_2}{\mathcal S}{\beta}{\alpha'}$ are derivations; or
f68726f @teg Finished comparison with CoS and finished definition of FuC
authored
89
8c3033c @teg Added index
authored
90 \item\label{definition:Derivation:item:Horizontal} a \emph{horizontal composition}\index{derivation!functorial calculus!horizontal composition}:
beb7e3e @teg Defined a formalism
authored
91 \[
dd1ccae @teg Merged Formalism and Classical Logic chapters
authored
92 \Psi\;=\;
93 \vls
94 (
d8f02e0 @teg Defined size of derivations and minor fixes
authored
95 \vlder{\Phi_1}{}{\beta_1}{\alpha_1}
dd1ccae @teg Merged Formalism and Classical Logic chapters
authored
96 \;\;.\;\;
97 \vlder{\Phi_2}{}{\beta_2}{\alpha_2}
98 )
99 \qquad\mbox{or}\qquad
100 \Psi\;=\;
101 \vls
102 [
103 \vlder{\Phi_1}{}{\beta_1}{\alpha_1}
104 \;\;.\;\;
105 \vlder{\Phi_2}{}{\beta_2}{\alpha_2}
b7b6849 @teg Trailing whitespace
authored
106 ]
d8f02e0 @teg Defined size of derivations and minor fixes
authored
107 \quad,
beb7e3e @teg Defined a formalism
authored
108 \]
dd1ccae @teg Merged Formalism and Classical Logic chapters
authored
109 where $\vlder{\Phi_1}{}{\beta_1}{\alpha_1}$ and $\vlder{\Phi_2}{}{\beta_2}{\alpha_2}$ are derivations, and $\alpha\equiv\vls[\alpha_1.\alpha_2]$ and $\beta\equiv\vls[\beta_1.\beta_2]$, or $\alpha\equiv\vls(\alpha_1.\alpha_2)$ and $\beta\equiv\vls(\beta_1.\beta_2)$, respectively.
beb7e3e @teg Defined a formalism
authored
110 \end{enumerate}
9fd7aef @teg Submitted
authored
111
112 A derivation with premiss $\ttt$ is, from now on, called a \emph{proof}.
113
0b6498c @teg Minor fixes
authored
114 The size of a derivation $\Psi$, denoted $\size{\Psi}$, is defined to be the sum of the size of the formulae appearing in $\Psi$.
beb7e3e @teg Defined a formalism
authored
115 \end{definition}
8e3c317 @teg Added informal text and minor cleanups
authored
116 %---------------
beb7e3e @teg Defined a formalism
authored
117
8e3c317 @teg Added informal text and minor cleanups
authored
118 %-------------------------------------------------------------
d846ecd @teg Started using Notation and Convention macros
authored
119 \begin{convention}\label{convention:DerAssociativeComposition}
2133672 @teg Remarked associativity of vertical composition of derivations
authored
120 Given derivations $\vlder{\Phi_1}{}{\beta_1}{\alpha_1}$, $\vlder{\Phi_2}{}{\beta_2}{\alpha_2}$ and $\vlder{\Phi_3}{}{\beta_3}{\alpha_3}$, and inference rule instances $\vlinf{\rho_1}{}{\alpha_2}{\beta_1}$ and $\vlinf{\rho_2}{}{\alpha_3}{\beta_2}$ we consider
121 \[
122 \vlinf{\rho_2}{}
123 {
124 \vlder{\Phi_3}{}
125 {
126 \beta_3
127 }
128 {
129 \alpha_3
130 }
131 }
132 {
133 \left(
134 \vlinf{\rho_1}{}
135 {
136 \vlder{\Phi_2}{}
137 {
138 \beta_2
139 }
140 {
141 \alpha_2
142 }
143 }
144 {
145 \vlder{\Phi_1}{}
146 {
147 \beta_1
148 }
149 {
150 \alpha_1
151 }
152 }
153 \right)
154 }
65497be @teg Minor fixes
authored
155 \qquad\mbox{and}\qquad
2133672 @teg Remarked associativity of vertical composition of derivations
authored
156 \vlinf{\rho_1}{}
157 {
158 \left(
159 \vlinf{\rho_2}{}
160 {
161 \vlder{\Phi_3}{}
162 {
163 \beta_3
164 }
165 {
166 \alpha_3
167 }
168 }
169 {
170 \vlder{\Phi_2}{}
171 {
172 \beta_2
173 }
174 {
175 \alpha_2
176 }
177 }
178 \right)
179 }
180 {
181 \vlder{\Phi_1}{}
182 {
183 \beta_1
184 }
185 {
186 \alpha_1
187 }
188 }
189 \]
0b6498c @teg Minor fixes
authored
190 to be equal, and we denote them both by
2133672 @teg Remarked associativity of vertical composition of derivations
authored
191 \[
192 \vlderivation
193 {
194 \vlde{\Phi_3}{}
195 {
196 \beta_3
197 }
198 {
199 \vlin{\rho_2}{}
200 {
201 \alpha_3
202 }
203 {
204 \vlde{\Phi_2}{}
205 {
206 \beta_2
207 }
208 {
209 \vlin{\rho_1}{}
210 {
211 \alpha_2
212 }
213 {
214 \vlde{\Phi_1}{}
215 {
216 \beta_1
217 }
218 {
219 \vlhy
220 {
221 \alpha_1
222 }
223 }
224 }
225 }
226 }
227 }
228 }
229 \quad.
230 \]
d846ecd @teg Started using Notation and Convention macros
authored
231 \end{convention}
8e3c317 @teg Added informal text and minor cleanups
authored
232 %---------------
ebcb22a @teg Added TODO's
authored
233
cc7c8e8 @teg Why not automatic normal form of derivation?
authored
234 %----------------------------------------------------
235 \begin{remark}\label{remark:NoNeedAssocOfComposition}
236 If desireable, Convention~\vref{convention:DerAssociativeComposition} could be made redundant by forcing associativity of horizontal composition in Definition~\vref{definition:Derivation}. The only reason we did not do this was for the sake of brevity of the following results.
237 \end{remark}
238 %-----------
239
240
8e3c317 @teg Added informal text and minor cleanups
authored
241 %--------------------------------------
f68726f @teg Finished comparison with CoS and finished definition of FuC
authored
242 \begin{lemma}\label{lemma:DerInContext}
0b6498c @teg Minor fixes
authored
243 Given a derivation $\vlder{\Phi}{}{\beta}{\alpha}$ and a context $\xi\vlhole$, a derivation $\vlder{\Psi}{}{\xi\{\beta\}}{\xi\{\alpha\}}$, with size $\size{\Phi}+\size{\xi\vlhole}$, can be constructed.
f68726f @teg Finished comparison with CoS and finished definition of FuC
authored
244 \end{lemma}
245
246 \begin{proof}
247 We proceed by structural induction on $\xi\vlhole$. The base case, $\xi\vlhole\equiv\vlhole$, is trivial.
248 For the inductive case, let
249 \[
a6481b2 @teg Fixed proof of context closure
authored
250 \xi\vlhole\equiv\vls(\xi'\vlhole.\gamma)\quad,\qquad\xi\vlhole\equiv\vls(\gamma.\xi'\vlhole)\quad,
f68726f @teg Finished comparison with CoS and finished definition of FuC
authored
251 \]
a6481b2 @teg Fixed proof of context closure
authored
252 \[
253 \xi\vlhole\equiv\vls[\xi'\vlhole.\gamma]\qquad\mbox{or}\qquad\xi\vlhole\equiv\vls[\gamma.\xi'\vlhole]\quad.
254 \]
255 for some formula $\gamma$ and a context $\xi'\vlhole$. By the inductive hypothesis we can construct the derivation $\vlder{\Psi'}{}{\xi'\{\beta\}}{\xi'\{\alpha\}}$, so the result follows by case (\ref{definition:Derivation:item:Horizontal}) of Definition~\vref{definition:Derivation}.
f68726f @teg Finished comparison with CoS and finished definition of FuC
authored
256 \end{proof}
8e3c317 @teg Added informal text and minor cleanups
authored
257 %----------
f68726f @teg Finished comparison with CoS and finished definition of FuC
authored
258
8e3c317 @teg Added informal text and minor cleanups
authored
259 %--------------------------------------------
d846ecd @teg Started using Notation and Convention macros
authored
260 \begin{notation}\label{notation:DerInContext}
0b6498c @teg Minor fixes
authored
261 Given a derivation $\vlder{\Phi}{}{\beta}{\alpha}$ and a context $\xi\vlhole$, the derivation $\vlder{}{}{\xi\{\beta\}}{\xi\{\alpha\}}$ constructed in the proof of Lemma~\vref{lemma:DerInContext} is denoted $\xi\{\Phi\}$.
d846ecd @teg Started using Notation and Convention macros
authored
262 \end{notation}
8e3c317 @teg Added informal text and minor cleanups
authored
263 %-------------
f68726f @teg Finished comparison with CoS and finished definition of FuC
authored
264
8e3c317 @teg Added informal text and minor cleanups
authored
265 %----------------------------------------
5c798ba @teg Use varioref and fix some references
authored
266 \begin{lemma}\label{lemma:DerComposition}
f68726f @teg Finished comparison with CoS and finished definition of FuC
authored
267 Given two derivations $\vlder{\Phi_1}{}{\beta}{\alpha}$ and $\vlder{\Phi_2}{}{\gamma}{\beta}$, a derivation $\vlder{\Psi}{}{\gamma}{\alpha}$, with size $\size{\Phi_1}+\size{\Phi_2}-\size{\beta}$, can be constructed.
beb7e3e @teg Defined a formalism
authored
268 \end{lemma}
269
8cf5447 @teg Changed names of derivations and started sketch of proof of composition
authored
270 \begin{proof}
dd1ccae @teg Merged Formalism and Classical Logic chapters
authored
271 We argue by structural induction on $\Phi_1$
8cf5447 @teg Changed names of derivations and started sketch of proof of composition
authored
272 \begin{enumerate}
7d910ed @teg Defined inference rule and changed notation of composition of derivat…
authored
273
dd1ccae @teg Merged Formalism and Classical Logic chapters
authored
274 \item\label{proof:DerComposition:item:Formula} if $\Phi_1=\beta$ then $\Psi=\Phi_2$, with size $\size{\Phi_1}+\size{\Phi_2}-\size{\beta}$;
7d910ed @teg Defined inference rule and changed notation of composition of derivat…
authored
275
dd1ccae @teg Merged Formalism and Classical Logic chapters
authored
276 \item\label{proof:DerComposition:item:Vertical} if
f68726f @teg Finished comparison with CoS and finished definition of FuC
authored
277 \[
278 \Phi_1\;=\;
d8f02e0 @teg Defined size of derivations and minor fixes
authored
279 \vlderivation
280 {
281 \vlde{\Phi''_1}{}
282 {
283 \beta
284 }
285 {
286 \vlin{\rho}{}
287 {
288 \alpha'
289 }
290 {
291 \vlde{\Phi'_1}{}
292 {
293 \beta'
294 }
295 {
296 \vlhy
297 {
298 \alpha
299 }
300 }
301 }
302 }
303 }\quad,
f68726f @teg Finished comparison with CoS and finished definition of FuC
authored
304 \]
305 then, by the inductive hypothesis, we can construct $\vlder{\Psi'}{}{\gamma}{\alpha'}$, with size $\size{\Phi''_1}+\size{\Phi_2}-\size{\beta}$, we can then build
306 \[
307 \Psi\;=\;
d8f02e0 @teg Defined size of derivations and minor fixes
authored
308 \vlderivation
309 {
310 \vlde{\Psi'}{}
311 {
312 \gamma
313 }
314 {
315 \vlin{\rho}{}
316 {
317 \alpha'
318 }
319 {
320 \vlde{\Phi'_1}{}
321 {
322 \beta'
323 }
324 {
325 \vlhy
326 {
327 \alpha
328 }
329 }
330 }
331 }
332 }\quad,
f68726f @teg Finished comparison with CoS and finished definition of FuC
authored
333 \]
334 with size $\size{\Phi'_1}+\size{\Psi'}=\size{\Phi'_1}+\size{\Phi''_1}+\size{\Phi_2}-\size{\beta}=\size{\Phi_1}+\size{\Phi_2}-\size{\beta}$;
7d910ed @teg Defined inference rule and changed notation of composition of derivat…
authored
335
b7b6849 @teg Trailing whitespace
authored
336 \item if
f68726f @teg Finished comparison with CoS and finished definition of FuC
authored
337 \[
338 \Phi_1\;=\;
dd1ccae @teg Merged Formalism and Classical Logic chapters
authored
339 \vls[\vlder{\Phi_{1,1}}{}{\beta_1}{\alpha_1}\;\;.\;\;\vlder{\Phi_{1,2}}{}{\beta_2}{\alpha_2}]
340 \qquad\mbox{or}\qquad
341 \Phi_1\;=\;
342 \vls(\vlder{\Phi_{1,1}}{}{\beta_1}{\alpha_1}\;\;.\;\;\vlder{\Phi_{1,2}}{}{\beta_2}{\alpha_2})
f68726f @teg Finished comparison with CoS and finished definition of FuC
authored
343 \]
dd1ccae @teg Merged Formalism and Classical Logic chapters
authored
344 we argue by structural induction on $\Phi_2$:
345 \begin{enumerate}
346 \item if $\Phi_2$ is a formula (resp., a vertical composition), the result follow by a symmetric argument to case \ref{proof:DerComposition:item:Formula} (resp., \ref{proof:DerComposition:item:Vertical}) above.
347 \item if
348 \[
349 \Phi_2\;=\;
350 \vls[\vlder{\Phi_{2,1}}{}{\gamma_1}{\beta_1}\;\;.\;\;\vlder{\Phi_{2,2}}{}{\gamma_2}{\beta_2}]
351 \qquad\mbox{or}\qquad
352 \Phi_2\;=\;
353 \vls(\vlder{\Phi_{2,1}}{}{\gamma_1}{\beta_1}\;\;.\;\;\vlder{\Phi_{2,2}}{}{\gamma_2}{\beta_2})
354 \]
355 then, by the first inductive hypothesis, we can construct
356 \[
357 \vlder{\Psi_1}{}{\gamma_1}{\alpha_1}
358 \qquad\mbox{and}\qquad
359 \vlder{\Psi_2}{}{\gamma_2}{\alpha_2}\quad,
360 \]
361 with size $\size{\Phi_{1,1}}+\size{\Phi_{2,1}}-\size{\beta_1}$ and $\size{\Phi_{1,2}}+\size{\Phi_{2,2}}-\size{\beta_2}$, respectively, we can then build
362 \[
363 \Psi\;=\;
364 \vls[\vlder{\Psi_1}{}{\gamma_1}{\alpha_1}\;\;.\;\;\vlder{\Psi_2}{}{\gamma_2}{\alpha_2}]
365 \qquad\mbox{or}\qquad
366 \Psi\;=\;
367 \vls(\vlder{\Psi_1}{}{\gamma_1}{\alpha_1}\;\;.\;\;\vlder{\Psi_2}{}{\gamma_2}{\alpha_2})
368 \]
369 with size $\size{\Psi_1}+\size{\Psi_2}=\size{\Phi_{1,1}}+\size{\Phi_{1,2}}+\size{\Phi_{2,1}}+\size{\Phi_{2,2}}-(\size{\beta_1}+\size{\beta_2})=\size{\Phi_1}+\size{\Phi_2}-\size{\beta}$.
370 \end{enumerate}
8cf5447 @teg Changed names of derivations and started sketch of proof of composition
authored
371 \end{enumerate}
372 \end{proof}
8e3c317 @teg Added informal text and minor cleanups
authored
373 %----------
8cf5447 @teg Changed names of derivations and started sketch of proof of composition
authored
374
8e3c317 @teg Added informal text and minor cleanups
authored
375 %----------------------------------------------
d846ecd @teg Started using Notation and Convention macros
authored
376 \begin{notation}\label{notation:DerComposition}
f68726f @teg Finished comparison with CoS and finished definition of FuC
authored
377 Given derivations $\vlder{\Phi_1}{}{\beta}{\alpha}$ and $\vlder{\Phi_2}{}{\gamma}{\beta}$, the derivation $\vlder{\Psi}{}{\gamma}{\alpha}$ constructed in the proof of Lemma~\vref{lemma:DerComposition} is denoted:
beb7e3e @teg Defined a formalism
authored
378 \[
379 \vlderivation
380 {
8cf5447 @teg Changed names of derivations and started sketch of proof of composition
authored
381 \vlde{\Phi_2}{}{\gamma}
beb7e3e @teg Defined a formalism
authored
382 {
8cf5447 @teg Changed names of derivations and started sketch of proof of composition
authored
383 \vlde{\Phi_1}{}{\beta}
beb7e3e @teg Defined a formalism
authored
384 {
385 \vlhy{\alpha}
386 }
387 }
388 }\quad.
389 \]
d846ecd @teg Started using Notation and Convention macros
authored
390 \end{notation}
8e3c317 @teg Added informal text and minor cleanups
authored
391 %-------------
beb7e3e @teg Defined a formalism
authored
392
8e3c317 @teg Added informal text and minor cleanups
authored
393 %=======================================================================
dd1ccae @teg Merged Formalism and Classical Logic chapters
authored
394 \section{The Calculus of Structures}\label{section:CalculusOfStructures}
beb7e3e @teg Defined a formalism
authored
395
b6d3376 @teg Major search/cond-replace
authored
396 We now present the calculus of structures and in Theorem~\vref{theorem:CoSToFunc} and Theorem~\vref{theorem:FuncToCoS} we show that the functorial calculus and the calculus of structures polynomially simulate each other.
8e3c317 @teg Added informal text and minor cleanups
authored
397
398 The intuition behind derivations in the calculus of structures is that we rewrite formulae by applying inference rules inside a context.
be3798e @teg Informal comments
authored
399
8e3c317 @teg Added informal text and minor cleanups
authored
400 %---------------------------------------
eadb5e0 @teg Defined CoS
authored
401 \begin{definition}\label{definition:CoS}
8c3033c @teg Added index
authored
402 Given a deductive system $\mathcal S$, a set of formulae, $\mathcal F$, and $\alpha$ and $\beta$ from $\mathcal F$; a \emph{calculus of structures derivation $\Psi$ in $\mathcal S$ from $\alpha$ to $\beta$}\index{derivation!calculus of structures}, denoted $\vlder{\Psi}{\mathcal{S}}{\beta}{\alpha}$, is defined to be
eadb5e0 @teg Defined CoS
authored
403 \begin{enumerate}
0b6498c @teg Minor fixes
authored
404 \item\label{definition:CoS:item:Formula} a formula: $\Psi\;=\;\alpha\equiv\beta$; or
eadb5e0 @teg Defined CoS
authored
405
8c3033c @teg Added index
authored
406 \item\label{definition:CoS:item:Vertical} a \emph{vertical composition}\index{derivation!calculus of structures!vertical composition}:
eadb5e0 @teg Defined CoS
authored
407 \[
0b6498c @teg Minor fixes
authored
408 \Psi\;=\;
409 \vlderivation
eadb5e0 @teg Defined CoS
authored
410 {
0b6498c @teg Minor fixes
authored
411 \vlde{\Phi_2}{}
eadb5e0 @teg Defined CoS
authored
412 {
d8f02e0 @teg Defined size of derivations and minor fixes
authored
413 \beta
eadb5e0 @teg Defined CoS
authored
414 }
415 {
0b6498c @teg Minor fixes
authored
416 \vlin{\rho}{}
417 {
418 \xi\{\alpha'\}
419 }
420 {
421 \vlde{\Phi_1}{}
422 {
423 \xi\{\beta'\}
424 }
425 {
426 \vlhy
427 {
428 \alpha
429 }
430 }
431 }
eadb5e0 @teg Defined CoS
authored
432 }
433 }
434 \quad,
435 \]
0b6498c @teg Minor fixes
authored
436 where $\vlinf{\rho}{}{\alpha'}{\beta'}$ is an instance of an inference rule from $\mathcal{S}$, and $\vlder{\Phi_1}{\mathcal S}{\xi\{\beta'\}}{\alpha}$ and $\vlder{\Phi_2}{\mathcal S}{\beta}{\xi\{\alpha'\}}$ are calculus of structures derivations.
eadb5e0 @teg Defined CoS
authored
437 \end{enumerate}
0b6498c @teg Minor fixes
authored
438 The size of a calculus of structures derivation $\Psi$, denoted $\size{\Psi}$, is defined to be the sum of the size of the formulae appearing in $\Psi$.
ebcb22a @teg Added TODO's
authored
439 \end{definition}
8e3c317 @teg Added informal text and minor cleanups
authored
440 %---------------
c7a7e30 @teg Made the section on threshold formulae compile
authored
441
8e3c317 @teg Added informal text and minor cleanups
authored
442 %---------------------------------------
443 \begin{theorem}\label{theorem:CoSToFunc}
f68726f @teg Finished comparison with CoS and finished definition of FuC
authored
444 A calculus of structures derivation $\vlder{\Phi}{}{\beta}{\alpha}$ can be transformed into a functorial calculus derivation $\vlder{\Psi}{}{\beta}{\alpha}$ such that $\size{\Psi}\le\size{\Phi}$.
9e79774 @teg Added Theorems comparing Formalism A to CoS
authored
445 \end{theorem}
446
e3fd71c @teg Started proof of equality between CoS and FC
authored
447 \begin{proof}
f68726f @teg Finished comparison with CoS and finished definition of FuC
authored
448 We argue by structural induction on $\Phi$. The base case is trivial; $\Phi=\alpha\equiv\beta=\Psi$. For the inductive case, consider the following calculus of structures derivation:
449 \[
450 \Phi\;=\;
451 \vlinf{\rho}{}
452 {
453 \vlder{\Phi_2}{}
454 {
455 \beta
456 }
457 {
458 \xi\{\alpha'\}
459 }
460 }
461 {
462 \vlder{\Phi_1}{}
463 {
464 \xi\{\beta'\}
465 }
466 {
467 \alpha
468 }
469 }
470 \quad.
471 \]
0b6498c @teg Minor fixes
authored
472 By the inductive hypothesis, there are functorial calculus derivations $\vlder{\Psi_1}{}{\xi\{\beta'\}}{\alpha}$ and $\vlder{\Psi_2}{}{\beta}{\xi\{\alpha'\}}$, such that $\size{\Psi_1}\le\size{\Phi_1}$ and $\size{\Psi_2}\le\size{\Phi_2}$. By Lemma~\vref{lemma:DerInContext}, there is a functorial calculus derivation $\xi\left\{\vlinf{\rho}{}{\alpha'}{\beta'}\right\}$, with size $\size{\xi\vlhole}+\size{\alpha'}+\size{\beta'}$. By Lemma~\vref{lemma:DerComposition}, we can combine the three functorial calculus derivations to create $\vlder{\Psi}{}{\beta}{\alpha}$, with size $\size{\Psi_1}+\size{\Psi_2}+\size{\xi\vlhole}+\size{\beta'}+\size{\alpha'}-\size{\xi\vlhole}-\size{\beta'}-\size{\xi\vlhole}-\size{\alpha'}=\size{\Psi_1}+\size{\Psi_2}-\size{\xi\vlhole}\le\size{\Phi_1}+\size{\Phi_2}=\size{\Phi}$.
e3fd71c @teg Started proof of equality between CoS and FC
authored
473 \end{proof}
dcc67dd @teg Proved translation from FuC to CoS
authored
474 %----------
475
932e7b3 @teg Added example of flows and derivations
authored
476 %--------------
477 \begin{example}
478 Figure~\vref{figure:ExampleAtomicFlows} has three examples of calculus of structures derivations transformed into functorial calculus derivations.
479 \end{example}
480 %------------
481
dcc67dd @teg Proved translation from FuC to CoS
authored
482 %--------------------------------------
483 \begin{lemma}\label{lemma:CoSDerInContext}
484 Given a calculus of structures derivation $\vlder{\Phi}{}{\beta}{\alpha}$ and a context $\xi\vlhole$, a calculus of structures derivation $\vlder{\Psi}{}{\xi\{\beta\}}{\xi\{\alpha\}}$ can be constructed, such that the number of inference rule instances in $\Psi$ is the same as the number of inference rule instances in $\Phi$, and the size of the largest formula in $\Psi$ is the sum of the largest formula in $\Phi$ and $\size{\xi\vlhole}$.
485 \end{lemma}
486
487 \begin{proof}
488 The statements follows by structural induction on $\Phi$.
489 \end{proof}
490 %----------
491
492 %----------------------------------------
493 \begin{lemma}\label{lemma:CoSDerComposition}
494 Given two calculus of structures derivations $\vlder{\Phi_1}{}{\beta}{\alpha}$ and $\vlder{\Phi_2}{}{\gamma}{\beta}$, a calculus of structures derivation $\vlupsmash{\vlder{\Psi}{}{\gamma}{\alpha}}$ can be constructed, such that the number of inference rule instances in $\Psi$ is the sum of the number of inference rule instances in $\Phi_1$ and $\Phi_2$ combined, and the largest formula in $\Psi$ is the largest formula of $\Phi_1$ or the largest formula of $\Phi_2$.
495 \end{lemma}
496
497 \begin{proof}
498 The statement follows by structural induction on $\Phi_1$.
499 \end{proof}
500 %----------
501
502 %---------------------------------------
503 \begin{theorem}\label{theorem:FuncToCoS}
504 A functorial calculus derivation $\vlder{\Phi}{}{\beta}{\alpha}$ can be transformed into a calculus of structures derivation $\vlder{\Psi}{}{\beta}{\alpha}$ such that the size of $\Psi$ depends at most quadratically on the size of $\Phi$.
505 \end{theorem}
506
507 \begin{proof}
508 We first show how to construct $\Psi$: The base cases, when $\Phi$ is a formula or a vertical composition, are trivial. For the inductive case, consider a conjunction of functorial calculus derivations (the argument for the disjunction is similar):
509 \[
510 \Phi\;=\;
511 \vls
512 (
513 \vlder{\Phi_1}{}{\beta_1}{\alpha_1}
514 \;\;.\;\;
515 \vlder{\Phi_2}{}{\beta_2}{\alpha_2}
516 )\quad.
517 \]
b6d3376 @teg Major search/cond-replace
authored
518 By the inductive hypothesis and Lemma~\vref{lemma:CoSDerInContext} there are calculus of structures derivations
dcc67dd @teg Proved translation from FuC to CoS
authored
519 \[
520 \vlder{\Psi_1}{}{\vls(\beta_1.\alpha_1)}{\vls(\alpha_1.\alpha_1)}
521 \qquad\mbox{and}\qquad
522 \vlder{\Psi_2}{}{\vls(\beta_1.\beta_2)}{\vls(\beta_1.\alpha_2)}
523 \quad,
524 \]
525 and by Lemma~\ref{lemma:CoSDerComposition} there exists a calculus of structures derivation $\vlder{\Psi}{}{\vls(\beta_1.\beta_2)}{\vls(\alpha_1.\alpha_2)}$.
526
527 To find an upper bound on the size of $\Psi$, we observe that it depends at most linearly on the number of inference rule instances in $\Psi$ multiplied by the size of the largest formula in $\Psi$. Furthermore, by the above Lemmata, the number of inference rules in $\Psi$ is the same as the number of inference rules in $\Phi$ and the size of the largest inference rule depends at most linearly on the size of $\Phi$, so the size of $\Psi$ depends at most quadratically on the size of $\Phi$.
528 \end{proof}
529 %----------
6e9a9e6 @teg Moved summary of results about CoS
authored
530
531 The calculus of structures is now well developed for classical \cite{Brun:03:Atomic-C:oz,Brun:06:Cut-Elim:cq,Brun:06:Locality:zh,BrunTiu:01:A-Local-:mz,Brun:06:Deep-Inf:qy}, intuitionistic \cite{Tiu:06:A-Local-:gf}, linear \cite{Stra:02:A-Local-:ul,Stra:03:MELL-in-:oy}, modal \cite{Brun::Deep-Seq:ay,GoreTiu:06:Classica:uq,Stou:06:A-Deep-I:rt} and commutative/non-commutative logics \cite{Gugl:06:A-System:kl,Tiu:06:A-System:ai,Stra:03:Linear-L:lp,Brus:02:A-Purely:wd,Di-G:04:Structur:wy,GuglStra:01:Non-comm:rp,GuglStra:02:A-Non-co:lq,GuglStra:02:A-Non-co:dq,Kahr:06:Reducing:hc,Kahr:07:System-B:fk}. The basic proof complexity properties of the calculus of structures are known \cite{BrusGugl:07:On-the-P:fk}. The calculus of structures promoted the discovery of a new class of proof nets for classical and linear logic \cite{LamaStra:05:Construc:qq,LamaStra:05:Naming-P:ov,LamaStra:06:From-Pro:et,StraLama:04:On-Proof:ec} (see also \cite{Guir:06:The-Thre:qt}). There exist implementations in Maude of deep-inference proof systems \cite{Kahr:07:Maude-as:lr}.
Something went wrong with that request. Please try again.