From 989ff2b034eeecf624fbedbb0d81a3b288744260 Mon Sep 17 00:00:00 2001 From: madvorak Date: Mon, 13 Nov 2023 11:57:30 +0100 Subject: [PATCH] scribe 6 in progress --- README.md | 6 ++-- class_6.tex | 78 +++++++++++++++++++++++++++++++++++++++++++++++++++ main.pdf | Bin 146049 -> 308484 bytes main.tex | 1 + preamble.tex | 2 ++ 5 files changed, 83 insertions(+), 4 deletions(-) create mode 100644 class_6.tex diff --git a/README.md b/README.md index 4394d2d..7eb69e7 100644 --- a/README.md +++ b/README.md @@ -1,9 +1,7 @@ -# formalisms_every_computer_scientist_should_know - -Formalisms Every Computer Scientist Should Know course at ISTA 2023. +# Formalisms Every Computer Scientist Should Know at ISTA 2023 This repo contains LaTeX transcription of the classes. -There are also [Lean transcriptions of the classes](https://github.com/madvorak/fecssk). +There are also [Lean formalizations of the classes](https://github.com/madvorak/fecssk). ## Contributions diff --git a/class_6.tex b/class_6.tex new file mode 100644 index 0000000..09b5ee5 --- /dev/null +++ b/class_6.tex @@ -0,0 +1,78 @@ +{{%Localize command definitions + +\chapter{Class 6} +% ChatGPT was used when working on this transcript. + +Formal system $F$ is a set of rules. +Rule is a finite set of (formulas) premises $p_0, \dots, p_k$ and (a formula called) conclusion $c$. +We usually have infinitely many rules but only finitely many different rule schemata. +For example, schema $\phi \rightarrow \phi$ gives infinitely many rules like $p_3 \rightarrow p_3$. +Axiom is a rule without premises.\bigskip\\ +Proof (derivation) is a finite sequence of formulas $\phi_0, \dots, \phi_n$ such that every formula in the sequence is +\begin{itemize} +\item either an axiom (which can be viewed as a special case of the following); +\item or the conclusion of a rule whose premises occur earlier in the sequence. +\end{itemize} +This is a linear view. + +Linear view is usually easier for proving meta theorems. +Tree view (inductive definition) is usually better in practice. + +Theorem is a formula that occurs in a proof. We distinguish the following: +\begin{itemize} +\item $\vdash \phi \;\dots\;$ ``$\phi$ is a theorem (of the formal system $F$)'' (has a proof) [syntax] +\item $\vDash \phi \;\dots\;$ ``$\phi$ is valid ($\phi$ is tautology)'' (is true in all models) [semantics] +\end{itemize} +Formal system equipped with semantics is called a logic. +Most of logic is about establishing $\vdash \phi$ iff $\vDash \phi$. + +Rule $R$ is sound iff [if all premises of $R$ are valid, then the conclusion of $R$ is valid]. +Formal system $F$ is sound iff all rules are sound (or equivalently, every theorem is valid). +Formal system $F$ is complete iff every valid formula is a theorem. +Formal system $F$ is consistent unless $\vdash \bot$ (or equivalently, there exists a formula that is not a theorem). +Rule $R$ is derivable in $F$ iff [for all formulas $\phi$, $\underset{F \cup \{R\}}\vdash \phi$ iff $\underset{F}\vdash \phi$]. +Rule $R$ is admissible in $F$ iff $F \cup \{R\}$ is still consistent. +Formula $\phi$ is expressible in a logic $L$ iff [there exists a formula $\psi$ of $L$ such that, for all interpretations $v$, $[[ \phi ]]_v = [[ \psi ]]_v$. %`⟦φ⟧ᵥ = ⟦ψ⟧ᵥ`]. +For example $\phi_1 \wedge \phi_2$ is expressible using only $\neg$ and $\vee$ (de Morgan) +as $\psi = \neg(\neg\phi_1 \wedge \neg\phi_2)$. + +We can enumerate all theorems by systematically enumerating all possible proofs. +The proof is a witness for validity. +Sound formal system gives a sound procedure for validity (but not necessarily complete). +Sound complete formal system gives a sound semi-complete procedure for validity (may not terminate on inputs +that represent a formula that is not valid). +To get a decision procedure (sound and complete procedure for validity), we need both +(1) sound complete formal system for validity, and +(2) sound complete formal system for satisfiability (to define a formal system for satisfiability, +replace ``formulas'' ($\phi$ is valid) by ``judgements'' ($\phi$ is satisfiable); +all axioms are satisfiable, all rules go from satisfiables to satisfiable). +For every input $\phi$, one of them will eventually terminate. +Conclude; either $\phi$ is valid, or $\neg\phi$ is satisfiable (which means that $\phi$ is not valid). +Recall that, if both a set and its complement are recursively-enumerable, the set is recursive (decidable). + +Example (formal system for unsatisfiability): +$$\inferrule +{\Gamma[\bot]\qquad\qquad\Gamma[\top]} +{\Gamma[p]} +$$ + +\section{Hilbert formal system for propositional logic} + +Hilbert system uses connectives $\rightarrow$ and $\neg$ only. +Hilbert system has three axioms and one rule -- modus ponens (MP): +$$\inferrule +{\phi\qquad\qquad\qquad\phi\rightarrow\psi} +{\psi} +$$ +Axioms: +\begin{itemize} +\item (K): $\qquad +\phi\rightarrow\psi\rightarrow\phi$ +\item (S): $\qquad +(\phi\rightarrow\psi\rightarrow\chi)\rightarrow((\phi\rightarrow\psi)\rightarrow(\phi\rightarrow\chi))$ +\item (EM): $\qquad +(\neg\phi\rightarrow\neg\psi)\rightarrow(\psi\rightarrow\phi)$ +\end{itemize} +TODO + +}} % End localization of command definitions diff --git a/main.pdf b/main.pdf index 26d5564a62e08454efa596fb12d84b46e9b604b7..914412e48ab15dc5ca9595c5596cc0f80b334a4f 100644 GIT binary patch literal 308484 zcmcG01z1(v);8VUAV`BCwKugvK)R8X?vU;-X{5VBP+B^qrAxX6LAphxQ{vy?@#r~v zujjk>{ydLM_GXPa#vJ3Fqvl+TT25G$5x~TbL_Iv-IfcYV#zJPTZ-&IfL&mH~#w=#7 zWKG5_Om@$}M9;>-$c~-~NJqxUhh$`Bcykx$A9vr=WaZ=l|KrqTVPO%vxmes>^lmQp ze_k|M*jR2p0O-EagZ>3Kdm{q}Q)??G7AC-t>;QmY@l(1P)B4ZLEob{bSpa~)rfAgK=&E*G{9KU7B`j=^O{fec`&7ij$bh!D=o3x?{t{!qRTt}@>SFy@ zfcujl`;XdU{UzRh%a7wbzu%+nuMB^QHt|38dW-E_Sl=4O?&c4-D*a}HCx7bv_HULq zA8)O7OZn}tTVvj?nA_iON4OpE_Wo~o{cWsoU*DSbcJy1@ez)@NShuGB4xrmxZY}?_ z6t^PX-eCOa1P%UF9oxTHNg8Z7AW$G<7BzLy`oZ0?eOpRAJV?yaMpniSCSNZ820+Rz9I>@4ja?2Pm*kzA$&<@aiV96>J|f-bSG3-75(28^cR`L5=x zXGzQV$PW?>^rAdwbnB9eq5lMjg&b(v$eu;gBjt#WQXa#`915X9i~!-#@_d28j>kCx zA;8&PR-gT-(SxLht4De6;T zO_Wwabs!R(gS#l(!6V&6!*>8fM}+Vh8N(?Zjdl)` 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