From 0e4ccf58f4c989fbd09cc9b5047cead740b70eec Mon Sep 17 00:00:00 2001 From: Janis Hutz Date: Tue, 5 May 2026 14:39:56 +0200 Subject: [PATCH] [FMFP] Axiomatic semantics --- ...methods-functional-programming-summary.pdf | Bin 580037 -> 600914 bytes ...methods-functional-programming-summary.tex | 10 +- .../03_rules-summary.tex | 184 ++++++++++++++++++ .../02_axiomatic-semantics/00_intro.tex | 45 +++++ .../01_hoare-logic/00_triples-assertions.tex | 17 ++ .../01_hoare-logic/01_derivation-systems.tex | 78 ++++++++ .../01_hoare-logic/02_total-correctness.tex | 12 ++ .../02_soundness-completeness.tex | 27 +++ 8 files changed, 370 insertions(+), 3 deletions(-) create mode 100644 semester4/fmfp/parts/03_language-semantics/01_operational-semantics/03_rules-summary.tex create mode 100644 semester4/fmfp/parts/03_language-semantics/02_axiomatic-semantics/00_intro.tex create mode 100644 semester4/fmfp/parts/03_language-semantics/02_axiomatic-semantics/01_hoare-logic/00_triples-assertions.tex create 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zk5ySh*$_aoj2&Q9L-Y?VGc~y=P^`odHrR}fVxsmrzdx)X0#-4r1hS$suOvU}oQ9SL LL{84k&K&YD-ipWa diff --git a/semester4/fmfp/formal-methods-functional-programming-summary.tex b/semester4/fmfp/formal-methods-functional-programming-summary.tex index 908fb68..97f6557 100644 --- a/semester4/fmfp/formal-methods-functional-programming-summary.tex +++ b/semester4/fmfp/formal-methods-functional-programming-summary.tex @@ -5,6 +5,7 @@ \usepackage{lmodern} \usepackage{ebproof} \usepackage{overarrows} +\usepackage{bm} \NewOverArrowCommand{\overrightharpoon}{% end=\rightharpoonup @@ -114,7 +115,6 @@ \input{parts/03_language-semantics/01_operational-semantics/00_big-step-semantics/05_unfolding-loops.tex} \input{parts/03_language-semantics/01_operational-semantics/00_big-step-semantics/06_deterministic-semantics.tex} \input{parts/03_language-semantics/01_operational-semantics/00_big-step-semantics/07_extensions-of-imp.tex} -% \input{parts/03_language-semantics/01_operational-semantics/00_big-step-semantics/} \input{parts/03_language-semantics/01_operational-semantics/01_small-step-semantics/00_structural-operational-semantics.tex} \input{parts/03_language-semantics/01_operational-semantics/01_small-step-semantics/01_rules.tex} \input{parts/03_language-semantics/01_operational-semantics/01_small-step-semantics/02_multi-step-derivation-seq.tex} @@ -122,9 +122,13 @@ \input{parts/03_language-semantics/01_operational-semantics/01_small-step-semantics/04_proofs.tex} \input{parts/03_language-semantics/01_operational-semantics/01_small-step-semantics/05_semantic-equivalence.tex} \input{parts/03_language-semantics/01_operational-semantics/01_small-step-semantics/06_extensions.tex} -% \input{parts/03_language-semantics/01_operational-semantics/01_small-step-semantics/} \input{parts/03_language-semantics/01_operational-semantics/02_equiv.tex} -% \input{parts/03_language-semantics/01_operational-semantics/} +\input{parts/03_language-semantics/01_operational-semantics/03_rules-summary.tex} +\input{parts/03_language-semantics/02_axiomatic-semantics/00_intro.tex} +\input{parts/03_language-semantics/02_axiomatic-semantics/01_hoare-logic/00_triples-assertions.tex} +\input{parts/03_language-semantics/02_axiomatic-semantics/01_hoare-logic/01_derivation-systems.tex} +\input{parts/03_language-semantics/02_axiomatic-semantics/01_hoare-logic/02_total-correctness.tex} +\input{parts/03_language-semantics/02_axiomatic-semantics/02_soundness-completeness.tex} % \input{parts/03_language-semantics/} diff --git a/semester4/fmfp/parts/03_language-semantics/01_operational-semantics/03_rules-summary.tex b/semester4/fmfp/parts/03_language-semantics/01_operational-semantics/03_rules-summary.tex new file mode 100644 index 0000000..2143887 --- /dev/null +++ b/semester4/fmfp/parts/03_language-semantics/01_operational-semantics/03_rules-summary.tex @@ -0,0 +1,184 @@ +\newpage +\subsubsection{Operational Semantics Rules Overview} +\paragraph{Big-Step Semantics} +\[ + \begin{prooftree} + \infer0[\textsc{Skip}$_{NS}$]{\langle \texttt{skip}, \sigma \rangle \rightarrow \sigma} + \end{prooftree} + \qquad + \begin{prooftree} + \infer0[\textsc{Ass}$_{NS}$]{\langle x := e, \sigma \rangle \rightarrow \sigma[x \mapsto \cA\llbracket e \rrbracket \sigma ]} + \end{prooftree} +\] + +\[ + \begin{prooftree} + \hypo{\langle s, \sigma \rangle \rightarrow \sigma'} + \hypo{\langle s', \sigma' \rangle \rightarrow \sigma''} + \infer2[\textsc{Seq}$_{NS}$]{\langle s;s', \sigma \rangle \rightarrow \sigma''} + \end{prooftree} +\] + +\[ + \begin{prooftree} + \hypo{\langle s, \sigma \rangle \rightarrow \sigma'} + \infer1[\textsc{IfT}$_{NS}$]{\langle \texttt{if}\; b \; \texttt{then} \; s \; \texttt{else} \; s' \; \texttt{end}, \sigma \rangle \rightarrow \sigma'} + \end{prooftree} + \qquad + \begin{prooftree} + \hypo{\langle s, \sigma \rangle \rightarrow \sigma'} + \infer1[\textsc{IfF}$_{NS}$]{\langle \texttt{if}\; b \; \texttt{then} \; s \; \texttt{else} \; s' \; \texttt{end}, \sigma \rangle \rightarrow \sigma'} + \end{prooftree} +\] + +\begin{align*} + \begin{prooftree} + \hypo{\langle s, \sigma \rangle \rightarrow \sigma'} + \hypo{\langle \texttt{while} \; b \; \texttt{do} \; s \; \texttt{end}, \sigma' \rangle \rightarrow \sigma''} + \infer2[\textsc{WhT}$_{NS}$]{\langle \texttt{while} \; b \; \texttt{do} \; s \; \texttt{end}, \sigma \rangle \rightarrow \sigma''} + \end{prooftree} + & & + \begin{prooftree} + \infer0[\textsc{WhF}$_{NS}$]{\langle \texttt{while} \; b \; \texttt{do} \; s \; \texttt{end}, \sigma \rangle \rightarrow \sigma} + \end{prooftree} \\ + \text{if } + \cB\llbracket b \rrbracket \sigma = \texttt{tt} + & & + \text{if } + \cB\llbracket b \rrbracket \sigma = \texttt{ff} +\end{align*} + +\[ + \begin{prooftree} + \infer0[\textsc{Ass}$_{NS}$]{\langle x := e, \sigma \rangle \rightarrow \sigma[x \mapsto \cA\llbracket e \rrbracket \sigma]} + \end{prooftree} + \qquad + \begin{prooftree} + \infer0[\textsc{Ass}$_{NS}$]{\langle v := v + 1, \sigma_\text{zero} \rangle \rightarrow \sigma[v \mapsto \cA\llbracket v + 1 \rrbracket \sigma_\text{zero}]} + \end{prooftree} +\] + +\paragraph{Small-Step Semantics} +\[ + \begin{prooftree} + \infer0[\textsc{Skip}$_{SOS}$]{\langle \texttt{skip}, \sigma \rangle \rightarrow_1 \sigma} + \end{prooftree} + \qquad + \begin{prooftree} + \infer0[\textsc{Ass}$_{SOS}$]{\langle x := e, \sigma \rangle \rightarrow_1 \sigma[x \mapsto \cA\llbracket e \rrbracket \sigma ]} + \end{prooftree} +\] + +\[ + \begin{prooftree} + \hypo{\langle s, \sigma \rangle \rightarrow_1 \sigma'} + \infer1[\textsc{Seq1}$_{SOS}$]{\langle s;s', \sigma \rangle \rightarrow_1 \langle s', \sigma' \rangle} + \end{prooftree} + \qquad + \begin{prooftree} + \hypo{\langle s', \sigma' \rangle \rightarrow_1 \langle s'', \sigma' \rangle} + \infer1[\textsc{Seq2}$_{SOS}$]{\langle s;s', \sigma \rangle \rightarrow_1 \langle s'';s', \sigma' \rangle} + \end{prooftree} +\] + +\[ + \begin{prooftree} + \infer0[\textsc{IfT}$_{SOS}$]{\langle \texttt{if}\; b \; \texttt{then} \; s \; \texttt{else} \; s' \; \texttt{end}, \sigma \rangle \rightarrow_1 \langle s', \sigma' \rangle} + \end{prooftree} + \qquad + \begin{prooftree} + \infer0[\textsc{IfF}$_{SOS}$]{\langle \texttt{if}\; b \; \texttt{then} \; s \; \texttt{else} \; s' \; \texttt{end}, \sigma \rangle \rightarrow_1 \langle s', \sigma' \rangle} + \end{prooftree} +\] + +\[ + \begin{prooftree} + \hypo{\langle s, \sigma \rangle \rightarrow_1 \sigma'} + \infer1[\textsc{IfT1}$_{SOS}$]{\langle \texttt{if}\; b \; \texttt{then} \; s \; \texttt{else} \; s' \; \texttt{end}, \sigma \rangle \rightarrow_1 \sigma'} + \end{prooftree} + \qquad + \begin{prooftree} + \hypo{\langle s, \sigma \rangle \rightarrow_1 \langle s'', \sigma' \rangle} + \infer1[\textsc{IfT2}$_{SOS}$]{\langle \texttt{if}\;b \; \texttt{then} \; s \; \texttt{else} \; s' \; \texttt{end}, \sigma \rangle \rightarrow_1 \langle s''', \sigma' \rangle} + \end{prooftree} +\] + +\[ + \begin{prooftree} + \infer0[\textsc{While}$_{SOS}$]{\langle \texttt{while} \; b \; \texttt{do} \; s \; \texttt{end}, \sigma \rangle \rightarrow_1 + \langle \texttt{if} \; b \; \texttt{then} \; s; \; \texttt{while} \; b \; \texttt{do} \; s \; \texttt{end else skip end}, \sigma \rangle} + \end{prooftree} +\] + +\paragraph{Extensions} +\subparagraph{Big-Step Semantics} +\[ + \begin{prooftree} + \hypo{\langle s, \sigma[x \mapsto \cA \llbracket e \rrbracket \sigma] \rangle \rightarrow \sigma' } + \infer1[\textsc{Loc}$_{NS}$]{\langle \texttt{var x := e in s end}, \sigma \rangle \rightarrow \sigma'[x \mapsto \sigma(x)]} + \end{prooftree} + \qquad + \begin{prooftree} + \hypo{\langle s, \sigma_\text{zero}[\overrightharpoon{x_i} \mapsto \overrightharpoon{\cA \llbracket e_i \rrbracket \sigma}] + [\overrightharpoon{y_j} \mapsto \overrightharpoon{\sigma(z_j)}] \rangle \rightarrow \sigma'} + \infer1[\textsc{Call}$_{NS}$]{\langle p(\overrightharpoon{e_i}; \overrightharpoon{z_j}), \sigma \rangle + \rightarrow \sigma[\overrightharpoon{z_j} \mapsto \overrightharpoon{\sigma'(y_j)}]} + \end{prooftree} +\] + +\[ + \begin{prooftree} + \hypo{\langle s, \sigma \rangle \rightarrow \sigma'} + \infer1[\textsc{ND1}$_{NS}$]{\langle s \bigbox s', \sigma \rangle \rightarrow \sigma'} + \end{prooftree} + \qquad + \begin{prooftree} + \hypo{\langle s', \sigma \rangle \rightarrow \sigma'} + \infer1[\textsc{ND2}$_{NS}$]{\langle s \bigbox s', \sigma \rangle \rightarrow \sigma'} + \end{prooftree} +\] + + +\subparagraph{Small-Step Semantics} +\[ + \begin{prooftree} + \infer0[\textsc{Loc}$_{SOS}$]{\langle \texttt{var x:= e in s end}, \sigma \rangle \rightarrow_1 \langle \texttt{s;restore }(x, \sigma(x)), + \sigma[x \mapsto \cA \llbracket e \rrbracket \sigma] \rangle} + \end{prooftree} +\] +\[ + \begin{prooftree} + \infer0[\textsc{Ret}$_{SOS}$]{\langle \texttt{restore }(x, \sigma(x)), \sigma \rangle \rightarrow_1 \sigma[x \mapsto v]} + \end{prooftree} +\] +\[ + \begin{prooftree} + \infer0[\textsc{ND1}$_{SOS}$]{\langle s \bigbox s', \sigma \rangle \rightarrow_1 \langle s, \sigma \rangle} + \end{prooftree} + \qquad + \begin{prooftree} + \infer0[\textsc{ND2}$_{SOS}$]{\langle s \bigbox s', \sigma \rangle \rightarrow_1 \langle s, \sigma' \rangle} + \end{prooftree} +\] +\[ + \begin{prooftree} + \hypo{\langle s, \sigma \rangle \rightarrow_1 \langle s'', \sigma' \rangle} + \infer1[\textsc{Par1}$_{SOS}$]{\langle s \texttt{ par } s', \sigma \rangle \rightarrow_1 \langle s'' \texttt{ par } s', \sigma' \rangle} + \end{prooftree} + \qquad + \begin{prooftree} + \hypo{\langle s, \sigma \rangle \rightarrow_1 \sigma'} + \infer1[\textsc{Par2}$_{SOS}$]{\langle s \texttt{ par } s', \sigma \rangle \rightarrow_1 \langle s', \sigma' \rangle} + \end{prooftree} +\] +\[ + \begin{prooftree} + \hypo{\langle s', \sigma \rangle \rightarrow_1 \langle s'', \sigma' \rangle} + \infer1[\textsc{Par3}$_{SOS}$]{\langle s \texttt{ par } s', \sigma \rangle \rightarrow_1 \langle s \texttt{ par } s'', \sigma' \rangle} + \end{prooftree} + \qquad + \begin{prooftree} + \hypo{\langle s', \sigma \rangle \rightarrow_1 \sigma'} + \infer1[\textsc{Par4}$_{SOS}$]{\langle s \texttt{ par } s', \sigma \rangle \rightarrow_1 \langle s, \sigma' \rangle} + \end{prooftree} +\] diff --git a/semester4/fmfp/parts/03_language-semantics/02_axiomatic-semantics/00_intro.tex b/semester4/fmfp/parts/03_language-semantics/02_axiomatic-semantics/00_intro.tex new file mode 100644 index 0000000..ecb15a4 --- /dev/null +++ b/semester4/fmfp/parts/03_language-semantics/02_axiomatic-semantics/00_intro.tex @@ -0,0 +1,45 @@ +\newpage +\subsection{Axiomatic Semantics} +\subsubsection{Program Correctness} +\inlinedefinition[Partial Correctness] \textit{if} a program terminates \textit{then} there will be a certain relationship between the initial and final state + +\inlinedefinition[Total Correctness] Program terminates \textit{and} is partially correct, i.e.\\ +\texttt{total correctness = partial correctness + termination} + +\inlinedefinition[Formal Specification] is used to expressed the aforementioned relationship between the initial and final state. +It does not include guarantees for termination and can thus only be used to provide a starting point for a prove of partial correctness. + +We typically express \bi{Formal Specifications} using Small- or Big-Step semantics. + +\inlineexample Consider the factorial statement +\begin{code}{text} + y := 1; + while not x = 1 do + y := y * x; + x := x - 1 + end +\end{code} +The specification that we want to prove is here given by ``The final value of \texttt{y} is the factorial of the initial value \texttt{x}''. +Do note that this statement is only partially correct, as it does not terminate for $x < 1$. + +We can express the specification formally as follows using big-step semantics: +\[ + \forall \sigma, \sigma'. \vdash \langle y := 1; \texttt{while not x = 1 do y := y * x; x := x - 1 end}, \sigma \rangle \rightarrow \sigma' + \implies \sigma'(\texttt{y}) = \sigma(\texttt{x})! \land \sigma(\texttt{x}) > 0 +\] + +% We then prove the partial correctness in three steps: +% +% \bi{Step 1}: The body of the loop +% \[ +% \forall \sigma, \sigma''. \vdash \langle \texttt{y := y * x; x := x - 1}, \sigma \rangle \rightarrow \sigma'' \land \sigma''(\texttt{x}) > 0 +% \implies \sigma(\texttt{y}) \times \sigma(\texttt{x})! = \sigma''(\texttt{y}) \times \sigma''(\texttt{x})! \land \sigma(\texttt{x}) > 0 +% \] +% +% \bi{Step 2}: The loop satisfies +% \[ +% \forall \sigma, \sigma''. \vdash \langle \texttt{y := y * x; x := x - 1}, \sigma \rangle \rightarrow \sigma'' \land \sigma''(\texttt{x}) > 0 +% \implies \sigma(\texttt{y}) \times \sigma(\texttt{x})! = \sigma''(\texttt{y}) \times \sigma''(\texttt{x})! \land \sigma(\texttt{x}) > 0 +% \] +Since these kinds of proofs take a lot of effort and get very complicated rather quickly, axiomatic semantics provide a nice and fast way of proving correctness, +because we can focus on the essential properties. diff --git a/semester4/fmfp/parts/03_language-semantics/02_axiomatic-semantics/01_hoare-logic/00_triples-assertions.tex b/semester4/fmfp/parts/03_language-semantics/02_axiomatic-semantics/01_hoare-logic/00_triples-assertions.tex new file mode 100644 index 0000000..8ffd66f --- /dev/null +++ b/semester4/fmfp/parts/03_language-semantics/02_axiomatic-semantics/01_hoare-logic/00_triples-assertions.tex @@ -0,0 +1,17 @@ +\newpage +\subsubsection{Hoare Logic} +\paragraph{Hoare Triples} +\inlinedefinition Properties are specified as \bi{Hoare Triples} $\{ \bm{P} \} \ s \ \{ \bm{Q} \}$, with statement $s$, $\bm{P}$ the precondition and $\bm{Q}$ the postcondition. +Here, $\bm{P}, \bm{Q}$ are assertions and are, together with $\bm{R}$, meta-variables over assertions. + +\inlinedefinition[Logical Variables] To keep the original value of a variable, we can ``reassign'' values in the precondition +(e.g. $x = N$ to then in the postcondition state something regarding $N$). + +Pre- and Postconditions are assertions, i.e. they are boolean expressions plus logical variables. +Often, quantification is used in assertions as well in practice, as well as other expressions such as $x!$ when it is convenient and we also assume that the \bi{substitution lemma} +still holds: +\[ + \cB \llbracket \bm{P}[x \mapsto e] \rrbracket \sigma = \cB \llbracket \bm{P} \rrbracket \sigma [x \mapsto \cA \llbracket e \rrbracket \sigma] +\] + +We use $P_1 \land P_2$ instead of $P_1 \texttt{and} P_2$, $P_1 \lor P_2$ instead of $P_1 \texttt{or} P_2$ and $\neg P$ instead of $\texttt{not} P$ diff --git a/semester4/fmfp/parts/03_language-semantics/02_axiomatic-semantics/01_hoare-logic/01_derivation-systems.tex b/semester4/fmfp/parts/03_language-semantics/02_axiomatic-semantics/01_hoare-logic/01_derivation-systems.tex new file mode 100644 index 0000000..6f42df8 --- /dev/null +++ b/semester4/fmfp/parts/03_language-semantics/02_axiomatic-semantics/01_hoare-logic/01_derivation-systems.tex @@ -0,0 +1,78 @@ +\paragraph{Derivation Systems} +We again use derivation trees, where their rules specify which triples can be derived for each statement. +The premises and conclusions of the derivation rules are Hoare Triples. + +Again, we write $\vdash \{ \bm{P} \} \ s \ \{ \bm{Q} \}$ if there exists a (finite) derivation tree ending in $\{ \bm{P} \} \ s \ \{ \bm{Q} \}$, and +\[ + \vdash \{ \bm{P} \} \ s \ \{ \bm{Q} \} \Leftrightarrow \exists T. \texttt{root}(T) \equiv \{ \bm{P} \} \ s \ \{ \bm{Q} \} +\] + + +\subparagraph{The rules} +\[ + \begin{prooftree} + \infer0[$\textsc{Skip}_{Ax}$]{\{ \bm{P} \} \ \texttt{skip} \ \{ \bm{P} \}} + \end{prooftree} + \qquad + \begin{prooftree} + \infer0[$\textsc{Ass}_{Ax}$]{\{ \bm{P}[x \mapsto e] \} \ \texttt{x := e} \ \{ \bm{P} \}} + \end{prooftree} +\] + +Sequential Composition \texttt{s;s'}, loop (\texttt{while b do s end}) and conditional statements (\texttt{if b then s1 else s2 end}) +\[ + \begin{prooftree} + \hypo{\{ \bm{P} \} \ s \ \{ \bm{Q} \}} + \hypo{\{ \bm{Q} \} \ s' \ \{ \bm{R} \}} + \infer2[$\textsc{Seq}_{Ax}$]{\{ \bm{P} \} \ \texttt{s;s'} \ \{ \bm{P} \}} + \end{prooftree} + \qquad + \begin{prooftree} + \hypo{\{ b \land \bm{P} \} \ s \ \{ \bm{Q} \}} + \hypo{\{ \neg b \land \bm{P} \} \ s' \ \{ \bm{Q} \}} + \infer2[$\textsc{If}_{Ax}$]{\{ \bm{P} \} \ \texttt{if b then s else s' end} \ \{ \bm{Q} \}} + \end{prooftree} +\] +\[ + \begin{prooftree} + \hypo{\{ b \land \bm{P} \} \ s \ \{ \bm{P} \}} + \infer1[$\textsc{Wh}_{Ax}$]{\{ \bm{P} \} \ \texttt{while b do s end} \ \{ \neg b \land \bm{P} \}} + \end{prooftree} +\] +With these rules, we can only evaluate \textit{syntactically}, thus expressions like $\{ x = 4 \land y = 5 \} \ \texttt{skip} \ \{ y = 5 \land x = 4 \}$ +are not an instance of the $\textsc{Skip}_{Ax}$ because the precondition and postcondition are not identical. Thus we often need to apply \textit{semantic} reasoning, +e.g. applying mathematical properties. + +\inlinedefinition[Semantic entailment] expresses the reasoning steps +``We write $\bm{P} \models \bm{Q}$ if and only if for all states $\sigma$, +$\cB \llbracket \bm{P} \rrbracket \sigma = \texttt{tt}$ implies $\cB \llbracket \bm{Q} \rrbracket \sigma = \texttt{tt}$'' + +This leads to the \bi{Rule of Consequence} +\[ + \begin{prooftree} + \hypo{\{ \bm{P}' \} \ s \ \{ \bm{Q}' \}} + \infer1[$\textsc{Cons}_{Ax}$]{\{ \bm{P} \} \ s \ \{ \bm{Q} \}} + \end{prooftree} + \qquad + \text{if } \bm{P} \models \bm{P}' \text{ and } \bm{Q}' \models \bm{Q} +\] +a rule, where we can \bi{strengthen preconditions} ($\bm{P}$ cannot be weaker than $\bm{P}'$) and \bi{weaken postconditions} ($\bm{Q}$ cannot be stronger than $\bm{Q}'$) + +Since the derivation trees often get quite large, we can group them around each line in the program text. + +\shade{gray}{Inline Notation} can be used to make the changes more easily legible. +For example, to express instances of $\textsc{Skip}_{Ax}$, instead of writing $\vdash \{ \bm{P} \} \ \texttt{skip} \ \{ \bm{P} \}$, we can write. +More examples on slides 167 - 170 (pages 30 - 33 in Slide Deck 4) +\rmvspace +\begin{align*} + & \{ \bm{P} \} \\ + & \quad \texttt{skip} \\ + & \{ \bm{P} \} +\end{align*} + +\shade{gray}{Forward Assignment} +\[ + \begin{prooftree} + \infer0[$\textsc{AssF}_{Ax}$]{\{ \bm{P} \} \ x := e \ \{ \exists V. \bm{P}[x \mapsto V] \land x = e[x \mapsto V] \}} + \end{prooftree} +\] diff --git a/semester4/fmfp/parts/03_language-semantics/02_axiomatic-semantics/01_hoare-logic/02_total-correctness.tex b/semester4/fmfp/parts/03_language-semantics/02_axiomatic-semantics/01_hoare-logic/02_total-correctness.tex new file mode 100644 index 0000000..0b482a0 --- /dev/null +++ b/semester4/fmfp/parts/03_language-semantics/02_axiomatic-semantics/01_hoare-logic/02_total-correctness.tex @@ -0,0 +1,12 @@ +\paragraph{Total Correctness} +We use a different form of Hoare triple $\{ \bm{P} \} \ s \ \{ \Downarrow \bm{Q} \}$, which describes total correctness. + +All rules for total correctness are equivalent to the ones for partial correctness, apart from the rule for loops. +\[ + \begin{prooftree} + \hypo{\{ b \land \bm{P} \land e = Z \} \ s \ \{ \Downarrow \bm{P} \land e < Z \}} + \infer1[$\textsc{WhTot}_{Ax}$]{\{ \bm{P} \} \ \texttt{while b do s end} \ \{ \Downarrow \neg b \land \bm{P} \}} + \end{prooftree} + \qquad + \text{if } b \land \bm{P} \models 0 \leq e +\] diff --git a/semester4/fmfp/parts/03_language-semantics/02_axiomatic-semantics/02_soundness-completeness.tex b/semester4/fmfp/parts/03_language-semantics/02_axiomatic-semantics/02_soundness-completeness.tex new file mode 100644 index 0000000..879a2db --- /dev/null +++ b/semester4/fmfp/parts/03_language-semantics/02_axiomatic-semantics/02_soundness-completeness.tex @@ -0,0 +1,27 @@ +\subsubsection{Soundness and Completeness} +\inlinedefinition[Soundness] If a property can be proven, then it holds. +As a result, an unsound derivation system does not provide any guarantees, as we may miss errors (we may have false negatives). + +\inlinedefinition[Completeness] If a property holds, then it can be proven. +As a result, we may have a correct program that we can't prove in an incomplete derivation system (we may have false positives) + +Both can be proven with regard to an operational semantics. + +\inlineexample The partial correctness triple $\{ \bm{P} \} \ s \ \{ \bm{Q} \}$ is valid, written $\models \{ \bm{P} \} \ s \ \{ \bm{Q} \}$ if and only if +\[ + \forall \sigma, \sigma'. \cB \llbracket \bm{P} \rrbracket \sigma = \texttt{tt} \land \vdash \langle s, \sigma \rangle \rightarrow \sigma' + \implies \cB \llbracket \bm{Q} \rrbracket \sigma' = \texttt{tt} +\] + +More generally: +\begin{itemize} + \item \bi{Soundness} $\vdash \{ \bm{P} \} \ s \ \{ \bm{Q} \} \implies \models \{ \bm{P} \} \ s \ \{ \bm{Q} \}$ + \item \bi{Soundness} $\models \{ \bm{P} \} \ s \ \{ \bm{Q} \} \implies \vdash \{ \bm{P} \} \ s \ \{ \bm{Q} \}$ +\end{itemize} + +\begin{theorem}[]{Soundness, Completeness} + For all partial correctness triples $\{ \bm{P} \} \ s \ \{ \bm{Q} \}$ of IMP we have + \[ + \vdash \{ \bm{P} \} \ s \ \{ \bm{Q} \} \Leftrightarrow \models \{ \bm{P} \} \ s \ \{ \bm{Q} \} + \] +\end{theorem}