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\subsection{Formal proofs}
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Given a language like $\cL = \{ \oplus, \otimes, +, \times \}$, and derivation rules
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\begin{multicols}{2}
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\begin{itemize}
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\item $\alpha$: If $+$, then $\otimes$
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\item $\beta$: If $+$, then $\times$
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\item $\gamma$: If $\otimes$ and $\times$, then $\oplus$
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\item $\delta$: $+$ holds
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\end{itemize}
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or displayed using graphical notation:
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\begin{align*}
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\frac{+}{\otimes} \; \alpha
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\qquad \frac{+}{\times} \; \beta \\
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\frac{\otimes \quad \times}{\oplus} \; \gamma
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\qquad \frac{}{+} \; \delta
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\end{align*}
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\end{multicols}
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Rules like $\delta$ above are also commonly referred to as an \textit{axiom}.
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To prove $\oplus$ in this language, we can either write the following or draw a derivation tree:
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\begin{multicols}{2}
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\begin{itemize}
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\item $+$ holds by $\delta$
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\item $\otimes$ holds by $\alpha$ with 1.
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\item $\times$ holds by $\beta$ with 1.
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\item $\oplus$ holds by $\gamma$ with 2 and 3
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\end{itemize}
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Or as derivation tree
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\[
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\begin{prooftree}
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% Left branch
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\infer0[$\delta$]{+}
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\infer1[$\alpha$]{\otimes}
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% Right branch
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\infer0[$\delta$]{+}
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\infer1[$\beta$]{\times}
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\infer2[$\gamma$]{ \oplus }
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\end{prooftree}
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\]
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\end{multicols}
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\subsection{Natural deduction}
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The rules from above here are used to construct derivations under assumptions, e.g.
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$A_1, \ldots, A_n \vdash A$, which is read as ``$A$ follows from $A_1, \ldots, A_n$''.
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The derivations are always represented as derivation trees and a \bi{proof} is a derivation whose root has no assumptions.
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Since we have to prove a statement, we have to draw the derivation trees from the bottom up, with the goal of reaching an axiom or a rule that is an assumption
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using the other rules of the rule set.
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\subsection{Propositional logic}
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\subsubsection{Syntax}
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\inlinedefinition For a set of variables $\cV$, the \bi{language of propositional logic} $\cL_P$ is the smallest set where
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\begin{multicols}{2}
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\begin{itemize}
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\item $X \in \cL_P$ if $X \in \cV$
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\item $\bot \in \cL_P$
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\item $A \land B \in \cL_P$ if $A \in \cL_P$ and $B \in \cL_P$
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\item $A \lor B \in \cL_P$ if $A \in \cL_P$ and $B \in \cL_P$
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\item $A \rightarrow B \in \cL_P$ if $A \in \cL_P$ and $B \in \cL_P$
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\end{itemize}
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\end{multicols}
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\subsubsection{Semantics}
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\inlinedefinition[Valuation] $\sigma : \cV \rightarrow \{ \texttt{True}, \texttt{False} \}$ maps variables to truth values.
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They are the simple models (i.e. \textit{interpretations}). \bi{Valuations} is the set of valuations.
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\inlinedefinition[Satisfiability] smallest relation $\models \; \subseteq$ \textit{Valuations} $\times \cL_P$ such that
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\begin{multicols}{2}
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\begin{itemize}
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\item $\sigma \models X$ if $\sigma(X) = \texttt{True}$
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\item $\sigma \models A \land B$ if $\sigma \models A$ and $\sigma \models B$
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\item $\sigma \models A \lor B$ if $\sigma \models A$ or $\sigma \models B$
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\item $\sigma \models A \rightarrow B$ if whenever $\sigma \models A$ then $\sigma \models B$
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\end{itemize}
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\end{multicols}
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\inlinedefinition[satisfiable formula] A formula $A \in \cL_P$ is \bi{satisfiable} if $\sigma \models A$ for \textbf{some} valuation $\sigma$
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\inlinedefinition[tautology, valid formula] A formula $A \in \cL_P$ is \bi{valid} (a \bi{tautology}) if $\sigma \models A$ for \textbf{all} valuations $\sigma$
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\inlinedefinition[Semantic entailment] $A_1, \ldots , A_n \models A$ if $\forall \sigma$ we have $\sigma \models A_1, \ldots, \sigma \models A_n$, then $\sigma \models A$
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\newpage
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\subsubsection{Requirements for a deductive system}
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The derivation rules (syntactic entailment) and truth tables (semantic entailment) should agree.
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For that we have two requirements, for $\Gamma = A_1, \ldots, A_n$ a collection of formulas:
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\begin{itemize}
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\item \bi{Soundness:} If $\Gamma \vdash A$ can be derived, then $\Gamma \models A$
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\item \bi{Completeness:} If $\Gamma \models A$, then $\Gamma \vdash A$ can be derived
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\end{itemize}
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\bi{Decidability} is also desirable, i.e. having checks of the attributes be of low complexity.
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\subsubsection{Natural deduction for propositional formulas}
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\inlinedefinition[Sequent] Is an assertion of form $A_1, \ldots, A_n \vdash A$, with $A, A_1, \ldots, A_n$ being propositional formulas.
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\inlineintuition $A$ follows from the $A_i$ and if the system is sound, the $A_i$ semantically entail $A$.
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\inlinedefinition[Axiom] is the starting point (usually the leaves) of the derivation trees and are usually of the form
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\[
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\begin{prooftree}
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\infer0[axiom]{\ldots, A, \ldots \vdash A}
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\end{prooftree}
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\]
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i.e. when coming up with a derivation tree for a \bi{proof}, we want to reach a leaf where $A$ is contained in $\Gamma$.
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\inlinedefinition[Proof] of $A$ is a derivation tree with \texttt{root} $\vdash A$. If a deductive system is \textit{sound}, then $A$ is a tautology.
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There are two kinds of rules, \bi{introduce} and \bi{eliminate} connectives. If you are confused about the order when applying them when coming up with a deduction tree,
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they are oriented top-down, so e.g. the introduction rule is inverted when coming up with the deduction tree.
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If all rules are sound (i.e. they preserve semantic entailment), then the logic is sound.
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\subsubsection{Derivation rules for propositional logic}
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Remember that $E$ means \textit{elimination} and $I$ means \textit{introduction}, with $L$ and $R$ being the side (so $ER$ means elimination on the right)
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\paragraph{Conjunction}
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\begin{align*}
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\begin{prooftree}
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\hypo{\Gamma \vdash A}
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\hypo{\Gamma \vdash B}
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\infer2[$\land$-I]{\Gamma \vdash A \land B}
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\end{prooftree}
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\qquad
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\begin{prooftree}
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\hypo{\Gamma \vdash A \land B}
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\infer1[$\land$-EL]{\Gamma \vdash A}
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\end{prooftree}
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\qquad
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\begin{prooftree}
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\hypo{\Gamma \vdash A \land B}
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\infer1[$\land$-ER]{\Gamma \vdash B}
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\end{prooftree}
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\end{align*}
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\paragraph{Disjunction}
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\begin{align*}
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\begin{prooftree}
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\hypo{\Gamma \vdash A}
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\infer1[$\lor$-IL]{\Gamma \vdash A \lor B}
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\end{prooftree}
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\qquad
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\begin{prooftree}
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\hypo{\Gamma \vdash B}
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\infer1[$\lor$-IR]{\Gamma \vdash A \lor B}
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\end{prooftree}
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\qquad
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\begin{prooftree}
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\hypo{\Gamma \vdash A \lor B}
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\hypo{\Gamma, A \vdash C}
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\hypo{\Gamma, B \vdash C}
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\infer3[$\lor$-E]{\Gamma \vdash C}
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\end{prooftree}
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\end{align*}
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\paragraph{Implication}
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\begin{align*}
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\begin{prooftree}
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\hypo{\Gamma, A \vdash B}
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\infer1[$\rightarrow$-I]{\Gamma \vdash A \rightarrow B}
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\end{prooftree}
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\qquad
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\begin{prooftree}
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\hypo{\Gamma \vdash A \rightarrow B}
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\hypo{\Gamma \vdash A}
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\infer2[$\rightarrow$-E]{\Gamma \vdash B}
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\end{prooftree}
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\end{align*}
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\paragraph{Others}
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\begin{align*}
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\begin{prooftree}
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\hypo{\Gamma \vdash \bot}
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\infer1[$\bot$-E]{\Gamma \vdash A}
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\end{prooftree}
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\qquad
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\begin{prooftree}
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\hypo{\Gamma \vdash \neg A}
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\hypo{\Gamma \vdash A}
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\infer2[$\neg$-E]{\Gamma \vdash B}
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\end{prooftree}
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\qquad
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\begin{prooftree}
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\hypo{\Gamma, \neg A \vdash \bot}
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\infer1[RAA]{\Gamma \vdash A}
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\end{prooftree}
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\qquad
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\begin{prooftree}
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\infer0[axiom]{\ldots, A, \ldots \vdash A}
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\end{prooftree}
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\end{align*}
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\newsection
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\subsection{First-Order Logic}
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\subsubsection{Syntax}
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\inlinedefinition[Signature] consists of a set of function symbols $\cF$ and a set of predicate symbols $\cP$, as well as their arities.
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We write $f^k$ (or $p^k$, for predicates) to indicate that it has \textit{arity} $k \in \N$.
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Constant functions have arity $0$, linear functions have arity $1$, thus, the arity of a given function (or predicate) is given by
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the number of parameters to uniquely describe it, minus one.
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\inlinedefinition[Term] is the terms of first-order logic and is the smallest set, where (with $\cV$ again a set of variables)
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\begin{enumerate}
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\item $x \in \textit{Term}$ if $x \in \cV$
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\item $f^n(t_1, \ldots, t_n) \in \textit{Term}$ if $f^n \in \cF$ and $t_i \in \textit{Term}$, $\forall 1 \leq i \leq n$ (description of form of formulas)
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\end{enumerate}
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\inlinedefinition[Form] is the formulas of first-order logic, is the smallest set where
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\begin{enumerate}
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\item $\bot \in \textit{Form}$
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\item $p^n(t_1, \ldots, t_n) \in \textit{Form}$ if $p^n \in \cP$ and $t_j \in \textit{Term}$, $\forall 1\leq j \leq n$ (description of form of predicates)
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\item $A \circ B \in \textit{Form}$ if $A \in \textit{Form}$, $B \in \textit{Form}$ and $\circ \in \{ \land, \lor, \rightarrow \}$ (i.e. formulas with logic symbols)
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\item $Qx.A \in \textit{Form}$ if $A \in \textit{Form}$, $x \in \cV$ and $Q \in \{ \forall, \exists \}$ (i.e. formulas with quantifiers)
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\end{enumerate}
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\paragraph{Binding}
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\inlinedefinition[Bound variable] A variable that occurs in a quantifier in scope ({\color{blue} blue in example below})
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\inlinedefinition[Free variable] A variable that is not bound by a quantifier in scope ({\color{red} red in example below})
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\inlineexample $(q({\color{red} x}) \lor \exists {\color{blue} x}.\forall {\color{blue} y}. p(f({\color{blue} x}), {\color{red} z}) \land q({\color{red} a}))
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\lor \forall {\color{blue} x}.r({\color{blue} x}, {\color{red} z}, g({\color{blue} x}))$
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\inlinedefinition[$\alpha$-conversion] A \bi{bound} variable can be renamed at any time, but must preserve binding structure.
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\paragraph{Precedences}
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$\neg > \land > \lor > \rightarrow$, with $>$ a total order of precedences.
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Quantifiers extend as far right as possible, bounded by the end of line or a going out of scope by closing parenthesis.
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\subsubsection{Semantics}
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\inlinedefinition[Structure] is a pair $\cS = \langle U_\cS, I_\cS \rangle$, where $U_\cS$ is the universe (non-empty set) and $I_\cS$ is a mapping:
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\begin{enumerate}
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\item $I_\cS(p^n)$ is an $n$-ary relation on $U_\cS$ for $p^n \in \cP$ (short $p^\cS$)
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\item $I_\cS(f^n)$ is an $n$-ary (total) function on $U_\cS$ for $f^n \in \cF$ short ($f^\cS$)
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\end{enumerate}
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\inlineintuition The $I_\cS$ is essentially assigning to each predicate and formula its definition in the universe of the structure, noted as a relation.
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\inlinedefinition[Interpretation] is a pair $\cI = \langle \cS, v \rangle$, with $v : \cV \rightarrow U_\cS$ a valuation\\
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\inlineintuition it assigns definitions to the formulas and predicates (through the structure), as well as values to the variables (through the valuation).
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\inlinedefinition[Value] of a term $t$ under $\cI$ is written as $\cI(t)$ and defined by $\cI(x) = v(x)$ for $x \in \cV$ and\\
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$\cI(f(t_1, \ldots, t_n)) = f^\cS(\cI(t_1), \ldots, \cI(t_n))$
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\inlinedefinition[Satisfiability] $\models \subseteq$ Interpretations $\times$ \textit{Form} is the smallest relation satisfying
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\begin{align*}
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& \langle \cS, v \rangle \models p(t_1, \ldots, t_n) & & \text{ if } (\cI(t_1), \ldots, \cI(t_n)) \in p^\cS \text{ where } \cI = \langle \cS, v \rangle \\
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& \langle \cS, v \rangle \models \forall x. A & & \text{ if } \langle \cS, v[x \mapsto a] \rangle \models A, \text{ for all } a \in \U_\cS \\
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& \langle \cS, v \rangle \models \exists x. A & & \text{ if } \langle \cS, v[x \mapsto a] \rangle \models A, \text{ for some } a \in \U_\cS \\
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\end{align*}
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\inlinedefinition[Model] When $\langle \cS, v \rangle \models A$, then $\langle \cS, v \rangle$ is a \bi{model} for $A$.
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If $A$ does not have free variables, the satisfaction does not depend on the valuation $v$ and we write $\cS \models A$
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\inlinedefinition[Validity] When every interpretation is a model, we write $\models A$, and we say that $A$ is \bi{valid}
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\inlinedefinition[Satisfiability] $A$ is \bi{satisfiable}, if there exists at least one model for $A$.
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\inlineexample Given $\forall x.p(x, s(x))$, we a model would be
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\begin{align*}
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U_\cS & = \N \\
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p^\cS & = \{ (m, n) \divider m, n \in U_\cS \text{ and } m < n \} \\
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s^\cS & = \text{ successor function on } U_\cS, \text{ i.e. } s^\cS(x) = x + 1
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\end{align*}
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\paragraph{Substitution}
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\inlinedefinition[Substitution] Replace all occurrences of a free variable $x$ with some term $t$ in $A$.
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To denote a substitution, we write $A[x \mapsto t]$. \hl{Important} All free variables in $t$ must still be free in $A[x \mapsto t]$.
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If that would not be true anymore, do a $\alpha$-conversion first.
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\subsubsection{Quantifiers}
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\paragraph{Universal quantification}
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Additional rules are needed for the universal quantifier. * side condition is that $x$ is not free in any assumption of $\Gamma$
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\begin{align*}
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\begin{prooftree}
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\hypo{\Gamma \vdash A}
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\infer1[$\forall$-I*]{\Gamma \vdash \forall x.A}
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\end{prooftree}
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\qquad
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\begin{prooftree}
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\hypo{\Gamma \vdash \forall x.A}
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\infer1[$\forall$-E]{\Gamma \vdash A[x \rightarrow t]}
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\end{prooftree}
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\end{align*}
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Again here be mindful not to capture free variables.
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\paragraph{Existential quantification}
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Additional rules are needed for the existential quantifier. ** side condition is that $x$ is neither free in B nor free in $\Gamma$
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\begin{align*}
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\begin{prooftree}
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\hypo{\Gamma \vdash A[x \mapsto t]}
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\infer1[$\exists$-I]{\Gamma \vdash \exists x.A}
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||||
\end{prooftree}
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||||
\qquad
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||||
\begin{prooftree}
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||||
\hypo{\Gamma \vdash \exists x.A}
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\hypo{\Gamma, A \vdash B}
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\infer2[$\exists$-E**]{\Gamma \vdash A[x \rightarrow t]}
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||||
\end{prooftree}
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||||
\end{align*}
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Again here be mindful not to capture free variables.
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\subsection{Equality}
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Since equality is such an important concept, it isn't just a predicate, but a separate First-Order Logic (FOL),
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called \bi{FOL with equality}.
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||||
The language is extended by $t_1 = t_2 \in \textit{Form}$ if $t_1, t_2 \in \textit{Term}$,
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the semantic entailment $\models$ is also extended by ``$\cI \models t_1 = t_2$ if $\cI(t_1) = \cI(t_2)$''.
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This definition is the exact intuition of equality of two terms, in that they are equal if their value under the interpretation is equal.
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Equality is an equivalence relation, so the following rules apply (ref = reflexivity, sym = symmetry, trans = transitivity):
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\begin{align*}
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\begin{prooftree}
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\infer0[ref]{\Gamma \vdash t = t}
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\end{prooftree}
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\qquad
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\begin{prooftree}
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\hypo{\Gamma \vdash t = s}
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\infer1[sym]{\Gamma \vdash s = t}
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||||
\end{prooftree}
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||||
\qquad
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||||
\begin{prooftree}
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||||
\hypo{\Gamma \vdash t = s}
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||||
\hypo{\Gamma \vdash s = r}
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\infer2[sym]{\Gamma \vdash s = r}
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||||
\end{prooftree}
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||||
\end{align*}
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||||
Equality is also a congruence on terms and (definable) relations
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\begin{align*}
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||||
\begin{prooftree}
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||||
\hypo{\Gamma \vdash t_1 = s_1}
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||||
\hypo{\dots}
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||||
\hypo{\Gamma \vdash t_n = s_n}
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||||
\infer3[$cong_1$]{\Gamma f(t_1, \ldots, t_n) = f(s_1, \ldots, s_n)}
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||||
\end{prooftree}
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||||
\\
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\begin{prooftree}
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||||
\hypo{\Gamma \vdash t_1 = s_1}
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||||
\hypo{\dots}
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||||
\hypo{\Gamma \vdash t_n = s_n}
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||||
\hypo{\Gamma \vdash p(t_1, \ldots, t_n)}
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||||
\infer4[$cong_2$]{\Gamma p(s_1, \ldots, s_n)}
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||||
\end{prooftree}
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||||
\end{align*}
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\newsection
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\subsection{Correctness}
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||||
For many programs, termination is an important aspect when talking about correctness,
|
||||
as is, of course, that the return value is ``correct''.
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||||
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\subsubsection{Termination}
|
||||
\inlinetheorem For a function $f$ is defined in terms of other functions $g_1, \ldots, g_k$, for all of which we have $g_i \neq f$ and each $g_i$ terminates, then so does $f$.
|
||||
|
||||
\inlinelemma Sufficient condition for termination: The arguments are smaller along a \bi{well-founded} order on the function's domain
|
||||
|
||||
\inlinedefinition[Well-Founded Order] An order $>$ on a set $S$ is \bi{well-founded} if and only if there is no infinite decreasing chain $x_1 > x_2 > \ldots$ for $x_i \in S$
|
||||
An example of such an order is $>$ on $\N$, denoted $>_\N$. Counter example: $>_\Z$ (not bounded from below)
|
||||
|
||||
|
||||
\inlinelemma Let $R \subseteq S \times S$ be a relation on $S$. Let $s_0, s_i \in S$ and $i \geq 1$.
|
||||
Then $s_0 \; R^i \; s_i$ if and only if $s_1, \ldots s_{i - 1} \in S$ such that $s_0 \; R \; s_1 \R \ldots R \; s_{i - 1} \; R \; s_i$
|
||||
|
||||
\inlinedefinition $R^+ \equiv \bigcup_{n \geq 1} R^n$, where $R^n \equiv R \circ R^{n - 1}$ for $n \geq 2$ and $R^1 \equiv R$
|
||||
|
||||
\inlinetheorem If $>$ is a well-founded order on $S$, then $>^+$ is also well-founded on $S$
|
||||
@@ -0,0 +1,44 @@
|
||||
\subsubsection{Correctness - Behaviour}
|
||||
We denote that two functions \texttt{fac} and \texttt{fac2} compute the same function usually as
|
||||
\[
|
||||
\forall n \in \N. \texttt{fac } n = \texttt{fac2 } (n, 1)
|
||||
\]
|
||||
The two functions are given as follows:
|
||||
\begin{multicols}{2}
|
||||
\begin{code}{haskell}
|
||||
fac :: Int -> Int
|
||||
fac 0 = 1
|
||||
fac n = n * fac (n - 1)
|
||||
\end{code}
|
||||
\begin{code}{haskell}
|
||||
fac2 :: (Int, Int) -> Int
|
||||
fac2 (0, a) = a
|
||||
fac2 (n, a) = fac2 (n - 1, n * a)
|
||||
\end{code}
|
||||
\end{multicols}
|
||||
An important fact to consider is that testing, while useful to find errors can't replace a formal proof to show that a function is correct!
|
||||
|
||||
These proofs are based on a simple idea: \bi{functions are equations} and thus, we can reason about them through equational reasoning, or more generally,
|
||||
proofs in first-order logic with equality.
|
||||
|
||||
Often, especially in Haskell programs, we have cases depending on values. Logically, to prove such a function, we also use case distinction, also referred to as reasoning by cases.
|
||||
|
||||
\inlineexample Consider the Haskell function below \rmvspace
|
||||
\begin{code}{haskell}
|
||||
maxi :: Int -> Int -> Int
|
||||
maxi n m
|
||||
| n >= m = n
|
||||
| otherwise = m
|
||||
\end{code}
|
||||
To prove that it is correct, we can use reasoning by cases:
|
||||
|
||||
We have $n \geq m \lor \neg(n \geq m)$.
|
||||
|
||||
We then show that the function is correct for both cases (i.e. LHS and RHS of OR):
|
||||
\begin{enumerate}[label=C\arabic*]
|
||||
\item $n \geq m$: Then $\texttt{maxi } n \ m = n$ and $n \geq n$
|
||||
\item $\neg(n \geq m)$: Then $\texttt{maxi } n \ m = m$. But $m > n$, so we have $\texttt{maxi } n\ m \geq n$
|
||||
\end{enumerate}
|
||||
In this proof we used the \textbf{TND} and \textbf{$\lor$-E} (here also called \bi{Case Split}) rules.
|
||||
|
||||
What we have to show, given $Q \lor R$ for any proposition $P$ with case split, is that \bi{(1)} $P$ follows from $Q$ and \bi{(2)} $P$ follows from $R$
|
||||
Reference in New Issue
Block a user