From c079e88fe9657061fdd82e88b4465fb53b0bd41d Mon Sep 17 00:00:00 2001 From: Janis Hutz Date: Sat, 28 Feb 2026 15:21:27 +0100 Subject: [PATCH] [FMFP] Mostly summarized formal reasoning --- ...methods-functional-programming-summary.pdf | Bin 253305 -> 359987 bytes ...methods-functional-programming-summary.tex | 19 ++++- .../01_formal-reasoning/00_formal-proofs.tex | 41 ++++++++++ .../01_natural-deduction.tex | 8 ++ .../02_propositional-logic/00_syntax.tex | 12 +++ .../02_propositional-logic/01_semantics.tex | 19 +++++ .../02_deductive-system.tex | 10 +++ .../03_natural-deduction-prop-logic.tex | 19 +++++ .../04_derivation-rules-overview.tex | 77 ++++++++++++++++++ .../03_first-order-logic/00_syntax.tex | 38 +++++++++ .../03_first-order-logic/01_semantics.tex | 40 +++++++++ .../03_first-order-logic/02_quantifiers.tex | 31 +++++++ .../04_equality.tex} | 0 13 files changed, 313 insertions(+), 1 deletion(-) create mode 100644 semester4/fmfp/parts/01_formal-reasoning/00_formal-proofs.tex create mode 100644 semester4/fmfp/parts/01_formal-reasoning/01_natural-deduction.tex create mode 100644 semester4/fmfp/parts/01_formal-reasoning/02_propositional-logic/00_syntax.tex create mode 100644 semester4/fmfp/parts/01_formal-reasoning/02_propositional-logic/01_semantics.tex create mode 100644 semester4/fmfp/parts/01_formal-reasoning/02_propositional-logic/02_deductive-system.tex create mode 100644 semester4/fmfp/parts/01_formal-reasoning/02_propositional-logic/03_natural-deduction-prop-logic.tex create mode 100644 semester4/fmfp/parts/01_formal-reasoning/02_propositional-logic/04_derivation-rules-overview.tex create mode 100644 semester4/fmfp/parts/01_formal-reasoning/03_first-order-logic/00_syntax.tex create mode 100644 semester4/fmfp/parts/01_formal-reasoning/03_first-order-logic/01_semantics.tex create mode 100644 semester4/fmfp/parts/01_formal-reasoning/03_first-order-logic/02_quantifiers.tex rename semester4/fmfp/parts/{01_natural-deduction/00_intro.tex => 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\alpha + \qquad \frac{+}{\times} \; \beta \\ + \frac{\otimes \quad \times}{\oplus} \; \gamma + \qquad \frac{}{+} \; \delta + \end{align*} +\end{multicols} + +Rules like $\delta$ above are also commonly referred to as an \textit{axiom}. + +To prove $\oplus$ in this language, we can either write the following or draw a derivation tree: +\begin{multicols}{2} + \begin{itemize} + \item $+$ holds by $\delta$ + \item $\otimes$ holds by $\alpha$ with 1. + \item $\times$ holds by $\beta$ with 1. + \item $\oplus$ holds by $\gamma$ with 2 and 3 + \end{itemize} + Or as derivation tree + \[ + \begin{prooftree} + % Left branch + \infer0[$\delta$]{+} + \infer1[$\alpha$]{\otimes} + % Right branch + \infer0[$\delta$]{+} + \infer1[$\beta$]{\times} + \infer2[$\gamma$]{ \oplus } + \end{prooftree} + \] +\end{multicols} diff --git a/semester4/fmfp/parts/01_formal-reasoning/01_natural-deduction.tex b/semester4/fmfp/parts/01_formal-reasoning/01_natural-deduction.tex new file mode 100644 index 0000000..ea30879 --- /dev/null +++ b/semester4/fmfp/parts/01_formal-reasoning/01_natural-deduction.tex @@ -0,0 +1,8 @@ +\subsection{Natural deduction} +The rules from above here are used to construct derivations under assumptions, e.g. +$A_1, \ldots, A_n \vdash A$, which is read as ``$A$ follows from $A_1, \ldots, A_n$''. + +The derivations are always represented as derivation trees and a \bi{proof} is a derivation whose root has no assumptions. + +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 +using the other rules of the rule set. diff --git a/semester4/fmfp/parts/01_formal-reasoning/02_propositional-logic/00_syntax.tex b/semester4/fmfp/parts/01_formal-reasoning/02_propositional-logic/00_syntax.tex new file mode 100644 index 0000000..fcbd28a --- /dev/null +++ b/semester4/fmfp/parts/01_formal-reasoning/02_propositional-logic/00_syntax.tex @@ -0,0 +1,12 @@ +\subsection{Propositional logic} +\subsubsection{Syntax} +\inlinedefinition For a set of variables $\cV$, the \bi{language of propositional logic} $\cL_P$ is the smallest set where +\begin{multicols}{2} + \begin{itemize} + \item $X \in \cL_P$ if $X \in \cV$ + \item $\bot \in \cL_P$ + \item $A \land B \in \cL_P$ if $A \in \cL_P$ and $B \in \cL_P$ + \item $A \lor B \in \cL_P$ if $A \in \cL_P$ and $B \in \cL_P$ + \item $A \rightarrow B \in \cL_P$ if $A \in \cL_P$ and $B \in \cL_P$ + \end{itemize} +\end{multicols} diff --git a/semester4/fmfp/parts/01_formal-reasoning/02_propositional-logic/01_semantics.tex b/semester4/fmfp/parts/01_formal-reasoning/02_propositional-logic/01_semantics.tex new file mode 100644 index 0000000..659e31c --- /dev/null +++ b/semester4/fmfp/parts/01_formal-reasoning/02_propositional-logic/01_semantics.tex @@ -0,0 +1,19 @@ +\subsubsection{Semantics} +\inlinedefinition[Valuation] $\sigma : \cV \rightarrow \{ \texttt{True}, \texttt{False} \}$ maps variables to truth values. +They are the simple models (i.e. \textit{interpretations}). \bi{Valuations} is the set of valuations. + +\inlinedefinition[Satisfiability] smallest relation $\models \; \subseteq$ \textit{Valuations} $\times \cL_P$ such that +\begin{multicols}{2} + \begin{itemize} + \item $\sigma \models X$ if $\sigma(X) = \texttt{True}$ + \item $\sigma \models A \land B$ if $\sigma \models A$ and $\sigma \models B$ + \item $\sigma \models A \lor B$ if $\sigma \models A$ or $\sigma \models B$ + \item $\sigma \models A \rightarrow B$ if whenever $\sigma \models A$ then $\sigma \models B$ + \end{itemize} +\end{multicols} + +\inlinedefinition[satisfiable formula] A formula $A \in \cL_P$ is \bi{satisfiable} if $\sigma \models A$ for \textbf{some} valuation $\sigma$ + +\inlinedefinition[tautology, valid formula] A formula $A \in \cL_P$ is \bi{valid} (a \bi{tautology}) if $\sigma \models A$ for \textbf{all} valuations $\sigma$ + +\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 $\sigma \models A$ diff --git a/semester4/fmfp/parts/01_formal-reasoning/02_propositional-logic/02_deductive-system.tex b/semester4/fmfp/parts/01_formal-reasoning/02_propositional-logic/02_deductive-system.tex new file mode 100644 index 0000000..1f9809c --- /dev/null +++ b/semester4/fmfp/parts/01_formal-reasoning/02_propositional-logic/02_deductive-system.tex @@ -0,0 +1,10 @@ +\newpage +\subsubsection{Requirements for a deductive system} +The derivation rules (syntactic entailment) and truth tables (semantic entailment) should agree. +For that we have two requirements, for $\Gamma = A_1, \ldots, A_n$ a collection of formulas: +\begin{itemize} + \item \bi{Soundness:} If $\Gamma \vdash A$ can be derived, then $\Gamma \models A$ + \item \bi{Completeness:} If $\Gamma \models A$, then $\Gamma \vdash A$ can be derived +\end{itemize} + +\bi{Decidability} is also desirable, i.e. having checks of the attributes be of low complexity. diff --git a/semester4/fmfp/parts/01_formal-reasoning/02_propositional-logic/03_natural-deduction-prop-logic.tex b/semester4/fmfp/parts/01_formal-reasoning/02_propositional-logic/03_natural-deduction-prop-logic.tex new file mode 100644 index 0000000..56295d5 --- /dev/null +++ b/semester4/fmfp/parts/01_formal-reasoning/02_propositional-logic/03_natural-deduction-prop-logic.tex @@ -0,0 +1,19 @@ +\subsubsection{Natural deduction for propositional formulas} +\inlinedefinition[Sequent] Is an assertion of form $A_1, \ldots, A_n \vdash A$, with $A, A_1, \ldots, A_n$ being propositional formulas. + +\inlineintuition $A$ follows from the $A_i$ and if the system is sound, the $A_i$ semantically entail $A$. + +\inlinedefinition[Axiom] is the starting point (usually the leaves) of the derivation trees and are usually of the form +\[ + \begin{prooftree} + \infer0[axiom]{\ldots, A, \ldots \vdash A} + \end{prooftree} +\] +i.e. when coming up with a derivation tree for a \bi{proof}, we want to reach a leaf where $A$ is contained in $\Gamma$. + +\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. + +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, +they are oriented top-down, so e.g. the introduction rule is inverted when coming up with the deduction tree. + +If all rules are sound (i.e. they preserve semantic entailment), then the logic is sound. diff --git a/semester4/fmfp/parts/01_formal-reasoning/02_propositional-logic/04_derivation-rules-overview.tex b/semester4/fmfp/parts/01_formal-reasoning/02_propositional-logic/04_derivation-rules-overview.tex new file mode 100644 index 0000000..5fb5a5e --- /dev/null +++ b/semester4/fmfp/parts/01_formal-reasoning/02_propositional-logic/04_derivation-rules-overview.tex @@ -0,0 +1,77 @@ +\subsubsection{Derivation rules for propositional logic} +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) +\paragraph{Conjunction} +\begin{align*} + \begin{prooftree} + \hypo{\Gamma \vdash A} + \hypo{\Gamma \vdash B} + \infer2[$\land$-I]{\Gamma \vdash A \land B} + \end{prooftree} + \qquad + \begin{prooftree} + \hypo{\Gamma \vdash A \land B} + \infer1[$\land$-EL]{\Gamma \vdash A} + \end{prooftree} + \qquad + \begin{prooftree} + \hypo{\Gamma \vdash A \land B} + \infer1[$\land$-ER]{\Gamma \vdash B} + \end{prooftree} +\end{align*} + +\paragraph{Disjunction} +\begin{align*} + \begin{prooftree} + \hypo{\Gamma \vdash A} + \infer1[$\lor$-IL]{\Gamma \vdash A \lor B} + \end{prooftree} + \qquad + \begin{prooftree} + \hypo{\Gamma \vdash B} + \infer1[$\lor$-IR]{\Gamma \vdash A \lor B} + \end{prooftree} + \qquad + \begin{prooftree} + \hypo{\Gamma \vdash A \lor B} + \hypo{\Gamma, A \vdash C} + \hypo{\Gamma, B \vdash C} + \infer3[$\lor$-E]{\Gamma \vdash C} + \end{prooftree} +\end{align*} + +\paragraph{Implication} +\begin{align*} + \begin{prooftree} + \hypo{\Gamma, A \vdash B} + \infer1[$\rightarrow$-I]{\Gamma \vdash A \rightarrow B} + \end{prooftree} + \qquad + \begin{prooftree} + \hypo{\Gamma \vdash A \rightarrow B} + \hypo{\Gamma \vdash A} + \infer2[$\rightarrow$-E]{\Gamma \vdash B} + \end{prooftree} +\end{align*} + +\paragraph{Others} +\begin{align*} + \begin{prooftree} + \hypo{\Gamma \vdash \bot} + \infer1[$\bot$-E]{\Gamma \vdash A} + \end{prooftree} + \qquad + \begin{prooftree} + \hypo{\Gamma \vdash \neg A} + \hypo{\Gamma \vdash A} + \infer2[$\neg$-E]{\Gamma \vdash B} + \end{prooftree} + \qquad + \begin{prooftree} + \hypo{\Gamma, \neg A \vdash \bot} + \infer1[RAA]{\Gamma \vdash A} + \end{prooftree} + \qquad + \begin{prooftree} + \infer0[axiom]{\ldots, A, \ldots \vdash A} + \end{prooftree} +\end{align*} diff --git a/semester4/fmfp/parts/01_formal-reasoning/03_first-order-logic/00_syntax.tex b/semester4/fmfp/parts/01_formal-reasoning/03_first-order-logic/00_syntax.tex new file mode 100644 index 0000000..63f0b4b --- /dev/null +++ b/semester4/fmfp/parts/01_formal-reasoning/03_first-order-logic/00_syntax.tex @@ -0,0 +1,38 @@ +\newsection +\subsection{First-Order Logic} +\subsubsection{Syntax} +\inlinedefinition[Signature] consists of a set of function symbols $\cF$ and a set of predicate symbols $\cP$, as well as their arities. + +We write $f^k$ (or $p^k$, for predicates) to indicate that it has \textit{arity} $k \in \N$. +Constant functions have arity $0$, linear functions have arity $1$, thus, the arity of a given function (or predicate) is given by +the number of parameters to uniquely describe it, minus one. + +\inlinedefinition[Term] is the terms of first-order logic is smallest set, where (with $\cV$ again a set of variables) +\begin{enumerate} + \item $x \in \textit{Term}$ if $x \in \cV$ + \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) +\end{enumerate} + +\inlinedefinition[Form] is the formulas of first-order logic, is the smallest set where +\begin{enumerate} + \item $\bot \in \textit{Form}$ + \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) + \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) + \item $Qx.A \in \textit{Form}$ if $A \in \textit{Form}$, $x \in \cV$ and $Q \in \{ \forall, \exists \}$ (i.e. formulas with quantifiers) +\end{enumerate} + + +\paragraph{Binding} +\inlinedefinition[Bound variable] A variable that occurs in a quantifier in scope ({\color{blue} blue in example below}) + +\inlinedefinition[Free variable] A variable that is not bound by a quantifier in scope ({\color{red} red in example below}) + +\inlineexample $(q({\color{red} x}) \lor \exists {\color{blue} x}.\forall {\color{blue} y}. p(f({\color{blue} x}), {\color{red} z}) \land q({\color{red} a})) +\lor \forall {\color{blue} x}.r({\color{blue} x}, {\color{red} z}, g({\color{blue} x}))$ + +\inlinedefinition[$\alpha$-conversion] A \bi{bound} variable can be renamed at any time, but must preserve binding structure. + + +\paragraph{Precedences} +$\neg > \land > \lor > \rightarrow$, with $>$ a total order of precedences. +Quantifiers extend as far right as possible, bounded by the end of line or a going out of scope by closing parenthesis. diff --git a/semester4/fmfp/parts/01_formal-reasoning/03_first-order-logic/01_semantics.tex b/semester4/fmfp/parts/01_formal-reasoning/03_first-order-logic/01_semantics.tex new file mode 100644 index 0000000..b9c63c5 --- /dev/null +++ b/semester4/fmfp/parts/01_formal-reasoning/03_first-order-logic/01_semantics.tex @@ -0,0 +1,40 @@ +\subsubsection{Semantics} +\inlinedefinition[Structure] is a pair $\cS = \langle U_\cS, I_\cS \rangle$, where $U_\cS$ is the universe and it is a non-empty set and $I_\cS$ is a mapping with +\begin{enumerate} + \item $I_\cS(p^n)$ is an $n$-ary relation on $U_\cS$ for $p^n \in \cP$ (short $p^\cS$) + \item $I_\cS(f^n)$ is an $n$-ary (total) function on $U_\cS$ for $f^n \in \cF$ short ($f^\cS$) +\end{enumerate} +\inlineintuition The $I_\cS$ is essentially assigning to each predicate and formula its definition in the universe of the structure, noted as a relation. + +\inlinedefinition[Interpretation] is a pair $\cI = \langle \cS, v \rangle$, with $v : \cV \rightarrow U_\cS$ a valuation\\ +\inlineintuition it assigns definitions to the formulas and predicates (through the structure), as well as values to the variables (through the valuation). + +\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\\ +$\cI(f(t_1, \ldots, t_n)) = f^\cS(\cI(t_1), \ldots, \cI(t_n))$ + +\inlinedefinition[Satisfiability] $\models \subseteq$ Interpretations $\times$ \textit{Form} is the smallest relation satisfying +\begin{align*} + & \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 \\ + & \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 \\ + & \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 \\ +\end{align*} + +\inlinedefinition[Model] When $\langle \cS, v \rangle \models A$, then $\langle \cS, v \rangle$ is a \bi{model} for $A$. +If $A$ does not have free variables, the satisfaction does not depend on the valuation $v$ and we write $\cS \models A$ + +\inlinedefinition[Validity] When every interpretation is a model, we write $\models A$, and we say that $A$ is \bi{valid} + +\inlinedefinition[Satisfiability] $A$ is \bi{satisfiable}, if there exists at least one model for $A$. + +\inlineexample Given $\forall x.p(x, s(x))$, we a model would be +\begin{align*} + U_\cS & = \N \\ + p^\cS & = \{ (m, n) \divider m, n \in U_\cS \text{ and } m < n \} \\ + s^\cS & = \text{ successor function on } U_\cS, \text{ i.e. } s^\cS(x) = x + 1 +\end{align*} + + +\paragraph{Substitution} +\inlinedefinition[Substitution] Replace all occurrences of a free variable $x$ with some term $t$ in $A$. +To denote a substitution, we write $A[x \mapsto t]$. \hl{Important} All free variables in $t$ must still be free in $A[x \mapsto t]$. +If that would not be true anymore, do a $\alpha$-conversion first. diff --git a/semester4/fmfp/parts/01_formal-reasoning/03_first-order-logic/02_quantifiers.tex b/semester4/fmfp/parts/01_formal-reasoning/03_first-order-logic/02_quantifiers.tex new file mode 100644 index 0000000..0d737a1 --- /dev/null +++ b/semester4/fmfp/parts/01_formal-reasoning/03_first-order-logic/02_quantifiers.tex @@ -0,0 +1,31 @@ +\subsubsection{Quantifiers} +\paragraph{Universal quantification} +Additional rules are needed for the universal quantifier. * side condition is that $x$ is not free in any assumption of $\Gamma$ +\begin{align*} + \begin{prooftree} + \hypo{\Gamma \vdash A} + \infer1[$\forall$-I*]{\Gamma \vdash \forall x.A} + \end{prooftree} + \qquad + \begin{prooftree} + \hypo{\Gamma \vdash \forall x.A} + \infer1[$\forall$-E]{\Gamma \vdash A[x \rightarrow t]} + \end{prooftree} +\end{align*} +Again here be mindful not to capture free variables. + +\paragraph{Existential quantification} +Additional rules are needed for the existential quantifier. ** side condition is that $x$ is neither free in B nor free in $\Gamma$ +\begin{align*} + \begin{prooftree} + \hypo{\Gamma \vdash A[x \mapsto t]} + \infer1[$\exists$-I]{\Gamma \vdash \exists x.A} + \end{prooftree} + \qquad + \begin{prooftree} + \hypo{\Gamma \vdash \exists x.A} + \hypo{\Gamma, A \vdash B} + \infer2[$\exists$-E**]{\Gamma \vdash A[x \rightarrow t]} + \end{prooftree} +\end{align*} +Again here be mindful not to capture free variables. diff --git a/semester4/fmfp/parts/01_natural-deduction/00_intro.tex b/semester4/fmfp/parts/01_formal-reasoning/04_equality.tex similarity index 100% rename from semester4/fmfp/parts/01_natural-deduction/00_intro.tex rename to semester4/fmfp/parts/01_formal-reasoning/04_equality.tex