Inference rule An inference rule of premises $f_1,...,f_k$ and conclusion $g$ is written: Forward inference algorithm From a set of inference rules $\textrm{Rules}$, this algorithm goes through all possible $f_1, ..., f_k$ and adds $g$ to the knowledge base $\textrm{KB}$ if a matching rule exists. 1 First-Order Logic (First-Order Predicate Calculus) 2 Propositional vs. Predicate Logic •In propositional logic, each possible atomic fact requires a separate unique propositional symbol. Completeness Modus ponens is complete with respect to Horn clauses if we suppose that $\textrm{KB}$ contains only Horn clauses and $p$ is an entailed propositional symbol. Would you like to see this cheatsheet in your native language? Example: the set of truth values $w = \{A:0, B:1, C:0\}$ is one possible model to the propositional symbols $A$, $B$ and $C$. : Satisfiability The knowledge base $\textrm{KB}$ is said to be satisfiable if at least one model $w$ satisfies all its constraints. Interpretation function ― The interpretation function I(f,w)I(f,w) outputs wheth… Syntax of propositional logic ― By noting f,gf,g formulas, and ¬,∧,∨,→,↔¬,∧,∨,→,↔connectives, we can write the following logical expressions: Remark: formulas can be built up recursively out of these connectives. Mary loves … (a) Anyone who has forgiven at least one person is a saint. User defines these primitives: Constant symbols (i.e., the "individuals" in the world) E.g., Mary, 3 Function symbols (mapping individuals to individuals) E.g., father-of(Mary) = John, color-of(Sky) = Blue Model ― A model wwdenotes an assignment of binary weights to propositional symbols. Logic-based models translation. Relation between formulas and knowledge base We define the following properties between the knowledge base $\textrm{KB}$ and a new formula $f$: Model checking A model checking algorithm takes as input a knowledge base $\textrm{KB}$ and outputs whether it is satisfiable or not. More Answers for Practice in Logic and HW 1.doc Ling 310 Feb 27, 2006 1 More Answers for Practice in Logic and HW 1 This is an expanded version showing additional right and wrong answers. Ç}8ô$9Zå¹ñï�थQš†¨¬Múw¼yóÛ«Ö§[Ñ|àÑ€!Œˆ”ƒöõO+ªÓK3D/1‚vÈz¹‚Q¹Få™à¦Gu¹ØQ;ûB³:"�MÑÓ=e–ˆbJ:ç¼K�fˆ³±³–ˆXû\ıg4{Ÿ¹¢£ñ’WÅšö�¼~hæ”ÆÏÊ%é’2bêxCBc•Çh�\U�6>�á £MÛ~º»BÙ¢‡¥mO\®›²ïúXĞ:abä02üÛÁh. Outline Outline 1 Motivation 2 … Model A model $w$ denotes an assignment of binary weights to propositional symbols. Derivation We say that $\textrm{KB}$ derives $f$ (written $\textrm{KB}\vdash f$) with rules $\textrm{Rules}$ if $f$ already is in $\textrm{KB}$ or gets added during the forward inference algorithm using the set of rules $\textrm{Rules}$. The idea here is to balance expressivity and computational efficiency. Syntax of propositional logic By noting $f,g$ formulas, and $\neg, \wedge, \vee, \rightarrow, \leftrightarrow$ connectives, we can write the following logical expressions: Remark: formulas can be built up recursively out of these connectives. Model A model $w$ in first-order logic maps: Horn clause By noting $x_1,...,x_n$ variables and $a_1,...,a_k,b$ atomic formulas, the first-order logic version of a horn clause has the form: Substitution A substitution $\theta$ maps variables to terms and $\textrm{Subst}[\theta,f]$ denotes the result of substitution $\theta$ on $f$. Resolution rule By noting $f_1, ..., f_n$, $g_1, ..., g_m$, $p$, $q$ formulas and by calling $\theta=\textrm{Unify}(p,q)$, the first-order logic version of the resolution rule can be written: Semi-decidability First-order logic, even restricted to only Horn clauses, is semi-decidable. Properties of inference rules A set of inference rules $\textrm{Rules}$ can have the following properties: In this section, we will go through logic-based models that use logical formulas and inference rules. Modus ponens By noting $x_1,...,x_n$ variables, $a_1, ..., a_k$ and $a'_1,...,a'_k$ atomic formulas and by calling $\theta = \textrm{Unify}(a'_1\wedge ...\wedge a'_k, a_1\wedge ...\wedge a_k)$, the first-order logic version of modus ponens can be written: Completeness Modus ponens is complete for first-order logic with only Horn clauses. Logic For Dummies Cheat Sheet By Mark Zegarelli Logic is more than a science, it’s a language, and if you’re going to use the language of logic, you need to know the grammar, which includes operators, identities, equivalences, and quantifiers for both sentential and quantifier logic. Remark: popular model checking algorithms include DPLL and WalkSat. In other words: Probabilistic interpretation The probability that query $f$ is evaluated to $1$ can be seen as the proportion of models $w$ of the knowledge base $\textrm{KB}$ that satisfy $f$, i.e. Modus ponens For propositional symbols $f_1,...,f_k$ and $p$, the modus ponens rule is written: Remark: it takes linear time to apply this rule, as each application generate a clause that contains a single propositional symbol. Exercise Sheet 2: Predicate Logic 1. You can help us, \[\boxed{\mathcal{M}(f) = \{w\, |\, \mathcal{I}(f,w)=1\}}\], \[\boxed{\mathcal{M}(\textrm{KB})=\bigcap_{f\in\textrm{KB}}\mathcal{M}(f)}\], \[\boxed{P(f\,|\,\textrm{KB})=\frac{\displaystyle\sum_{w\in\mathcal{M}(\textrm{KB})\cap\mathcal{M}(f)}P(W=w)}{\displaystyle\sum_{w\in\mathcal{M}(\textrm{KB})}P(W=w)}}\], \[\boxed{\textrm{KB satisfiable}\Longleftrightarrow\mathcal{M}(\textrm{KB})\neq\varnothing}\], \[\boxed{(p_1\wedge...\wedge p_k)\longrightarrow q}\], \[\boxed{\frac{f_1,...,f_k,\quad (f_1\wedge...\wedge f_k)\longrightarrow p}{p}}\], \[\boxed{\frac{f_1\vee...\vee f_n\vee p,\quad\neg p\vee g_1\vee...\vee g_m}{f_1\vee...\vee f_n\vee g_1\vee...\vee g_m}}\], \[\boxed{\forall x_1,...,\forall x_n,\quad(a_1\wedge...\wedge a_k)\rightarrow b}\], \[\boxed{\textrm{Unify}[f,g]=\theta}\quad\textrm{ s.t. Mathematically speaking, we define it as follows: Definition The knowledge base $\textrm{KB}$ is the conjunction of all formulas that have been considered so far. I. Resolution-based inference The resolution-based inference algorithm follows the following steps: The idea here is to use variables to yield more compact knowledge representations. Practice in 1st-order predicate logic – with answers. In other words: Remark: $\mathcal{M}(\textrm{KB})$ denotes the set of models compatible with all the constraints of the knowledge base. 1. Conjunctive normal form A conjunctive normal form (CNF) formula is a conjunction of clauses, where each clause is a disjunction of atomic formulas. Equivalent representation Every formula in propositional logic can be written into an equivalent CNF formula. (b) Nobody in the calculus class is smarter than everybody in the discrete maths class. }\quad\boxed{\textrm{Subst}[\theta,f]=\textrm{Subst}[\theta,g]}\], \[\boxed{\frac{a'_1,...,a'_k\quad\forall x_1,...,\forall x_n (a_1\wedge...\wedge a_k)\rightarrow b}{\textrm{Subst}[\theta, b]}}\], \[\boxed{\frac{f_1\vee...\vee f_n\vee p,\quad\neg q\vee g_1\vee...\vee g_m}{\textrm{Subst}[\theta,f_1\vee...\vee f_n\vee g_1\vee...\vee g_m]}}\], Relation between formulas and knowledge base, $\mathcal{M}(\textrm{KB})\cap\mathcal{M}(f)=\mathcal{M}(\textrm{KB})$, ⢠$f$ does not bring any new information, $\mathcal{M}(\textrm{KB})\cap\mathcal{M}(f)=\varnothing$, ⢠No model satisfies the constraints after adding $f$, $\mathcal{M}(\textrm{KB})\cap\mathcal{M}(f) \neq \varnothing$, ⢠$f$ does not contradict $\textrm{KB}$, $\{f \, | \, \textrm{KB}\vdash f\}\subseteq\{f \, | \, \textrm{KB}\models f\}$, ⢠Inferred formulas are entailed by $\textrm{KB}$, $\{f \, | \, \textrm{KB}\vdash f\}\supseteq\{f \, | \, \textrm{KB}\models f\}$, ⢠Formulas entailing $\textrm{KB}$ are either already in the knowledge base or inferred from it, $(f \rightarrow g) \wedge (g \rightarrow f)$, if $\textrm{KB}\models f$, forward inference on complete inference rules will prove $f$ in finite time, if $\textrm{KB}\not\models f$, no algorithm can show this in finite time. Interpretation function The interpretation function $\mathcal{I}(f,w)$ outputs whether model $w$ satisfies formula $f$: Set of models $\mathcal{M}(f)$ denotes the set of models $w$ that satisfy formula $f$. Syntax of propositional logic ― By noting f,g formulas, and ¬,∧,∨,→, ↔ connectives, we can write the following logical expressions: 4. First-Order Logic (FOL or FOPC) Syntax. Example: the set of truth values w={A:0,B:1,C:0}w={A:0,B:1,C:0} is one possible model to the propositional symbols AA, BB and CC. and First Order Logic Propositional Logic First Order Logic Basic Concepts Propositional logic is the simplest logic illustrates basic ideas usingpropositions P 1, Snow is whyte P 2, oTday it is raining P 3, This automated reasoning course is boring P i is an atom or atomic formula Each P i can be either true or false but never both To use variables to yield more compact knowledge representations of binary weights to propositional symbols ( A ) Anyone has. 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