Computer Science Argumentation First-Order Argumentation

 24 June 14:01   

    In propositional logic, we advised formulas create about diminutive objects, which could alone be either true or false. First-order logic, the affair of this chapter, builds aloft propositional argumentation and allows you to attending central the altar discussed in formulas. We can accommodate this added aesthetic akin of granularity by discussing altar as elements of sets that can be beyond than just the set $$, and aswell cover arbitrarily circuitous relationships with anniversary other.

    We activate first by defining the syntax of first-order (FO) logic, and then accord these structures meaning.

    The area of address for first adjustment argumentation is first-order structures or models. A first-order anatomy contains

    # Relations,

    # Functions, and

    # Constants (functions of arity 0).

    The cant of first-order argumentation is

    # a set of affiliation symbols with associated arities, and

    # a set of action symbols with associated arities.

    Here are some archetype first-order argumentation vocabularies:

    # A graph

    ### Arithmetic

    ##

    Here, a blueprint is a set of vertices $V$ and a set of edges $E$. For the addition set agenda that the use of $+$ and $imes$ is absolutely syntactic and we could accept acclimated the symbols additional and times instead. We havent provided the symbols with any acceptation yet.

    A appellation denotes an aspect in the first-order structure. A appellation is acclimated to accredit to the elements in our area of discourse. Actuality are the rules that call what a appellation is:

    For example, if $f$ is a bifold action and $g$ is a ternary action then the afterward are all terms: $x,\; a,\; f(x,\; y),\; g(x,\; f(x,\; y),\; a)$.

    An diminutive blueprint is

    :$R(t\_,\; t\_,\; ldots,\; t\_)$,

    where $R$ is a $k$-ary affiliation and $t\_,\; t\_,\; ldots,\; t\_$ are terms. If examination $R$ as a set, then this is just addition way of adage that the tuple

    :$(t\_,\; t\_,\; ldots,\; t\_)in\; R$.

    A appropriate affiliation $=$ agency according and cannot be interpreted otherwise. For example, $t\_\; =\; t\_$ stands for

    :$=(t\_,\; t\_)$.

    A first-order blueprint is an announcement congenital using a accustomed first-order cant and variables and the symbols $(,\; ),$
eg, vee, wedge,
ightarrow, exists, forall. The set of first-order formulas is the minimum set acceptable the following:

    #$(phi\; vee\; psi)$

    #$(phi\; wedge\; psi)$

    #$($
eg phi)

    #$(phi$
ightarrow psi)

    #$(exists\; x\; phi)$

    #$(forall\; x\; phi)$

    A first-order anatomy over a accustomed cant consists of

    # A domain, which is a set of elements (also accepted as a universe)

    # A mapping advertence to every $k$-ary affiliation attribute in the cant a $k$-ary affiliation over the domain, and to every $k$-ary action attribute a $k$-ary action over the domain.

    These apparatus accord acceptation to the symbols.

    Example:

    :Relation Set =

    :Function Set =

    A first-order anatomy over this cant is:

    # Domain: the set of integers

    # Mapping : $+$
ightarrow addition, $imes$
ightarrow multiplication, $uparrow$
ightarrow exponentiation,
ightarrow ordering, $mathit$
ightarrow i+1

    In this structure, the formula

    :$exists\; x\; forall\; y\; forall\; z[\; imes\; (y,\; z,\; x)$
ightarrow ((y = 1) vee (z = 1))]

    expresses the account there exists a prime amount (the amount 1 aswell satisfies this statement).

    Note actuality that $imes\; (y,\; z,\; x)$ is agnate to $(x\; =\; y\; imes\; z)$.

    In the blueprint $(forall\; x\; phi)$ or $(exists\; x\; phi)$, $phi$ is referred to as the ambit of altitude of the capricious $x$. An accident of a capricious in a blueprint is apprenticed if it is in the ambit of altitude of that variable, contrarily the accident is said to be free. You can accede a quantifier use as a capricious declaration.

    A book is a blueprint with no chargeless variables (i.e., all variables are bound). A book is either true or false.

    A blueprint with chargeless variable(s) can be advised as anecdotic the backdrop of the chargeless variable(s). For example, $phi(x)$ denotes a blueprint with $x$ occuring chargeless and describes the backdrop of $x$.

    If a book $phi$ evaluates to true over a anatomy $I$, we say $I$ satisfies $phi$ and denote this by $I\; modelsphi$.

    Note: $phi(x\; leftarrow\; c)$ denotes the aftereffect of substituting $c$ for every chargeless accident of $x$ in $phi$. Barter is covered added after in this chapter.

    The axioms are a set of sentences meant to analyze adapted models from others. But typically, aswell accept causeless models, which are alleged non-standard models.

    Examples: Accede the cant , area the symbols accept the accepted acceptation (defined by FO sentences over this vocabulary). The accepted estimation for this set of axioms (see the end of this chapter) is the integers, but these are aswell annoyed by the set of integers modulo $p$. This achievability is disqualified out by abacus the afterward book to the axioms:

    

    Question: Can all non-standard models be axiomatized away? The acknowledgment is no. Accede the archetypal apparent in [TODO: acceptation figure] for the cant absolute alone (all elements in the high band are beyond than those on the lower line).

    There is no set of first-order sentences that can analyze this archetypal from the accustomed numbers. Intuitively, the cause is that we cannot backtrack arbitrarily (towards $0$) in a first-order sentence. A agnate non-standard archetypal can be acquired for the abounding cant above.

    Given a first-order anatomy $I$ and a FO book $phi$, can we acquaint if $I\; models\; phi$?

    This problem is decidable for bound structures. For example, accept $E$ is a bifold affiliation apery the edges of a graph, and the accustomed book is

    :$exists\; x\_\; exists\; x\_\; exists\; x\_(E(x\_,\; x\_)\; wedge\; E(x\_,\; x\_)\; wedge\; E(x\_,\; x\_)\; wedge\; x\_$
eq x_ wedge x_
eq x_ wedge x_
eq x_ ).

    We can appraise the accuracy of this book by aggravating all accessible ethics for $x\_,\; x\_$ and $x\_$. (Naive evaluation: nested loop.) This is polynomial time (in area size) but exponential in the admeasurement of the formula.

    On absolute domains, we may or may not be able to appraise the accuracy of a sentence. With the cant , area $N$ is the set of accustomed numbers, it is decidable if a book is true. (This is accepted as Presburger arithmetic.) About if we cover multiplication, the accuracy of a book becomes undecidable.

    A set $Delta$ of FO sentences is

    Given a set of FO sentences $Sigma$ and a book $phi$

    The axioms for authority are from three categories:

    # Axioms for boolean validity. These are affiliated from propositional logic.

    # Axioms for equality.

    # Axioms for quantification.

    Given a FO blueprint $varphi$, a boolean anatomy of $varphi$ a propositional blueprint $psi$ such that $varphi$ is acquired from $psi$ by replacing anniversary propositional capricious in $psi$ by a subformula of $varphi$.

    The set of Boolean forms of $varphi$ is denoted $BF(varphi)$.

    Examples:

    Claim: If $psi\; in\; BF(varphi)$ and $psi$ is valid, then $varphi$ is valid.

    Let $t,\; t\_,\; ldots,\; t\_,\; t\_,\; ldots,\; t\_$ be terms. The afterward are accurate formulas.

    Given a blueprint $varphi$ in which capricious $x$ occurs chargeless (denoted by $varphi(x)$) and a appellation $t$, we ascertain the barter of $t$ for $x$ in $varphi$, denoted $varphi(x\; leftarrow\; t)$, as the blueprint that after-effects from replacing every

    free accident of $x$ in $varphi$ by $t$, accountable to the coercion that $t$ contains no capricious $y$ quantified in $varphi$ such that $x$ occurs chargeless aural the ambit of altitude of $y$. If $x$ does not action chargeless $varphi$, then $varphi(x\; leftarrow\; t)$ is authentic as $varphi$.

    Example: Let $varphi$ be

    :$((x\; =\; 1)$
ightarrow exists x (x = y)),

    then

    #$forall\; x\; varphi$
ightarrow varphi(x leftarrow t), area $t$ is a appellation and $varphi(x\; leftarrow\; t)$ is a accurate substitution

    #$(forall\; x\; (varphi$
ightarrow psi))
ightarrow ((forall x varphi)
ightarrow (forall x psi))

    #$varphi$
ightarrow forall x varphi

    #$exists\; x\; varphi\; leftrightarrow$
eg forall x
eg varphi

    In summary, the axioms for authority are $\}$:

    #Axioms for boolean validity. These are affiliated from propositional logic.

    #Axioms for equality.

    #Axioms for quantification.

    Definition: A affidavit is a arrangement $phi\_,\; phi\_,\; ldots,\; phi\_$ of FO sentences such that for anniversary $i\; in$ either $phi\_\; in\}$ or such that $phi\_\; equiv\; psi$ and $phi\_\; equiv\; (psi$
ightarrow phi_).

    Notation: If there is a affidavit of $phi$ using $\}$, we denote this actuality by $vdash\_\}phi$.

    Fact (Soundness): If $vdash\_\}\; phi$, then $phi$ is valid.

    Remark: The set of formulas which can be accurate accurate is recursively enumerable.

    Example: Affidavit of the blueprint $forall\; x\; (phi\; land\; psi)$
ightarrow (forall x phi land forall x psi)

    :$forall\; x\; (phi\; land\; psi)$

    :$forall\; x\; (phi\; land\; psi)$
ightarrow (phi land psi) (x leftarrow x)

    :$phi\; land\; psi$

    :$phi$

    :$forall\; x\; (phi\; land\; psi)$
ightarrow phi

    :$forall\; x\; (forall\; x\; (phi\; land\; psi)$
ightarrow phi)

    :$forall\; x\; forall\; x\; (phi\; land\; psi)$
ightarrow forall x phi

    :$forall\; x\; (phi\; land\; psi)$
ightarrow forall x phi

    :$forall\; x\; phi$

    The affidavit of $forall\; x\; psi$ is symmetrical.

    :$forall\; x\; phi\; land\; forall\; x\; psi$

    :$forall\; x\; (phi\; land\; psi)$
ightarrow (forall x phi land forall x psi)

    Claim: Every FO book is agnate to a FO book of the anatomy $Q\_\; x\_\; cdots\; Q\_\; x\_\; phi$, area $Q\_\; in$ and $phi$ is quantifier free.

    This is accepted using the distributive backdrop of quantifiers. It may be all-important to rename variables, to abstain ambiguities. Although you can consistently move all quantifiers to the left, the consistent blueprint can infact be exponentially beyond than the aboriginal one.

    Example:

    :$phi\; equiv\; forall\; x\; (G(x,x)\; land\; (forall\; y\; G(x,y)\; lor\; exists\; y$
eg G(y,y))) land G(x,0)

    ::$equiv\; (forall\; x\; (G(x,x)\; land\; (forall\; y\; G(x,y)\; lor\; exists\; y$
eg G(y,y))) land G(w,0)

    ::$equivforall\; x\; (G(x,x)\; land\; (forall\; y\; G(x,y)\; lor\; exists\; y$
eg G(y,y)) land G(w,0))

    ::$equivforall\; x\; (G(x,x)\; land\; (forall\; y\; G(x,y)\; lor\; exists\; z$
eg G(z,z)) land G(w,0))

    ::$equivforall\; x\; (G(x,x)\; land\; forall\; y\; (G(x,y)\; lor\; exists\; z$
eg G(z,z)) land G(w,0))

    ::$equivforall\; x\; forall\; y\; (G(x,x)\; land\; (G(x,y)\; lor\; exists\; z$
eg G(z,z)) land G(w,0))

    ::$equivforall\; x\; forall\; y\; (G(x,x)\; land\; exists\; z\; (G(x,y)\; lor$
eg G(z,z)) land G(w,0))

    ::$equivforall\; x\; forall\; y\; exists\; z\; (G(x,x)\; land\; (G(x,y)\; lor$
eg G(z,z)) land G(w,0))

    # Call the adapted archetypal as carefully as accessible using a (possibly absolute but recursive) set $Delta$ of axioms (FO sentences).

    #Prove things using deduction.

    Notation: We say $phi$ is a accurate aftereffect of $Delta$ (denoted $Delta\; models\; phi$), if every archetypal acceptable $Delta$ aswell satisfies $phi$.

    Fact: $Sigma\; models\; phi$ iff $Sigma\; cup$ is unsatisfiable.

    The afterward fourteen first-order axioms call the backdrop of addition and numbers, i.e. accession ($+$), multiplication ($imes$), exponentiation ($uparrow$, adequation ($=$), acclimation (), almsman action ($sigma$) and butt (mod). This archetype shows the alive ability of first-order statements, which were originally hoped to accommodate a base for the one true mathematics.

    Question: Why are they alleged nonlogical axioms?

    :NT1: $forall\; x(sigma(x)$
e 0)

    :NT2: $forall\; x\; forall\; y\; (sigma(x)\; =\; sigma(y)$
ightarrow x=y)

    :NT3: $forall\; x\; (x=0\; vee\; exists\; y\; sigma(y)\; =\; x)$

    :NT4: $forall\; x\; (x+0=x)$

    :NT5: $forall\; x\; forall\; y(x+sigma(y)\; =\; sigma(x+y))$

    :NT6: $forall\; x\; (x\; imes\; 0\; =\; 0)$

    :NT7: $forall\; x\; forall\; y\; (x\; imes\; sigma(y)\; =\; (x\; imes\; y\; )\; +\; x)$

    :NT8: $forall\; x\; (\; x\; uparrow\; 0\; =\; sigma(0))$

    :NT9: $forall\; x\; forall\; y\; (x\; uparrow\; sigma(y)\; =\; (x\; uparrow\; y\; )\; imes\; x\; )$

    :NT10:

    :NT11:
ightarrow (sigma (x)le y)

    :NT12: $forall\; x\; forall\; y\; ($
eg (x

    :NT13:

    :NT14: $forall\; x\; forall\; y\; forall\; z\; forall\; z\; (mod(x,y,z)wedge\; mod(x,y,z)$
ightarrow z = z)