atomic formula

Let $\Sigma$ be a signature   and $T(\Sigma)$ the set of terms over $\Sigma$. The set $S$ of symbols for $T(\Sigma)$ is the disjoint union  of $\Sigma$ and $V$, a countably infinite  set whose elements are called variables. Now, adjoin $S$ the set $\{=,(,)\}$, assumed to be disjoint from $S$. An atomic formula $\varphi$ over $\Sigma$ is any one of the following:

1. 1.

either $(t_{1}=t_{2})$, where $t_{1}$ and $t_{2}$ are terms in $T(\Sigma)$,

2. 2.

or $(R(t_{1},...,t_{n}))$, where $R\in\Sigma$ is an $n$-ary relation symbol, and $t_{i}\in T(\Sigma)$.

Remarks.

1. 1.

Using atomic formulas, one can inductively build formulas   using the logical connectives $\vee$, $\neg$, $\exists$, etc… In this sense, atomic formulas are formulas that can not be broken down into simpler formulas; they are the building blocks of formulas.

2. 2.

A literal is a formula that is either atomic or of the form $\neg\varphi$ where $\varphi$ is atomic. If a literal is atomic, it is called a positive literal. Otherwise, it is a negative literal.

3. 3.

A finite disjunction  of literals is called a clause. In other words, a clause is a formula of the form $\varphi_{1}\vee\varphi_{2}\vee\cdots\vee\varphi_{n}$, where each $\varphi_{i}$ is a literal.

4. 4.

A qunatifier-free formula is a formula that does not contain the symbols $\exists$ or $\forall$.

5. 5.

If we identify a formula $\varphi$ with its double negation $\neg(\neg\varphi)$, then it can be shown that any quantifier-free formula can be identified with a formula that is in conjunctive normal form  , that is, a finite conjunction  of clauses. For a proof, see this link (http://planetmath.org/EveryPropositionIsEquivalentToAPropositionInDNF)

 Title atomic formula Canonical name AtomicFormula Date of creation 2013-03-22 12:42:54 Last modified on 2013-03-22 12:42:54 Owner CWoo (3771) Last modified by CWoo (3771) Numerical id 10 Author CWoo (3771) Entry type Definition Classification msc 03C99 Classification msc 03B10 Synonym quantifier free formula Related topic TermsAndFormulas Related topic CNF Related topic DNF Defines literal Defines clause Defines quantifier-free formula Defines positive literal Defines negative literal