CS 486: Spring 2001 Cornell University Prof. Robert Constable

Feb 13 Lecture Notes: Finishing definition of a tableau

Smullyan p.24 translated into formalism

Recall that SignedFormula == ℕ × F orm. We defined a (“tableau tree”) T T ree as

T T ree = SignedFormula + SignedFormula × T T ree + SignedFormula × T T ree × T T ree.

If we replace SignedFormula by ℕ × 𝔹 × F orm × ℕ, we get a LabeledTTree or LT T ree.

Tableaux_{X} = {t : T T ree | root(t) =< F, X > &

for all interior nodes of t if α occurs on P_{s} then successor of s is α_{1} or α_{2}; and

if β occurs on P_{s} then β_{1}, β_{2} are successors of s}

This defines a collection of analytic tableaux for X. Each element, t, is a particular tableau. We might denote it by t_{X}. Note, Smullyan writes P_{Y} for P_{s}.

We added the definition of a labeled tableau.

LabeledTableaux_{X} = {t : LT T ree | root(t) =< 1, F, X, 0 > &

for all interior nodes s ∈ nodes(t).s =< n, S, Y, k >, for some m < n. if t@m is an α on P_{s} then the successor of the node is < n + 1, α_{i}, m > if t@m is a β on P_{s} then the successors of the node are

< n′, β_{1}, m >, < n′ + 1, β_{1}, m > for some n′ > n}

Let step be the systematic procedure defined in Smullyan p.18 and specified more precisely in class on Thurs, Feb 8.

step: LT T ree × ℕ → LT T ree

Recall the definition and theorem.

Def : step(t_{i}, l_{i}) = t_{i+1} is a systematic tableau development for i = 0, 1, 2, …, l_{i} a sequence of labels.

Note: A particular systematic tableau is uniquely determined by a sequence of numbers l_{0}, l_{1}, …, l_{k} that select the line used in the next step. In a completely systematic development, the line used is the least unused label.

Theorem: For any X ∈ F orm, all systematic tableau developments starting from t_{0} =< 1, F, X, 0 >

terminate (in at most 2^{|X|} steps).

Def : Let step_{L}*(t) be the final tableau in a systematic development, for L a sequence of labels

resulting in a finished tableau.

Def : SystematicLabeledTableaux_{X} == {t : LT T ree | root(t) =< 1, F, X, 0 > & t =step_{L}*(root(t))}