Decision Problems with TM’s
Lecture 32: Decidability • Last time showed all below undecidable:
• ATM = {⟨M,w⟩ | M is a TM and w ∈ L(M)} CSCI 81 Spring, 2012 • HTM = {⟨M,w⟩ | M is a TM which halts on input w}
• TOTALTM = {M | M halts on all inputs} Kim Bruce • ETM ={M | M is a TM and L(M)=∅}
Semi-decidable Another Example
• ATM and HTM both semi-decidable using UTM. • Lε0 = {
• If
Entscheidungsproblem Decidable & Semidecidable
• More on Turing’s solution last step: SD
• Given TM M, can write a logical formula ψ of predicate D logic such that ψ is true iff M writes 0 on ε input. Context-Free • Let ψ be statement: In some configuration of M starting with ε, Languages some square s contains the symbol o Regular Languages • Let φ1,...,φn be axioms for M.
• Then formula is φ1∧... ∧φn→ ψ
• Thus M writes 0 on input ε iff φ1∧... ∧φn→ ψ is provable in predicate calculus. • Therefore provability undecidable! The Hierarchy Distinguishing D and SD
• Most obvious languages in SD also in D • Theorem: The set of context-free languages is a • AnBnCn = {anbncn | n ≥ 0} proper subset of D. • {wcw | w ∈ {a, b}*} Proof: Every context-free language is decidable, so the • {ww | w ∈ {a, b}*} context-free languages are a subset of D. • {w of form x∗y=z: x,y,z ∈ {0, 1}* and, when x, y, and z are There is at least one language, AnBnCn, that is decidable • • viewed as binary numbers, xy = z} but not context-free.
• But already found some in gap, e.g. HTM
Outside of SD Closure Properties
• Theorem: D is closed under complement • Proof: Let L ε D. Build TM deciding L • Uncountably many languages outside of SD • ... • Proof depends on TM deterministic and always halts. • Complement of HTM • What about SD?
• Not true for HTM Equivalences to SD SD & Turing Enumerable
• A TM M enumerates the language L iff, for some • Theorem: A language is SD iff it is Turing fixed state p of M, enumerable. L = {w : (s, ε) |- * (p, w)}. M • Proof: Spose L is Turing enumerable. Show L is SD. • Potentia"y infinite computation. • Let w be input. Start enumerating L. Every time enter state p, check to see if contents of tape is w. If yes then halt and stop. • A language is Turing-enumerable iff there is a Otherwise keep going. Turing machine that enumerates it. • “Obvious” proof in other direction not work!