First-order logic is not decidable in general; in particular, the set of logical validities in any signature that includes equality and at least one other predicate with two or more arguments is not decidable. Logical systems extending first-order logic, such as second-order logic and type theory, are also undecidable.

Example: The halting problem is partially computable. To determine HALTS(P,D), simply call P(D). Then, HALTS(P,D) halts and outputs Yes if P(D) halts, and loops otherwise.

An unreasonable time algorithm is a problem that would take a massive amount of computing power to solve. The amount of time it would take to calculate a solution would be unreasonable, hence the name.

D rejects (D), but then H accepted (D,(D)) and hence D accepted (D), contradiction! So D cannot exist, so H cannot exist either (D was built from H). This means that ATM is undecidable. Theorem The language ATM is recognizable.

Like Turing reducibility, mapping reducibility is transitive.

Recall that a language L is Turing recognizable if there is a Turing machine that accepts exactly the words in L, but can either reject or loop indefinitely on an input that’s not in L. We will show that ATM , the complement of ATM , is not Turing-recognizable.

Thus, if the machine accepts a string, it must do so in a finite number of steps! A Language of a Turing Machine is simply the set of all strings that are accepted by the Turing Machine. Thus, L is a Turing Recognizable Language (since the TM M recognizes it).

▶ Recall that EQDFA is decidable. ▶ EQTM isn’t, as can be proved via reduction from ETM. ▶ Suppose that EQTM is decidable and let R be a decider for it. ▶ We construct a TM S to decide ETM as follows.

A Turing machine can be used as a transducer. The most obvious way to do this is to treat the entire nonblank portion of the initial tape as input, and to treat the entire nonblank portion of the tape when the machine halts as output.

Statement: Turing Machine can change symbols on its tape, whereas the FA cannot change symbols on tape. Explanation: The following mentioned is the difference between 2-way FA and TM. Another instance is that TM has a read/write tape head while FA doesn’t.

A Turing machine consists of (a) a finite control, (b) one tape, representing the memory, that has a left margin and is divided into an infinite number of cells, and (c) a moving read/write head. The finite control can be in any one of a finite set Q of states.

An LBA differs from a Turing machine in that while the tape is initially considered to have unbounded length, only a finite contiguous portion of the tape, whose length is a linear function of the length of the initial input, can be accessed by the read/write head; hence the name linear bounded automaton.

For any linear-bounded grammar G there exists a linear- bounded automaton which accepts the language L(G) generated by G.

called a linear-bounded automaton (LBA). A linear bounded automaton is a non-deterministic Turing machine M = (Q, Σ, Γ, δ, s, t, r ) such that: * There are two special tape symbols < and >(the left end marker and right end marker).

Definition. A Turing Machine (TM) is a mathematical model which consists of an infinite length tape divided into cells on which input is given. After reading an input symbol, it is replaced with another symbol, its internal state is changed, and it moves from one cell to the right or left.