Why is propositional logic decidable?

For example, propositional logic is decidable, because the truth-table method can be used to determine whether an arbitrary propositional formula is logically valid.

What does decidable mean in logic?

capable of being decided

: capable of being decided specifically : capable of being decided as following or not following from the axioms of a logical system Was logic complete … ? And was it decidable, in the sense that there was a method that demonstrated the truth or falsity of every statement? —

Why is predicate logic not decidable?

It’s hard to say what the “cause” is – mathematical phenomena have proofs, not causes. But the key reason for the undecidability is that predicate logic is too powerful; it’s powerful enough to describe the algorithm you might try to use, so it can circumvent it.

Why is Decidability important?

If a programming language is decidable, then it will always be possible to decide whether a program is a valid program for that language or not. But even if a program is a valid program for that language, it remains undecidable whether that program may incur a buffer overflow or a deadlock.

Is first-order logic decidable?

First-order logic is complete because all entailed statements are provable, but is undecidable because there is no algorithm for deciding whether a given sentence is or is not logically entailed.

What does it mean for a problem to be decidable?

(definition) Definition: A decision problem that can be solved by an algorithm that halts on all inputs in a finite number of steps. The associated language is called a decidable language.

Is modal logic decidable?

In general, the decidability of modal logic is very, very robust (cf. HM92, Var89]). As a rule of thumb, the validity problem for a modal logic is typically decidable; one has to make an e ort to nd a modal logic with an undecidable validity problem (cf.

What is the difference between completeness and Decidability?

Completeness means that either a proof or disproof exists. Decidability means that there’s an algorithm for finding a proof or disproof. In nice cases, they are equivalent, since in a complete theory, you can just iterate over every possible proof until you find one that either proves or disproves the statement.

How do you know if a problem is decidable?

A problem is said to be Decidable if we can always construct a corresponding algorithm that can answer the problem correctly. We can intuitively understand Decidable problems by considering a simple example. Suppose we are asked to compute all the prime numbers in the range of .

What is Decidability in theory of computation?

A language is called Decidable or Recursive if there is a Turing machine which accepts and halts on every input string w. Every decidable language is Turing-Acceptable. A decision problem P is decidable if the language L of all yes instances to P is decidable.

What is the difference between decidable and undecidable problems?

Decidable problem is one for which we can design an algorithm (doesn’t matter whether it is polynomial time or not). Whereas for undecidable problem we can’t give an algorithm. One of the famous undecidable problem is Turing machine Halting problem.

Which of the following is decidable?

Which of the following are decidable? Explanation: (A) Intersection of two regular languages is regular and checking if a regular language is infinite is decidable.

Is the halting problem decidable?

The halting problem is theoretically decidable for linear bounded automata (LBAs) or deterministic machines with finite memory. A machine with finite memory has a finite number of configurations, and thus any deterministic program on it must eventually either halt or repeat a previous configuration: …

Are all finite languages decidable?

If A is finite, it is decidable because all finite languages are decidable (just hardwire each of the strings into the TM). If A is infinite, a TM M that decides A operates as follows.

Why is halting problem semi-decidable?

A language is said to be Semi-decidable if there exists a Turing machine which halts if a word belongs to the language (YES cases) and may reject or go into infinite loop if the word doesn’t belong to the language (NO case).

Is recursive language decidable?

Equivalently, a formal language is recursive if there exists a total Turing machine (a Turing machine that halts for every given input) that, when given a finite sequence of symbols as input, accepts it if it belongs to the language and rejects it otherwise. Recursive languages are also called decidable.

How do you prove a language is decidable?

Prove that the language it recognizes is equal to the given language and that the algorithm halts on all inputs. To prove that a given language is Turing-recognizable: Construct an algorithm that accepts exactly those strings that are in the language. It must either reject or loop on any string not in the language.

What does it mean if a language is decidable?

(definition) Definition: A language for which membership can be decided by an algorithm that halts on all inputs in a finite number of steps — equivalently, can be recognized by a Turing machine that halts for all inputs.

Is regular language decidable?

1. (a) True, since every regular language is context-free, every context-free language is decidable, and every decidable language is Turing-recognizable.

Is emptiness of CFL decidable?

CFL: It is decidable for emptiness problem, finiteness problem, and membership problem.

Do all regular languages have a Turing machine?

Nope! If you have a regular language, you can get a DFA for it, then convert that DFA into a Turing machine by slightly adjusting the transitions so that they mechanically move the tape head forward. As a result, that language is also Turing-recognizable.

Is completeness problem decidable for CFL?

So this is how we decide: Given a language L, take its complement L’ and check if L’ is empty => L is complete. Although emptiness is decidable for both DCFLs and CFLs, however CFLs are not closed under complementation. Hence, decidable in case of DCFLs.

Which of the following problem is un Decidable?

Which of the following problems is undecidable? Deciding if a given context-free grammar is ambiguous. Deciding if a given string is generated by a given context-free grammar. Deciding if the language generated by a given context-free grammar is empty.

Which of the following problem is undecidable problem of CFL?

To check whether a given CFL is regular or not in undecidable. To determine whether an FA halts on all inputs or not is decidable because we have final and non-final states in FAs that indicate whether the input string is accepted or not. Membership problem for type 0 (Recursively Enumerable) grammars is undecidable.