Understanding Decidability for Context Sensitive Languages

Let's analyze the two statements given regarding the decidability of problems for Context Sensitive Languages (CSLs). Decidability in formal language theory refers to whether an algorithm exists that can always determine the answer to a yes/no question about a language or a related property in a finite amount of time.

Analyzing Statement I: Emptiness of Intersection for CSLs

Statement I says: "The problem 'Is L\(_1\) ∧ L\(_2\) = ϕ?' is undecidable for context sensitive languages L\(_1\) and L\(_2\)."

This statement concerns the emptiness problem for the intersection of two Context Sensitive Languages, L\(_1\) and L\(_2\). First, let's consider the properties of CSLs:

• Context Sensitive Languages are closed under intersection. This means if L\(_1\) and L\(_2\) are CSLs, then their intersection, L\(_1\) \(\cap\) L\(_2\), is also a CSL.
• The emptiness problem for a language asks whether the language contains any strings (i.e., is it equal to the empty set \(\emptyset\)). For CSLs, the emptiness problem is known to be undecidable. There is no general algorithm that can take any Context Sensitive Grammar or LBA and determine in finite time if the language it describes is empty.

Since L\(_1\) \(\cap\) L\(_2\) is a CSL, determining if L\(_1\) \(\cap\) L\(_2\) is empty is equivalent to solving the emptiness problem for a CSL. As the emptiness problem for CSLs is undecidable, the problem "Is L\(_1\) \(\cap\) L\(_2\) = \(\emptyset\)?" for CSLs L\(_1\) and L\(_2\) is indeed undecidable.

Therefore, Statement I is true.

Analyzing Statement II: Membership Problem for CSLs

Statement II says: "The problem 'Is WϵL?' is decidable for context sensitive language L, (where W is a string)."

This statement concerns the membership problem for a Context Sensitive Language L and a given string W. The membership problem asks whether a specific string W belongs to a language L.

For Context Sensitive Languages, the membership problem is known to be decidable. This means there exists an algorithm that can take a CSL (represented by a Context Sensitive Grammar or a Linear Bounded Automaton) and a string W, and determine in a finite number of steps whether W is generated by the grammar or accepted by the automaton.

A standard way to show this decidability is using a Linear Bounded Automaton (LBA), which is the automaton model equivalent to Context Sensitive Grammars. An LBA has a tape whose size is proportional to the length of the input string. Given an input string W, an LBA can simulate all possible derivations of length up to a certain bound related to the string length and grammar size. Since CSGs are non-contracting (productions do not decrease string length, except possibly for productions involving the empty string \(\varepsilon\), but these are handled carefully), if a string W can be derived, it can be derived through a sequence of steps where intermediate strings are not significantly longer than W itself. The LBA can explore the finite (though potentially large) state space and tape configurations within the bounds set by the input string length to determine acceptance or rejection.

Therefore, Statement II is true.

Conclusion on Both Statements

Based on the analysis:

• Statement I, concerning the undecidability of the emptiness of intersection for CSLs, is true because the intersection is a CSL and the emptiness problem for CSLs is undecidable.
• Statement II, concerning the decidability of the membership problem for CSLs, is true because an algorithm (like simulation by an LBA) exists to solve this problem in finite time.

Thus, both Statement I and Statement II are true.

Revision Table: Decidability of Problems for Language Classes

Additional Information on Context Sensitive Languages

Context Sensitive Languages (CSLs) are positioned in the Chomsky hierarchy between Context-Free Languages (CFLs) and Recursively Enumerable Languages (RELs). They are formally defined by Context Sensitive Grammars (CSGs).

• Context Sensitive Grammars (CSGs): In a CSG, productions are of the form \(\alpha A \beta \to \alpha \gamma \beta\), where A is a non-terminal, \(\alpha\) and \(\beta\) are strings of terminals and non-terminals, and \(\gamma\) is a non-empty string of terminals and non-terminals. The 'context' \(\alpha\) and \(\beta\) must be present for the production to apply. An exception is allowed for the start symbol S, which can have a production \(S \to \varepsilon\) if S does not appear on the right-hand side of any production. CSGs are called non-contracting because they do not decrease the length of the sentential form in any step (except possibly the \(S \to \varepsilon\) rule).
• Linear Bounded Automata (LBAs): LBAs are the automaton equivalent of CSGs. They are non-deterministic Turing machines whose tape is bounded by a constant multiple of the length of the input string. The fact that LBAs only need tape proportional to the input size is key to the decidability of the membership problem for CSLs. While LBAs are powerful enough to recognize CSLs, determining properties like emptiness or equivalence requires analyzing the full state space which, for LBAs, becomes complex enough to lead to undecidability for certain problems.

Understanding the decidability of problems like membership and emptiness is crucial in formal language theory as it tells us what kinds of questions about these languages can be solved algorithmically.