Can every arbitrary problem be expressed as a language?
In the domain of computational complexity theory, the concept of expressing problems as languages is fundamental. To address this question we need to consider theoretical underpinnings of computation and formal languages.
A "language" in computational complexity theory is a set of strings over a finite alphabet. It is a formal construct that can be recognized by a computational model, such as a Turing machine. This concept is rooted in formal language theory, which studies the syntactic properties of formal languages and their classification.
To understand whether every arbitrary problem can be expressed as a language, we must first clarify what is meant by an "arbitrary problem." In computational terms, a problem typically refers to a decision problem or a function problem. A decision problem asks a yes/no question about an input, while a function problem requires computing a specific output for a given input.
For decision problems, the connection to languages is straightforward. A decision problem can indeed be expressed as a language. Consider a decision problem ( P ) that asks whether a given input ( x ) belongs to a set ( S ). This problem can be represented by the language ( L ) consisting of all strings ( x ) for which the answer to ( P(x) ) is "yes." Therefore, the decision problem ( P ) is equivalent to the language ( L ).
For instance, let us consider the decision problem of determining whether a given binary string represents an even number. The language corresponding to this problem is the set of all binary strings that end with a '0', as a binary number is even if and only if its least significant bit is 0. Thus, the problem can be expressed as the language ( L = { x in {0,1}^* mid x text{ ends with } 0 } ).
Function problems, on the other hand, are more nuanced. A function problem requires computing a value rather than making a binary decision. To express a function problem as a language, one approach is to consider the graph of the function. The graph of a function ( f ) is a set of ordered pairs ( (x, f(x)) ). This set can be seen as a language over the alphabet that includes symbols for both inputs and outputs.
For example, consider the function problem of computing the square of an integer. The graph of this function is the set of pairs ( { (x, x^2) mid x in mathbb{Z} } ). This set can be encoded as a language over an appropriate alphabet. However, this representation is more complex than the straightforward representation of decision problems.
The ability to express problems as languages is a powerful concept because it allows the use of formal language theory and automata theory to analyze computational problems. The classes of languages recognized by different computational models, such as regular languages, context-free languages, and recursively enumerable languages, correspond to different classes of decision problems.
Regular languages, for example, are those that can be recognized by finite automata. They correspond to decision problems that can be solved with a finite amount of memory. Context-free languages, recognized by pushdown automata, can represent problems that require a stack-like memory structure. Recursively enumerable languages, recognized by Turing machines, correspond to problems that can be solved by a general-purpose computer with unlimited memory.
To illustrate the connection between problems and languages further, consider the famous problem of determining whether a given string is a palindrome. A palindrome is a string that reads the same forward and backward. The decision problem of checking whether a string is a palindrome can be expressed as the language ( L = { x in Sigma^* mid x = x^R } ), where ( x^R ) denotes the reverse of ( x ).
Moreover, the concept of reductions between problems in computational complexity theory relies on expressing problems as languages. A reduction from problem ( A ) to problem ( B ) is a way of transforming instances of ( A ) into instances of ( B ) such that solving ( B ) allows us to solve ( A ). This transformation is often expressed as a function that maps strings in the language of ( A ) to strings in the language of ( B ).
For example, the problem of determining whether a graph is bipartite can be reduced to the problem of determining whether a graph has a proper 2-coloring. This reduction can be expressed as a function that transforms instances of the bipartiteness problem into instances of the 2-coloring problem. The language corresponding to the bipartiteness problem is the set of all graph encodings that represent bipartite graphs, and the language corresponding to the 2-coloring problem is the set of all graph encodings that have a proper 2-coloring.
Additionally, the theory of NP-completeness relies heavily on expressing problems as languages. A problem is NP-complete if it is in NP and every problem in NP can be reduced to it in polynomial time. The class NP consists of decision problems for which a "yes" answer can be verified by a nondeterministic polynomial-time algorithm. Expressing these problems as languages allows the use of formal methods to prove NP-completeness.
For instance, the SAT problem, which asks whether a given boolean formula is satisfiable, is NP-complete. The language corresponding to SAT is the set of all encodings of satisfiable boolean formulas. Proving that another problem, such as the Clique problem, is NP-complete involves showing that SAT can be reduced to Clique. This reduction is expressed as a function that transforms boolean formulas into graph encodings.
While decision problems can be directly expressed as languages, function problems require a more complex representation, often involving the graph of the function. The ability to express problems as languages is a cornerstone of computational complexity theory, enabling the use of formal language theory to analyze and classify problems. This approach underpins many fundamental concepts, such as reductions, NP-completeness, and the classification of languages recognized by different computational models.
Other recent questions and answers regarding EITC/IS/CCTF Computational Complexity Theory Fundamentals:
- Please describe the example in the answer where is binary string with even 1 symbols recognizing FSM." …the input string “1011”, the FSM does not reach the final state and gets stuck in S0 after processing the first three symbols."
- How does nondeterminism impact transition function?
- Are regular languages equivalent with Finite State Machines?
- Is PSPACE class not equal to the EXPSPACE class?
- Is algorithmically computable problem a problem computable by a Turing Machine accordingly to the Church-Turing Thesis?
- What is the closure property of regular languages under concatenation? How are finite state machines combined to represent the union of languages recognized by two machines?
- Is P complexity class a subset of PSPACE class?
- Does every multi-tape Turing machine has an equivalent single-tape Turing machine?
- What are the outputs of predicates?
- Are lambda calculus and turing machines computable models that answers the question on what does computable mean?
View more questions and answers in EITC/IS/CCTF Computational Complexity Theory Fundamentals