In logic, a true/false decision problem is decidable if there exists an effective method for deriving the correct answer. Zeroth-order logic (propositional logic) is decidable, whereas first-order and higher-order logic are not. Logical systems are decidable if membership in their set of logically valid formulas (or theorems) can be effectively determined. A theory (set of sentences closed under logical consequence) in a fixed logical system is decidable if there is an effective method for determining whether arbitrary formulas are included in the theory. Many important problems are undecidable, that is, it has been proven that no effective method for determining membership (returning a correct answer after finite, though possibly very long, time in all cases) can exist for them.
Each logical system comes with both a syntactic component, which among other things determines the notion of provability, and a semantic component, which determines the notion of logical validity. The logically valid formulas of a system are sometimes called the theorems of the system, especially in the context of first-order logic where Gödel's completeness theorem establishes the equivalence of semantic and syntactic consequence. In other settings, such as linear logic, the syntactic consequence (provability) relation may be used to define the theorems of a system.
A logical system is decidable if there is an effective method for determining whether arbitrary formulas are theorems of the logical system. For example, propositional logic is decidable, because the truth-table method can be used to determine whether an arbitrary propositional formula is logically valid.
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.
The validities of monadic predicate calculus with identity are decidable, however.
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In logic, a true/false decision problem is decidable if there exists an effective method for deriving the correct answer. Zeroth-order logic (propositional logic) is decidable, whereas first-order and higher-order logic are not. Logical systems are decidable if membership in their set of logically valid formulas (or theorems) can be effectively determined. A theory (set of sentences closed under logical consequence) in a fixed logical system is decidable if there is an effective method for determining whether arbitrary formulas are included in the theory.
Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or logical truths. It studies how conclusions follow from premises due to the structure of arguments alone, independent of their topic and content. Informal logic is associated with informal fallacies, critical thinking, and argumentation theory. It examines arguments expressed in natural language while formal logic uses formal language.
In mathematical logic, a theory (also called a formal theory) is a set of sentences in a formal language. In most scenarios a deductive system is first understood from context, after which an element of a deductively closed theory is then called a theorem of the theory. In many deductive systems there is usually a subset that is called "the set of axioms" of the theory , in which case the deductive system is also called an "axiomatic system". By definition, every axiom is automatically a theorem.
Ce cours couvre les fondements des systèmes numériques. Sur la base d'algèbre Booléenne et de circuitscombinatoires et séquentiels incluant les machines d'états finis, les methodes d'analyse et de syn
Ce cours couvre les fondements des systèmes numériques. Sur la base d'algèbre Booléenne et de circuits combinatoires et séquentiels incluant les machines d'états finis, les methodes d'analyse et de sy
With every smartphone and computer now boasting multiple processors, the use of functional ideas to facilitate parallel programming is becoming increasingly widespread. In this course, you'll learn th
With every smartphone and computer now boasting multiple processors, the use of functional ideas to facilitate parallel programming is becoming increasingly widespread. In this course, you'll learn th
We explore the border between decidability and undecidability of verification problems related to message passing systems that admit unbounded creation of threads and name mobility. Inspired by use ca
Automated termination provers often use the following schema to prove that a program terminates: construct a relational abstraction of the program's transition relation and then show that the relation
Springer-Verlag New York, Ms Ingrid Cunningham, 175 Fifth Ave, New York, Ny 10010 Usa2011
Motivated by the problem of deciding verification conditions for the verification of functional programs, we present new decision procedures for automated reasoning about functional lists. We first sh