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First-order logic also satisfies several metalogical theorems that make it amenable to analysis in proof theory, such as the Löwenheim–Skolem theorem and the compactness theorem. First-order logic is the standard for the formalization of mathematics into axioms, and is studied in the foundations of mathematics.
In artificial intelligence, a fluent is a condition that can change over time. In logical approaches to reasoning about actions, fluents can be represented in first-order logic by predicates having an argument that depends on time.
Even later, in the 1990s, statistical relational learning, an approach that combines probability with logical formulas, allowed probability to be combined with first-order logic, e.g., with either Markov Logic Networks or Probabilistic Soft Logic. Other, non-probabilistic extensions to first-order logic to support were also tried.
SPASS is a first-order logic theorem prover with equality. This is developed by the research group Automation of Logic, Max Planck Institute for Computer Science . The Theorem Prover Museum [ 27 ] is an initiative to conserve the sources of theorem prover systems for future analysis, since they are important cultural/scientific artefacts.
First Order Logic (FOL), with its high expressive power and ability to formalise much of mathematics, is a standard for comparing the expressibility of knowledge representation languages. Arguably, FOL has two drawbacks as a knowledge representation formalism in its own right, namely ease of use and efficiency of implementation.
In first-order syntactic unification, variables range over first-order terms and equivalence is syntactic. This version of unification has a unique "best" answer and is used in logic programming and programming language type system implementation, especially in Hindley–Milner based type inference algorithms.
In a first-order logic system, additional axioms are required to make inferences about the environment (for example, that a block cannot change position unless it is physically moved). The frame problem is the problem of finding adequate collections of axioms for a viable description of a robot environment.
There are three common ways of handling this in first-order logic: Use first-order logic with two types. Use ordinary first-order logic, but add a new unary predicate "Set", where "Set(t)" means informally "t is a set". Use ordinary first-order logic, and instead of adding a new predicate to the language, treat "Set(t)" as an abbreviation for ...
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