A well-formed formula (wff) is a sentence containing no "free" variables. I.e., all variables are "bound" by universal or existential quantifiers. E.g., ... Automated inference using FOL is harder than using PL because variables can take on potentially an infinite number of possible values from their domain. Hence there are potentially an ...
This demonstrates how FOL enables logical reasoning to derive new knowledge from given facts. Advanced Concepts in FOL. Unification: Finding substitutions that make two expressions identical, used in automated reasoning. Resolution: A rule of inference for theorem proving, used to derive contradictions and validate statements.
Valid Formula Schema H is valid iff valid for any FOL formula Fi obeying the side conditions Example: H1 and H2 are valid. 2- 17 Substitution σ of H σ : {F1 → ,...,Fn → } mapping place holders Fi of H to FOL formulae, (obeying the side conditions of H) Proposition (Formula Schema) If H is valid formula schema and
Inference in FOL: Inference rules • Is the Inference rule approach a viable approach for the FOL? • Yes. • The inference rules represent sound inference patterns one can apply to sentences in the KB • What is derived by inference rules follows from the KB • Caveat: we need to add rules for handling quantifiers M. Hauskrecht Inference ...
FOL inference rules for quantifier: First-order logic has inference rules similar to propositional logic, therefore here are some basic inference rules in FOL: ... from the formula given in the form of x P(x). The only constraint with this rule is that c must be a new word for which P(c) is true. ...
FOL inference rules for quantifier: As propositional logic we also have inference rules in first-order logic, so following are some basic inference rules in FOL: ... This rule states that one can infer P(c) from the formula given in the form of ∃x P(x) for a new constant symbol c. The restriction with this rule is that c used in the rule must ...
322 B.C. Aristotle “Syllogisms” (inference rules), quantifiers 1867 Boole PL 1879 Frege FOL 1922 Wittgenstein proof by truth tables 1930 Gödel 9complete algorithm for FOL 1930 Herbrand complete algorithm for FOL 1931 Gödel :9complete algorithm for arithmetic 1960 Davis/Putnam “practical” algorithm for PL (DP/DPLL)
The formula is false: $\mathbf{o_2}$ and $\mathbf{o_3}$ provide the counterexample: ... The following inferences are _in_valid in FOL. For each, describe a counter-model, where the premises are true but the conclusion is false. You may use a diagram as in Exercise 1 to give your model. a)
Inference Rules for FOL Inference rules for PL apply to FOL as well. E.g., Modus Ponens: If p is true and if p => q is true, then q is true. Conjunction (And-Introduction): If p is true and if q is true, then p ^ q is true. Simplification (And-Elimination): If p ^ q is true, then p is true. . . . Resolution:
Inference in FOL: Inference rules • Is the Inference rule approach a viable approach for the FOL? • Yes. • The inference rules represent sound inference patterns one can apply to sentences in the KB • What is derived follows from the KB • Caveat: we need to add rules for handling quantifiers
The language of FOL makes use of all of the connectives of TFL. So proofs in FOL will use all of the basic and derived rules from part IV. We will also use the proof-theoretic notions (particularly, the symbol ‘ ⊢ ’) introduced there. However, we will also need some new basic rules to govern the quantifiers, and to govern the identity sign.
Reduction to Propositional Inference (akapropositionalization)) Idea: Given a FOL closed KB Γ and query α,Convert (Γ ∧¬α) to PL =⇒use a PL SAT solver to check PL (un)satisfiability Trick: replace variables with ground terms, creating all possible instantiations of quantified sentences convert atomic sentences into propositional symbols
Automated Inference in FOL • Harder than in PL: – Variables can take on potentially an infinite number of possible values from their domain. – Hence there are potentially an infinite number of ways to apply the Universal-Elimination rules of inference • Goedel's Completeness Theorem: FOL entailment is semidecidable.
Challenges in Handling Uncertainty: FOL requires facts to be definitively true or false, making it difficult to represent probabilistic or uncertain information. High Computational Demands: When dealing with extensive knowledge bases, logical inference in FOL can become slow and resource-intensive.
This makes inference in FOL a hard topic, which we’ll look into in the next chapter. Just like with Gödel’s theorem, the proof of Turing’s theorem is out of scope. But what we’ll be able to appreciate are the pitfalls that naive inference algorithms in FOL can fall into, such as infinite loops in proof search, for example.
Decidability of FOL I FOL is undecidable (Turing & Church) There does not exist an algorithm for deciding if a FOL formula F is valid, i.e. always halt and says \yes" if F is valid or say \no" if F is invalid. I FOL is semi-decidable There is a procedure that always halts and says \yes" if F is valid, but may not halt if F is invalid. On the ...
Inference in FOL: Inference rules • Is the Inference rule approach a viable approach for the FOL? • Yes. • The inference rules represent sound inference patterns one can apply to sentences in the KB • What is derived by inference rules follows from the KB • Caveat: we need to add rules for handling quantifiers M. Hauskrecht
The article “Bayesian Inference Methods and Formula Explained” was originally published on QuantInsti blog. This post on Bayesian inference is the second of a multi-part series on Bayesian statistics and methods used in quantitative finance. In my previous post, I gave a leisurely introduction to Bayesian statistics and while doing so ...
