Complexity of Recursive Normal Default Logic 1. Introduction
... propositional logic nonmonotonic formalisms, the basic results are found in [BF91, MT91, Got92, Van89]. In case of formalisms admitting variables and, more generally, infinite recursive propositional nonmonotonic formalisms, a number of results has been found. These include basic complexity results ...
... propositional logic nonmonotonic formalisms, the basic results are found in [BF91, MT91, Got92, Van89]. In case of formalisms admitting variables and, more generally, infinite recursive propositional nonmonotonic formalisms, a number of results has been found. These include basic complexity results ...
Document
... • A argument in proposi:onal logic is a sequence of proposi:ons. All but the final proposi:on are called premises. The last statement is the conclusion. • The argument is valid if the premises imply the conclusion. An argument form is an argument that is valid no maMer what proposi:ons ar ...
... • A argument in proposi:onal logic is a sequence of proposi:ons. All but the final proposi:on are called premises. The last statement is the conclusion. • The argument is valid if the premises imply the conclusion. An argument form is an argument that is valid no maMer what proposi:ons ar ...
Classical Logic and the Curry–Howard Correspondence
... must for this reason reject certain principles of classical logic, such as the excluded middle P ∨ ¬P and the double-negation rule ¬¬P → P . Constructive logic, therefore, is a proper subsystem of classical logic, and an object worthy of study in its own right. In computer science, the Curry–Howard ...
... must for this reason reject certain principles of classical logic, such as the excluded middle P ∨ ¬P and the double-negation rule ¬¬P → P . Constructive logic, therefore, is a proper subsystem of classical logic, and an object worthy of study in its own right. In computer science, the Curry–Howard ...
Elements of Set Theory
... G such that H is equinumerous with G0 and G is equinumerous with H 0 . Based on these conditions we prove that H is equinumerous with G. Let f be a one-to-one correspondence between H and G0 ; and let g be a one-to-one correspondence between G and H 0 . Our strategy is to partition H and G into smal ...
... G such that H is equinumerous with G0 and G is equinumerous with H 0 . Based on these conditions we prove that H is equinumerous with G. Let f be a one-to-one correspondence between H and G0 ; and let g be a one-to-one correspondence between G and H 0 . Our strategy is to partition H and G into smal ...
The Emergence of First
... he employed some richer form of logic. I have distinguished between a logician's use of first-order logic (where quantifiers range only over individuals), second-order logic (where quantifiers can also range over sets or relations),
... he employed some richer form of logic. I have distinguished between a logician's use of first-order logic (where quantifiers range only over individuals), second-order logic (where quantifiers can also range over sets or relations),
CS311H: Discrete Mathematics Cardinality of Infinite Sets and
... First list those with p + q = 2, then p + q = 3, . . . ...
... First list those with p + q = 2, then p + q = 3, . . . ...
G - Courses
... terms according to the equalities between them in some structure satisfying the FO-sentence at hand. Here, we used the resolution procedure only for formulas of propositional logic. The resolution procedure can be extended to FO-formulas using unification of terms. There are other proofs of Göde ...
... terms according to the equalities between them in some structure satisfying the FO-sentence at hand. Here, we used the resolution procedure only for formulas of propositional logic. The resolution procedure can be extended to FO-formulas using unification of terms. There are other proofs of Göde ...
PDF
... numbers using first-order logic axioms. We have noted that weak first-order theories such as Q allow many non-standard interpretations of the numbers. There is a deep theorem of classical logic, the Lowenheim-Skolem theorem, that tells us that first-order logic is not adequate to define the standard ...
... numbers using first-order logic axioms. We have noted that weak first-order theories such as Q allow many non-standard interpretations of the numbers. There is a deep theorem of classical logic, the Lowenheim-Skolem theorem, that tells us that first-order logic is not adequate to define the standard ...
full text (.pdf)
... The argument for alternative 3 is symmetric. For the lower bound, we reduce the totality problem for oracle Turing machines with oracle A, a well-known Π2A -complete problem, to the equivalence problem (ii). The totality problem is to determine whether a given machine halts on all inputs. As above, ...
... The argument for alternative 3 is symmetric. For the lower bound, we reduce the totality problem for oracle Turing machines with oracle A, a well-known Π2A -complete problem, to the equivalence problem (ii). The totality problem is to determine whether a given machine halts on all inputs. As above, ...
slides
... Don’t get confused! The symbol |= is used in two different ways: I |= F F1 , . . . , Fn |= G In the first the left-hand-side is an interpretation, in the second it is a sequence (or set) of formulas. ...
... Don’t get confused! The symbol |= is used in two different ways: I |= F F1 , . . . , Fn |= G In the first the left-hand-side is an interpretation, in the second it is a sequence (or set) of formulas. ...
CHAPTER 14 Hilbert System for Predicate Logic 1 Completeness
... Step 2 We show that every maximal finitely consistent set is consistent by constructing its model. Step 3 We show that every finitely consistent set S can be extended to a maximal finitely consistent set S ∗ . I.e we show that for every finitely consistent set S there is a set S ∗ , such that S ⊂ S ...
... Step 2 We show that every maximal finitely consistent set is consistent by constructing its model. Step 3 We show that every finitely consistent set S can be extended to a maximal finitely consistent set S ∗ . I.e we show that for every finitely consistent set S there is a set S ∗ , such that S ⊂ S ...
Techniques for proving the completeness of a proof system
... Sometimes it is easier to show the truth of a formula than to derive the formula. The completeness result shows that nothing is missing in a proof system. The completeness result formalizes what a proof system achieves. With a completeness result, a paper about a proof system has more chances to get ...
... Sometimes it is easier to show the truth of a formula than to derive the formula. The completeness result shows that nothing is missing in a proof system. The completeness result formalizes what a proof system achieves. With a completeness result, a paper about a proof system has more chances to get ...
Logic and Proof
... • Quantifiers are duals of one another • Let be a formula. The following holds: not(x()) x(not()) not(x ()) x(not()) • Proof using a model: Domain = Bill, Jill Model = Bill is happy. It is not the case that everyone is happy ?=? Someone is not happy It is not the case that someone is h ...
... • Quantifiers are duals of one another • Let be a formula. The following holds: not(x()) x(not()) not(x ()) x(not()) • Proof using a model: Domain = Bill, Jill Model = Bill is happy. It is not the case that everyone is happy ?=? Someone is not happy It is not the case that someone is h ...
On Elkan`s theorems: Clarifying their meaning
... omitted from the first version of Elkan’s theorem. As to the rest of the assumptions, both t~A ∧ B! ⫽ min$t~A!, t~B!% and t~¬A! ⫽ 1 ⫺ t~A! are quite reasonable and, in fact, are often used in applications of fuzzy logic. Let us now concentrate on the last assumption, that is, on t~A! ⫽ t~B! if A and ...
... omitted from the first version of Elkan’s theorem. As to the rest of the assumptions, both t~A ∧ B! ⫽ min$t~A!, t~B!% and t~¬A! ⫽ 1 ⫺ t~A! are quite reasonable and, in fact, are often used in applications of fuzzy logic. Let us now concentrate on the last assumption, that is, on t~A! ⫽ t~B! if A and ...
p - Erwin Sitompul
... A formal proof is a set of proofs which follows logically from the set of premises. Formal proofs allow us to infer new true statements from known true statements. A proposition or its part can be transformed using a sequence of logical equivalence until some conclusions can be reached. Exam ...
... A formal proof is a set of proofs which follows logically from the set of premises. Formal proofs allow us to infer new true statements from known true statements. A proposition or its part can be transformed using a sequence of logical equivalence until some conclusions can be reached. Exam ...
com.1 The Compactness Theorem
... Problem com.2. In the standard model of arithmetic N, there is no element k ∈ |N| which satisfies every formula n < x (where n is 0...0 with n 0’s). Use the compactness theorem to show that the set of sentences in the language of arithmetic which are true in the standard model of arithmetic N are a ...
... Problem com.2. In the standard model of arithmetic N, there is no element k ∈ |N| which satisfies every formula n < x (where n is 0...0 with n 0’s). Use the compactness theorem to show that the set of sentences in the language of arithmetic which are true in the standard model of arithmetic N are a ...
Reaching transparent truth
... response to the well-known paradoxes that inevitably arise. Many of these (eg [Priest, 2006b, Kremer, 1988, Beall, 2009, Field, 2008]) are based in some way on the work in [Kripke, 1975], and our approach is no different. As such, this section first briefly reviews the so-called ‘Kripke construction ...
... response to the well-known paradoxes that inevitably arise. Many of these (eg [Priest, 2006b, Kremer, 1988, Beall, 2009, Field, 2008]) are based in some way on the work in [Kripke, 1975], and our approach is no different. As such, this section first briefly reviews the so-called ‘Kripke construction ...
The Compactness Theorem 1 The Compactness Theorem
... Theorem. This will play an important role in the second half of the course when we study predicate logic. This is due to our use of Herbrand’s Theorem to reduce reasoning about formulas of predicate logic to reasoning about infinite sets of formulas of propositional logic. Before stating and proving ...
... Theorem. This will play an important role in the second half of the course when we study predicate logic. This is due to our use of Herbrand’s Theorem to reduce reasoning about formulas of predicate logic to reasoning about infinite sets of formulas of propositional logic. Before stating and proving ...
An Introduction to Modal Logic VII The finite model property
... A normal modal logic Λ has the finite model property if and only if it has the finite frame property. Clearly the f.f.p. implies the f.m.p. On the other hand, suppose now that Λ has the f.m.p. and let ϕ∈ / Λ; by f.m.p., there is a finite model M where ϕ is not valid; consider M∼ , it is differentiat ...
... A normal modal logic Λ has the finite model property if and only if it has the finite frame property. Clearly the f.f.p. implies the f.m.p. On the other hand, suppose now that Λ has the f.m.p. and let ϕ∈ / Λ; by f.m.p., there is a finite model M where ϕ is not valid; consider M∼ , it is differentiat ...
