On the Construction of Analytic Sequent Calculi for Sub
... a) sub-classical logic. Various important and useful non-classical logics can be formalized in this way, with the most prominent example being intuitionistic logic. In general, the resulting logics come at first with no semantics. They might be also unusable for computational purposes, since the new ...
... a) sub-classical logic. Various important and useful non-classical logics can be formalized in this way, with the most prominent example being intuitionistic logic. In general, the resulting logics come at first with no semantics. They might be also unusable for computational purposes, since the new ...
Logic and Resolution
... Consider the formula ∀x∃y∃zP (f (y, z), x) Given the structure S , this formula is clearly true Note, however, that this would not be the case if we had, for instance, interpreted P as ‘less than’ ...
... Consider the formula ∀x∃y∃zP (f (y, z), x) Given the structure S , this formula is clearly true Note, however, that this would not be the case if we had, for instance, interpreted P as ‘less than’ ...
Upper-Bounding Proof Length with the Busy
... It is commonly taught that checking examples never suffices to establish the truth of a hypothesis, no matter how many examples one has checked, and that this is why a proof is needed. However, Chaitin (1984) has shown an upper bound on the smallest counterexample (should it exist), based on the len ...
... It is commonly taught that checking examples never suffices to establish the truth of a hypothesis, no matter how many examples one has checked, and that this is why a proof is needed. However, Chaitin (1984) has shown an upper bound on the smallest counterexample (should it exist), based on the len ...
The Compactness Theorem 1 The Compactness Theorem
... For the induction step, suppose that we have constructed assignments A0 , . . . , An such that An satisfies (∗). Consider the two assignments B, B 0 that extend An with dom(B) = dom(B 0 ) = {p1 , p2 , . . . , pn+1 } (say B[[pn+1 ]] = 0 and B 0 [[pn+1 ]] = 1.) Since any proper extension of An is an ...
... For the induction step, suppose that we have constructed assignments A0 , . . . , An such that An satisfies (∗). Consider the two assignments B, B 0 that extend An with dom(B) = dom(B 0 ) = {p1 , p2 , . . . , pn+1 } (say B[[pn+1 ]] = 0 and B 0 [[pn+1 ]] = 1.) Since any proper extension of An is an ...
Lecture 9 Notes
... In the tableaux calculus there is one rule for a connective with sign T one for the sign F . In refinement logic one rule operates on the assumptions (left) and one on the conclusion (right). If we compare the rules, we notice a strong similarity if we associate the T -formulas with assumptions and ...
... In the tableaux calculus there is one rule for a connective with sign T one for the sign F . In refinement logic one rule operates on the assumptions (left) and one on the conclusion (right). If we compare the rules, we notice a strong similarity if we associate the T -formulas with assumptions and ...
First-Order Logic with Dependent Types
... for the sort S. o is the type of formulas. The remainder of the signature encodes the usual grammar for FOL formulas. Higher-order abstract syntax is used, i.e., λ is used to bind the free variables in a formula, and quantifiers are operators taking a λ expression as an argument.2 Quantifiers and th ...
... for the sort S. o is the type of formulas. The remainder of the signature encodes the usual grammar for FOL formulas. Higher-order abstract syntax is used, i.e., λ is used to bind the free variables in a formula, and quantifiers are operators taking a λ expression as an argument.2 Quantifiers and th ...
Propositional Calculus
... • Programs are functions and their semantics involve function application. Programs may also produce function by returning functions as values. In pure functional programming, this is it, there are no variables, side effects, nor loops. This simplifies semantics but does not reduce computational pow ...
... • Programs are functions and their semantics involve function application. Programs may also produce function by returning functions as values. In pure functional programming, this is it, there are no variables, side effects, nor loops. This simplifies semantics but does not reduce computational pow ...
ECE 301 – Digital Electronics
... Logic gates are the basic building blocks for (combinational and sequential) logic circuits. They are, however, abstractions. ...
... Logic gates are the basic building blocks for (combinational and sequential) logic circuits. They are, however, abstractions. ...
Combinatorial Circuits
... up the inputs and outputs and the ports that interface the device with those signal lines. A signal line is any connection that can transmit one bit at a time. An entity can be referred to as a module consisting of specific tasks. • A functional specification that specifies how the outputs are deter ...
... up the inputs and outputs and the ports that interface the device with those signal lines. A signal line is any connection that can transmit one bit at a time. An entity can be referred to as a module consisting of specific tasks. • A functional specification that specifies how the outputs are deter ...
Slide 1
... If f is in this form, there are two ways to implement the logic gate for the logic function. 1. expand the logic function through de Morgan rule and direct implementation on the expanded function. f = (A+B`C)` = A`(B`C)`= A`(B``+C`) = A`(B+C`) Implement the logic gate with the previous method, the i ...
... If f is in this form, there are two ways to implement the logic gate for the logic function. 1. expand the logic function through de Morgan rule and direct implementation on the expanded function. f = (A+B`C)` = A`(B`C)`= A`(B``+C`) = A`(B+C`) Implement the logic gate with the previous method, the i ...
Curry–Howard correspondence
In programming language theory and proof theory, the Curry–Howard correspondence (also known as the Curry–Howard isomorphism or equivalence, or the proofs-as-programs and propositions- or formulae-as-types interpretation) is the direct relationship between computer programs and mathematical proofs. It is a generalization of a syntactic analogy between systems of formal logic and computational calculi that was first discovered by the American mathematician Haskell Curry and logician William Alvin Howard. It is the link between logic and computation that is usually attributed to Curry and Howard, although the idea is related to the operational interpretation of intuitionistic logic given in various formulations by L. E. J. Brouwer, Arend Heyting and Andrey Kolmogorov (see Brouwer–Heyting–Kolmogorov interpretation) and Stephen Kleene (see Realizability). The relationship has been extended to include category theory as the three-way Curry–Howard–Lambek correspondence.