Martin-Löf`s Type Theory
... It is also possible to view a set as a problem description in a way similar to Kolmogorov’s explanation of the intuitionistic propositional calculus [25]. In particular, a set can be seen as a specification of a programming problem; the elements of the set are then the programs that satisfy the spec ...
... It is also possible to view a set as a problem description in a way similar to Kolmogorov’s explanation of the intuitionistic propositional calculus [25]. In particular, a set can be seen as a specification of a programming problem; the elements of the set are then the programs that satisfy the spec ...
Mathematical Logic Fall 2004 Professor R. Moosa Contents
... Mathematical Logic is the study of the type of reasoning done by mathematicians. (i.e. proofs, as opposed to observation) Axioms are the first unprovable laws. They are statements about certain “basic concepts” (undefined first concepts). There is usually some sort of “soft” justification for believ ...
... Mathematical Logic is the study of the type of reasoning done by mathematicians. (i.e. proofs, as opposed to observation) Axioms are the first unprovable laws. They are statements about certain “basic concepts” (undefined first concepts). There is usually some sort of “soft” justification for believ ...
Proof Theory: From Arithmetic to Set Theory
... The natural deduction calculus and the sequent calculus were both invented by Gentzen in 1934. Both calculi are pretty illustrations of the symmetries of logic. In this course I shall focus on the sequent calculus since it is a central tool in ordinal analysis and allows for generalizations to infin ...
... The natural deduction calculus and the sequent calculus were both invented by Gentzen in 1934. Both calculi are pretty illustrations of the symmetries of logic. In this course I shall focus on the sequent calculus since it is a central tool in ordinal analysis and allows for generalizations to infin ...
Modal logic and the approximation induction principle
... in terms of observations. That is, a process semantics is captured by means of a sublogic of HennessyMilner logic; two states in an LTS are equivalent if and only if they make true exactly the same formulas in this sublogic. In particular, Hennessy-Milner logic itself characterizes bisimulation equi ...
... in terms of observations. That is, a process semantics is captured by means of a sublogic of HennessyMilner logic; two states in an LTS are equivalent if and only if they make true exactly the same formulas in this sublogic. In particular, Hennessy-Milner logic itself characterizes bisimulation equi ...
EVERYONE KNOWS THAT SOMEONE KNOWS
... An example of a universally true formula in our language is ∀x (2x ∃y 2y φ → 2x φ), where variable y does not occur in formula φ. Informally, this statement means “if agent x knows that somebody knows φ, then agent x herself knows φ”. We show that this statement is derivable in our logical system in ...
... An example of a universally true formula in our language is ∀x (2x ∃y 2y φ → 2x φ), where variable y does not occur in formula φ. Informally, this statement means “if agent x knows that somebody knows φ, then agent x herself knows φ”. We show that this statement is derivable in our logical system in ...
Ultrasheaves
... objects X in C. For such a “Grothendieck topology” a sheaf is a functor which respects the topology on C in a appropriate manner. The sheaves on a category C with a Grothendieck topology J (a “site”) form a “Grothendieck topos” Sh(C, J). The interest for toposes in logic has as a starting point Lawv ...
... objects X in C. For such a “Grothendieck topology” a sheaf is a functor which respects the topology on C in a appropriate manner. The sheaves on a category C with a Grothendieck topology J (a “site”) form a “Grothendieck topos” Sh(C, J). The interest for toposes in logic has as a starting point Lawv ...
On interpretations of arithmetic and set theory
... holds as can easily be checked. (It is somewhat annoying that the convenient notation of writing interpretation-applications as superscripts is at odds with the usual mapson-left convention for morphisms.) As usual, two morphisms f : T → S and g : S → T are said to be inverse to each other if fg = 1 ...
... holds as can easily be checked. (It is somewhat annoying that the convenient notation of writing interpretation-applications as superscripts is at odds with the usual mapson-left convention for morphisms.) As usual, two morphisms f : T → S and g : S → T are said to be inverse to each other if fg = 1 ...
3) FPGA Based Systems Design
... When the logic blocks are larger and more complex, the architecture is called coarse-grained. The I/O blocks are on the outer edges of the structure and provide individually selectable input, output, or bidirectional access to the outside world. The distributed programmable interconnection mat ...
... When the logic blocks are larger and more complex, the architecture is called coarse-grained. The I/O blocks are on the outer edges of the structure and provide individually selectable input, output, or bidirectional access to the outside world. The distributed programmable interconnection mat ...
Factoring out the impossibility of logical aggregation
... deductively closed in the same relative sense, i.e., for all ∈ ∗ , if B, then ∈ B. It follows in particular that ∈ B ⇔ ∈ B when ↔ , and that ∈ B when . Deductive closure and its consequences raise the “logical omniscience problem” that is widely discussed in epistemic logic (see, ...
... deductively closed in the same relative sense, i.e., for all ∈ ∗ , if B, then ∈ B. It follows in particular that ∈ B ⇔ ∈ B when ↔ , and that ∈ B when . Deductive closure and its consequences raise the “logical omniscience problem” that is widely discussed in epistemic logic (see, ...
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.