A Proof Theory for Generic Judgments: An extended abstract
... proof, and to prove the formula B[c/x] instead. In natural deduction and sequent calculus proofs, such new variables are called eigenvariables. In Gentzen’s original presentation of the sequent calculus [5], eigenvariables were immutable: reading proofs bottom-up, once an eigenvariable is introduced ...
... proof, and to prove the formula B[c/x] instead. In natural deduction and sequent calculus proofs, such new variables are called eigenvariables. In Gentzen’s original presentation of the sequent calculus [5], eigenvariables were immutable: reading proofs bottom-up, once an eigenvariable is introduced ...
Propositional Logic
... The problem of finding at least one model of the set of formulas that is also a model of the formula , is known as the propositonal satisfiability (PSAT) problem. An exhaustive procedure for solving the PSAT problem is to try systematically all of the ways to assign True and False to the atoms in ...
... The problem of finding at least one model of the set of formulas that is also a model of the formula , is known as the propositonal satisfiability (PSAT) problem. An exhaustive procedure for solving the PSAT problem is to try systematically all of the ways to assign True and False to the atoms in ...
Lecture 11 Artificial Intelligence Predicate Logic
... • Representing knowledge using logic is appealing because you can derive new knowledge from old mathematical deduction. • In this formalism you can conclude that a new statement is true if by proving that it follows from the statement that are already known. • It provides a way of deducing new state ...
... • Representing knowledge using logic is appealing because you can derive new knowledge from old mathematical deduction. • In this formalism you can conclude that a new statement is true if by proving that it follows from the statement that are already known. • It provides a way of deducing new state ...
Semi-constr. theories - Stanford Mathematics
... generated as follows: (i) 0 is a t.s., and (ii) if σ, τ are t.s. then so also is σ → τ. These theories have infinitely many variables xτ, yτ, zτ, … of each type τ; type superscripts are suppressed when there is no ambiguity. We occasionally use other kinds of letters like f, g, … n, m,… as well as c ...
... generated as follows: (i) 0 is a t.s., and (ii) if σ, τ are t.s. then so also is σ → τ. These theories have infinitely many variables xτ, yτ, zτ, … of each type τ; type superscripts are suppressed when there is no ambiguity. We occasionally use other kinds of letters like f, g, … n, m,… as well as c ...
completeness theorem for a first order linear
... for the rst order temporal logics with since and until over linear time and rationals were given in [16]. In the case of FOLTL (and similarly when the ow of time is isomorphic to reals or integers) the set of valid formulas is not recursively enumerable, and there is no recursive axiomatization of ...
... for the rst order temporal logics with since and until over linear time and rationals were given in [16]. In the case of FOLTL (and similarly when the ow of time is isomorphic to reals or integers) the set of valid formulas is not recursively enumerable, and there is no recursive axiomatization of ...
Lecture 14 Notes
... In propositional logic we could use booleanization to relate the truth table semantics of tableaux to the evidence semantics of refinement logic. This technique can be extended to first-order logic if the domains under consideration are finite. In this case a universally quantified formula (∀x)B cor ...
... In propositional logic we could use booleanization to relate the truth table semantics of tableaux to the evidence semantics of refinement logic. This technique can be extended to first-order logic if the domains under consideration are finite. In this case a universally quantified formula (∀x)B cor ...
Knowledge Representation
... • Semantic nets, frames and objects all allow you to define relations between objects, including class relations (X isa Y). • Only restricted inference supported by the methods - that based on inheritance. ...
... • Semantic nets, frames and objects all allow you to define relations between objects, including class relations (X isa Y). • Only restricted inference supported by the methods - that based on inheritance. ...
Lesson 12
... There is another property of calculi. To illustrate it, let’s raise a question: having a formula , does the calculus decide ? In other words, is there an algorithm that would answer Yes or No, having as input and answering the question whether is logically valid or no? If there is such an algo ...
... There is another property of calculi. To illustrate it, let’s raise a question: having a formula , does the calculus decide ? In other words, is there an algorithm that would answer Yes or No, having as input and answering the question whether is logically valid or no? If there is such an algo ...
(A B) |– A
... There is another property of calculi. To illustrate it, let’s raise a question: having a formula , does the calculus decide ? In other words, is there an algorithm that would answer Yes or No, having as input and answering the question whether is logically valid or no? If there is such an algo ...
... There is another property of calculi. To illustrate it, let’s raise a question: having a formula , does the calculus decide ? In other words, is there an algorithm that would answer Yes or No, having as input and answering the question whether is logically valid or no? If there is such an algo ...
Modal Logic
... The canonical frame for System K is the pair Fk = (Wk,Rk) where (1) Wk = {X | X is an MCS } (2) If X and Y are MCSs, then X Rk Y iff {❏X} Y. The canonical model for System K is given by Mk = (Fk,Vk) where for each X Wk, Vk(X) = X P. Lemma For each MCS X Wk and for each formula ,Mk ...
... The canonical frame for System K is the pair Fk = (Wk,Rk) where (1) Wk = {X | X is an MCS } (2) If X and Y are MCSs, then X Rk Y iff {❏X} Y. The canonical model for System K is given by Mk = (Fk,Vk) where for each X Wk, Vk(X) = X P. Lemma For each MCS X Wk and for each formula ,Mk ...
Glossary
... the simplest logic gate. It has one input and one output. The output is always the opposite of the input. ...
... the simplest logic gate. It has one input and one output. The output is always the opposite of the input. ...
Lecture 15: The Lambda Calculus
... The Lambda Calculus • What is the lambda calculus then? “A formal system for function definition, function application and recursion.” ...
... The Lambda Calculus • What is the lambda calculus then? “A formal system for function definition, function application and recursion.” ...
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.