Knowledge Representation: Logic
... divided into point-like, line-like and so on. Addition of a new component may be then achieved by adding a new subclass, but it can be impossible, for example for street names. We may as well express the content of the map using logic and add new components by introducing new predicates. Wojciech Ja ...
... divided into point-like, line-like and so on. Addition of a new component may be then achieved by adding a new subclass, but it can be impossible, for example for street names. We may as well express the content of the map using logic and add new components by introducing new predicates. Wojciech Ja ...
Parametric Polymorphism and Abstract Models of Storage
... [Declarative languages] are an interesting subset, but... inconvenient... We need them because at the moment we don’t know how to construct proofs with ... imperatives and ...
... [Declarative languages] are an interesting subset, but... inconvenient... We need them because at the moment we don’t know how to construct proofs with ... imperatives and ...
03_boolean algebra
... Conversion of English Sentences to Boolean Equations The first step in designing a logic network: − Translate English sentences to Boolean Eqns. − We must break down each sentence into phrases − And associate a Boolean variable with each phrase (possible if a phrase can have a “true”/”false” valu ...
... Conversion of English Sentences to Boolean Equations The first step in designing a logic network: − Translate English sentences to Boolean Eqns. − We must break down each sentence into phrases − And associate a Boolean variable with each phrase (possible if a phrase can have a “true”/”false” valu ...
The semantics of propositional logic
... The formal language of formulas (1.3) Our formulas can be viewed as strings over an alphabet composed of our atoms (p, q, r, p1 , p2 , . . .), plus the symbols ¬, ∧, ∨, →, and the open and close parentheses, ( and ). (⊥ is a convenience in our proofs, and shouldn’t appear in formulas.) In the previ ...
... The formal language of formulas (1.3) Our formulas can be viewed as strings over an alphabet composed of our atoms (p, q, r, p1 , p2 , . . .), plus the symbols ¬, ∧, ∨, →, and the open and close parentheses, ( and ). (⊥ is a convenience in our proofs, and shouldn’t appear in formulas.) In the previ ...
A SHORT AND READABLE PROOF OF CUT ELIMINATION FOR
... disjunction property, etc. Logicians have turned their attention to first-order modal logics in the search for a predicate provability logic. Results of Vardanyan (in [3]) and Montagna [7] showed that the “natural” first-order extension of GL, known as Quantified GL (QGL), is not as “nice”: Its Gent ...
... disjunction property, etc. Logicians have turned their attention to first-order modal logics in the search for a predicate provability logic. Results of Vardanyan (in [3]) and Montagna [7] showed that the “natural” first-order extension of GL, known as Quantified GL (QGL), is not as “nice”: Its Gent ...
Bounded Functional Interpretation
... where A is any formula, provided that we read the relation ≤1 intensionally (more on this below). This is a blatantly false principle in classical mathematics. Intuitionistic mathematics accepts it due to reasons of continuity: If one warrants intuitionistically the antecedent ∀g ≤1 f ∃nA(g, n), the ...
... where A is any formula, provided that we read the relation ≤1 intensionally (more on this below). This is a blatantly false principle in classical mathematics. Intuitionistic mathematics accepts it due to reasons of continuity: If one warrants intuitionistically the antecedent ∀g ≤1 f ∃nA(g, n), the ...
HOARE`S LOGIC AND PEANO`S ARITHMETIC
... Thus, we may observe that equations (3)-(6) alone define N under initial algebra semantics and so we may consider (1) and (2) as additions, making a first refinement of the standard algebraic specification for arithmetic, designed to rule out finite models. The theoretical objective of adding the in ...
... Thus, we may observe that equations (3)-(6) alone define N under initial algebra semantics and so we may consider (1) and (2) as additions, making a first refinement of the standard algebraic specification for arithmetic, designed to rule out finite models. The theoretical objective of adding the in ...
Using linear logic to reason about sequent systems
... ! P ` dBe is provable, then there is an object-level cut-free proof of the Forum sequent P; · −→ dBe; ·. Theorem: Determining whether or not a canonical proof system is coherent is decidable. In particular, determining duality of a right and left introduction rule connective can be done by bounding ...
... ! P ` dBe is provable, then there is an object-level cut-free proof of the Forum sequent P; · −→ dBe; ·. Theorem: Determining whether or not a canonical proof system is coherent is decidable. In particular, determining duality of a right and left introduction rule connective can be done by bounding ...
Introduction, Scheme basics (expressions, values)
... Evaluation of Expressions The value of a numeral: number The value of a built-in operator: machine instructions to execute The value of any name: the associated value in the environment To Evaluate a combination: (as opposed to special form) a. Evaluate all of the sub-expressions in some order b. A ...
... Evaluation of Expressions The value of a numeral: number The value of a built-in operator: machine instructions to execute The value of any name: the associated value in the environment To Evaluate a combination: (as opposed to special form) a. Evaluate all of the sub-expressions in some order b. A ...
Local Normal Forms for First-Order Logic with Applications to
... First-order (FO) logic and its extensions play an important role in many branches of (theoretical) computer science. Examples that will be considered in this paper are automata theory and descriptive complexity. Since Büchi’s and Elgot’s famous characterization of the regular string languages as th ...
... First-order (FO) logic and its extensions play an important role in many branches of (theoretical) computer science. Examples that will be considered in this paper are automata theory and descriptive complexity. Since Büchi’s and Elgot’s famous characterization of the regular string languages as th ...
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... These three themes are a logical follow up of the well-rounded study of a calculus, its semantics/models and its usages/applications. In short, the untyped λ-calculus is a computational model used for the formalisation of the foundations of Mathematics. Based on the untyped λ-calculus (typed version ...
... These three themes are a logical follow up of the well-rounded study of a calculus, its semantics/models and its usages/applications. In short, the untyped λ-calculus is a computational model used for the formalisation of the foundations of Mathematics. Based on the untyped λ-calculus (typed version ...
1 Preliminaries 2 Basic logical and mathematical definitions
... When considering logic programs there exists a particular class of interpretations which are relevant, namely Herbrand interpretations. Let assume that the first order language L is defined on a signature Σ which contains at least one 0-ary function symbol. The set τ (Σ) of the ground terms is calle ...
... When considering logic programs there exists a particular class of interpretations which are relevant, namely Herbrand interpretations. Let assume that the first order language L is defined on a signature Σ which contains at least one 0-ary function symbol. The set τ (Σ) of the ground terms is calle ...
The modal logic of equilibrium models
... H (‘here’) and T (‘there’) such that H ⊆ T . The logical language to talk about such models has connectives ⊥, ∧, ∨, and ⇒. The latter is interpreted in a non-classical way and is therefore different from the material implication →: H, T |= ϕ ⇒ ψ iff H, T |= ϕ → ψ and T, T |= ϕ → ψ where → is interp ...
... H (‘here’) and T (‘there’) such that H ⊆ T . The logical language to talk about such models has connectives ⊥, ∧, ∨, and ⇒. The latter is interpreted in a non-classical way and is therefore different from the material implication →: H, T |= ϕ ⇒ ψ iff H, T |= ϕ → ψ and T, T |= ϕ → ψ where → is interp ...
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.