Modal logic and the approximation induction principle
... Milner 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 equivalence. For several process semantics, mainly in the realm of simulation, van Glabbeek introduces th ...
... Milner 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 equivalence. For several process semantics, mainly in the realm of simulation, van Glabbeek introduces th ...
Chapter X: Computational Complexity of Propositional Fuzzy Logics
... the language {&, →, ∧, 0} has axioms (A1)–(A3), (A4a) ϕ ∧ ψ → ϕ (A4b) ϕ ∧ ψ → ψ ∧ ϕ (A4c) ϕ & (ϕ → ψ) → ϕ ∧ ψ (A5)–(A7), and deduction rule modus ponens. Uppercase Latin characters are used for logics: MTL, BL, SBL, Ł, G, Π stand for monoidal t-norm logic, basic logic, strict basic logic, Łukasiewic ...
... the language {&, →, ∧, 0} has axioms (A1)–(A3), (A4a) ϕ ∧ ψ → ϕ (A4b) ϕ ∧ ψ → ψ ∧ ϕ (A4c) ϕ & (ϕ → ψ) → ϕ ∧ ψ (A5)–(A7), and deduction rule modus ponens. Uppercase Latin characters are used for logics: MTL, BL, SBL, Ł, G, Π stand for monoidal t-norm logic, basic logic, strict basic logic, Łukasiewic ...
Proof for functional programming - University of Kent School of
... The de ning forms of languages are more complex than simple equations. Section 3 looks at conditional de nitions (using `guards'), pattern matching and local de nitions (in let and where clauses) each of which adds complications, not least when the features interact. Reasoning cannot be completely e ...
... The de ning forms of languages are more complex than simple equations. Section 3 looks at conditional de nitions (using `guards'), pattern matching and local de nitions (in let and where clauses) each of which adds complications, not least when the features interact. Reasoning cannot be completely e ...
Introduction to Logic
... without changing its value. In Aristotle this meant simply that the pairs he determined could be exchanged. The intuition might have been that they “essentially mean the same”. In a more abstract, and later formulation, one would say that “not to affect a proposition” is “not to change its truth val ...
... without changing its value. In Aristotle this meant simply that the pairs he determined could be exchanged. The intuition might have been that they “essentially mean the same”. In a more abstract, and later formulation, one would say that “not to affect a proposition” is “not to change its truth val ...
λ-definition of Function(al)s by Normal Forms
... recursion, defining any partial functions on any data structures? [9] answered positively both questions. The use of Böhm-tree [14, 15, 4] proved that the combinators representing the constructors of any homogeneous terms algebras are also a basis for the full combinatory logic and indeed for nf’s. ...
... recursion, defining any partial functions on any data structures? [9] answered positively both questions. The use of Böhm-tree [14, 15, 4] proved that the combinators representing the constructors of any homogeneous terms algebras are also a basis for the full combinatory logic and indeed for nf’s. ...
HKT Chapters 1 3
... A strict partial order is a binary relation < that is irreflexive and transitive. Any strict partial order < has an associated partial order ≤ defined by a ≤ b if a < b or a = b. Any preorder ≤ has an associated strict partial order defined by a < b if a ≤ b but b ≤ a. For partial orders ≤, these two ...
... A strict partial order is a binary relation < that is irreflexive and transitive. Any strict partial order < has an associated partial order ≤ defined by a ≤ b if a < b or a = b. Any preorder ≤ has an associated strict partial order defined by a < b if a ≤ b but b ≤ a. For partial orders ≤, these two ...
Intuitionistic Type Theory - The collected works of Per Martin-Löf
... What we shall do is also mathematical logic in the first sense, but certainly not in the third. The principal problem that remained after Principia Mathematica was completed was, according to its authors, that of justifying the axiom of reducibility (or, as we would now say, the impredicative compre ...
... What we shall do is also mathematical logic in the first sense, but certainly not in the third. The principal problem that remained after Principia Mathematica was completed was, according to its authors, that of justifying the axiom of reducibility (or, as we would now say, the impredicative compre ...
Belief closure: A semantics of common knowledge for
... partitions, and then explains that this definition can be rephrased into more intuitive terms, using the notion of a 'reachable' state of the world. Very roughly speaking, definition 1 is of the circular kind, and definition 2 of the iterate kind. However, in view of the immediate mathematical equiv ...
... partitions, and then explains that this definition can be rephrased into more intuitive terms, using the notion of a 'reachable' state of the world. Very roughly speaking, definition 1 is of the circular kind, and definition 2 of the iterate kind. However, in view of the immediate mathematical equiv ...
PDF
... Gurevich (1988) formalized the notion of a logic capturing PTIME and conjectured a negative answer to the above question. We saw above that in the presence of a linear order, FO + IFP captures PTIME. It can also be seen that this built-in order is necessary. On arbitrary nite structures, FO + IFP c ...
... Gurevich (1988) formalized the notion of a logic capturing PTIME and conjectured a negative answer to the above question. We saw above that in the presence of a linear order, FO + IFP captures PTIME. It can also be seen that this built-in order is necessary. On arbitrary nite structures, FO + IFP c ...
type system is a tractable syntactic method for proving the absence
... • Strong vs. Weak – Assignment is only permitted if types are consistent – E.g., Java (strong) and C (weak) ...
... • Strong vs. Weak – Assignment is only permitted if types are consistent – E.g., Java (strong) and C (weak) ...
Introduction to Logic
... without changing its value. In Aristotle this meant simply that the pairs he determined could be exchanged. The intuition might have been that they “essentially mean the same”. In a more abstract, and later formulation, one would say that “not to affect a proposition” is “not to change its truth val ...
... without changing its value. In Aristotle this meant simply that the pairs he determined could be exchanged. The intuition might have been that they “essentially mean the same”. In a more abstract, and later formulation, one would say that “not to affect a proposition” is “not to change its truth val ...
What is a chip?
... edge over an FPGA. One thing to note is that an ASIC is designed to be fully optimized in terms of gates and logic. All the internal structures are used for customer facing or mission specific applications or functions. So, while an ASIC may consume more power per unit die size than an FPGA, this po ...
... edge over an FPGA. One thing to note is that an ASIC is designed to be fully optimized in terms of gates and logic. All the internal structures are used for customer facing or mission specific applications or functions. So, while an ASIC may consume more power per unit die size than an FPGA, this po ...
Strong Logics of First and Second Order
... The first task is to render precise the relevant notion of “absoluteness”. This will be the main purpose of Sections 1 and 2. In Section 1 we shall investigate the many facets of the absoluteness of first-order logic. In Section 2 we shall start by investigating two traditional strong logics (ω-log ...
... The first task is to render precise the relevant notion of “absoluteness”. This will be the main purpose of Sections 1 and 2. In Section 1 we shall investigate the many facets of the absoluteness of first-order logic. In Section 2 we shall start by investigating two traditional strong logics (ω-log ...
Two classes of Boolean functions for dependency analysis
... functions for groundness, finiteness, and suspension analysis. In Section 3 we discuss in more detail two classes of Boolean functions that lend themselves naturally to this. In Section 4 we consider a variety of ways to represent Boolean functions so that their manipulation can be made efficient. I ...
... functions for groundness, finiteness, and suspension analysis. In Section 3 we discuss in more detail two classes of Boolean functions that lend themselves naturally to this. In Section 4 we consider a variety of ways to represent Boolean functions so that their manipulation can be made efficient. 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.