Verifikation reaktiver Systeme - Universität Kaiserslautern

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– Compound types: Ifσ 1 ,...σ n are types, (σ 1 ,...,σ n )op is a type denoting the
set resulting from the application of op to set sets σ 1 ,...,σ n . For example,
prod is the type operator of arity 2 which denotes the cartesian product
operation. (σ 1 ,σ 2 )prod is usually written as σ 1 × σ 2 .
– Function types: Ifσ 1 and σ 2 are types, then σ 1 → σ 2 is the function type
with domain σ 1 and range σ 2 . It denotes the set of total functions from the
set denoted by its domain to the set denoted by its range. Syntactically,
→ is simply a distinguished type operator. As it always denotes the same
operation in any model of the HOL theory, it is singled out in the definition
of HOL types.
– Type variables: They stand for arbitrary sets in the universe.
Terms The terms in the HOL logic are expressions that denote elements of the
sets denoted by types. Thus, each term is associated with a unique type which
will be expressed by t σ . (The type subscript may be omitted if it is clear from
the context.) As the definition of types in HOL is relative to a particular type
structure Ω, the formal definition of terms is relative to a given collection of
typed constants over Ω. A signature over Ω is just a set Σ Ω of such constants.
The set Terms ΣΩ of terms over Σ Ω is defined to be the smallest set closed under
the formation rules:
– Constants: Aconstantc σ over Ω is a pair (c, σ) wherec ∈ Names and
σ ∈ T ypes ω . (Assume that an infinite set Names of names is given.)
– Variables: Ifx ∈ Names and σ ∈ Types Ω then the variable var x σ is a term
over Σ Ω .
– Lambda-abstractions: The lambda-abstraction (which denotes a function)
λx σ .t σ2 σ 1 → σ2 is a term if var x σ1 ∈ Terms ΣΩ and t σ2 ∈ Terms ΣΩ .
– Function applications:Ift σ′ →σ ∈ Terms ΣΩ and t‘ σ‘ ∈ Terms ΣΩ then t σ′ →σt‘ σ ′ σ ∈
Terms.Anapplicationtt‘ denotes the result of applying the function denoted
by t to the value denoted by t ′ .
Standard structures A standard type structure Ω contains the atomic types B
of boolean values and I of individuals. Logical formulas are then identified with
terms of type bool. In addition, for being standard a signature must contain
various logical constants with interpretation given by a standard model M:
– Implication: ⇒ B→B→B represents the implication. M(⇒, B → B → B): b ⇒ b‘
is 0, if b = 1 and b‘ =0; otherwise b ⇒ b‘ is1.
– Equality: = α→α→B denotes equality on the set denoted by α. The intended
interpretation M(=,α → α → B) is as follows: x = X x ′ is 1 if and only if
x = x ′ ; otherwise it is 0.
– Choice function: ɛ(α → B) → α is a choice function. M(ɛ, (α → B) →
α) ∈ Π X∈U (X → B) → X is the function assigning to each X ∈ U the
choice function sending f ∈ (X → B) toch X (f) =ch(f 1 1)iff 1 2=0ch(X)
otherwise.