By Dale Miller
Formal platforms that describe computations over syntactic buildings take place usually in computing device technological know-how. good judgment programming presents a traditional framework for encoding and animating such structures. although, those platforms frequently embrace variable binding, a concept that has to be handled rigorously at a computational point. This ebook goals to teach programming language in keeping with a easily typed model of higher-order common sense presents a sublime, declarative capability for delivering the sort of therapy. 3 large issues are lined in pursuit of this aim. First, a proof-theoretic framework that helps a basic view of good judgment programming is pointed out. moment, an exact language known as λProlog is constructed via utilising this view to higher-order common sense. ultimately, a technique for programming with standards is uncovered by means of displaying how a number of computations over formal gadgets reminiscent of logical formulation, useful courses, and λ-terms and π-calculus expressions could be encoded in λProlog
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Extra info for Programming with Higher-Order Logic
Both forms of quantiﬁcation range over explicit domains speciﬁed by types. In mathematical notation, the universal and existential quantiﬁcation of x over the formula F at the type τ are written as ∀τ x F and ∃τ x F , respectively. Both quantiﬁers bind a variable and establish a scope for its binding. In concrete syntax, the depicted expressions are written as (pi (x:T)\ F) and (sigma (x:T)\ F), respectively, where F, T, and x are themselves the concrete syntax renditions of the formula F , the type τ , and the variable x.
However, the ﬁrst-order fragment that we are considering currently requires two conditions to be satisﬁed: The type expressions must be of order at most 1, and the sort o must not be used in them. Notice that these restrictions are satisﬁed by all the built-in constants we have considered up to this point. The λProlog language allows the user to identify newly declared constants as operators of particular ﬁxity and precedence. This is done via operator declarations that have the form ﬁxity c1,.
It is apparent from the representation of quantiﬁcation that the backslash token plays the role of a binding operator. This role will become precise in Chapter 4 when it will be identiﬁed ofﬁcially as an inﬁx operator representing λ-abstraction. In that context, pi and sigma will be recognized to be polymorphic constants of the type (A -> o) -> o. For the moment, however, we treat pi x\ and sigma x\ simply as unanalyzed, concrete syntax forms for the quantiﬁers depicted by ∀x and ∃x in our metalinguistic discourse.
Programming with Higher-Order Logic by Dale Miller