computer science

ΛProlog

ΛProlog is a computer science topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand ΛProlog rather than just read about it. In short: λProlog, also written lambda Prolog, is a logic programming language featuring polymorphic typing, modular programming, and higher-order programming. These extensions to Prolog are derived from the higher-order hereditary Harrop formulas used to justify the foundations of λProlog.

Key takeaways

ΛProlog belongs to computer science; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect ΛProlog to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of ΛProlog from memory before moving on to harder problems.

Reference excerpt

λProlog, also written lambda Prolog, is a logic programming language featuring polymorphic typing, modular programming, and higher-order programming. These extensions to Prolog are derived from the higher-order hereditary Harrop formulas used to justify the foundations of λProlog. Higher-order quantification, simply typed λ-terms, and higher-order unification gives λProlog the basic supports needed to capture the λ-tree syntax approach to higher-order abstract syntax, an approach to representing syntax that maps object-level bindings to programming language bindings. Programmers in λProlog need not deal with bound variable names: instead various declarative devices are available to deal with binder scopes and their instantiations.

History Since 1986, λProlog has received numerous implementations. As of 2023, the language and its implementations are still actively being developed. The Abella theorem prover has been designed to provide an interactive environment for proving theorems about the declarative core of λProlog.

Programming in λProlog Two unique features of λProlog include implications and universal quantification. Implication is used for local scoping of predicate definitions while universal quantification is used for local scoping of variables, as in the following implementation of reverse depending on an auxiliary rev predicate:

A common use of these scoping constructs is to simulate scope often seen in an inference-rule presentation of a logic. For example, proof search (and proof checking) in natural deduction may be encoded as follows:

See also Curry's paradox#Lambda calculus — about inconsistency problems caused by combining (propositional) logic and untyped lambda calculus Comparison of Prolog implementations Prolog syntax and semantics

References

Tutorials and texts Dale Miller and Gopalan Nadathur have written the book Programming with higher-order logic, published by Cambridge University Press in June 2012. Amy Felty has written in a 1997 tutorial on lambda Prolog and its Applications to Theorem Proving. John Hannan has written a tutorial on Program Analysis in lambda Prolog for the 1998 PLILP Conference. Olivier Ridoux has written Lambda-Prolog de A à Z... ou presque (163 pages, French). It is available as PostScript, PDF, and html.

External links λProlog homepage Entry at the Software Preservation Group.

Implementations The Teyjus λProlog compiler is currently the oldest implementation still being maintained. This compiler project is led by Gopalan Nadathur and various of his colleagues and students. ELPI: an Embeddable λProlog Interpreter has been developed by Enrico Tassi and Claudio Sacerdoti Coen. It is implemented in OCaml and is available online. The system is described in a paper that appeared LPAR 2015. ELPI is also available as a Coq plugin: see Enrico Tassi's tutorial on this plugin. The Abella prover can be used to prove theorems about λProlog programs and specifications.

Worked examples

Example 1 — a first encounter with ΛProlog

Start with the simplest possible case. Write down what ΛProlog claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In computer science, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to ΛProlog before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about ΛProlog ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of ΛProlog

In research: ΛProlog appears in computer science research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses ΛProlog in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: ΛProlog is common in secondary-school and first-year university syllabi. It links to neighbouring topics Logic in computer science, Programming language topic stubs, Prolog programming language family, so understanding it makes those chapters shorter.
In everyday life: Look for ΛProlog outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study ΛProlog in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what ΛProlog means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain ΛProlog out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is ΛProlog in simple terms?

Why does ΛProlog matter?

Because it connects several computer science ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study ΛProlog?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on ΛProlog.