scholarly journals Superposition with Lambdas

2021 ◽  
Author(s):  
Alexander Bentkamp ◽  
Jasmin Blanchette ◽  
Sophie Tourret ◽  
Petar Vukmirović ◽  
Uwe Waldmann
Keyword(s):  
Higher Order ◽  
Equivalence Classes ◽  
Order Logic ◽  
Inference Rules ◽  
Higher Order Logic ◽  
Superposition Calculus

AbstractWe designed a superposition calculus for a clausal fragment of extensional polymorphic higher-order logic that includes anonymous functions but excludes Booleans. The inference rules work on $$\beta \eta $$ β η -equivalence classes of $$\lambda $$ λ -terms and rely on higher-order unification to achieve refutational completeness. We implemented the calculus in the Zipperposition prover and evaluated it on TPTP and Isabelle benchmarks. The results suggest that superposition is a suitable basis for higher-order reasoning.

scholarly journals Making Higher-Order Superposition Work

2021 ◽  
pp. 415-432
Author(s):  
Petar Vukmirović ◽  
Alexander Bentkamp ◽  
Jasmin Blanchette ◽  
Simon Cruanes ◽  
Visa Nummelin ◽  
...  
Keyword(s):  
Higher Order ◽  
Order Logic ◽  
Theorem Prover ◽  
First Order Logic ◽  
Inference Rules ◽  
Higher Order Logic ◽  
First Order ◽  
New Challenges

AbstractSuperposition is among the most successful calculi for first-order logic. Its extension to higher-order logic introduces new challenges such as infinitely branching inference rules, new possibilities such as reasoning about formulas, and the need to curb the explosion of specific higher-order rules. We describe techniques that address these issues and extensively evaluate their implementation in the Zipperposition theorem prover. Largely thanks to their use, Zipperposition won the higher-order division of the CASC-J10 competition.

Extended natural semantics

1993 ◽  
Vol 3 (2) ◽  
pp. 123-152 ◽  
Author(s):  
John Hannan
Keyword(s):  
Logic Programming ◽  
Programming Language ◽  
Functional Programming ◽  
Higher Order ◽  
Order Logic ◽  
Inference Rules ◽  
Abstract Syntax ◽  
Higher Order Logic ◽  
First Order ◽  
Definition Of

AbstractWe extend the definition of natural semantics to include simply typed λ-terms, instead of first-order terms, for representing programs, and to include inference rules for the introduction and discharge of hypotheses and eigenvariables. This extension, which we call extended natural semantics, affords a higher-level notion of abstract syntax for representing programs and suitable mechanisms for manipulating this syntax. We present several examples of semantic specifications for a simple functional programming language and demonstrate how we achieve simple and elegant manipulations of bound variables in functional programs. All the examples have been implemented and tested in λProlog, a higher-order logic programming language that supports all of the features of extended natural semantics.

scholarly journals Superposition with First-class Booleans and Inprocessing Clausification

2021 ◽  
pp. 378-395
Author(s):  
Visa Nummelin ◽  
Alexander Bentkamp ◽  
Sophie Tourret ◽  
Petar Vukmirović
Keyword(s):  
State Of The Art ◽  
Higher Order ◽  
The State ◽  
Order Logic ◽  
Theorem Prover ◽  
First Order Logic ◽  
Higher Order Logic ◽  
First Order ◽  
Superposition Calculus

AbstractWe present a complete superposition calculus for first-order logic with an interpreted Boolean type. Our motivation is to lay the foundation for refutationally complete calculi in more expressive logics with Booleans, such as higher-order logic, and to make superposition work efficiently on problems that would be obfuscated when using clausification as preprocessing. Working directly on formulas, our calculus avoids the costly axiomatic encoding of the theory of Booleans into first-order logic and offers various ways to interleave clausification with other derivation steps. We evaluate our calculus using the Zipperposition theorem prover, and observe that, with no tuning of parameters, our approach is on a par with the state-of-the-art approach.

scholarly journals Closed Structure

2021 ◽  
Author(s):  
Peter Fritz ◽  
Harvey Lederman ◽  
Gabriel Uzquiano
Keyword(s):  
Formal Language ◽  
Syntactic Structure ◽  
Natural Extension ◽  
Propositional Attitudes ◽  
Higher Order ◽  
Order Logic ◽  
Bertrand Russell ◽  
Structured Propositions ◽  
Higher Order Logic ◽  
Closed Structure

AbstractAccording to the structured theory of propositions, if two sentences express the same proposition, then they have the same syntactic structure, with corresponding syntactic constituents expressing the same entities. A number of philosophers have recently focused attention on a powerful argument against this theory, based on a result by Bertrand Russell, which shows that the theory of structured propositions is inconsistent in higher order-logic. This paper explores a response to this argument, which involves restricting the scope of the claim that propositions are structured, so that it does not hold for all propositions whatsoever, but only for those which are expressible using closed sentences of a given formal language. We call this restricted principle Closed Structure, and show that it is consistent in classical higher-order logic. As a schematic principle, the strength of Closed Structure is dependent on the chosen language. For its consistency to be philosophically significant, it also needs to be consistent in every extension of the language which the theorist of structured propositions is apt to accept. But, we go on to show, Closed Structure is in fact inconsistent in a very natural extension of the standard language of higher-order logic, which adds resources for plural talk of propositions. We conclude that this particular strategy of restricting the scope of the claim that propositions are structured is not a compelling response to the argument based on Russell’s result, though we note that for some applications, for instance to propositional attitudes, a restricted thesis in the vicinity may hold some promise.

Adapting functional programs to higher order logic

2008 ◽  
Vol 21 (4) ◽  
pp. 377-409 ◽  
Author(s):  
Scott Owens ◽  
Konrad Slind
Keyword(s):  
Higher Order ◽  
Order Logic ◽  
Higher Order Logic
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