scholarly journals Making Automatic Theorem Provers more Versatile

10.29007/n6j7 ◽  
2018 ◽  
Author(s):  
Simon Cruanes
Keyword(s):  
Higher Order ◽  
Order Logic ◽  
Higher Order Logic ◽  
Research Directions ◽  
First Order ◽  
Theorem Provers ◽  
Smt Solvers ◽  
Automatic Theorem Provers ◽  
Automatic Theorem

We argue that automatic theorem provers should become more versatile and should be able to tackle problems expressed in richer input formats. Salient research directions include (i) developing tight combinations of SMT solvers and first-order provers; (ii) adding better handling of theories in first-order provers; (iii) adding support for inductive proving; (iv) adding support for user-defined theories and functions; and (v) bringing to the provers some basic abilities to deal with logics beyond first-order, such as higher-order logic.

scholarly journals Leo-III Version 1.1 (System description)

10.29007/grmx ◽  
2018 ◽  
Author(s):  
Christoph Benzmüller ◽  
Alexander Steen ◽  
Max Wisniewski
Keyword(s):  
Higher Order ◽  
Order Logic ◽  
Theorem Prover ◽  
Higher Order Logic ◽  
System Description ◽  
Agent Based ◽  
First Order ◽  
Theorem Provers ◽  
Automated Theorem Prover ◽  
Ordering Constraints

Leo-III is an automated theorem prover for (polymorphic) higher-order logic which supports all common TPTP dialects, including THF, TFF and FOF as well as their rank-1 polymorphic derivatives. It is based on a paramodulation calculus with ordering constraints and, in tradition of its predecessor LEO-II, heavily relies on cooperation with external first-order theorem provers.Unlike LEO-II, asynchronous cooperation with typed first-order provers and an agent-based internal cooperation scheme is supported. In this paper, we sketch Leo-III's underlying calculus, survey implementation details and give examples of use.

scholarly journals Superposition for Full Higher-order Logic

2021 ◽  
pp. 396-412
Author(s):  
Alexander Bentkamp ◽  
Jasmin Blanchette ◽  
Sophie Tourret ◽  
Petar Vukmirović
Keyword(s):  
Higher Order ◽  
Order Logic ◽  
First Order Logic ◽  
Stepping Stones ◽  
Higher Order Logic ◽  
First Order ◽  
Theorem Provers ◽  
New Challenges

AbstractWe recently designed two calculi as stepping stones towards superposition for full higher-order logic: Boolean-free $$\lambda $$ λ -superposition and superposition for first-order logic with interpreted Booleans. Stepping on these stones, we finally reach a sound and refutationally complete calculus for higher-order logic with polymorphism, extensionality, Hilbert choice, and Henkin semantics. In addition to the complexity of combining the calculus’s two predecessors, new challenges arise from the interplay between $$\lambda $$ λ -terms and Booleans. Our implementation in Zipperposition outperforms all other higher-order theorem provers and is on a par with an earlier, pragmatic prototype of Booleans in Zipperposition.

scholarly journals Extending a brainiac prover to lambda-free higher-order logic

2021 ◽  
Author(s):  
Petar Vukmirović ◽  
Jasmin Blanchette ◽  
Simon Cruanes ◽  
Stephan Schulz
Keyword(s):  
Data Structures ◽  
Theorem Proving ◽  
State Of The Art ◽  
Higher Order ◽  
The State ◽  
Order Logic ◽  
Automatic Theorem Proving ◽  
Higher Order Logic ◽  
First Order ◽  
Automatic Theorem

AbstractDecades of work have gone into developing efficient proof calculi, data structures, algorithms, and heuristics for first-order automatic theorem proving. Higher-order provers lag behind in terms of efficiency. Instead of developing a new higher-order prover from the ground up, we propose to start with the state-of-the-art superposition prover E and gradually enrich it with higher-order features. We explain how to extend the prover’s data structures, algorithms, and heuristics to $$\lambda $$ λ -free higher-order logic, a formalism that supports partial application and applied variables. Our extension outperforms the traditional encoding and appears promising as a stepping stone toward full higher-order logic.

Resolution in type theory

1971 ◽  
Vol 36 (3) ◽  
pp. 414-432 ◽  
Author(s):  
Peter B. Andrews
Keyword(s):  
Set Theory ◽  
Type Theory ◽  
Higher Order ◽  
Order Logic ◽  
First Order Logic ◽  
Higher Order Logic ◽  
First Order ◽  
Different Types ◽  
Axiomatic Set Theory ◽  
Order Predicate Calculus

In [8] J. A. Robinson introduced a complete refutation procedure called resolution for first order predicate calculus. Resolution is based on ideas in Herbrand's Theorem, and provides a very convenient framework in which to search for a proof of a wff believed to be a theorem. Moreover, it has proved possible to formulate many refinements of resolution which are still complete but are more efficient, at least in many contexts. However, when efficiency is a prime consideration, the restriction to first order logic is unfortunate, since many statements of mathematics (and other disciplines) can be expressed more simply and naturally in higher order logic than in first order logic. Also, the fact that in higher order logic (as in many-sorted first order logic) there is an explicit syntactic distinction between expressions which denote different types of intuitive objects is of great value where matching is involved, since one is automatically prevented from trying to make certain inappropriate matches. (One may contrast this with the situation in which mathematical statements are expressed in the symbolism of axiomatic set theory.).

scholarly journals Formalization of Linear Space Theory in the Higher-Order Logic Proving System

2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
Jie Zhang ◽  
Danwen Mao ◽  
Yong Guan
Keyword(s):  
Linear Space ◽  
Theorem Proving ◽  
Predicate Logic ◽  
Higher Order ◽  
Order Logic ◽  
First Order Logic ◽  
Space Theory ◽  
Higher Order Logic ◽  
First Order ◽  
Important Approach

Theorem proving is an important approach in formal verification. Higher-order logic is a form of predicate logic that is distinguished from first-order logic by additional quantifiers and stronger semantics. Higher-order logic is more expressive. This paper presents the formalization of the linear space theory in HOL4. A set of properties is characterized in HOL4. This result is used to build the underpinnings for the application of higher-order logic in a wider spectrum of engineering applications.

scholarly journals Antirealism/realism of Hintikka’s game-theoretical semantics

2018 ◽  
Vol 26 (3) ◽  
Author(s):  
Heda Festini
Keyword(s):  
Higher Order ◽  
Order Logic ◽  
First Order Logic ◽  
Higher Order Logic ◽  
First Order ◽  
Theory Of Meaning ◽  
Truth Conditional ◽  
Context Dependent ◽  
Extended Notion

Hintikka’s game-theoretical semantics (GTS) is presented as an anti-Tarskian semantical approach to the context-dependent fragments of Englisch, which overcomes the usual notion of semantical realism. Analysing Hintikka’s critique of Tarski’s interpretation of the truth-conditional theory of meaning, its recursive fashion and the narrow notion of realism, Hintikka’s basic conception is presented in the following manner:1. the Context-Principle vs. the Frege Principle,2.First-order logic together with higher-order logic vs. the primacy of first-order logic,3.verificationist/falsificationist theory vs. Taraski’s narrow truth- conditional theory.Comparing some reviews of Hintikka’s GTS (M. Dummett, E. Itkonen, E. Saarinen, M. Hand) with a short examination of the antirealistic/realistic controversis by C. Wright and M. Dummett, the following was reached:Hintikka’s GTS introduces a new, more extended notion of realism, which embraces Taraski-type realistic semantics, Hintikka’s GTS and with this the question of the possibility to also include Dummett’s neoverificationism or other orientations, remains open.

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.

Nitpick: A Counterexample Generator for Isabelle/HOL Based on the Relational Model Finder Kodkod

10.29007/6shf ◽  
2018 ◽  
Author(s):  
Jasmin Christian Blanchette
Keyword(s):  
Higher Order ◽  
Order Logic ◽  
Experimental Results ◽  
Relational Model ◽  
Finite Model ◽  
Higher Order Logic ◽  
First Order ◽  
Recursive Functions

Nitpick is a counterexample generator for Isabelle/HOL that builds on Kodkod, a SAT-based first-order relational model finder. Nitpick supports unbounded quantification, (co)inductive predicates and datatypes, and (co)recursive functions. Fundamentally a finite model finder, it approximates infinite types by finite subsets. Our experimental results on Isabelle theories and the TPTP library indicate that Nitpick generates more counterexamples than other model finders for higher-order logic, without restrictions on the form of the formulas to falsify.

scholarly journals HOL-λσ: an intentional first-order expression of higher-order logic

2001 ◽  
Vol 11 (1) ◽  
pp. 21-45 ◽  
Author(s):  
GILLES DOWEK ◽  
THERESE HARDIN ◽  
CLAUDE KIRCHNER
Keyword(s):  
Order Theory ◽  
Higher Order ◽  
Search Method ◽  
Order Logic ◽  
Automated Deduction ◽  
Higher Order Logic ◽  
First Order ◽  
Proof Search ◽  
First Order Theory ◽  
Explicit Substitutions

We give a first-order presentation of higher-order logic based on explicit substitutions. This presentation is intentionally equivalent to the usual presentation of higher-order logic based on λ-calculus, that is, a proposition can be proved without the extensionality axioms in one theory if and only if it can be in the other. We show that the Extended Narrowing and Resolution first-order proof-search method can be applied to this theory. In this way we get a step-by-step simulation of higher-order resolution. Hence, expressing higher-order logic as a first-order theory and applying a first-order proof search method is a relevant alternative to a direct implementation. In particular, the well-studied improvements of proof search for first-order logic could be reused at no cost for higher-order automated deduction. Moreover, as we stay in a first-order setting, extensions, such as equational higher-order resolution, may be easier to handle.

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