scholarly journals Partiality and recursion in interactive theorem provers – an overview

2014 ◽  
Vol 26 (1) ◽  
pp. 38-88 ◽  
Author(s):  
ANA BOVE ◽  
ALEXANDER KRAUSS ◽  
MATTHIEU SOZEAU
Keyword(s):  
Software Development ◽  
Type Theory ◽  
Theoretical Basis ◽  
Order Logic ◽  
Higher Order Logic ◽  
Intensive Research ◽  
Recursive Functions ◽  
Constructive Type Theory ◽  
Theorem Provers ◽  
Interactive Theorem Provers

The use of interactive theorem provers to establish the correctness of critical parts of a software development or for formalizing mathematics is becoming more common and feasible in practice. However, most mature theorem provers lack a direct treatment of partial and general recursive functions; overcoming this weakness has been the objective of intensive research during the last decades. In this article, we review several techniques that have been proposed in the literature to simplify the formalization of partial and general recursive functions in interactive theorem provers. Moreover, we classify the techniques according to their theoretical basis and their practical use. This uniform presentation of the different techniques facilitates the comparison and highlights their commonalities and differences, as well as their relative advantages and limitations. We focus on theorem provers based on constructive type theory (in particular, Agda and Coq) and higher-order logic (in particular Isabelle/HOL). Other systems and logics are covered to a certain extent, but not exhaustively. In addition to the description of the techniques, we also demonstrate tools which facilitate working with the problematic functions in particular theorem provers.

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.

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.).

On nonstandard models in higher order logic

1984 ◽  
Vol 49 (1) ◽  
pp. 204-219
Author(s):  
Christian Hort ◽  
Horst Osswald
Keyword(s):  
Model Theory ◽  
Type Theory ◽  
Order Logic ◽  
Elementary Embedding ◽  
Simple Type ◽  
Order Model ◽  
Higher Order Logic ◽  
First Order ◽  
Simple Type Theory ◽  
Nonstandard Models

There are two concepts of standard/nonstandard models in simple type theory.The first concept—we might call it the pragmatical one—interprets type theory as a first order logic with countably many sorts of variables: the variables for the urelements of type 0,…, the n-ary relational variables of type (τ1, …, τn) with arguments of type (τ1,…,τn), respectively. If A ≠ ∅ then 〈Aτ〉 is called a model of type logic, if A0 = A and . 〈Aτ〉 is called full if, for every τ = (τ1,…,τn), . The variables for the urelements range over the elements of A and the variables of type (τ1,…, τn) range over those subsets of which are elements of . The theory Th(〈Aτ〉) is the set of all closed formulas in the language which hold in 〈Aτ〉 under natural interpretation of the constants. If 〈Bτ〉 is a model of Th(〈Aτ〉), then there exists a sequence 〈fτ〉 of functions fτ: Aτ → Bτ such that 〈fτ〉 is an elementary embedding from 〈Aτ〉 into 〈Bτ〉. 〈Bτ〉 is called a nonstandard model of 〈Aτ〉, if f0 is not surjective. Otherwise 〈Bτ〉 is called a standard model of 〈Aτ〉.This first concept of model theory in type logic seems to be preferable for applications in model theory, for example in nonstandard analysis, since all nice properties of first order model theory (completeness, compactness, and so on) are preserved.

scholarly journals Checkable Proofs for First-Order Theorem Proving

10.29007/s6d1 ◽  
2018 ◽  
Author(s):  
Giles Reger ◽  
Martin Suda
Keyword(s):  
Theorem Proving ◽  
Critical Mass ◽  
Current Situation ◽  
Boolean Satisfiability ◽  
Order Logic ◽  
First Order Logic ◽  
First Order ◽  
Theorem Provers ◽  
Interactive Theorem Provers ◽  
Theory Reasoning

Inspired by the success of the DRAT proof format for certification of boolean satisfiability (SAT),we argue that a similar goal of having unified automatically checkable proofs should be soughtby the developers of automated first-order theorem provers (ATPs). This would not onlyhelp to further increase assurance about the correctness of prover results,but would also be indispensable for tools which rely on ATPs,such as ``hammers'' employed within interactive theorem provers.The current situation, represented by the TSTP format is unsatisfactory,because this format does not have a standardised semantics and thus cannot be checked automatically.Providing such semantics, however, is a challenging endeavour. One would ideallylike to have a proof format which covers only-satisfiability-preserving operations such as Skolemisationand is versatile enough to encompass various proving methods (i.e. not just superposition)or is perhaps even open ended towards yet to be conceived methods or at least easily extendable in principle.Going beyond pure first-order logic to theory reasoning in the style of SMT orbeyond proofs to certification of satisfiability are further interesting challenges.Although several projects have already provided partial solutions in this direction,we would like to use the opportunity of ARCADE to further promote the idea andgather critical mass needed for its satisfactory realisation.

scholarly journals POLYMORPHISM AND THE OBSTINATE CIRCULARITY OF SECOND ORDER LOGIC: A VICTIMS’ TALE

2018 ◽  
Vol 24 (1) ◽  
pp. 1-52
Author(s):  
PAOLO PISTONE
Keyword(s):  
Type Theory ◽  
Order Type ◽  
Higher Order ◽  
Second Order ◽  
Order Logic ◽  
Higher Order Logic ◽  
Related Notion ◽  
Foundational Research ◽  
The One ◽  
Second Order Logic

AbstractThe investigations on higher-order type theories and on the related notion of parametric polymorphism constitute the technical counterpart of the old foundational problem of the circularity (or impredicativity) of second and higher-order logic. However, the epistemological significance of such investigations has not received much attention in the contemporary foundational debate.We discuss Girard’s normalization proof for second order type theory or System F and compare it with two faulty consistency arguments: the one given by Frege for the logical system of the Grundgesetze (shown inconsistent by Russell’s paradox) and the one given by Martin-Löf for the intuitionistic type theory with a type of all types (shown inconsistent by Girard’s paradox).The comparison suggests that the question of the circularity of second order logic cannot be reduced to Russell’s and Poincaré’s 1906 “vicious circle” diagnosis. Rather, it reveals a bunch of mathematical and logical ideas hidden behind the hazardous idea of impredicative quantification, constituting a vast (and largely unexplored) domain for foundational research.

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 Completeness theorems for first-order logic analysed in constructive type theory

2021 ◽  
Vol 31 (1) ◽  
pp. 112-151
Author(s):  
Yannick Forster ◽  
Dominik Kirst ◽  
Dominik Wehr
Keyword(s):  
Type Theory ◽  
Order Logic ◽  
First Order Logic ◽  
First Order ◽  
Constructive Type Theory ◽  
Constructive Type ◽  
The Standard Model ◽  
Coq Proof Assistant ◽  
Game Theoretic ◽  
Completeness Theorems

Abstract We study various formulations of the completeness of first-order logic phrased in constructive type theory and mechanised in the Coq proof assistant. Specifically, we examine the completeness of variants of classical and intuitionistic natural deduction and sequent calculi with respect to model-theoretic, algebraic, and game-theoretic semantics. As completeness with respect to the standard model-theoretic semantics à la Tarski and Kripke is not readily constructive, we analyse connections of completeness theorems to Markov’s Principle and Weak K̋nig’s Lemma and discuss non-standard semantics admitting assumption-free completeness. We contribute a reusable Coq library for first-order logic containing all results covered in this paper.

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.

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