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Induction with Recursive Definitions in Superposition [article]

Marton Hajdu, Petra Hozzova, Laura Kovacs, Andrei Voronkov, Michael W. Whalen, Ruzica Piskac
2021
Our experimental results show that handling recursive definitions in superposition reasoning significantly improves automated reasoning with induction.  ...  In this paper we propose new extensions and generalizations to automate induction with recursive functions in saturation-based first-order theorem proving, using the superposition calculus.  ...  Rewriting with Recursive Function Definitions We extend superposition reasoning with two inference rules in support of rewriting recursive functions by their definitions.  ... 
doi:10.34727/2021/isbn.978-3-85448-046-4_34 fatcat:jlqgdyqu4bacngrerqwbqmke2a

Superposition with Structural Induction [chapter]

Simon Cruanes
2017 Lecture Notes in Computer Science  
We present a new extension of typed superposition that can perform structural induction. Several inductive goals can be attempted within a single saturation loop, by leveraging AVATAR [1] .  ...  Superposition-based provers have been successfully used to discharge proof obligations stemming from proof assistants. However, many such obligations require induction to be proved.  ...  out a lot of related works and limitations in several occasions).  ... 
doi:10.1007/978-3-319-66167-4_10 fatcat:jzgkbftdsvdsthbtkxg4ipmeqm

A recursion planning analysis of inductive completion

Richard Barnett, David Basin, Jane Hesketh
1993 Annals of Mathematics and Artificial Intelligence  
Recursion analysis chooses induction schemas and variables and rippling controls rewriting in explicit induction proofs.  ...  Our study includes an experimental comparison of Clam, an explicit induction theorem prover, with an implementation of Huet and Hullot's inductionless induction.  ...  In particular, when superposition is used to select induction schemata, then these are limited to those recursions present in the given equations.  ... 
doi:10.1007/bf01530798 fatcat:dqcrm6gmnrba3imwxi6vzdbdtq

An overview of Rewrite Rule Laboratory (RRL)

D. Kapur, H. Zhang
1995 Computers and Mathematics with Applications  
the use of formal methods in hardware and software design.  ...  RRL (Rewrite Rule Laboratory) was originally developed as an environment for experimenting with automated reasoning algorithms for equational logic based on rewrite techniques.  ...  The cover set method is similar to Boyer and Moore's approach in which inductive inference rules are designed from recursive definitions of functions.  ... 
doi:10.1016/0898-1221(94)00218-a fatcat:mqgk7ygjerd3poaldoyq3qbcpu

An overview of Rewrite Rule Laboratory (RRL) [chapter]

Deepak Kapur, Hantao Zhang
1989 Lecture Notes in Computer Science  
the use of formal methods in hardware and software design.  ...  RRL (Rewrite Rule Laboratory) was originally developed as an environment for experimenting with automated reasoning algorithms for equational logic based on rewrite techniques.  ...  The cover set method is similar to Boyer and Moore's approach in which inductive inference rules are designed from recursive definitions of functions.  ... 
doi:10.1007/3-540-51081-8_138 fatcat:qwdmeuudbrhzjklqdzzgcxcp24

A Lambda-Free Higher-Order Recursive Path Order [chapter]

Jasmin Christian Blanchette, Uwe Waldmann, Daniel Wand
2017 Lecture Notes in Computer Science  
This new order fully coincides with the standard RPO on first-order terms also in the presence of currying, distinguishing it from previous work.  ...  It appears promising as the basis of a higher-order superposition calculus.  ...  to Stephan Merz, Tobias Nipkow, and Christoph Weidenbach for making this research possible; to Heiko Becker and Dmitriy Traytel for proving some theorems on the lexicographic and multiset extensions in  ... 
doi:10.1007/978-3-662-54458-7_27 fatcat:tyswf6bl3japhgppwbknvg546u

A Transfinite Knuth–Bendix Order for Lambda-Free Higher-Order Terms [chapter]

Heiko Becker, Jasmin Christian Blanchette, Uwe Waldmann, Daniel Wand
2017 Lecture Notes in Computer Science  
The order appears promising as the basis of a higher-order superposition calculus.  ...  The restriction of this new order to first-order terms coincides with the traditional KBO.  ...  and Christoph Weidenbach for making this research possible; to Mathias Fleury and Dmitriy Traytel for helping us formalize the syntactic ordinals; to Andrei Popescu and Christian Sternagel for advice with  ... 
doi:10.1007/978-3-319-63046-5_27 fatcat:t3y2q3czrrawzixu5ffoiyhbei

Problems and Prospects for Quantum Computational Speed-up [chapter]

E. V. Krishnamurthy
2003 Lecture Notes in Computer Science  
In particular, we relate the primitive recursive function approach used in computability theory with the harmonic oscillator basis used in quantum physics.  ...  Hence transfinite induction is not applicable to reason about quantum programs.  ...  If the devices are not error-prone and there is no loss of energy, then such a superposition is identifiable with a primitive recursive computable function,if the coefficients in the linear combination  ... 
doi:10.1007/3-540-44864-0_80 fatcat:pb3s7iq5l5fvhkkneep3gdkhym

Superposition with completely built-in Abelian groups

Guillem Godoy, Robert Nieuwenhuis
2004 Journal of symbolic computation  
A new technique is presented for superposition with first-order clauses with built-in Abelian groups (AG).  ...  Although with arbitrary free symbols the decision problem is NP-complete in all three cases, AG unification behaves better in practice.  ...  Conclusions A new technique has been presented for superposition with first-order clauses with built-in AG.  ... 
doi:10.1016/s0747-7171(03)00070-1 fatcat:re4woj42undrhmwzgvprvdkm2m

Linear algorithms for recognizing and parsing superpositional graphs

Ahti Peder, Härmel Nestra, Jaan Raik, Mati Tombak, Raimund Ubar
2011 Facta universitatis - series Electronics and Energetics  
Structurally synthesized binary decision diagrams (SSBDD) are a special type of BDDs that are generated by superposition according to the structure of propositional formula.  ...  This paper presents linear time algorithms for testing whether a binary graph is a superpositional graph and for restoring the history of its generating process.  ...  The authors thank their colleagues Peeter Laud and Reimo Palm from University of Tartu for their help in improving the superpositionality test algorithms.  ... 
doi:10.2298/fuee1103325p fatcat:gvmnlqof3fdx5bjcwjhbni6dpi

Quantum Recursion and Second Quantisation [article]

Mingsheng Ying
2014 arXiv   pre-print
This paper introduces a new notion of quantum recursion of which the control flow of the computation is quantum rather than classical as in the notions of recursion considered in the previous studies of  ...  A typical example is recursive quantum walks, which are obtained by slightly modifying the construction of the ordinary quantum walks.  ...  The first version of this paper is the text of the third part of my talk "Quantum programming: from superposition of data to superposition of programs" at the Tsinghua Software Day, April 21-22, 2014 (  ... 
arXiv:1405.4443v2 fatcat:at52ae6ssjhmpjjpdpsmefpeuy

A strong restriction of the inductive completion procedure

Laurent Fribourg
1989 Journal of symbolic computation  
This verification is done by called (recursively) PROC or any other inductive proof procedure. The normalised equation is then oriented as &,,Mm = N,. ~ G = D, and then is included in R,+~ at step 3.  ...  Nevertheless, as in the unconditional case, inductive reducibility and complete superposability are both fulfilled when P satisfies the Huet-Hullot principle of definition.  ...  The Pa-superposition class contains three equations S1, $2 and $3 which correspond to superposition with (ql), (q2) and (q3), respectively.  ... 
doi:10.1016/s0747-7171(89)80069-0 fatcat:4rgttgj3abcgblolbwvs36ybxe

A Lambda Calculus for Quantum Computation

André van Tonder
2004 SIAM journal on computing (Print)  
The calculus turns out to be closely related to the linear lambda calculi used in the study of Linear Logic.  ...  It provides a computational model equivalent to the Turing machine, and continues to be of enormous benefit in the classical theory of computation.  ...  Let us call a substate definite with respect to the computational basis if it is textually the same in all branches of the superposition.  ... 
doi:10.1137/s0097539703432165 fatcat:w5qgqihfdbeajp4b6zkfwnpafa

Approximative versions of Kolmogorov's superposition theorem, proved constructively

Manuela Nees
1994 Journal of Computational and Applied Mathematics  
The proofs are constructive in the sense that only rationals, expressible as finite decimal sums, are used. The approximations consist in algorithms with error bounds.  ...  Concerning their complexity, the two results differ in the way the occurring sequences are constructed: recursively, respectively explicitly.  ...  With the aid of the subintervals Z$ in Definition 3.1, we construct recursively functions c$$ on I by modifying the construction of the 6, in [3] and replace simultaneously the irrational A by a sequence  ... 
doi:10.1016/0377-0427(94)90179-1 fatcat:ytu2fvmf4rayfpvkaviacekntu

Physics and metaphysics looks at computation [article]

Karl Svozil
2005 arXiv   pre-print
With the strong Artificial Intelligence assumption, human consciousness is just a function of the organs (maybe in a very wide sense and not only restricted to neuronal brain activity), and thus the question  ...  In dualistic models of the mind, human thought transcends symbolic paper-and-pencil operations.  ...  In a very broad sense, this is the induction problem plaguing inductive science from its very start. Induction is "bottom-up."  ... 
arXiv:physics/0508207v1 fatcat:vjiw3qvvona7tlyhzp3jf5fxya
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