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Separation Logic with Linearly Compositional Inductive Predicates and Set Data Constraints [article]

Chong Gao, Taolue Chen, Zhilin Wu
2018 arXiv   pre-print
We integrate DBS into separation logic with linearly compositional inductive predicates, obtaining a logic thereof where set data constraints of linear data structures can be specified.  ...  We show that the satisfiability of this logic is decidable.  ...  Linearly compositional SLID with set data constraints In this section, we introduce separation logic with linearly compositional inductive predicates and set data constraints, denoted by SLID S LC rP s  ... 
arXiv:1811.00699v1 fatcat:wmu53znpejfhto72wctn6hb724

TRUTHS, INDUCTIVE DEFINITIONS, AND KRIPKE-PLATEK SYSTEMS OVER SET THEORY

KENTARO FUJIMOTO
2018 Journal of Symbolic Logic (JSL)  
AbstractIn this article we study the systems KF and VF of truth over set theory as well as related systems and compare them with the corresponding systems over arithmetic.  ...  Let L ∈ = {∈} be the language of firstorder set theory with the membership relation ∈ as its only non-logical symbol. ZF stands for Zermelo-Fraenkel set theory over L ∈ .  ...  We write I(L) for the set of L-inductive operator forms.  ... 
doi:10.1017/jsl.2017.77 fatcat:lx5fovnfmnh53nwaih7rpl42ca

The Type Theoretic Interpretation of Constructive Set Theory: Inductive Definitions [chapter]

Peter Aczel
1986 Studies in Logic and the Foundations of Mathematics  
We present a generalisation of the type-theoretic interpretation of constructive set theory into Martin-Löf type theory.  ...  Date: October 12th, 2005. 2000 Mathematics Subject Classification. 03F25 (Relative consistency and interpretations), 03F50 (Metamathematics of constructive systems).  ...  The structural axiom schemes of CZF are Extensionality (1) and Set Induction (2) .  ... 
doi:10.1016/s0049-237x(09)70683-4 fatcat:skeuy4t4cbd3nf6kpenyv5vmd4

Discovering relational knowledge from two disjoint sets of literatures using inductive Logic Programming

Supphachai Thaicharoen, Tom Altman, Katheleen Gardiner, Krzysztof J. Cios
2009 2009 IEEE Symposium on Computational Intelligence and Data Mining  
In this paper, we report an application of an Inductive Logic Programming (ILP), specifically the WARMR algorithm, to the field of literature-based discovery.  ...  The discovery approach is based on Swanson's discovery models where indirect connections between two disjoint sets of literatures A and C could be found through a set of common terms B extracted from A  ...  It could be typically attained by the use of Inductive Logic Programming (ILP). ILP is an inductive machine learning technique at the intersection of machine learning and logic programming [9] .  ... 
doi:10.1109/cidm.2009.4938661 dblp:conf/cidm/ThaicharoenAGC09 fatcat:ilepo6mtirdhbd2ciwms6maafq

A Framework for Set-Oriented Computation in Inductive Logic Programming and Its Application in Generalizing Inverse Entailment [chapter]

Héctor Corrada Bravo, David Page, Raghu Ramakrishnan, Jude Shavlik, Vitor Santos Costa
2005 Lecture Notes in Computer Science  
We propose a new approach to Inductive Logic Programming that systematically exploits caching and offers a number of advantages over current systems.  ...  It avoids redundant computation, is more amenable to the use of set-oriented generation and evaluation of hypotheses, and allows relational DBMS technology to be more easily applied to ILP systems.  ...  Introduction The goal of Inductive Logic Programming (ILP) [1] is to autonomously learn first-order logic programs that model relational data.  ... 
doi:10.1007/11536314_5 fatcat:ypdqsuvlevaobbysz4e3s3zm2e

Set Theory INC# ∞# Based on Infinitary Intuitionistic Logic with Restricted Modus Ponens Rule (Part.II) Hyper Inductive Definitions

Jaykov Foukzon
2021 Journal of Advances in Mathematics and Computer Science  
In this paper intuitionistic set theory INC# ∞# in infinitary set theoretical language is considered. External induction principle in nonstandard intuitionistic arithmetic were derived.  ...  Infinitary and Hyperinfinitary Logics Classical infinitary logic By a vocabulary, we mean a set L of constant symbols, and relation and operation symbols with finitely many argument places.  ...  The syntax of bivalent hyperinfinitary first-order logics L # ∞ # consists of a (ordered) set of sorts and a set of function and relation symbols, these latter together with the corresponding type, which  ... 
doi:10.9734/jamcs/2021/v36i430359 fatcat:egi7kk2mcjcdvh7zz35kzqhxsi

Set Theory INC# ∞# Based on Innitary Intuitionistic Logic with Restricted Modus Ponens Rule. Hyper Inductive Denitions. Application in Transcendental Number Theory. Generalized Lindemann-Weierstrass Theorem

Jaykov Foukzon
2021 Journal of Advances in Mathematics and Computer Science  
In this paper intuitionistic set theory INC#∞# in infinitary set theoretical language is considered. External induction principle in nonstandard intuitionistic arithmetic were derived.  ...  Main results are: (i) number ee is transcendental; (ii) the both numbers e + π and e − π are irrational.  ...  Infinitary Logic Classical infinitary logic By a vocabulary, we mean a set L of constant symbols, and relation and operation symbols with finitely many argument places.  ... 
doi:10.9734/jamcs/2021/v36i830394 fatcat:tqshlqhh25hyfjbl5hlbizpplu

Definitions and notations [chapter]

Thomas Forster
Logic, Induction and Sets  
The words "cycle" and "path" mean simple cycle and simple path in a graph.  ...  Definitions and notations All the graphs under consideration are supposed to be without isolated vertices, multiple edges and loops.  ...  of construction of sets A and B).  ... 
doi:10.1017/cbo9780511810282.003 fatcat:byjecik5mvezjp3xbhwiqy5mi4

Statistical Relational Learning: An Inductive Logic Programming Perspective [chapter]

Luc De Raedt
2005 Lecture Notes in Computer Science  
shall start from inductive logic programming and study how inductive logic programming formalisms, settings and techniques can be extended to deal with probabilistic issues.  ...  settings of the field of inductive logic programming [3] .  ...  This work is part of the EU IST FET project APRIL II (Application of Probabilistic Inductive Logic Programming II).  ... 
doi:10.1007/11564126_3 fatcat:vtoipejcmjcg7ow6piwy4eiyjm

Statistical Relational Learning: An Inductive Logic Programming Perspective [chapter]

Luc De Raedt
2005 Lecture Notes in Computer Science  
shall start from inductive logic programming and study how inductive logic programming formalisms, settings and techniques can be extended to deal with probabilistic issues.  ...  settings of the field of inductive logic programming [3] .  ...  This work is part of the EU IST FET project APRIL II (Application of Probabilistic Inductive Logic Programming II).  ... 
doi:10.1007/11564096_3 fatcat:lwxyiiawqfbopjvgy5evuj2yzi

Inductive equivalence in clausal logic and nonmonotonic logic programming

Chiaki Sakama, Katsumi Inoue
2010 Machine Learning  
Conditions of inductive equivalence change depending on the logic of representation languages and the logic of induction or inductive logic programming (ILP).  ...  and logic programming.  ...  abductive and inductive logic programming.  ... 
doi:10.1007/s10994-010-5189-4 fatcat:vxaidrnzg5brle4rxdazclg4pu

Inductive Equivalence of Logic Programs [chapter]

Chiaki Sakama, Katsumi Inoue
2005 Lecture Notes in Computer Science  
P1 and P2 are weakly equivalent if they have the same declarative meaning.  ...  P1 and P2 are strongly equivalent if P1∪R and P2∪R have the same declarative meaning for any program R. † These equivalence relations compare capabilities of deductive reasoning between programs.  ...  Two theories B1 and B2 are inductively equivalent under a logic L if it holds that B1 ∪ H |= L E iff B2 ∪ H |= L E for any set E of examples and for any hypothesis H such that B1∪H and B2∪H are consistent.Results  ... 
doi:10.1007/11536314_19 fatcat:ytly42423rd5hbxn2wzr4x5n74

Equivalence of Logic Programs Under Updates [chapter]

Katsumi Inoue, Chiaki Sakama
2004 Lecture Notes in Computer Science  
P1 and P2 are weakly equivalent if they have the same declarative meaning.  ...  P1 and P2 are strongly equivalent if P1∪R and P2∪R have the same declarative meaning for any program R. † These equivalence relations compare capabilities of deductive reasoning between programs.  ...  Two theories B1 and B2 are inductively equivalent under a logic L if it holds that B1 ∪ H |= L E iff B2 ∪ H |= L E for any set E of examples and for any hypothesis H such that B1∪H and B2∪H are consistent.Results  ... 
doi:10.1007/978-3-540-30227-8_17 fatcat:jnclh6ctkjbfbgpkbdflkzbzci

Extending Classical Logic with Inductive Definitions [article]

Marc Denecker
2000 arXiv   pre-print
The goal of this paper is to extend classical logic with a generalized notion of inductive definition supporting positive and negative induction, to investigate the properties of this logic, its relationships  ...  to other logics in the area of non-monotonic reasoning, logic programming and deductive databases, and to show its application for knowledge representation by giving a typology of definitional knowledge  ...  An abstract inductive definition (ID) D in this logic defines a set Def ined(D) of symbols, called the set of defined symbols, by a set of rules of the form p ← B where p is a defined atom and B a set  ... 
arXiv:cs/0003019v1 fatcat:svyilctko5hprckbrzjvpcc7ee

Induction and Confirmation Theory: An Approach based on a Paraconsistent Nonmonotonic Logic

Ricardo Sousa Silvestre
2011 Princípios  
We pick up one of the most wide-spread nonmonotonic formalisms – default logicand analyze to what extent and under what adjustments it could work as a logic of induction in the philosophical sense.  ...  This paper is an effort to realize and explore the connections that exist between nonmonotonic logic and confirmation theory.  ...  A pseudo-inductive logic of plausibility P is a pair <M*, *> where M* = < ⊱ ,> is a pseudo-inductive modal logic and *( ⊱ ) is a set of modal operators.  ... 
doaj:fce609c5faa948e3b3a56efa667ff8e3 fatcat:kpcogg5l2vhqdkcbe76a4dbfyq
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