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### Justification by Infinite Loops

David Atkinson, Jeanne Peijnenburg
2010 Notre Dame Journal of Formal Logic
In the present paper we demonstrate that the same holds if the justification takes the form of a closed loop.  ...  This suggests that infinitism sits more comfortably with a coherentist view of justification than with an approach in which justification is portrayed as a linear process.  ...  In Section 4 we extend our study to loops of infinite size. We demonstrate that justification by an infinite linear chain is usually indistinguishable from justification by an infinite loop.  ...

### Loops and Networks [chapter]

David Atkinson, Jeanne Peijnenburg
We show that generally, i.e. in what we have called the usual class, infinite loops yield the same value for the target as do infinite chains; it is only in the exceptional class that the values differ  ...  The first treats loops rather than chains.  ...  However, from the fact that a finite loop differs from an infinite 'loop', it does not follow that an infinite loop is in fact an infinite chain.  ...

### Pluralism in Probabilistic Justification [chapter]

David Atkinson, Jeanne Peijnenburg
2012 Probabilities, Laws, and Structures
Thus it makes sense to say that a target proposition, E 0 , can be epistemically justified not only by an infinite one-dimensional chain or an infinite one-dimensional loop, but also by an infinite network  ...  A target proposition, E 0 , can be probabilistically justified by a chain or by a loop, and both the chain and the loop can be finite or infinite. 2 Moreover, in each of these four cases the conditional  ...  Appendix When the special equality (16) does not hold, the quadratic iteration (15) can be put into canonical form by means of the substitution With this transformation, (15) becomes 9 where The conditional  ...

### Loop checking and negation

Roland N. Bol
1993 The Journal of Logic Programming
By replacing in T every justification J by its corresponding J', we obtain a tree T".  ...  However, the (loopchecked) interpreter need not be effective: in general, traversing infinite justifications is required.  ...  Partial support by Esprit BRA-project 3020 Integration is gratefully acknowledged.  ...

### Exploiting Game Theory for Analysing Justifications [article]

Simon Marynissen, Bart Bogaerts, Marc Denecker
2020 arXiv   pre-print
In this paper, we continue the study of justification theory by means of three major contributions. The first is studying the relation between justification theory and game theory.  ...  The third contribution is establishing a precise criterion of when a semantics induced by justification theory yields consistent results.  ...  If M is a set of loops, M * denotes the set of (finite) paths obtained by a finite iteration of loops in M; M ω denotes the set of infinite paths obtained by infinite iterations of loops in M.  ...

### Justifications and a Reconstruction of Parity Game Solving Algorithms [article]

Ruben Lapauw, Maurice Bruynooghe, Marc Denecker
2021 arXiv   pre-print
By carefully ordering Justify steps, we reconstruct three algorithms well known from the literature.  ...  Parity games are infinite two-player games played on directed graphs. Parity game solvers are used in the domain of formal verification.  ...  The function starts its main loop by attracting nodes for level p.  ...

### Exploiting Game Theory for Analysing Justifications

SIMON MARYNISSEN, BART BOGAERTS, MARC DENECKER
2020 Theory and Practice of Logic Programming
The third contribution is establishing a precise criterion of when a semantics induced by justification theory yields consistent results.  ...  why a property holds (or does not hold) in a model.In this paper, we continue the study of justification theory by means of three major contributions.  ...  If M is a set of loops, M * denotes the set of (finite) paths obtained by a finite iteration of loops in M; M ω denotes the set of infinite paths obtained by infinite iterations of loops in M.  ...

### Extensions to Justification Theory [article]

Simon Marynissen
2019 arXiv   pre-print
Knowledge representation languages covered by justification theory include logic programs, argumentation frameworks, inductive definitions, and nested inductive and coinductive definitions.  ...  In addition, justifications are also used for implementation purposes.  ...  We can also view a justification as a partial function from F to subsets of F by mapping a defined fact to the body of the rule in the justification if it exists.  ...

### A Compact Linear Translation for Bounded Model Checking

Paul B. Jackson, Daniel Sheridan
2007 Electronical Notes in Theoretical Computer Science
A restriction function | k maps (k, l) loop paths and infinite loop-case denotations to their finite representations.  ...  We call such infinite paths (k, l) loop paths. Similarly, a finite loop-case denotatioṅ a = a 0 , . . . , a k−1 where a i ∈ B represents the infinite denotation a 0 . . . a l−1 (a l . . . a k−1 ) ω .  ...  for the loop case and l = − for the prefix case.  ...

### Fractal Patterns in Reasoning

David Atkinson, Jeanne Peijnenburg
2012 Notre Dame Journal of Formal Logic
All three papers emphasize that a proposition can be justified by an infinite regress, on condition that epistemic justification is interpreted probabilistically.  ...  The first two papers showed this for one-dimensional chains and for one-dimensional loops of propositions, each proposition being justified probabilistically by its precursor.  ...  The matter becomes even more interesting if we replace the infinite onedimensional structure (a chain or a loop) by an infinite, many-dimensional probabilistic network.  ...

### Loop Quantization of the Schwarzschild Black Hole

Rodolfo Gambini, Jorge Pullin
2013 Physical Review Letters
We can construct explicitly the exact solutions of the physical Hilbert space annihilated by all constraints.  ...  This allows the completion of the Dirac quantization procedure using loop quantum gravity techniques.  ...  until stopped by any other external force.  ...

### QUANTUM GRAVITY IN CYCLIC (EKPYROTIC) AND MULTIPLE (ANTHROPIC) UNIVERSES WITH STRINGS AND/OR LOOPS

T. J. CHUNG
2008 The Eleventh Marcel Grossmann Meeting
The anthropic principle can be studied by means of string theory [5] [6] [7] [8] .  ...  It is postulated that either string theory and /or loop quantum gravity might be able to substantiate these hypotheses. Recently, the cyclic or ekpyrotic model has been reported [1-2].  ...  The anthropic principle can be studied by means of string theory [5] [6] [7] [8] .  ...

### Finite Reasons Without Foundations

Ted Poston
2014 Metaphilosophy
A view of reasons that holds that the 'reason for' relation is constrained by PAA and PAC can avoid an infinite regress if the 'reason for' relation is contextual.  ...  Moreover, such a view of reasons can maintain that skepticism is false by the maintaining that there is more to epistemic justification than what can be expressed in any reasoning session.  ...  What is problematic about circularity is the idea that a simple loop captures all there is to epistemic justification.  ...

### Justification by an Infinity of Conditional Probabilities

David Atkinson, Jeanne Peijnenburg
2009 Notre Dame Journal of Formal Logic
In this paper we show that the assumption invalidates some venerable attacks on infinitism: once we accept that epistemic justification is gradual, an infinitist stance makes perfect sense.  ...  It is only without the assumption that infinitism runs into difficulties.  ...  A variant of the latter argument has recently been put forward by Gillet, who holds that an infinite regress of deferred justification cannot yield an unconditional, determinate justification for the target  ...

### On Nested Justification Systems (full version) [article]

Simon Marynissen, Jesse Heyninck, Bart Bogaerts, Marc Denecker
2022 arXiv   pre-print
Nested justification systems, first introduced by Denecker et al. (2015), allow for the composition of justification systems.  ...  Furthermore, we show how nested justification systems allow representing fixpoint definitions (Hou and Denecker 2009).  ...  So to eliminate x from the justification, we can replace this loop with the fact f . On the left side, w can be replaced by a and b.  ...
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