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On Strong Normalization in Proof-Graphs for Propositional Logic

Marcela Quispe-Cruz, Edward Haeusler, Lew Gordeev
2016 Electronical Notes in Theoretical Computer Science  
In proof-graphs, the main reason for obtaining the strong normalization property using such simple complexity measure is a direct consequence of the fact that each formula occurs only once in the proof-graph  ...  In this work, we consider full minimal propositional logic and show how to reduce (eliminating maximal formulas) these representations such that strong normalization theorem can be proved by simply counting  ...  Proof-graphs for full propositional logic In [6] we considered implication as the only logic connective.  ... 
doi:10.1016/j.entcs.2016.06.012 fatcat:f4wwpgry4fc5rmowu6ew2a2hlu

Proof-graphs for Minimal Implicational Logic

Marcela Quispe-Cruz, Edward Hermann Haeusler, Lew Gordeev
2014 Electronic Proceedings in Theoretical Computer Science  
In the present paper we give a precise definition of proof-graphs for the minimal implicational logic, together with a normalization procedure for these proof-graphs.  ...  We present the formalism of proof-graphs for purely implicational logic, which are graphs of a specific shape that are intended to capture the logical structure of a deduction.  ...  [1] , that use graphs for representing proofs, pointing out that proof-graphs offer a better way to facilitate the visualisation and understanding of proofs in the underlying logic.  ... 
doi:10.4204/eptcs.144.2 fatcat:stbdnq2enneoliegqxe3dokhey

Basic proof theory

Lev Gordeev
1998 Science of Computer Programming  
Second-order logic. 1. Intuitionistic N-system. Normal deductions. 2. Typed polymorphic lambda calculus. 3. Girard's strong normalization proof. 4-5.  ...  Normalization for the N-system with the omega rule. 4. Schuette's proof that TI for ordinal notations below epsilon 0 is underivable in PA. 5.  ... 
doi:10.1016/s0167-6423(98)00002-1 fatcat:zaujpt3oybegxplm2wqphdgchq

Some remarks on lengths of propositional proofs

Samuel R. Buss
1995 Archive for Mathematical Logic  
Some open problems in propositional proof length and in logical flow graphs are discussed.  ...  We survey the best known lower bounds on symbols and lines in Frege and extended Frege proofs.  ...  Cycles in the Logical Flow Graph The logical flow graph of a proof P is a directed graph on occurences of subformulas in P indicating the 'implicational flow' of the formulas in the proof.  ... 
doi:10.1007/s001530050030 fatcat:zs3xnde2g5evdgxmb7kjgs5q4u

Some remarks on lengths of propositional proofs

Samuel R. Buss
1995 Archive for Mathematical Logic  
Some open problems in propositional proof length and in logical flow graphs are discussed.  ...  We survey the best known lower bounds on symbols and lines in Frege and extended Frege proofs.  ...  Cycles in the Logical Flow Graph The logical flow graph of a proof P is a directed graph on occurences of subformulas in P indicating the 'implicational flow' of the formulas in the proof.  ... 
doi:10.1007/bf02391554 fatcat:mo2k5kyitngj5lzehnjlv66ibu

Identity of Proofs Based on Normalization and Generality

Kosta Došen
2003 Bulletin of Symbolic Logic  
This proposal seems more promising than the other one for the general proof theory of classical logic.  ...  Prawitz proposed to analyze identity of proofs in terms of the equivalence relation based on reduction to normal formin natural deduction.  ...  A plenary address at the Logic Colloquium 2002 in Miinster was based on the present paper. I am grateful to the organizers for inviting me to deliver this talk.  ... 
doi:10.2178/bsl/1067620091 fatcat:2k56fbyssveqvfdafld4w2oovm

Identity of Proofs Based on Normalization and Generality [article]

K. Dosen
2004 arXiv   pre-print
The first approach, which stems from Prawitz, Kreisel and Lambek, and is based on normalization of proofs, gives good results in intuitionistic, but not in classical logic.  ...  In general proof theory there are two approaches to the question of identity criteria for proofs.  ...  A plenary address at the Logic Colloquium 2002 in Münster was based on the present paper. I am grateful to the organizers for inviting me to deliver this talk.  ... 
arXiv:math/0208094v12 fatcat:mttiom5k25bjdiad4ld4bu6d54

Page 7604 of Mathematical Reviews Vol. , Issue 2004j [page]

2004 Mathematical Reviews  
Unfortunately, as they admit in the corrigendum, there are prob- lems with their proof, but they claim it still provides a strong normalization proof for Au-calculus. G. E.  ...  This is an interesting and very well written paper on the com- plexity of proofs of propositional formulas.  ... 

A Strongly Normalising Curry-Howard Correspondence for IZF Set Theory [chapter]

Alexandre Miquel
2003 Lecture Notes in Computer Science  
in IZF.  ...  We propose a method for realising the proofs of Intuitionistic Zermelo-Fraenkel set theory (IZF) by strongly normalising λ-terms.  ...  Notice that in order to prove the strong normalisation invariant for system F ω.2 ++ , we first have to check the soundness of propositional subtyping: Proposition 8 -If Σ A ≤ B, then for all ρ ∈ Σ one  ... 
doi:10.1007/978-3-540-45220-1_35 fatcat:xmkngrzs4jh2pnfzmzw3ssmixe

Priority Structures in Deontic Logic

Johan van Benthem, Davide Grossi, Fenrong Liu
2013 Theoria  
In this two-level setting, we also offer novel takes on vexed issues such as the Chisholm paradox and modeling strong permission.  ...  This paper proposes a systematic application of recent developments in the logic of preference to a number of topics in deontic logic.  ...  The paper has also greatly benefited from invaluable discussions with Jörg Hansen on the preference-based tradition in deontic logic, and from many helpful comments from Marek Sergot and the participants  ... 
doi:10.1111/theo.12028 fatcat:xzzggofia5ebvoynp73fvd74ty

A self-dual language for graph theory

T.A McKee
1979 Journal of combinatorial theory. Series B (Print)  
A sense is considered in which the properties expressible in this language are precisely the self-dual properties of graph theory. 00958956/79/07$2.OO/O  ...  A symbolic language is developed around the primitive notions of edge and spanning tree, with each spanning tree being interpreted as a binary relation on edges.  ...  (This is called disjunctive normal form in logic.)  ... 
doi:10.1016/0095-8956(79)90068-6 fatcat:ezjvaa2l65b7rprdigwgb5mdmm

Some Notes on Proofs with Alpha Graphs [chapter]

Frithjof Dau
2006 Lecture Notes in Computer Science  
It is well-known that Peirce's Alpha graphs correspond to propositional logic (PL). Nonetheless, Peirce's calculus for Alpha graphs differs to a large extent to the common calculi for PL.  ...  on the number of propositional variables in the formula.  ...  Peirce's calculus for Alpha graphs differs to a large extent to the common calculi for propositional logic (PL).  ... 
doi:10.1007/11787181_13 fatcat:46v34wpfabhrvkegv4tusoch6a

Autoepistemic logic

Wiktor Marek, Mirosław Truszczyński
1991 Journal of the ACM  
Autoepistemic logic is one of the principa~modes of nonmonotonic reasoning. It umties several other modes of nonmonotonic reasoning and has important applications in logic programming.  ...  Finally, connections with stable model semantics for logic programs with negation is discussed. In particular, it is proven that the problem of existence of stable models is NP-complete.  ...  The proof of the main result of Section 5 relies heavily on the analysis of the normal form of Proposition 3.5.  ... 
doi:10.1145/116825.116836 fatcat:ngrivoxco5czdfx36etcwcjmoy

Two Paradigms of Logical Computation in Affine Logic? [chapter]

Gianluigi Bellin
2003 Logic for Concurrency and Synchronisation  
We prove that such a reduction has the strong normalization and Church-Rosser properties.  ...  A notion of irrelevance in a proof-net is defined and the possibility of cancelling the irrelevant parts of a proof-net without erasing the entire net is taken as one of the correctness conditions; therefore  ...  (ii) The cut-elimination process for proof-nets modulo irrelevance for MAL + Mix has the strong normalization and Church-Rosser property. PROOF.  ... 
doi:10.1007/0-306-48088-3_3 fatcat:xg6bf74pyzgrjl4dhdapxnw7oq

Creating Büchi Automata for Multi-valued Model Checking [chapter]

Stefan J. J. Vijzelaar, Wan J. Fokkink
2017 Lecture Notes in Computer Science  
This paper presents a method to create Büchi automata for multi-valued model checking using quasi-Boolean logics. It allows for multi-valued propositions as well as multi-valued transitions.  ...  In explicit state model checking of linear temporal logic properties, a Büchi automaton encodes a temporal property.  ...  Together with the steering logic that was considered in Section 4, this will allow us to investigate execution steering based on abstract models [11] , building on our previous implementation of multi-valued  ... 
doi:10.1007/978-3-319-60225-7_15 fatcat:glawsc2gljbpnidpqyc2azkieq
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