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Predicative proof theory of PDL and basic applications
[article]

2021
*
arXiv
*
pre-print

Hence (by the cutfree subformula property) such

arXiv:1904.05131v2
fatcat:v6wuedrlyneqzp6youdraxryvq
*predicative*extension*of*PA proves that any given [ P^∗]-free sequent is valid in*PDL*iff it is deducible in Seq_ω^*pdl*by a finite cut-*and*omega-rule-free ... The former also implies standard Herbrand-style conclusions, which eventually leads to PSPACE-decidability*of**PDL*-validity*of*S, provided that P is atomic*and*A is in a suitable*basic*conjunctive normal ... As usual in*proof**theory*, our cutfree sequent calculi provide useful help in the verification*of**PDL*-(un)provability*of*concrete formulas*of*simple shapes. ...##
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An arithmetical hierarchy in propositional dynamic logic

1989
*
Information and Computation
*

We prove that any alternation

doi:10.1016/0890-5401(89)90033-3
fatcat:2ie55fyggjbxbijjb7ibortosy
*of*modalities in*PDL*adds to its expressive power. The*proof*uses Turing machine models where*PDL*formulas define the arithmetical hierarchy*of*sets. ... As a by-product, we obtain a theorem*of*Berman*and*Paterson. Syntax. The*basic*objects are two sets*of*primitives: (/)0 the*basic*formulas,*and*.d 0 the*basic*programs. ... We refer to (Harel, 1984) for*basic*facts concerning*PDL**theory*. 4. if p*and*q are formulas*and*a is a program then p 1\ q, ,p, (tJ.) p are formulas. ...##
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Page 3184 of Mathematical Reviews Vol. , Issue 2003e
[page]

2003
*
Mathematical Reviews
*

) extended with the axioms

*of*“no learning” (NL)*and*“perfect recall” (PR),*and*the full*and*weak test-free versions*of*the {NL, PR}-extensions*of*the*basic*product*PDL*©@S5. ... The combinations*of**PDL*(propositional dynamic logic)*and**of*the modal logic $5 (accepted as an adequate representation for the notion*of*“knowledge”) may have various*applications*in program verification ...##
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Page 3763 of Mathematical Reviews Vol. , Issue 86h
[page]

1986
*
Mathematical Reviews
*

The

*proof**of*Theorem A is obtained by reduction to the emptiness problem for some tree automata*and*the*proof**of*Theorem B results from an*application**of*the recurring domino problem. ... Computation*and**proof**theory*(Aachen, 1983), 163-174, Lecture Notes in Math., 1104, Springer, Berlin-New York, 1984. ...##
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Page 3950 of Mathematical Reviews Vol. , Issue 84j
[page]

1984
*
Mathematical Reviews
*

The author proposes a semantic

*theory*that incorporates two sorts*of*dependence*of*truth on features*of*context: context-dependence*and*84):03065 ... A complete treatise*of*this analysis will appear in the forthcoming Ph.D. Thesis*of*Janssen [“Foundations*and**applications**of*Montague grammar”, Ph.D. Thesis, Univ. ...##
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An essay in combinatory dynamic logic

1991
*
Information and Computation
*

PASSY

doi:10.1016/0890-5401(91)90026-x
fatcat:vgjeqzuasrdxlbkhcdryr6r67y
*AND*TINCHEV PREFACE The*theory*developed here, although deriving its motivation*and*parts*of*its terminology from programming*theory*, can be viewed as a*theory*for reasoning about action in general ... (Hare], 1979) Syntactically, modal logic in general,*and*dynamic logic in particular, lacks the "static" notion*of*possible world, or execution state, which is the essence*of*its semantics. ... Theses.*and*thus we first*of*all would like to repeat our thanks to Dimiter Skordev-our teacher*and*supervisor. ...##
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A near-optimal method for reasoning about action

1980
*
Journal of computer and system sciences (Print)
*

We broaden the scope

doi:10.1016/0022-0000(80)90061-6
fatcat:j57aj7gnbfa7jgtrhfmytexnyq
*of**applicability**of*the Hintikka approach with the help*of*the notion*of*a H&i&z StrUc&e, which has two functions*and*four*predicates*over a domain*of**theories*. ... Our*basic*test is that r is satisfiable if*and*only if the union*of*the*theories*along some path*of*a tableau for Y is consistent. ...##
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The gamut of dynamic logics
[chapter]

2006
*
Handbook of the History of Logic
*

Acknowledgement Thanks to Albert Visser, Johan van Benthem, Balder ten Cate

doi:10.1016/s1874-5857(06)80033-6
fatcat:cxjzfgmh4bgiviw7hharvgmao4
*and*Marc Pauly for useful feedback on various drafts*of*this paper. BIBLIOGRAPHY ... Discourse Representation*Theory*Now we briefly introduce a very streamlined*and**basic*version*of*Discourse Representation*Theory*(DRT). ...*Proof*. The*proof*system for*PDL*is complete,*and*every formula in the language*of**PDL*DEL is provably equivalent to a*PDL*formula. ...##
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Fork Algebras as a Sufficiently Rich Universal Institution
[chapter]

2006
*
Lecture Notes in Computer Science
*

Tarlecki goes even further when he writes: "... this suggests that we should strive at a development

doi:10.1007/11784180_19
fatcat:xtppciowrbca3kfwjqhymqcoay
*of*a convenient to use*proof**theory*(with support tools!) ... condition: Since the language*of*fork algebras is algebraic, the only*predicate*is equality. ... From here on, we assume the reader has a nodding acquaintance with*basic*concepts from category*theory*such as the notions*of*category, functor, natural transformation*and*colimits. ...##
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Page 4430 of Mathematical Reviews Vol. , Issue 84k
[page]

1984
*
Mathematical Reviews
*

They constitute

*basic*sets*of*tools useful in proving properties*of*program schemes. Propositional*theories**of*data structures, e.g., the arithmetic*of*natural numbers*and*stacks, are constructed. ... He remarks that this technique is*applicable*to only a finite number*of*such calculi,*and*that the problem*of*finding all*of*them is open. ...##
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Cyclic Proofs, Hypersequents, and Transitive Closure Logic
[article]

2022
*
arXiv
*
pre-print

We show that previously proposed sequent systems are cut-free incomplete for

arXiv:2205.08616v1
fatcat:rxajdbswovahxakthd4vlr7bm4
*basic*validities from Kleene Algebra (KA)*and*Propositional Dynamic Logic (*PDL*), over standard translations. ... A peculiarity*of*our system is its richer correctness criterion, exhibiting 'alternating traces'*and*necessitating a more intricate soundness argument than for traditional cyclic*proofs*. ... This result can be seen as a formal interpretation*of*cyclic modal*proof**theory*within cyclic*predicate**proof**theory*, in the spirit*of*[21, 20] . ...##
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The Logic of Games and its Applications
[chapter]

1985
*
North-Holland Mathematics Studies
*

In this the operator (α) resembles the modalities α

doi:10.1016/s0304-0208(08)73078-0
fatcat:sx5gzznzgrd6vazda73cpxebpu
*and*[α]*of**PDL*which are also monotonic. However, these modalities*of**PDL*are disjunctive ... We develop a Logic in which the*basic*objects*of*concern are games, or equivalently, monotone*predicate*transforms. ... Specifically, we have a language with a finite number*of**basic*symbols a i for atomic programs,*and*P j for atomic*predicates*. We then define programs*and*formulas as follows. ...##
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Descriptive dynamic logic and its application to reflective architectures

1996
*
Future generations computer systems
*

The reasoning dynamics

doi:10.1016/0167-739x(96)00007-6
fatcat:dmvbsqzyejaqjjdxgk3d2p43m4
*of*a knowledge base on top*of*a MLA can therefore be described by how the local*theories**of*the units evolve during execution. ... The aim*of*this paper is to propose dynamic logic as a cotmnon logical framework to describe*and*identify the most relevant formal characteristics*of*multi-language logical architectures (MLA) in order ...*of*this paper. ...##
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Modal Kleene Algebra Applied to Program Correctness
[chapter]

2016
*
Lecture Notes in Computer Science
*

Modal Kleene algebras are relatives

doi:10.1007/978-3-319-48989-6_19
fatcat:mcbdfeuc4zbrze64ndtt4yr4km
*of*dynamic logics that support program construction*and*verification by equational reasoning. ... Modularity*of*the approach is demonstrated by variants that capture program termination*and*recursion, memory models for programs with pointers,*and*program trace semantics. ... Over the last decade, the mathematics*of*MKA has been well investigated, models relevant to computing have been constructed, extensions*and*variations introduced,*applications*from game*theory*to termination ...##
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Bisimulation Safe Fixed Point Logic

2014
*
Advances in Modal Logic
*

We define

dblp:conf/aiml/ZaidGJ14
fatcat:mlgku77fbzbe5edqe53ggbyvfq
*and*investigate a new modal fixed-point logic, called bisimulation safe fixed-point logic BSFP, which is a calculus*of*binary relations that extends both*PDL**and*the modal µ-calculus. ... We identify a fragment*of*BSFP that is equivalent to the extension*of**PDL*by context-free grammars. ... Let τ be a vocabulary*of*monadic*predicates*P i*and*binary action*predicates*E a ,*and*let Z 1 , Z 2 . . . be a collection*of*binary*predicate*variables. ...
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