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### Predicative proof theory of PDL and basic applications [article]

Lev Gordeev
2021 arXiv   pre-print
Hence (by the cutfree subformula property) such 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.  ...

### An arithmetical hierarchy in propositional dynamic logic

Sorin Istrail
1989 Information and Computation
We prove that any alternation 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.  ...

### 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  ...

### 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.  ...

### 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.  ...

### An essay in combinatory dynamic logic

Solomon Passy, Tinko Tinchev
1991 Information and Computation
PASSY 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.  ...

### A near-optimal method for reasoning about action

Vaughan R. Pratt
1980 Journal of computer and system sciences (Print)
We broaden the scope 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.  ...

### The gamut of dynamic logics [chapter]

Jan van Eijck, Martin Stokhof
2006 Handbook of the History of Logic
Acknowledgement Thanks to Albert Visser, Johan van Benthem, Balder ten Cate 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.  ...

### Fork Algebras as a Sufficiently Rich Universal Institution [chapter]

Carlos G. Lopez Pombo, Marcelo F. Frias
2006 Lecture Notes in Computer Science
Tarlecki goes even further when he writes: "... this suggests that we should strive at a development 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.  ...

### 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.  ...

### Cyclic Proofs, Hypersequents, and Transitive Closure Logic [article]

Anupam Das, Marianna Girlando
2022 arXiv   pre-print
We show that previously proposed sequent systems are cut-free incomplete for 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] .  ...

### The Logic of Games and its Applications [chapter]

Rohit Parikh
1985 North-Holland Mathematics Studies
In this the operator (α) resembles the modalities α 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.  ...

### Descriptive dynamic logic and its application to reflective architectures

Carles Sierra, Lluis Godo, Ramon López de Màntaras, Mara Manzano
1996 Future generations computer systems
The reasoning dynamics 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.  ...

### Modal Kleene Algebra Applied to Program Correctness [chapter]

Victor B. F. Gomes, Georg Struth
2016 Lecture Notes in Computer Science
Modal Kleene algebras are relatives 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  ...

### Bisimulation Safe Fixed Point Logic

Faried Abu Zaid, Erich Grädel, Stephan Jaax
2014 Advances in Modal Logic
We define 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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