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Axiomatic constraint systems for proof search modulo theories [article]

Damien Rouhling, Mahfuza Farooque, Stéphane Graham-Lengrand, Assia Mahboubi, Jean-Marc Notin
2015 arXiv   pre-print
Proving soundness and completeness of the calculus leads to an axiomatisation that identifies the conditions under which abstract constraints can be generated and propagated in the same way unifiers usually  ...  This paper investigates a generalisation of such mechanisms whereby theory-specific constraints are produced instead of substitutions.  ...  The views and conclusions contained herein are those of the authors and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of DARPA  ... 
arXiv:1412.6790v2 fatcat:hncvb4njkjfpbokd3m3xu5t2oi

Axiomatic Constraint Systems for Proof Search Modulo Theories [chapter]

Damien Rouhling, Mahfuza Farooque, Stéphane Graham-Lengrand, Assia Mahboubi, Jean-Marc Notin
2015 Lecture Notes in Computer Science  
Proving soundness and completeness of the calculus leads to an axiomatisation that identifies the conditions under which abstract constraints can be generated and propagated in the same way unifiers usually  ...  This paper investigates a generalisation of such mechanisms whereby theory-specific constraints are produced instead of substitutions.  ...  Although constraint-based unification algorithms or universal unification algorithms could provide theory reasoners for our framework, our paper is not about using constraints to specify unification modulo  ... 
doi:10.1007/978-3-319-24246-0_14 fatcat:cs2xb26p4vaqxkyl25oyefwuf4

Differential Dynamic Logic for Hybrid Systems

André Platzer
2008 Journal of automated reasoning  
Our main result proves that this calculus axiomatises the transition behaviour of hybrid systems completely relative to differential equations.  ...  The calculus is compositional, i.e., it reduces properties of hybrid programs to properties of their parts.  ...  DFG) as part of the Transregional Collaborative Research Center "Automatic Verification and Analysis of Complex Systems" (SFB/TR 14 AVACS).  ... 
doi:10.1007/s10817-008-9103-8 fatcat:rr2zob6u6jdnzftqjysgybtvyq

Polarities & Focussing: a journey from Realisability to Automated Reasoning [article]

Stéphane Graham-Lengrand
2014 arXiv   pre-print
We use these concepts to describe the main algorithm of SAT-solvers and SMT-solvers: DPLL. We then describe the implementation of a proof-search engine called Psyche.  ...  Its architecture, based on the concept of focussing, offers a platform where smart techniques from automated reasoning (or a user interface) can safely and trustworthily be implemented via the use of an  ...  For this we extended the focussed sequent calculus LKF into LK p (T ), equipped with the ability to polarise atoms on-the-fly and call a decision procedure specific to the background theory T .  ... 
arXiv:1412.6781v1 fatcat:sj4zl2ayu5bzfjaq76znvllsb4

Differential Dynamic Logic dℒ [chapter]

André Platzer
2010 Logical Analysis of Hybrid Systems  
and axiomatisable theory [288] .  ...  Our main result proves that this calculus axiomatises the transition behaviour of hybrid systems completely relative to differential equations.  ...  Let formula φ be an instance of ψ, with ψ being a formula of the base theory, i.e., φ is ψ θ 1 z 1 . . . θ n z n for some variables z i and arbitrary terms θ i .  ... 
doi:10.1007/978-3-642-14509-4_2 fatcat:awfwcd5vqvc4lhmcfgpysr65bm

A Complete Axiomatisation for Quantifier-Free Separation Logic [article]

Stéphane Demri, Étienne Lozes, Alessio Mansutti
2021 arXiv   pre-print
The calculus is divided in three parts: the axiomatisation of core formulae where Boolean combinations of core formulae capture the expressivity of the whole logic, axioms and inference rules to simulate  ...  a bottom-up elimination of separating connectives, and finally structural axioms and inference rules from propositional calculus and Boolean BI with the magic wand.  ...  We would like to thank the anonymous reviewers for their numerous remarks and suggestions that help us to improve the quality of the document.  ... 
arXiv:2006.05156v3 fatcat:djrnvmhq4nbcbalij652qcrn6u

Modular Labelled Sequent Calculi for Abstract Separation Logics [article]

Zhé Hóu, Ranald Clouston, Rajeev Goré, Alwen Tiu
2018 arXiv   pre-print
This paper non-trivially improves upon this previous work by giving a general framework of calculi on which any new axiom in the logic satisfying a certain form corresponds to an inference rule in our  ...  this paper, which developed a modular proof theory for various PASLs using cut-free labelled sequent calculi.  ...  We thank the anonymous reviewers of this paper for their helpful comments. Clouston was supported by a research grant (12386) from Villum Fonden.  ... 
arXiv:1710.10805v3 fatcat:lxjqcf4qjvehvegzcsowiebaya

Subject index volumes 1–200

1999 Theoretical Computer Science  
meanings, 1792 SOhtiOnS to systems of set constraints, 3272 spaces ordinary -, 2994 To -, 581 zero-dimensional -, 2525 space-time bound, 34 11 complexity classes of Turing machines, 525  ...  modulo a semigroup, 3186 sums, 2030,2095,2123,2153,2711 of games, 1696 digital -, 2356 harmonic -, 27 11 S-unimodal systems. 2525 super exponential algorithms, 2469 Turing theories. 3289 supercompiler  ... 
doi:10.1016/s0304-3975(98)00319-3 fatcat:s22ud3iiqjht7lfbtc3zctk7zm

4 Computational modal logic [chapter]

Ian Horrocks, Ullrich Hustadt, Ulrike Sattler, Renate Schmidt
2007 Studies in Logic and Practical Reasoning  
because we can forget the constraints concerning an "old" world once we procede to the next one. • labelled tableaux are closely related to propositional tableaux: they are sets of labelled formulae that  ...  Similarly, KB n is axiomatised by adding the axiom schema B to the axiomatisation of K n . Finally, for KB4 n , we add both axiom schemas B and 4.  ... 
doi:10.1016/s1570-2464(07)80007-3 fatcat:zrhdgzcgovfc3lxuuqnjpsa5bi

1 Modal logic: a semantic perspective [chapter]

Patrick Blackburn, Johan van Benthem
2007 Studies in Logic and Practical Reasoning  
It has been used as a tool for reasoning about time, beliefs, computational systems, necessity and possibility, and much else besides.  ...  As we shall see, this concept holds the key to modal model theory and characterises the link with first-order logic. This chapter has two main pedagogical goals.  ...  We might then point out that the set of all modal validities (that is, the minimal modal logic) can be axiomatised by a Hilbert-style proof system called K.  ... 
doi:10.1016/s1570-2464(07)80004-8 fatcat:unpkk66lxjdibjbv4epdapt6yu

Relevant and substructural logics [chapter]

Greg Restall
2006 Handbook of the History of Logic  
This is a history of relevant and substructural logics, written for the Handbook of the History and Philosophy of Logic, edited by Dov Gabbay and John Woods. 1  ...  This more recent work is applicable to Orlov's system, because Orlov has axiomatised the implication and negation fragment of the now well-known relevant logic R.  ...  Neither time nor space allows me to consider tableaux for substructural logics, save for this reference.  ... 
doi:10.1016/s1874-5857(06)80030-0 fatcat:rzftxqhxpnacveqe5guz2s6sse

Fuzzy-Set Based Logics — an History-Oriented Presentation of their Main Developments [chapter]

Didier Dubois, Francesc Esteva, Lluís Godo, Henri Prade
2007 Handbook of the History of Logic  
This issue can be related to a misunderstanding regarding the law of excluded middle and the law of non-contradiction, and the connections between many-valued logics and modal logics.  ...  This new trend has also put formal logic, originally developed in connection with the foundations of mathematics, in a completely new perspective, as a tool for processing information on computers.  ...  Finally, let us comment that other proof search oriented calculi include a tableaux calculus for Lukasiewicz logic [Hähnle, 1994] , decomposition proof systems for Gödel logic [Avron and Konikowska,  ... 
doi:10.1016/s1874-5857(07)80009-4 fatcat:s4i5zfomgzbxnfbuif5ffsesea

The two Faces of Deduction Logic Axioms

Thomas Ball, Jürgen Giesl, Reiner Hähnle, Tobias Nipkow, Thomas Ball, Jürgen Giesl, Reiner Hähnle, Tobias Nipkow, Franz Baader, T Dresden, Franz Baader, Silvio Ghilardi (+28 others)
2008 Lecture Notes in Articial Intelligence   unpublished
Many applications of automated deduction require reasoning modulo some form of integer arithmetic or other theories.  ...  We present combinatory array logic, CAL, using a small, but powerful core of combinators, and reduce it to the theory of uninterpreted functions.  ...  See http://symbolaris.com/info/KeYmaera.html Keywords: Dynamic logic, dierential equations, sequent calculus, axiomatisation, automated theorem proving, verication of hybrid systems Full Paper: http://  ... 
fatcat:ulyf2gmvrfb43ig574cwyxnqri

An Old Discipline with a New Twist: The Course "Logic in Action"

Johan van Benthem
2017 IfColog journal of logics and their applications (FLAP)  
We thank all the authors who have contributed to this special issue. Each submission for this special issue were assigned to two reviewers. We are immensely grateful to  ...  We hope that this volume will inspire teachers so that the way logic is taught to a wide range of students from undergraduate to postgraduate gets improved, and thereby make those audiences aware of the  ...  We would like to also thank again all the reviewers for the conference TTL 2015 as well as all the members of the organizing committee of TTL2015, M.  ... 
dblp:journals/flap/Benthem17 fatcat:e5d54ekdhrbmjcs3ywmfzhaslm

Satisfiability Games for Branching-Time Logics

Oliver Friedmann, Martin Lange, Markus Latte, Colin Stirling
2013 Logical Methods in Computer Science  
Some decision procedures for these logics are known; they use tree automata, tableaux or axiom systems.  ...  We then obtain a reduction to the problem of solving parity or Büchi games. The worst-case complexity of the obtained algorithms matches the known lower bounds for these logics.  ...  Reynolds' proof system [Rey01] is an approach at giving a sound and complete finite axiomatisation for CTL * .  ... 
doi:10.2168/lmcs-9(4:5)2013 fatcat:dvodt4haq5hrpisnvb7ztodthy
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