This paper presents a calculus of Socratic proofs for
Propositional Linear-Time Logic (PLTL) and discusses potential
automation of its proof search

Bolotov, A.

Grigoriev, O.

Shangin, V.

Urbanski, M.

WestminsterResearch

WestminsterResearchhttp://www.westminster.ac.uk/westminsterresearch Socratic Proofs for Propositional Linear-Time LogicUrbanski, M., Bolotov, A., Shangin, V. and Grigoriev, O. This is an electronic version of a paper presented at the Joint Automated Reasoning Workshop and Deduktionstreffen, Vienna 23 to 24th July 2014.The conference proceedings are available online at:http://arw.csc.liv.ac.uk/year/2014/ARW2014proceedings.pdfThe WestminsterResearch online digital archive at the University of Westminster aims to make the research output of the University available to a wider audience. Copyright and Moral Rights remain with the authors and/or copyright owners.Whilst further distribution of specific materials from within this archive is forbidden, you may freely distribute the URL of WestminsterResearch: ((http://westminsterresearch.wmin.ac.uk/).In case of abuse or copyright appearing without permission e-mail repository@westminster.ac.ukSocratic Proofs for Propositional Linear-Time LogicMariusz Urbanski1 Alexander Bolotov2 Vasilyi Shangin3 Oleg Grigoriev31 Adam Mickiewicz University, Poznan´, PolandMariusz.Urbanski@amu.edu.pl2 University of Westminster, UKA.Bolotov@wmin.ac.uk3 Moscow State University, Russiashangin@philos.msu.ru, grigoriev@philos.msu.ruAbstract:This paper presents a calculus of Socratic proofs forPropositional Linear-Time Logic (PLTL) and discusses po-tential automation of its proof search.1 IntroductionPropositional Linear-Time Logic (PLTL) [6] gained vari-ous deductive constructions: axiomatic [5], tableau [11],resolution [4], and natural deduction [1]. In this paperwe present a calculus of Socratic proofs for PropositionalLinear-Time Logic (PLTL) [6], [3], [7] abbreviated asPLTLSP. The calculus is based upon the hypersequent cal-culus [2] and it fits into the framework of Socratic proofsby Wisniewski (cf. [8], [10] and [9]).2 Logic PLTLTWe utilise the language of PLTL which extends the lan-guage of Classical Propositional Calculus (CPC) by tem-poral operators: U (until)  (at the next moment in time),2 (always in the future), and3 (at sometime in the future oreventually). The semantics for the temporal part of the logicPLTLT is defined in the standard way over linear sequenceof states, finite in the past, infinite in the future.In order to formulate PLTLT we need to extend the lan-guage of PLTL with the following signs: `, ?, 1 and 2. Intu-itively, ` stands for derivability relation and ? is a question-forming operator. The numerals 1 and 2 will be used toencode tree-structure of a Socratic transformation.There are two disjoint categories of wffs: declarativewffs (d-wffs) and eroteticwffs (e-wffs), or questions. Thereare also two types of d-wffs: atomic d-wffs and indexed d-wffs. Atomic d-wffs are expressions of the form S ` A,where S is a finite sequence (possibly with repetitions) ofPLTL-wffs, and A is a PLTL-wff, and if A is an empty for-mula, then S is a non-empty sequence. Indexed d-wffs areexpressions of the form S `n A or of the form T `n, whereS ` A and T ` are atomic d-wffs of and n is a sequenceof 1’s or 2’s, starting with 1. E-wffs, or questions are ex-pressions of the form ?(), where  is a non-empty finitesequence of indexed atomic d-wffs (constituents of ).In the formulation of rules we shall use the followingclassification of PLTL formulae to  and  types: 1 2A ^B A B:(A _B) :A :B:(A! B) A :B2A A 2A:3A :A 2:A:(AUB) :B :(A ^(AUB)) 1 2 1:(A ^B) :A :B AA _B A B :AA! B :A B A:2A :A 3:A A3A A 3A :AAUB B A ^(AUB) :BRules for PLTLT :L :? (;S 0  0 T `n C; 	)? (;S 0 1 0 2 0 T `n C; 	)L :? (;S 0  0 T ` C; 	)? (;S 0 1 0 T `n1 C;S 0 2 0 T `n2 C; 	)L:: :? (;S 0 ::A 0 T `n C; 	)? (;S 0 A 0 T `n C; 	)L: :? (;S 0 :A 0 T `n C; 	)? (;S 0 :A 0 T `n C; 	)R :? (;S `n ; 	)? (;S `n1 1;S `n2 2; 	)R :? (;S `n ; 	)? (;S 0 1 `n 2; 	)R:: :? (;S `n ::A; 	)? (;S `n A; 	)R: :? (;S `n :A; 	)? (;S `n :A; 	)If none of the above rules is applicable to a PLTL formulaB, such a formula is called marked. If all PLTL-formulaswithin an indexed formula S `n A are marked, such a for-mula is called a state.The following is a state-prestate rule:S P : ? (;S `n A; 	)? (;S `n A; 	)where S `n A is a state and S (resp. A) results from S(resp. A) by replacing all the formulas of the formB withB and deleting all the remaining formulas. Every formulaof the form S `m A, where n is an initial subsequence ofm or m is an initial subsequence of n, is called a pre-state(cf. [11]).Definition 1. Let q = hQ1; : : : ; Qri be a finite sequence ofquestions ofP. LetQg; Qh 1; Qh (1  g < h 1  r) beelements of the sequence q. Let Sj `n Aj be a constituentof Qg and let Sk `m Ak be a constituent of Qh such thatSj = Sk, Aj = Ak and the sequence n is an initial sub-sequence of the sequence m. Let Sl `i Al be a constituentof Qh 1 such that Sk `m Ak is obtained from Sl `i Al byapplication of a PT-rule. Then Sj `n Aj ; : : : ; Sl `i Alform a loop (a sequence of atomic d-wffs of P : : : etc.),and Sk `m Ak is called a loop-generating formula.Socratic transformations are sequences of questions thataim at deciding derivability of formuls from sets of formuls.Definition 2. A finite sequence hQ1; : : : ; Qri of questionsofP is a Socratic transformation of S ` A iff the followingconditions hold: (i)Q1 =?(S `1 A); (ii)Qi (where i =2; : : : ; r) results from Qi 1 by applying a PT-rule.Definition 3. A constituent  of a question Qi is calledsuccessful iff one of the following holds: (a)  is of the formT 0B0U `n B, or (b)  is of the form T 0B0U 0:B0W `n C,or (c)  is of the form T 0:B0U 0B0W `n C.Definition 4. A Socratic transformation hQ1; : : : ; Qri ofS ` A is completed iff the for each constituent  of Qr atleast one of the following conditions hold: (a) no rule isapplicable to PLTL-formulas in , or (b)  is successful, or(c)  is a loop-generating formula.Definition 5. A formula B is called an eventuality in S `nA iff one of the following holds: (i) B is a term of S andthere exists a PLTL-formula C such that B = 3C, or (ii)there exists a PLTL-formula C such that B = A = 2C.Definition 6. A completed Socratic transformation q =hQ1; : : : ; Qri is a Socratic proof of S ` A iff: (a) allthe constituents of Qn are successful, or (b) for each non-successful constituent  of Qn,  is a loop-generating for-mula and the loop generated by  contains a pre-state withan unfulfilled eventuality.The presented system is sound and complete. Proofs ofthese theorems involve construction of a canonical modelwith maximal consistent sets of formulae as its states.3 ExamplesIn the examples below by highlighting we indicate a for-mula which is analyzed at the current step. Double under-lining of a formula reflects that it is a state. The questionfollowing the one containing a state is obtained by state-prestate rule.Example 11. ?(`1 2p! p )2. ?( 2p `1 p)3. ?(p;2p `1 p)Example 21. ?(`1 2p!p )2. ?( 2p `1 p)3. ?(p;2p `1 p)4. ?( 2p `1 p)5. ?(p;2p `1 p)References[1] O. Grigoryev A. Basukoski, A. Bolotov andV. Shangin. Natural deduction system for linear timetemporal logic . Logical investigations, (13):71–95,2004.[2] A. Avron. The Method of Hypersequents in the ProofTheory of Propositional Non-Classical Logics, pages1–32. Logic: Foundations to Applications. ClarendonPress, 1996.[3] M. Finger, D. M. Gabbay, and M. Reynolds. Ad-vanced Tense Logic, volume 7 of Handbook of Philo-sophical Logic, pages 43–203. Springer, 2002.[4] Michael Fisher, Clare Dixon, and Martin Peim.Clausal temporal resolution. ACM Trans. Comput.Log., 2(1):12–56, 2001.[5] D. M. Gabbay, A. Pnueli, S. Shelah, and J. Stavi. Onthe temporal analysis of fairness. In 7th ACM Sympo-sium on Principles of Programming Languages, pages163–173, 1980.[6] A. Pnueli. The temporal logic of programs. In Pro-ceedings of the 18th Symposium on Foundations ofComputer Science, pages 46–57, 1977.[7] A. P. Sistla and E. M. Clarke. The complexity ofPropositional Linear Temporal Logic. Journal of theAssociation for Computing Machinery, 32(3):733–749, 1985.[8] A. Wis´niewski. Socratic Proofs. Journal of Philo-sophical Logic, 33(2):299–326, 2004.[9] A. Wis´niewski and V. Shangin. Socratic proofsfor quantifiers. Journal of Philosophical Logic,35(2):147–178, 2006.[10] A. Wis´niewski, G. Vanackere, and D. Leszczyn´ska.Socratic Proofs and Paraconsistency. A Case Study.Studia Logica, 80:431–466, 2005.[11] P. Wolper. The tableau method for temporal logic: Anoverview. Logique et Analyse, 28:119–136, 1985.

Socratic Proofs for Propositional Linear-Time Logic

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Socratic Proofs for Propositional Linear-Time Logic

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