Proceedings of the Twenty-Sixth International Joint Conference on Artificial Intelligence (IJCAI-17)
nanoCoP: Natural Non-clausal Theorem Proving
Jens Otten Department of Informatics, University of Oslo, Norway jeotten@ifi.uio.no
Abstract has the disjunctive normal form (quantifiers are eliminated) Most efficient fully automated theorem provers im- ¬P (a) ∨ (P (y)∧¬P (g(y))∧Q∧R) ∨ (P (y)∧¬P (g(y))∧Q∧¬R) plement proof search calculi that require the in- ∨ (P (y)∧¬P (g(y))∧¬Q∧R) ∨ (P (y)∧¬P (g(y))∧¬Q∧¬R) put formula to be in a clausal form, i.e. disjunc- ∨ P (g(g(a))) . tive or conjunctive normal form. The translation While the use of a clausal form technically simplifies the into clausal form introduces a significant overhead proof calculi and their implementations, it has some funda- to the proof search and modifies the structure of mental disadvantages. The standard translation into clausal the original formula. Translating a proof in clausal form as well as the definitional translation [Plaisted and form back into a more readable non-clausal proof Greenbaum, 1986; Eder, 1992], which introduces definitions of the original formula is not straightforward. This for subformulae, cause a significant overhead for the proof paper presents a non-clausal automated theorem search [Otten, 2010]. For example, the disjunctive normal prover for classical first-order logic. It is based on form of F#, in which some parts of the formula are copied, a non-clausal connection calculus and implemented has more than twice the size of the original formula. Further- with a few lines of Prolog code. Working entirely more, a translation into clausal form modifies the structure on the original structure of the input formula yields of the formula, hence, a translation of the clausal proof back not only a speed up of the proof search, but the re- into one of the original formula is not straightforward [Reis, sulting non-clausal proofs are also shorter. 2015]. On the other hand, proof search with more “natural” non-clausal calculi, such as sequent or standard tableau cal- 1 Introduction culi [Gentzen, 1935; Hahnle,¨ 2001] is less efficient. This paper describes the non-clausal prover nanoCoP for Automated Theorem Proving (ATP) in classical first-order classical first-order logic [Otten, 2016]. By performing the logic is a core research area in the field of Automated Rea- proof search on the original structure of the input formula, soning (see [Robinson and Voronkov, 2001] for an overview). it combines the advantages of more natural non-clausal cal- It is concerned with the question whether a conjecture H is culi with the efficiency of a goal-oriented connection-based a logical consequence of a given set of axioms G1,...,Gn, proof search. The prover is based on a non-clausal connec- written G1,...,Gn |= H, in which G1,...,Gn,H are first- tion calculus [Otten, 2011] (Section 2) that generalizes the order formulae. More specifically the objective in ATP is clausal connection calculus [Bibel, 1983; 1987] and is im- to find a proof for the validity of a given formula F .A plemented in a very compact way (Section 3). It follows formula F is valid if and only if it evaluates to true for all the lean methodology already used for the clausal connec- possible interpretation of its function and predicate symbols. tion prover leanCoP, whose minimal Prolog source code is According to the Deduction Theorem (for classical logic), shown in Figure 1. An experimental evaluation (Section 4) G1,...,Gn |= H holds if and only if there is a proof for indicates a solid performance of nanoCoP. the validity of the formula (G1 ∧ ... ∧ Gn) ⇒ H.A formal description of the proof search algorithm is usually specified prove(I,S):- \+member(scut,S) -> prove([-(#)],[],I,[],S) ; in form of a proof (search) calculus consisting of axioms and lit(#,C,_) -> prove(C,[-(#)],I,[],S). rules. Implementations of such proof calculi are called ATP prove(I,S):- member(comp(L),S), I=L -> prove(1,[]) ; (member(comp(_),S);retract(p)) -> J is I+1, prove(J,S). systems or (automated theorem) provers. prove([],_,_,_,_). Most efficient fully automated theorem provers implement prove([L|C],P,I,Q,S):- \+ (member(A,[L|C]), member(B,P), proof calculi that require the input formula to be in a clausal A==B), (-N=L;-L=N) -> ( member(D,Q), L==D ; member(E,P), unify_with_occurs_check(E,N) ; lit(N,F,H), form, i.e. disjunctive or conjunctive normal form. Formulae (H=g -> true ; length(P,K), K true ; that are not in clausal form are translated into clausal form in \+p -> assert(p), fail), prove(F,[L|P],I,Q,S) ), (member(cut,S) -> ! ; true), prove(C,P,I,[L|Q],S). a preprocessing step. For example, the formula F# P (a)∧(∀y(P (y)⇒P (g(y)))∨(Q⇒Q)∨(R⇒R)) ⇒ P (g(g(a))) Figure 1: Source code of the leanCoP 2.0 core prover
4924 Proceedings of the Twenty-Sixth International Joint Conference on Artificial Intelligence (IJCAI-17) 2 The Non-clausal Connection Calculus copy z }| { The standard notation for first-order formulae is used. Terms P (y)0 P (y0)0 (denoted by t) are built up from functions (f, g, h, i), con- P (g(y))1 P (g(y0))1 1 0 stants (a, b, c), and variables (x, y, z). An atomic formula (A) P (a) 0 1 P (g(g(a))) Q Q is built up from predicate symbols (P, Q, R, S) and terms; a (first-order) formula (F, G, H) is built up from atomic formu- R0 R1 lae, the connectives ¬, ∧, ∨, ⇒, and the first-order quantifiers ∀ and ∃.A literal L has the form A or ¬A. Its complement L is A if L is of the form ¬A; otherwise L is ¬L. Figure 2: Graphical representation of a non-clausal matrix and its A term substitution σ assigns terms to variables. A for- non-clausal connection proof mula in clausal form has the form ∃x1 ... ∃xn(C1 ∨...∨Cn), where each clause Ci is a conjunction of literals L1,...,Lmi . A clause C contains a literal L if and only if (iff) L ∈ C or It is represented as a set of clauses {C1,...,Cn}, called a C0 contains L for M 0 ∈ C, C0∈ M 0. A clause C is α-related (clausal) matrix.A polarity 0 or 1 is used to represent nega- L M 0 {C0,C00}⊆M 0 0 to a literal iff it contains a matrix with tion, i.e. literals of the form A and ¬A are represented by A C0,C00 C0 L C00 1 0 1 for clauses , such that contains and contains and A , respectively, A connection is a set {A ,A } of liter- C (or C=C00). A copy of the clause C in the matrix M is als with the same predicate symbol but different polarities. made by renaming all free variables in C. M[C1\C2] denotes In a non-clausal matrix, a clause consists of literals and 0 the matrix M, in which the clause C1 is replaced by C2. C (sub)matrices. Let F be a formula and pol be a polarity. The parent clause C M 0 ∈ C0 C ∈ M 0 M 0 pol pol is a of iff and for some . non-clausal matrix M(F ) of a formula F is a set of Let M be a matrix and P ath be a set of literals. C is an clauses, in which a clause is a set of literals and matrices, and extension clause (e-clause) of the matrix M with respect to a is defined according to Table 1. In Table 1, x∗ is a new vari- ∗ ∗ ∗ set of literals P ath iff either (a) C contains a literal of P ath, able, t is the skolem term f (x1, . . . , xn) in which f is a or (b) C is α-related to all literals of P ath occurring in M and new function symbol and x1, . . . , xn are the free variables in if C has a parent clause, it contains a literal of P ath. In the β- ∀xG or ∃xG. In G[x\t] all free occurrences of x in G are clause of C2 with respect to L2, denoted by β-clauseL (C2), replaced by t. The non-clausal matrix M(F ) of a formula 2 0 L2 and clauses that are α-related to L2 are deleted from C2. F is the matrix M(F ). In the graphical representation its The non-clausal connection calculus [Otten, 2011], which clauses are arranged horizontally, while the literals and ma- is sound and complete, is shown in Figure 3. Therein, M is trices of each clause are arranged vertically. For example, the a non-clausal matrix, C is a (subgoal) clause or ε (a special formula F# of Section 1 has the simplified (i.e. redundant empty symbol) and (the active) P ath is a set of literals or ε; brackets are removed) non-clausal matrix M# = M(F#): σ is a term substitution. A non-clausal connection proof of {{P (a)1}, {{{P (y)0,P (g(y))1}}, {{Q0}, {Q1}}, M is a non-clausal connection proof of ε, M, ε. {{R0}, {R1}}}, {P (g(g(a)))0}} . The analytic, i.e. bottom-up, proof search is carried out in the same way as in the clausal calculus. Additional backtrack- The graphical representation of the matrix M is depicted # ing might be required when selecting C1 in the decomposition in Figure 2. It already contains a clause copy with the fresh M 0 rule, but no backtracking is required when selecting 1. The variable y and represents a non-clausal connection proof us- rigid σ 0 , i.e. single, term substitution is calculated whenever a ing the term substitution σ with σ(y) = a and σ(y ) = g(a); connection is identified in a reduction or extension rule. On literals of each connection are connected with a line. formulae in clausal form, the non-clausal connection calculus Compared to the formal clausal connection calculus [Otten coincides with the clausal connection calculus. and Bibel, 2003], a decomposition rule is added to the non- clausal calculus and the extension rule is slightly modified. Axiom (A) {}, M, P ath type F pol M(F pol) pol pol atomic A {{A }} C2,M, {} Start (S) and C2 is copy of C1∈M α (¬G)pol M(G1−pol) ε, M, ε (G ∧ H)1 {{M(G1)}, {M(H1)}} 0 0 0 C, M, P ath∪{L2} (G ∨ H) {{M(G )}, {M(H )}} Reduction (R) and σ(L1)=σ(L2) C∪{L1}, M, P ath∪{L2} (G ⇒ H)0 {{M(G1)}, {M(H0)}} 0 0 0 β (G ∧ H) {{M(G ),M(H )}} C3,M[C1\C2], P ath∪{L1} C, M, P ath 1 1 1 Extension (E) (G ∨ H) {{M(G ),M(H )}} C∪{L1}, M, P ath (G ⇒ H)1 {{M(G0),M(H1)}} and C3:=β-clauseL2 (C2), C2 is copy of C1, C1 is e-clause γ (∀xG)1 M(G[x\x∗]1) of M wrt. P ath∪{L1}, C2 contains L2 with σ(L1)=σ(L2) (∃xG)0 M(G[x\x∗]0) δ (∀xG)0 M(G[x\t∗]0) C ∪ C1, M, P ath Decomposition (D) and C1∈M1 (∃xG)1 M(G[x\t∗]1) C∪{M1}, M, P ath
Table 1: The definition of the non-clausal matrix Figure 3: The non-clausal connection calculus
4925 Proceedings of the Twenty-Sixth International Joint Conference on Artificial Intelligence (IJCAI-17)
3 The nanoCoP Theorem Prover non-clausal connection calculus of Figure 3. Cla, Mat, and The implementation of the non-clausal connection calcu- Path represent the subgoal clause C, the (indexed) matrix lus of Figure 3 follows the lean methodology [Beckert and M and the (active) P ath. The term substitution σ is stored Posegga, 1995], which is already used for the clausal con- implicitly by Prolog. The indexed path PathI contains the nection prover leanCoP [Otten and Bibel, 2003]. It uses indices of all clauses and matrices that contain literals of very compact Prolog code to implement the basic calcu- Path; it is used for calculating extension clauses. The list lus and adds a few essential optimization techniques in Lem is used for the lemmata rule and contains all literals [ ] order to prune the search space. The resulting natural that have already been “solved” Otten, 2010 . This prove nonclausal connection prover nanoCoP is available under predicate succeeds iff there is a connection proof for the the GNU General Public License and can be downloaded at tuple Cla, Mat, Path with |Path| < PathLim. In this case Proof Mat http://www.leancop.de/nanocop/ . returns a connection proof. The input matrix has In a first step, the input formula F is translated into a non- to be stored in Prolog’s database as explained above. clausal (indexed) matrix M(F ) according to Table 1; redun- Finally, the last predicate prove_ec(ClaB,Cla1,Mat, dant brackets are removed [Otten, 2011]. Additionally, ev- PathI,ClaB1,Mat1) is used to calculate extension clauses. ery (sub-)clause (I,V ): C and (sub-)matrix J : M is marked Starting with the (largest possible) extension clause Cla1, its with a unique index I or J; clause C is also marked with a set β-clause ClaB, the current (indexed) matrix Mat, and the in- of variables V that are newly introduced in C but not in any dexed path PathI, it returns an appropriate extension clause subclause of C. Atomic formulae are represented by Prolog Cla, copies it into Mat and returns its β-clause ClaB1 and atoms, term variables by Prolog variables and the polarity 1 the new (indexed) matrix Mat1. by “-”. Sets, e.g. clauses and matrices, are represented by nanoCoP uses additional optimization techniques that are Prolog lists (representing multisets). For example, the matrix already used in the clausal connection prover leanCoP: regu- larity, lemmata, and restricted backtracking [Otten, 2010]. M# from Section 2 is represented by the Prolog term [(0ˆK)ˆ[]:[-(p(a))], (1ˆK)ˆ[]:[2ˆK:[(3ˆK)ˆ[Y]:[p(Y),-(p(g(Y)))]], % start rule 4ˆK:[(5ˆK)ˆ[]:[q],(6ˆK)ˆ[]:[-q]], prove(Mat,PathLim,Set,[(Iˆ0)ˆV:Cla1|Proof]) :- member((Iˆ0)ˆV:Cla,Mat), positiveC(Cla,Cla1), 7ˆK:[(8ˆK)ˆ[]:[r],(9ˆK)ˆ[]:[-r]]], prove(Cla1,Mat,[],[Iˆ0],PathLim,[],Set,Proof). (10ˆK)ˆ[]:[p(g(g(a)))]] prove(Mat,PathLim,Set,Proof) :- retract(pathlim) -> in which the Prolog variable K is used to enumerate clause ( member(comp(PathLim),Set) -> prove(Mat,1,[],Proof) ; copies. In the second step, the matrix M = M(F ) is written PathLim1 is PathLim+1, prove(Mat,PathLim1,Set,Proof) ) ; into Prolog’s database. For every literal Lit in M the fact member(comp(_),Set) -> prove(Mat,1,[],Proof). % axiom lit(Lit,ClaB,ClaC,Grnd) prove([],_,_,_,_,_,_,[]).
% decomposition rule is asserted into the database where ClaC ∈ M is the (largest) prove([J:Mat1|Cla],MI,Path,PI,PathLim,Lem,Set,Proof) :- !, clause in which Lit occurs and ClaB is the β-clause of ClaC member(Iˆ_:Cla1,Mat1), prove(Cla1,MI,Path,[I,J|PI],PathLim,Lem,Set,Proof1), with respect to Lit. Grnd is set to g if the smallest clause prove(Cla,MI,Path,PI,PathLim,Lem,Set,Proof2), in which Lit occurs is ground, i.e. does not contain any vari- append(Proof1,Proof2,Proof). ables; otherwise Grnd is set to n. Storing literals of M in the % reduction and extension rules prove([Lit|Cla],MI,Path,PI,PathLim,Lem,Set,Proof) :- database in this way is called lean Prolog technology [Otten, Proof=[[IˆV:[NegLit|ClaB1]|Proof1]|Proof2], 2010] and integrates the advantages of the Prolog technology \+ (member(LitC,[Lit|Cla]), member(LitP,Path), LitC==LitP), (-NegLit=Lit;-Lit=NegLit) -> approach [Stickel, 1988] into the lean theorem proving frame- ( member(LitL,Lem), Lit==LitL, ClaB1=[], Proof1=[] work. No other modifications or simplifications of the origi- ; member(NegL,Path), unify_with_occurs_check(NegL,NegLit), nal formula are done during these two preprocessing steps. ClaB1=[], Proof1=[] ; The nanoCoP source code is shown in Figure 4. As an lit(NegLit,ClaB,Cla1,Grnd1), optimization, it uses positive start clauses [Otten, 2011], cal- ( Grnd1=g -> true ; length(Path,K), K
4926 Proceedings of the Twenty-Sixth International Joint Conference on Artificial Intelligence (IJCAI-17) 4 Experimental Evaluation 5 Conclusion The following evaluations were conducted on a 3.4 GHz Formal reasoning is a fundamental task in mathematics, com- Xeon system with 4 GB of RAM running Linux 3.13.0 and puter science and many related fields. Since Frege published ECLiPSe Prolog 5.10. The CPU time limit for all proof at- the first formal calculus for first-order logic in his Begriffs- tempts was set 100 seconds. schrift [Frege, 1879], the development and implementation The following formula Fn is a slightly extended version of of efficient proof calculi has made significant progress. But n n n n the formula F# of Section 1, where f , g , h , and i are up to now, most —if not all— efficient fully automated the- abbreviations for n nested applications of these functions: orem provers for classical first-order logic translate the input formula into a clausal form. By using the standard translation n Fn ≡ P (a) ∧ ( ¬((Q(f (c)) ∧ ∀x(Q(f(x)) ⇒ Q(x))) ⇒ Q(c)) the size of the resulting formula can grow exponentially with ∨ ¬((R(hn(c)) ∧ ∀x(R(h(x))⇒R(x))) ⇒ R(c)) respect to the size of the original formula. Even a definitional ∨ ¬((S(in(c)) ∧ ∀x(S(i(x))⇒ S(x))) ⇒ S(c)) translation increases the size of the formula, which results in a ∨ ∀y(P (y) ⇒ P (g(y))) ) ⇒ ∃z P (gn(z)) . significant overhead for the proof search [Otten, 2010]. Both clausal form translations modify the structure of the formula, Table 2 shows the timings on this (valid) formula class making it difficult to translate the (clausal) proof back into a for n=10, n=30, and n=90 for the following provers: proof of the original formula. [ the lean (non-clausal) tableau prover leanTAP Beckert and This paper introduced nanoCoP, the first efficient non- ] [ Posegga, 1995 , the resolution prover Prover9 McCune, clausal connection prover for classical first-order logic. Us- ] [ ] 2005 , the superposition prover E Schulz, 2002 (using ing non-clausal matrices the proof search works directly on options “--auto --tptp3-format”), the clausal connec- the original structure of the input formula and, thus, avoids a [ ] tion prover leanCoP Otten and Bibel, 2003; Otten, 2010 , translation into any clausal form. This combines the advan- and nanoCoP. The leanCoP core prover with the stan- tages of more natural non-clausal sequent or tableau provers dard (“[nodef]”) and the definitional (“[def]”) translation into with the goal-oriented efficiency of connection provers. clausal form were tested. For nanoCoP restricted backtrack- Even though the non-clausal inferences introduce a slight ing was switched off (Set=[ ]). Times are given in seconds; overhead, nanoCoP outperforms both clausal form transla- the size of the returned proof tree, i.e. the number of nodes of tions of the leanCoP core prover on a large set of TPTP prob- the proof tree, is given in brackets. lems. About half of the returned non-clausal proofs are up to The last line of Table 2 shows the number of proved prob- 96% shorter than their clausal counterparts. lems of all 5051 first-order (FOF) problems in the TPTP li- nanoCoP returns a compact non-clausal connection proof. brary v3.7.0 [Sutcliffe, 2009]. For leanTAP, leanCoP, and A connection corresponds to a closed branch in the tableau nanoCoP, the required equality axioms were added in a pre- calculus [Hahnle,¨ 2001] or an axiom in the sequent calcu- processing step (which is included in the timings). The lus [Gentzen, 1935]. Hence, the translation into, e.g., a se- nanoCoP core prover performs significantly better than both quent proof is straightforward. The compact size of nanoCoP clausal form translations of the leanCoP core prover. makes it also a suitable tool for the development of verifiably Furthermore, 40%/51% of the proofs found by nanoCoP correct software [Otten and Bibel, 2017], as its correctness are shorter than those of leanCoP [nodef]/[def], respectively; can be proven much more easily than that of an ATP system as many of the problems in the TPTP library are “mostly” in a consisting of several thousand lines of source code. clausal form, 56%/47% of the proofs have the same size. The Only few research work investigates non-clausal connec- proofs of nanoCoP are up to 96%/74% shorter than those of tion calculi and implementations. Other approaches [An- the leanCoP versions [nodef]/[def], respectively. drews, 1981; Bibel, 1987; Hahnle¨ et al., 2004; Kreitz and The “full” leanCoP 2.2 prover, which additionally uses a Otten, 1999] work (efficiently) only for ground formulae or strategy scheduling [Otten, 2010], proves 1710 TPTP prob- their implementation is not available anymore [Issar, 1990]. lems. The version of nanoCoP using a restricted backtrack- Another important application of nanoCoP is its usage ing strategy, i.e. Set=[cut,comp(6)], was tested as well; it within non-classical logics, such as (first-order) intuitionistic proves 1485 TPTP problems. or modal logic, for which the use of a clausal form is either Tests have also shown that the classical version of the not desirable or not possible. Hence, future work includes the non-clausal connection prover JProver [Schmitt et al., 2001], combination of the non-clausal approach with the prefix (uni- which is the only published non-clausal connection prover so fication) technique for some non-classical logics, as already far, has a lower performance than leanTAP. done for leanCoP [Otten, 2008; 2014]. In order to improve performance, further optimization techniques need to be in- leanTAP Prover9 E leanCoP 2.2 nanoCoP tegrated into nanoCoP, such as strategy scheduling [Otten, 2.3 2009-02A 1.9 [nodef] [def] [ ] 2010], learning [Kaliszyk and Urban, 2015] or variable split- F10 0.17 (128) – 1.22 (2916) – – 0.09 (45) ting [Antonsen and Waaler, 2007]. F30 – – 84.57 (57628) – – 0.12 (125) F90 ––––– 0.42 (365) TPTP 404 1611 2782 1134 1065 1232 Acknowledgements The author would like to thank Wolfgang Bibel for his helpful Table 2: Results on formula class Fn and the TPTP library comments on a preliminary version of this paper.
4927 Proceedings of the Twenty-Sixth International Joint Conference on Artificial Intelligence (IJCAI-17)
References [Otten, 2008] Jens Otten. leanCoP 2.0 and ileanCoP 1.2: High performance lean theorem proving in classical and [Andrews, 1981] Peter B. Andrews. Theorem proving via intuitionistic logic. In Alessandro Armando, Peter Baum- general matings. J. ACM, 28(2):193–214, 1981. gartner, and Gilles Dowek, editors, IJCAR 2008, volume [Antonsen and Waaler, 2007] Roger Antonsen and Arild 5195 of Lecture Notes in Artificial Intelligence, pages Waaler. Liberalized variable splitting. J. Autom. Reason- 283–291. Springer, 2008. ing, 38(1–3):3–30, 2007. [Otten, 2010] Jens Otten. Restricting backtracking in con- [Beckert and Posegga, 1995] Bernhard Beckert and Joachim nection calculi. AI Commun., 23(2–3):159–182, 2010. Posegga. leanTAP: Lean tableau-based deduction. J. Au- [Otten, 2011] Jens Otten. A non-clausal connection calculus. tom. Reasoning, 15(3):339–358, 1995. In Kai Brunnler¨ and George Metcalfe, editors, TABLEAUX [Bibel, 1983] Wolfgang Bibel. Matings in matrices. Com- 2011, volume 6793 of Lecture Notes in Artificial Intelli- mun. ACM, 26(11):844–852, 1983. gence, pages 226–241. Springer, 2011. [Bibel, 1987] Wolfgang Bibel. Automated theorem proving. [Otten, 2014] Jens Otten. MleanCoP: A connection prover Artificial intelligence. F. Vieweg und Sohn, Wiesbaden, for first-order modal logic. In Stephane´ Demri, Deepak 2nd edition, 1987. Kapur, and Christoph Weidenbach, editors, IJCAR 2014, volume 8562 of Lecture Notes in Artificial Intelligence, [Eder, 1992] Elmar Eder. Relative Complexities of First Or- pages 269–276. Springer, 2014. der Calculi. Vieweg, Braunschweig, 1992. [Otten, 2016] Jens Otten. nanoCoP: A non-clausal connec- [Frege, 1879] Gottlob Frege. Begriffsschrift: Eine der tion prover. In Nicola Olivetti and Ashish Tiwari, editors, arithmetischen nachgebildete Formelsprache des reinen IJCAR 2016, volume 9706 of Lecture Notes in Artificial Denkens. L. Nebert, Halle, 1879. Intelligence, pages 302–312. Springer, 2016. [Gentzen, 1935] Gerhard Gentzen. Untersuchungen uber¨ das [Plaisted and Greenbaum, 1986] David A. Plaisted and Logische Schließen. Mathematische Zeitschrift, 39:176– Steven Greenbaum. A structure-preserving clause form 210, 405–431, 1935. translation. J. Symbolic Computation, 2(3):293–304, [Hahnle¨ et al., 2004] Reiner Hahnle,¨ Neil. V. Murray, and 1986. Erik Rosenthal. Linearity and regularity with negation nor- [Reis, 2015] Giselle Reis. Importing SMT and connection mal form. Theoretical Computer Science, 328:325–354, proofs as expansion trees. In Proof Exchange for Theorem 2004. Proving (PxTP), volume 186 of EPTCS, pages 3–10, 2015. [Hahnle,¨ 2001] Reiner Hahnle.¨ Tableaux and related meth- [Robinson and Voronkov, 2001] John Alan Robinson and ods. In Alan Robinson and Andrei Voronkov, editors, Andrei Voronkov, editors. Handbook of Automated Rea- Handbook of Automated Reasoning, volume I, chapter 3, soning. Elsevier Science Publishers, Amsterdam, 2001. pages 101–178. Elsevier Science B.V., 2001. [Schmitt et al., 2001] Stephan Schmitt, Lori Lorigo, [Issar, 1990] Sunil Issar. Path-focused duplication: A search Christoph Kreitz, and Aleksey Nogin. JProver: Integrat- procedure for general matings. In T. S. W. Dietterich, edi- ing connection-based theorem proving into interactive tor, AAAI-90, pages 221–226. MIT Press, 1990. proof assistants. In Rajeev Gore,´ Alexander Leitsch, and Tobias Nipkow, editors, IJCAR 2001, volume 2083 of [Kaliszyk and Urban, 2015] Cezary Kaliszyk and Josef Ur- Lecture Notes in Artificial Intelligence, pages 421–426. ban. FeMaLeCoP: Fairly efficient machine learning Springer, 2001. connection prover. In Martin Davis, Ansgar Fehnker, Annabelle McIver, and Andrei Voronkov, editors, LPAR- [Schulz, 2002] Stephan Schulz. E – a brainiac theorem 20, volume 9450 of Lecture Notes in Artificial Intelligence, prover. AI Commun., 15(2–3):111–126, 2002. pages 88–96. Springer, 2015. [Stickel, 1988] Mark E. Stickel. A PROLOG technology the- [Kreitz and Otten, 1999] Christoph Kreitz and Jens Otten. orem prover: Implementation by an extended PROLOG Connection-based theorem proving in classical and non- compiler. J. Autom. Reasoning, 4(4):353–380, 1988. classical logics. J. UCS, 5(3):88–112, 1999. [Sutcliffe, 2009] Geoff Sutcliffe. The TPTP problem library [McCune, 2005] William McCune. Release of Prover9. and associated infrastructure: The FOF and CNF parts, 2005. Mile high conference on quasigroups, loops and v3.5.0. J. Autom. Reasoning, 43(4):337–362, 2009. nonassociative systems. [Otten and Bibel, 2003] Jens Otten and Wolfgang Bibel. leanCoP: lean connection-based theorem proving. Jour- nal of Symbolic Computation, 36(1–2):139–161, 2003. [Otten and Bibel, 2017] Jens Otten and Wolfgang Bibel. Ad- vances in connection-based automated theorem proving. In J. Bowen, M. Hinchey, and E.-R. Olderog, editors, Provably Correct Systems. Springer, 2017.
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