Using Gentzen system techniques with existing automated theorem provers and rewriting tools
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1 Using Gentzen system techniques with existing automated theorem provers and rewriting tools Peter Jipsen Chapman Univ, Orange, CA March 6, 2008 Peter Jipsen (Chapman Univ, Orange, CA) Implementing Gentzen systems March 6, / 17
2 Outline Background Gentzen systems Term rewriting systems Maude Theorem provers Prover9 Peter Jipsen (Chapman Univ, Orange, CA) Implementing Gentzen systems March 6, / 17
3 Outline Background Gentzen systems Term rewriting systems Maude Theorem provers Prover9 Peter Jipsen (Chapman Univ, Orange, CA) Implementing Gentzen systems March 6, / 17
4 Outline Background Gentzen systems Term rewriting systems Maude Theorem provers Prover9 Peter Jipsen (Chapman Univ, Orange, CA) Implementing Gentzen systems March 6, / 17
5 Outline Background Gentzen systems Term rewriting systems Maude Theorem provers Prover9 Peter Jipsen (Chapman Univ, Orange, CA) Implementing Gentzen systems March 6, / 17
6 Outline Background Gentzen systems Term rewriting systems Maude Theorem provers Prover9 Peter Jipsen (Chapman Univ, Orange, CA) Implementing Gentzen systems March 6, / 17
7 Outline Background Gentzen systems Term rewriting systems Maude Theorem provers Prover9 Peter Jipsen (Chapman Univ, Orange, CA) Implementing Gentzen systems March 6, / 17
8 Background In the late 80s I learned about term-rewriting from the algebraic perspective: normal form, Church-Rosser property (confluence), termination, critical pairs, Knuth-Bendix completion... Applies to (free) monoids, groups, boolean algebras, (some) finitely presented monoids, groups,... Used in algebraic specification languages, in optimization of functional programs, in equational theorem provers,... Peter Jipsen (Chapman Univ, Orange, CA) Implementing Gentzen systems March 6, / 17
9 Background In the late 90s I learned about sequents and Gentzen systems from the logical perspective: Developed for classical logic (LK), intuitionistic logic (LJ), linear logic (MALL, IMALL),... For substructural logics in general Gentzen systems provide decision procedures for many logics Related to natural deduction, tableau methods, resolution theorem provers,... Peter Jipsen (Chapman Univ, Orange, CA) Implementing Gentzen systems March 6, / 17
10 Aims of this talk 1. Show how a standard term-rewriting system (e.g. Maude2) can be used to implement a Gentzen calculus = program to decide for RL, CRL, IRL, ICRL, InRL,... equations extend this approach to Hypersequent calculi 2. Adding sequent rules to an automated theorem prover (e.g. Prover9) can improve its performance Peter Jipsen (Chapman Univ, Orange, CA) Implementing Gentzen systems March 6, / 17
11 Term-rewriting Given a term t and an equation r = s, a rewrite step consists of matching r to a subterm of t i.e. find a substitution σ such that σ(r) is a subterm of t then replace this subterm with σ(s) to get a new term t. This is denoted by t t or t r=s t Note that if r = s is an identity in a theory, then t = t holds in this theory. E.g. x = (x y) x is used to rewrite x x ((x y) x) x then (y x) x = x is used to rewrite ((x y) x) x x Peter Jipsen (Chapman Univ, Orange, CA) Implementing Gentzen systems March 6, / 17
12 Term-rewriting tools A term-rewriting tool is a program that implements the process of term-rewriting for a given algebraic signature Takes a set of rewrite rules (i.e. equations r = s) and a term t as input and applies the rules repeatedly (from left to right) and fairly to t i.e. the tool computes the transitive closure of t t This can be a very efficient way of doing calculations on a computer The set of rewrite rules is the program, the rewriting tool is the interpreter that runs the program The process is easily parallelized and distributed over many processors Dozens of term-rewriting systems have been developed, e.g. for prototyping algebraic specifications, implementing model checkers,... Peter Jipsen (Chapman Univ, Orange, CA) Implementing Gentzen systems March 6, / 17
13 Maude Maude is an order-sorted conditional term-rewriting tool with search capabilities Developed by SRI International and University of Illinois at Champaign Urbana Open source (GPL, free download), well documented (450 page manual) Efficient and flexible, allows the use of convenient syntax fmod DLAT-TERMS is sorts Term. ops V ˆ : Term Term -> Term [assoc comm]. eq (x V y) ˆ x = x. eq (x ˆ y) V x = x. eq x V (y ˆ z) = (x V y) ˆ (x V z). endfm Peter Jipsen (Chapman Univ, Orange, CA) Implementing Gentzen systems March 6, / 17
14 Gentzen sequent calculi formulas φ, ψ, θ, sequences of formulas Γ,, Σ, sequents Γ Σ sequent rules e.g. Γ, φ, Σ Γ, ψ, Σ ( l) Γ, φ ψ, Σ Main idea is to view sequent rules as rewrite rules (upwards): (Γ, φ ψ, Σ) (Γ, φ, Σ Γ, ψ, Σ) Prefer to view this algebraically (not essential): u(x y)v z (uxv z uyv z) Here u, v, x, y, z can be instantiated with arbitrary terms uv is an associative operation symbol Peter Jipsen (Chapman Univ, Orange, CA) Implementing Gentzen systems March 6, / 17
15 Deciding equations in RL RL is short for Residuated Lattice (lattice with monoid operation + residuals) Suppose someone has a huge RL-term t and asks if t = 1 is true in RL Implement the cut-free sequent calculus for RL. But how? On my homepage is a JavaScript implementation. But is it correct? Efficient? Rather use the sequent rules as rewrite rules and write a short Maude module that is easy to check Same approach works for CRL, IRL, InCRL, FL, FL e, FL ew,... Peter Jipsen (Chapman Univ, Orange, CA) Implementing Gentzen systems March 6, / 17
16 Deciding equations in RL RL is short for Residuated Lattice (lattice with monoid operation + residuals) Suppose someone has a huge RL-term t and asks if t = 1 is true in RL Implement the cut-free sequent calculus for RL. But how? On my homepage is a JavaScript implementation. But is it correct? Efficient? Rather use the sequent rules as rewrite rules and write a short Maude module that is easy to check Same approach works for CRL, IRL, InCRL, FL, FL e, FL ew,... Peter Jipsen (Chapman Univ, Orange, CA) Implementing Gentzen systems March 6, / 17
17 Deciding equations in RL RL is short for Residuated Lattice (lattice with monoid operation + residuals) Suppose someone has a huge RL-term t and asks if t = 1 is true in RL Implement the cut-free sequent calculus for RL. But how? On my homepage is a JavaScript implementation. But is it correct? Efficient? Rather use the sequent rules as rewrite rules and write a short Maude module that is easy to check Same approach works for CRL, IRL, InCRL, FL, FL e, FL ew,... Peter Jipsen (Chapman Univ, Orange, CA) Implementing Gentzen systems March 6, / 17
18 Deciding equations in RL RL is short for Residuated Lattice (lattice with monoid operation + residuals) Suppose someone has a huge RL-term t and asks if t = 1 is true in RL Implement the cut-free sequent calculus for RL. But how? On my homepage is a JavaScript implementation. But is it correct? Efficient? Rather use the sequent rules as rewrite rules and write a short Maude module that is easy to check Same approach works for CRL, IRL, InCRL, FL, FL e, FL ew,... Peter Jipsen (Chapman Univ, Orange, CA) Implementing Gentzen systems March 6, / 17
19 Deciding equations in RL RL is short for Residuated Lattice (lattice with monoid operation + residuals) Suppose someone has a huge RL-term t and asks if t = 1 is true in RL Implement the cut-free sequent calculus for RL. But how? On my homepage is a JavaScript implementation. But is it correct? Efficient? Rather use the sequent rules as rewrite rules and write a short Maude module that is easy to check Same approach works for CRL, IRL, InCRL, FL, FL e, FL ew,... Peter Jipsen (Chapman Univ, Orange, CA) Implementing Gentzen systems March 6, / 17
20 Demo of Gentzen systems implemented in Maude The files boolean-algebra.maude and RL-sequent.maude (and other files) can be found at jipsen/maude/files Peter Jipsen (Chapman Univ, Orange, CA) Implementing Gentzen systems March 6, / 17
21 Deciding equations in BA BA is short for Boolean algebras Suppose someone has a huge BA-term t and asks if t = 1 is true in BA Implement the sequent calculus for LK? Rather use an off-the-shelf SAT-solver (tests satisfiability of Boolean formulas) Highly optimized, used for hardware verification Same approach works for testing whether an equation holds in a finitely generated variety and for checking tautologies in propositional logics with a finite matrix of truth values Peter Jipsen (Chapman Univ, Orange, CA) Implementing Gentzen systems March 6, / 17
22 Deciding equations in BA BA is short for Boolean algebras Suppose someone has a huge BA-term t and asks if t = 1 is true in BA Implement the sequent calculus for LK? Rather use an off-the-shelf SAT-solver (tests satisfiability of Boolean formulas) Highly optimized, used for hardware verification Same approach works for testing whether an equation holds in a finitely generated variety and for checking tautologies in propositional logics with a finite matrix of truth values Peter Jipsen (Chapman Univ, Orange, CA) Implementing Gentzen systems March 6, / 17
23 Deciding equations in BA BA is short for Boolean algebras Suppose someone has a huge BA-term t and asks if t = 1 is true in BA Implement the sequent calculus for LK? Rather use an off-the-shelf SAT-solver (tests satisfiability of Boolean formulas) Highly optimized, used for hardware verification Same approach works for testing whether an equation holds in a finitely generated variety and for checking tautologies in propositional logics with a finite matrix of truth values Peter Jipsen (Chapman Univ, Orange, CA) Implementing Gentzen systems March 6, / 17
24 Deciding equations in BA BA is short for Boolean algebras Suppose someone has a huge BA-term t and asks if t = 1 is true in BA Implement the sequent calculus for LK? Rather use an off-the-shelf SAT-solver (tests satisfiability of Boolean formulas) Highly optimized, used for hardware verification Same approach works for testing whether an equation holds in a finitely generated variety and for checking tautologies in propositional logics with a finite matrix of truth values Peter Jipsen (Chapman Univ, Orange, CA) Implementing Gentzen systems March 6, / 17
25 Deciding equations in BA BA is short for Boolean algebras Suppose someone has a huge BA-term t and asks if t = 1 is true in BA Implement the sequent calculus for LK? Rather use an off-the-shelf SAT-solver (tests satisfiability of Boolean formulas) Highly optimized, used for hardware verification Same approach works for testing whether an equation holds in a finitely generated variety and for checking tautologies in propositional logics with a finite matrix of truth values Peter Jipsen (Chapman Univ, Orange, CA) Implementing Gentzen systems March 6, / 17
26 Deciding equations in BA BA is short for Boolean algebras Suppose someone has a huge BA-term t and asks if t = 1 is true in BA Implement the sequent calculus for LK? Rather use an off-the-shelf SAT-solver (tests satisfiability of Boolean formulas) Highly optimized, used for hardware verification Same approach works for testing whether an equation holds in a finitely generated variety and for checking tautologies in propositional logics with a finite matrix of truth values Peter Jipsen (Chapman Univ, Orange, CA) Implementing Gentzen systems March 6, / 17
27 Deciding equations in abelian l-groups Ab-LG is short for abelian lattice ordered groups Suppose someone has a huge Ab-LG-term t and asks if t = 1 is true in Ab-LG Implement the hypersequent calculus for abelian logic (Metcalfe-Olivetti-Gabbay 2004) Can be done with a term rewriting tool Alternatively use a SMT-solver (symbolic model checking modulo theories) Need to compare these approaches to see what works better in practise Same works whenever the logic has a terminating hypersequent calculus (product logic, MV-algebras, linear Heyting algebras,...) Peter Jipsen (Chapman Univ, Orange, CA) Implementing Gentzen systems March 6, / 17
28 Deciding equations in abelian l-groups Ab-LG is short for abelian lattice ordered groups Suppose someone has a huge Ab-LG-term t and asks if t = 1 is true in Ab-LG Implement the hypersequent calculus for abelian logic (Metcalfe-Olivetti-Gabbay 2004) Can be done with a term rewriting tool Alternatively use a SMT-solver (symbolic model checking modulo theories) Need to compare these approaches to see what works better in practise Same works whenever the logic has a terminating hypersequent calculus (product logic, MV-algebras, linear Heyting algebras,...) Peter Jipsen (Chapman Univ, Orange, CA) Implementing Gentzen systems March 6, / 17
29 Deciding equations in abelian l-groups Ab-LG is short for abelian lattice ordered groups Suppose someone has a huge Ab-LG-term t and asks if t = 1 is true in Ab-LG Implement the hypersequent calculus for abelian logic (Metcalfe-Olivetti-Gabbay 2004) Can be done with a term rewriting tool Alternatively use a SMT-solver (symbolic model checking modulo theories) Need to compare these approaches to see what works better in practise Same works whenever the logic has a terminating hypersequent calculus (product logic, MV-algebras, linear Heyting algebras,...) Peter Jipsen (Chapman Univ, Orange, CA) Implementing Gentzen systems March 6, / 17
30 Deciding equations in abelian l-groups Ab-LG is short for abelian lattice ordered groups Suppose someone has a huge Ab-LG-term t and asks if t = 1 is true in Ab-LG Implement the hypersequent calculus for abelian logic (Metcalfe-Olivetti-Gabbay 2004) Can be done with a term rewriting tool Alternatively use a SMT-solver (symbolic model checking modulo theories) Need to compare these approaches to see what works better in practise Same works whenever the logic has a terminating hypersequent calculus (product logic, MV-algebras, linear Heyting algebras,...) Peter Jipsen (Chapman Univ, Orange, CA) Implementing Gentzen systems March 6, / 17
31 Deciding equations in abelian l-groups Ab-LG is short for abelian lattice ordered groups Suppose someone has a huge Ab-LG-term t and asks if t = 1 is true in Ab-LG Implement the hypersequent calculus for abelian logic (Metcalfe-Olivetti-Gabbay 2004) Can be done with a term rewriting tool Alternatively use a SMT-solver (symbolic model checking modulo theories) Need to compare these approaches to see what works better in practise Same works whenever the logic has a terminating hypersequent calculus (product logic, MV-algebras, linear Heyting algebras,...) Peter Jipsen (Chapman Univ, Orange, CA) Implementing Gentzen systems March 6, / 17
32 Deciding equations in abelian l-groups Ab-LG is short for abelian lattice ordered groups Suppose someone has a huge Ab-LG-term t and asks if t = 1 is true in Ab-LG Implement the hypersequent calculus for abelian logic (Metcalfe-Olivetti-Gabbay 2004) Can be done with a term rewriting tool Alternatively use a SMT-solver (symbolic model checking modulo theories) Need to compare these approaches to see what works better in practise Same works whenever the logic has a terminating hypersequent calculus (product logic, MV-algebras, linear Heyting algebras,...) Peter Jipsen (Chapman Univ, Orange, CA) Implementing Gentzen systems March 6, / 17
33 Deciding equations in commutative CGBL CGBL is short for commutative generalized basic logic algebras = RL with xy = yx and divisibility: x y x = y(y\x) = (x/y)y Suppose someone has a CGBL-term t and asks if t = 1 is true in CGBL Theorem (J - Montagna 2008) The varieties ICGBL and CGBL have the finite model property (FEP), hence their universal theory is decidable Currently there is no reasonable proof procedure Can use an automated theorem prover and finite model finder E.g. Prover9 and Mace4 Same works (in principle) whenever we have FEP or FMP but not really effective Peter Jipsen (Chapman Univ, Orange, CA) Implementing Gentzen systems March 6, / 17
34 Deciding equations in commutative CGBL CGBL is short for commutative generalized basic logic algebras = RL with xy = yx and divisibility: x y x = y(y\x) = (x/y)y Suppose someone has a CGBL-term t and asks if t = 1 is true in CGBL Theorem (J - Montagna 2008) The varieties ICGBL and CGBL have the finite model property (FEP), hence their universal theory is decidable Currently there is no reasonable proof procedure Can use an automated theorem prover and finite model finder E.g. Prover9 and Mace4 Same works (in principle) whenever we have FEP or FMP but not really effective Peter Jipsen (Chapman Univ, Orange, CA) Implementing Gentzen systems March 6, / 17
35 Deciding equations in commutative CGBL CGBL is short for commutative generalized basic logic algebras = RL with xy = yx and divisibility: x y x = y(y\x) = (x/y)y Suppose someone has a CGBL-term t and asks if t = 1 is true in CGBL Theorem (J - Montagna 2008) The varieties ICGBL and CGBL have the finite model property (FEP), hence their universal theory is decidable Currently there is no reasonable proof procedure Can use an automated theorem prover and finite model finder E.g. Prover9 and Mace4 Same works (in principle) whenever we have FEP or FMP but not really effective Peter Jipsen (Chapman Univ, Orange, CA) Implementing Gentzen systems March 6, / 17
36 Deciding equations in commutative CGBL CGBL is short for commutative generalized basic logic algebras = RL with xy = yx and divisibility: x y x = y(y\x) = (x/y)y Suppose someone has a CGBL-term t and asks if t = 1 is true in CGBL Theorem (J - Montagna 2008) The varieties ICGBL and CGBL have the finite model property (FEP), hence their universal theory is decidable Currently there is no reasonable proof procedure Can use an automated theorem prover and finite model finder E.g. Prover9 and Mace4 Same works (in principle) whenever we have FEP or FMP but not really effective Peter Jipsen (Chapman Univ, Orange, CA) Implementing Gentzen systems March 6, / 17
37 Deciding equations in commutative CGBL CGBL is short for commutative generalized basic logic algebras = RL with xy = yx and divisibility: x y x = y(y\x) = (x/y)y Suppose someone has a CGBL-term t and asks if t = 1 is true in CGBL Theorem (J - Montagna 2008) The varieties ICGBL and CGBL have the finite model property (FEP), hence their universal theory is decidable Currently there is no reasonable proof procedure Can use an automated theorem prover and finite model finder E.g. Prover9 and Mace4 Same works (in principle) whenever we have FEP or FMP but not really effective Peter Jipsen (Chapman Univ, Orange, CA) Implementing Gentzen systems March 6, / 17
38 Deciding equations in commutative CGBL CGBL is short for commutative generalized basic logic algebras = RL with xy = yx and divisibility: x y x = y(y\x) = (x/y)y Suppose someone has a CGBL-term t and asks if t = 1 is true in CGBL Theorem (J - Montagna 2008) The varieties ICGBL and CGBL have the finite model property (FEP), hence their universal theory is decidable Currently there is no reasonable proof procedure Can use an automated theorem prover and finite model finder E.g. Prover9 and Mace4 Same works (in principle) whenever we have FEP or FMP but not really effective Peter Jipsen (Chapman Univ, Orange, CA) Implementing Gentzen systems March 6, / 17
39 Deciding equations in GBL? GBL is short for generalized basic logic algebras Suppose someone has a GBL-term t and asks if t = 1 is true in GBL Currently don t know if GBL has a decidable equational theory Can still use an automated theorem prover and finite model finder but there are equations that have no finite counter model E.g. every finite GBL-algebra is commutative [J - Montagna 2006] More effective to send t to Nick Galatos Peter Jipsen (Chapman Univ, Orange, CA) Implementing Gentzen systems March 6, / 17
40 Deciding equations in GBL? GBL is short for generalized basic logic algebras Suppose someone has a GBL-term t and asks if t = 1 is true in GBL Currently don t know if GBL has a decidable equational theory Can still use an automated theorem prover and finite model finder but there are equations that have no finite counter model E.g. every finite GBL-algebra is commutative [J - Montagna 2006] More effective to send t to Nick Galatos Peter Jipsen (Chapman Univ, Orange, CA) Implementing Gentzen systems March 6, / 17
41 Deciding equations in GBL? GBL is short for generalized basic logic algebras Suppose someone has a GBL-term t and asks if t = 1 is true in GBL Currently don t know if GBL has a decidable equational theory Can still use an automated theorem prover and finite model finder but there are equations that have no finite counter model E.g. every finite GBL-algebra is commutative [J - Montagna 2006] More effective to send t to Nick Galatos Peter Jipsen (Chapman Univ, Orange, CA) Implementing Gentzen systems March 6, / 17
42 Deciding equations in GBL? GBL is short for generalized basic logic algebras Suppose someone has a GBL-term t and asks if t = 1 is true in GBL Currently don t know if GBL has a decidable equational theory Can still use an automated theorem prover and finite model finder but there are equations that have no finite counter model E.g. every finite GBL-algebra is commutative [J - Montagna 2006] More effective to send t to Nick Galatos Peter Jipsen (Chapman Univ, Orange, CA) Implementing Gentzen systems March 6, / 17
43 Deciding equations in GBL? GBL is short for generalized basic logic algebras Suppose someone has a GBL-term t and asks if t = 1 is true in GBL Currently don t know if GBL has a decidable equational theory Can still use an automated theorem prover and finite model finder but there are equations that have no finite counter model E.g. every finite GBL-algebra is commutative [J - Montagna 2006] More effective to send t to Nick Galatos Peter Jipsen (Chapman Univ, Orange, CA) Implementing Gentzen systems March 6, / 17
44 Questions What algorithm should be used to decide quasi-equations in CGBL? Theorem (J - Montagna 2008) The variety of GBL-algebras has undecidable quasiequational theory Is the equational theory of GBL decidable? Is there a cut-free sequent (or hypersequent) system for CGBL (or GBL)? Peter Jipsen (Chapman Univ, Orange, CA) Implementing Gentzen systems March 6, / 17
45 Questions What algorithm should be used to decide quasi-equations in CGBL? Theorem (J - Montagna 2008) The variety of GBL-algebras has undecidable quasiequational theory Is the equational theory of GBL decidable? Is there a cut-free sequent (or hypersequent) system for CGBL (or GBL)? Peter Jipsen (Chapman Univ, Orange, CA) Implementing Gentzen systems March 6, / 17
46 Questions What algorithm should be used to decide quasi-equations in CGBL? Theorem (J - Montagna 2008) The variety of GBL-algebras has undecidable quasiequational theory Is the equational theory of GBL decidable? Is there a cut-free sequent (or hypersequent) system for CGBL (or GBL)? Peter Jipsen (Chapman Univ, Orange, CA) Implementing Gentzen systems March 6, / 17
47 Questions What algorithm should be used to decide quasi-equations in CGBL? Theorem (J - Montagna 2008) The variety of GBL-algebras has undecidable quasiequational theory Is the equational theory of GBL decidable? Is there a cut-free sequent (or hypersequent) system for CGBL (or GBL)? Peter Jipsen (Chapman Univ, Orange, CA) Implementing Gentzen systems March 6, / 17
48 Questions What algorithm should be used to decide quasi-equations in CGBL? Theorem (J - Montagna 2008) The variety of GBL-algebras has undecidable quasiequational theory Is the equational theory of GBL decidable? Is there a cut-free sequent (or hypersequent) system for CGBL (or GBL)? Peter Jipsen (Chapman Univ, Orange, CA) Implementing Gentzen systems March 6, / 17
49 Questions What algorithm should be used to decide quasi-equations in CGBL? Theorem (J - Montagna 2008) The variety of GBL-algebras has undecidable quasiequational theory Is the equational theory of GBL decidable? Is there a cut-free sequent (or hypersequent) system for CGBL (or GBL)? Peter Jipsen (Chapman Univ, Orange, CA) Implementing Gentzen systems March 6, / 17
50 References [N. Galatos, P. Jipsen, T. Kowalski, H. Ono, 2007] Residuated Lattices: An algebraic glimpse at substrutural logics, Studies in Logic, vol 151, Elsevier, 2007, xxi+509 pp [P. Jipsen, F. Montagna, 2006] On the structure of generalized BL-algebras, Algebra Universalis 55 (2006), [P. Jipsen, F. Montagna, 2008] The Blok-Ferreirim theorem for normal GBL-algebras and its application, Algebra Universalis, to appear [P. Jipsen, C. Tsinakis, 2002] A survey on residuated lattices, in: J. Martinez (Ed.), Ordered Algebraic Structures, Kluwer, 2002, Peter Jipsen (Chapman Univ, Orange, CA) Implementing Gentzen systems March 6, / 17
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