## An Implementation of G4ip in Pizza

Warning: This page is now rather old! While you might still be interested in the algorithms, Robert Macdonald reported that Pizza and the current Java implementation (version 1.3.0) do not work together. This means you need to install an older Java version if you want to recompile the files given below. I am happy to answer all question concerning the prover, but be aware that currently for any kind of Java stuff I am using MLJ, which as of writing this note has not yet been made available for the general audience (maybe in the future also its OCaml equivalent). So I am not very fluent in Pizza anymore. Update Pizza development is continued and starting from version 0.40 it should work with recent Java implementations.

#### Introduction

A convenient representation of intuitionistic logic is Getzen's sequent calculus LJ (also G1i). A sequent of LJ can be proved by applying inference rules until one reaches axioms, or can make no further progress in which case one must backtrack or even abandon the search. Unfortunately an interpreter for LJ using this depth-first strategy cannot guarantee termination of the proof search. Several modifications can be made to LJ's inference rules without loss of soundness and completeness. As result an efficient depth-first proof search can be designed for the propositional fragment of intuitionistic logic. The name G4ip has been assigned to the corresponding calculus in [Troelstra and Schwichtenberg, 1996]. This calculus is also known as LJT which has been studied thoroughly in [Dyckhoff, 1992]. The inference rules of G4ip are given here.

It is not very complicated to implement an interpreter for G4ip using a logic programming language (backtracking is directly supported within the language). Our first implementation is written in the logic programming language Lambda Prolog and can be found here. Another implementation by Hodas and Miller written in Lolli can be found here (see [Hodas and Miller, 1994]). These are simple and straightforward implementations of G4ip's rules. On the other hand it seems that imperative languages need a rather high overhead of code when implementing a logic calculus. For example choice points are usually implemented with stacks. We shall demonstrate the implementation technique of success continuations which provides an equally simple method for implementing logic calculi in imperative languages. This technique is not new: it has been introduced in [Carlsson, 1984]. This paper presents a rather technical implementation of Prolog in LISP. Later an excellent paper, [Elliot and Pfenning, 1991], appeared which describes a full-fledged implementation of Lambda Prolog in SML. We demonstrate the technique of success continuations for G4ip in Pizza.

Pizza is an object-oriented programming language and an attractive extension of Java. Although Pizza is a superset of Java, Pizza programs can be translated into Java or compiled into ordinary Java Byte Code (see [Odersky and Wadler, 1997] for a technical introduction to Pizza). We make use of the following two new features of Pizza:

• higher-order functions, i.e. functions may be passed as parameters or returned from methods,
• class cases and pattern matching: this allows much simpler and more readable code.
These features are not directly present in Java, but Pizza makes them accessible by translating them into Java. Pizza provides the programmer with the same extensive libraries for graphic and network applications as Java. The higher-order functions are essential for the technique of success continuations. The success continuations are functions passed as parameters or returned as values.

#### The Representation of Formulae and Sequents

Amongst the new language features of Pizza are class cases and pattern matching, which provide a very pleasant syntax for algebraic data types. The formulae of G4ip are specified by the following grammar:

`F ::= false | A | F & F | F v F | F -> F`

The class cases allow a straightforward implementation of this specification; it is analogous to the SML implementation of Lambda Prolog's formulae in [Elliot and Pfenning, 1991]. The class of formulae for G4ip is given below:

public class Form {
case False();
case Atm(String c);
case And(Form c1,Form c2);
case Or(Form c1,Form c2);
case Imp(Form c1,Form c2);
}
Two examples that illustrate the use of the representation are as follows:

`          p -> p  ` is represented as `   Imp(Atm("p"),Atm("p"))`
`a v (a -> false)  ` is represented as `   Or(Atm("a"),Imp(Atm("a"),False()))`

The class cases of Pizza also support an implementation of formulae specified by a mutually recursive grammar. This is required, for example, when implementing hereditary Harrop formulae.

The sequents of G4ip, which have the form `Gamma=>G`, are represented by means of the class below. The left-hand side of each sequent is specified by a multiset of formulae. Therefore, we do not need to worry about the order in which the formulae occur.

public class Sequent {
Form G;
Context Gamma;
public Sequent(Context _Gamma, Form _G) {...};
}
We have a constructor for generating new sequents during proof search. `Context` is a class which represents multisets; it is a simple extension of the class `Vector` available in the Java libraries. This class provides methods for adding elements to a multiset (`add`), taking out elements from a multiset (`removeElement`) and testing the membership of an element in a multiset (`includes`).

#### The Technique of Success Continuations

We have to distinguish between the concepts of proof obligations (which must be proved) and choice points (which can be tried out to construct a proof). The first argument of the method `prove` is the sequent being proved; the second argument is an anonymous function. The function `prove` is now of the form `prove(sequent,sc)`. Somewhat simplified the first argument is the leftmost premise and the second argument `sc`, the success continuation, represents the other proof obligations. In case we succeed in proving the first premise we then can attempt to prove the other premises. The technique of success continuations will be explained using the following proof (each sequent is marked with a number): The inference rules fall into three groups:

• inference rules with a single premise (e.g. ->_R, &_L),
• inference rules with two premises (e.g. v_L) and
• inference rules without premises (e.g. Axiom).
The following picture shows the order in which the sequents are being proved. Suppose we have called `prove` with a sequent s and a success continuation is. The inference rules of the first group manipulate s obtaining s' and call `prove` again with the new sequent s' and the current success continuation (Steps 1-2, 3-4 and 5-6). The inference rules of the second group have two premises, s1 and s2. These rules call `prove` with s1 and a new success continuation `prove(s2,is)` (Step 2-3). The third group of inference rules only invoke the success continuation if the rule was applicable (Steps 4-5 and 6-7).

We are going to give a detailed description of the code for the rules: &_L, ->_R, v_Ri, v_L and Axiom. The function `prove` receives as arguments a sequent `Sequent(Gamma,G)` and a success continuation `sc`. It enumerates all formulae as being principal and two switch statements select a corresponding case depending on the form and the occurrence of the principal formula.

The &_L rule is in the first group; it modifies the sequent being proved and calls `prove` again with the current success continuation sc. The code is as follows (`Gamma` stands for the set of formulae on the left-hand side of a sequent excluding the principal formula; `G` stands for the goal formula of a sequent; `B` and `C` stand for the two components of the principal formula).

case And(Form B, Form C):
The code for the ->_R rule is similar:

case Imp(Form B, Form C):
The v_Ri rule is an exception in the first group. It breaks up a goal formula of the form `B1 v B2` and proceeds with one of its component. Since we do not know in advance which component leads to a successful proof we have to try both. Therefore this rule acts as a choice point, which is encoded by a recursive call of `prove` for each case.
case Or(Form B1,Form B2):
prove(new Sequent(Gamma,B1),sc);
prove(new Sequent(Gamma,B2),sc); break;
The v_L rule falls into the second group where the current success continuation, sc, is modified. It calls `prove` with the first premise, `B,Gamma=>G`, and wraps up the success continuation with the new proof obligation, `C,Gamma=>G`. The construction `fun()->void {...}` defines an anonymous function: the new success continuation. In case the sequent `B,Gamma=>G` can be proved, this function is invoked.
case Or(Form B,Form C):
); break
The Axiom rule falls into the third group. It first checks if the principal formula (which is an atom) matches with the goal formula and then invokes the success continuation sc in order to prove all remaining proof obligations.
case Atm(String c):
if (G instanceof Atm) {
if (G.c.compareTo(c) == 0) { sc(); }
} break;
The proof search is started with an initial success continuation is. This initial success continuation is invoked when a proof has been found. In this case we want to give some response to the user, an example for the initial success continuation could be as follows:
public void initial_sc() { System.out.println("Provable!"); }
Suppose we attempt to start the proof search with `prove(p,p => p,is)`. We would find that the prover responds twice with `"Provable!"`, because it finds two proofs. In our implementation this problem is avoided by encoding the proof search as a thread. Whenever a proof is found, the initial success continuation displays the proof and suspends the thread. The user can decide to resume with the proof search or abandon the search.

#### Conclusion

The implementation cannot be considered as optimal in terms of speed. A much more efficient algorithm for G4ip (but less clear) has been implemented by Dyckhoff in Prolog. Similar ideas can be encoded in our Pizza implementation; but our point was not the efficiency but the clarity of the implementation using success continuations. The technique is applicable elsewhere whenever backtracking is required. We compared the code of our implementation with an implementation in Lambda Prolog: the ratio of code is approximately 2 to 1. (see LambdaProlog code and Pizza code). This result is partly due to the fact that we had to implement a class for multisets. In a future version of Java, we could have accessed a package in the library. The technique of success continuation can also be applied to a first-order calculus as shown in [Elliot and Pfenning, 1991], but the required mechanism of substitution needs to be implemented separately. However, we think the technique of success continuations provides a remarkable simple implementation for logic calculi.

We had to make some compromises in order to support as many platforms as possible. This should change with the release of new browsers and a stable Java-specification (resp. Pizza-specification).

A paper about the implementation appeared in the LNAI series No 1397, Automated Reasoning with Analytic Tableaux and Related Methods, ed. Harry de Swart, International Conference Tableaux'98 in Oisterwijk, The Netherlands. The title is: Implementation of Proof Search in the Imperative Programming Language Pizza (pp. 313-319). The paper can be found here: DVI, Postscript (© Springer-Verlag LNCS).

Acknowledgements: I am very grateful for Dr Roy Dyckhoff's constant encouragement and many comments on my work. I thank Dr Gavin Bierman who helped me to test the prover applet.

#### Implementation

Prover Applet
Jar Version (slightly faster, but requires Netscape 4 or MS Explorer 4).

References
• [Carlsson, 1984]
M. Carlsson, On Implementing Prolog in Functional Programming, New Generation Computing, pp 347-359.
• [Dyckhoff, 1992]
R. Dyckhoff, Contraction-Free Sequent Calculi for Intuitionistic Logic, Journal of Symbolic Logic 57(3), pp 795-807.
• [Elliot and Pfenning, 1991]
C. Elliot, F. Pfenning, A Semi-Functional Implementation of a Higher-Order Programming Language, In Peter Lee, editor, Topics in Advanced Language Implementation, MIT Press, pp 289-352. Available electronically.
• [Hodas and Miller, 1994]
J. Hodas, D. Miller, Logic Programming in a Fragment of Intuitionistic Linear Logic, Information and Computation 110(2), pp 327-365. Available electronically.