Symbolic Negotiation Revisited Peep K¨ ungas Norwegian University of Science and Technology Department of Computer and Information Science Trondheim, Norway peep@idi.ntnu.no Mihhail Matskin Royal Institute of Technology Department of Microelectronics and Information Technology Kista, Sweden misha@imit.kth.se ABSTRACT In this paper we propose a formalism for symbolic negotia- tion. We regard symbolic negotiation as cooperative prob- lem solving (CPS), which is based on symbolic reasoning and is extended with negotiation-specific rules. The underlying CPS formalism was previously presented in [3]. Here we ex- tend the results and position symbolic negotiation according to other distributed problem solving mechanisms. Categories and Subject Descriptors I.2.11 [Distributed Artificial Intelligence]: Multiagent sys- tems; I.2.3 [Deduction and Theorem Proving]: Deduction; F.4.1 [Mathematical Logic]: Computational logic General Terms Algorithms, Theory Keywords Symbolic reasoning, partial deduction, cooperative problem solving, negotiation 1. INTRODUCTION Symbolic negotiation is regarded in the field of computer science as a process, where parties try to reach an agree- ment on the high-level means for achieving their goals by applying symbolic reasoning techniques. Formalisation of symbolic negotiation could contribute both to the studies of human behaviour and multi-agent systems. In the former field the rational part of human reasoning could be mod- elled, while in the latter field self-organising systems with adaptive behaviour could be implemented. Previously, K¨ ungas and Matskin proposed [2] that sym- bolic negotiation could be formalised as partial deduction (PD) for linear logic (LL). In [3] a formalisation of PD for LL was presented and its soundness and completeness was Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. To copy otherwise, to republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. AAMAS’06 May 8–12 2006, Hakodate, Hokkaido, Japan. Copyright 2006 ACM 1-59593-303-4/06/0005 ...$5.00. proved. The paper regarded symbolic negotiation as inter- leaved cooperative problem solving (CPS) and plan modi- fication. However, the paper formalised only the CPS part of the process. In this paper we extend the proposed CPS formalism with plan modification operators and then anal- yse symbolic negotiaton as a whole. We also formalise the coalition formation process and analyse its effect to CPS and symbolic negotiation. Anyway, due to the limited space we do not present proofs for propositions and theorems in this paper. We would like to underline that from a computational point of view, we can regard CPS as AI planning and sym- bolic negotiation as plan reuse/repair. It has been shown [5] that from problem solving point of view in general neither planning from scratch nor plan repair has an advantage over each-other. Therefore we expect both CPS and symbolic ne- gotiation to be computationally equivalent. Moreover, both CPS and symbolic negotiation lead to the same results as we prove in this paper. However, compared to CPS, symbolic negotiation pro- vides a more human-like way of problem solving, which can be more naturally followed by human participants. In ad- dition, symbolic negotiation may encode a sort of search heuristics, which would make CPS computationally less de- manding. These heuristics, however, are not discussed in this paper. 2. LINEAR LOGIC AND PARTIAL DEDUC- TION 2.1 Linear logic LL is a refinement of classical logic introduced by J.-Y. Gi- rard to provide means for keeping track of “resources”. In LL two assumptions of a propositional constant A are distin- guished from a single assumption of A. This does not apply in classical logic, since there the truth value of a fact does not depend on the number of copies of the fact. Indeed, LL is not about truth, it is about computation. We consider the !-Horn fragment of LL (HLL) [1] con- sisting of multiplicative conjunction (⊗), linear implication (⊸) and “of course” operator (!). In terms of resource ac- quisition the logical expression A ⊗ B ⊢ C ⊗ D means that resources C and D are obtainable only if both A and B are obtainable. After the sequent has been applied, A and B are consumed and C and D are produced. While implication A ⊸ B as a computability statement clause in HLL could be applied only once, !(A ⊸ B) may be used an unbounded number of times. When A ⊸ B is